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Mathematics Chapter 13 Limits And Derivatives

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    NCERT Solution For Class 11 Mathematics

    Limits And Derivatives Here is the CBSE Mathematics Chapter 13 for Class 11 students. Summary and detailed explanation of the lesson, including the definitions of difficult words. All of the exercises and questions and answers from the lesson's back end have been completed. NCERT Solutions for Class 11 Mathematics Limits And Derivatives Chapter 13 NCERT Solutions for Class 11 Mathematics Limits And Derivatives Chapter 13 The following is a summary in Hindi and English for the academic year 2021-2022. You can save these solutions to your computer or use the Class 11 Mathematics.

    Question 1
    CBSEENMA11013108
    Wired Faculty App

    Evaluate the following :

    space space space space space space space limit as straight x rightwards arrow of left parenthesis 3 minus straight x squared right parenthesis
    Solution
    space space space space space space space space space space space space space space space space limit as straight x rightwards arrow 1 of left parenthesis 3 minus straight x squared right parenthesis space equals limit as straight x rightwards arrow 1 of 3 minus limit as straight x rightwards arrow 1 of straight x squared equals 3 minus left parenthesis 1 right parenthesis squared equals 3 minus 1 equals 2
    Question 2
    CBSEENMA11013110
    Wired Faculty App

    Evaluate the following :

    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Solution
    space space space space space space space space space space space space space space space limit as straight x rightwards arrow 4 of left square bracket straight x left parenthesis straight x plus 1 right parenthesis right square bracket space equals space open parentheses limit as straight x rightwards arrow 4 of space straight x close parentheses. open parentheses limit as straight x rightwards arrow 4 of left parenthesis straight x plus 1 right parenthesis close parentheses equals 4 open parentheses limit as straight x rightwards arrow 4 of straight x plus limit as straight x rightwards arrow 4 of 1 close parentheses equals 4 left parenthesis 4 plus 1 right parenthesis equals 4 left parenthesis 5 right parenthesis equals 20
    Question 3
    CBSEENMA11013112
    Wired Faculty App

    Evaluate the following :

    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Solution
    space space space space space limit as space space straight x rightwards arrow negative 1 of left parenthesis 1 plus straight x plus straight x squared plus...... space plus space straight x squared right parenthesis space equals stack lim space space left parenthesis 1 right parenthesis with straight x rightwards arrow negative 1 below plus limit as straight x rightwards arrow negative 1 of left parenthesis straight x right parenthesis space plus space limit as straight x rightwards arrow negative 1 of straight x squared plus space...... plus space limit as straight x rightwards arrow negative 1 of straight x to the power of 12
            = 1 + (-1) + (-1)2 + .... + (-1)12 = 1 + (-1) + 1 + ..... + 1 = 1
    Question 4
    CBSEENMA11013113
    Wired Faculty App

    Evaluate  <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Solution
    space space space space space space space space space space space space space space space space space space limit as straight x rightwards arrow 4 of fraction numerator 4 straight x plus 3 over denominator straight x minus 2 end fraction equals fraction numerator 4 left parenthesis 4 right parenthesis plus 3 over denominator 4 minus 2 end fraction equals fraction numerator 16 plus 3 over denominator 2 end fraction equals 19 over 2
    Question 5
    CBSEENMA11013116
    Wired Faculty App

    Evaluate    space space space space space space space limit as straight x rightwards arrow negative 1 of fraction numerator straight x to the power of 10 plus straight x to the power of 5 plus 1 over denominator straight x minus 1 end fraction
    Solution
    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Question 6
    CBSEENMA11013117
    Wired Faculty App

    Evaluate     <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>.
    Solution

    Evaluating the function at x =2, it is of the form space 0 over 0.
    Therefor, space space space space space space space space space space space space space space space space space space space space space space space space limit as straight x rightwards arrow 2 of fraction numerator straight x cubed minus 4 straight x squared plus 4 straight x over denominator straight x squared minus 4 end fraction equals limit as straight x rightwards arrow 2 of fraction numerator straight x left parenthesis straight x squared minus 4 straight x plus 4 right parenthesis over denominator straight x squared minus 2 squared end fraction equals limit as straight x rightwards arrow 2 of fraction numerator straight x left parenthesis straight x minus 2 right parenthesis squared over denominator left parenthesis straight x minus 2 right parenthesis left parenthesis straight x plus 2 right parenthesis end fraction equals limit as straight x rightwards arrow 2 of fraction numerator straight x left parenthesis straight x minus 2 right parenthesis over denominator straight x plus 2 end fraction

                                                                                                                                                                                                                                                                                                                                                              space space space space space space space space space space space space space space space space space space space space space open parentheses because space straight x not equal to 2 rightwards double arrow negative 2 not equal to 0 close parentheses
              space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator 3 left parenthesis 2 right parenthesis plus 5 over denominator 2 plus 2 end fraction equals 11 over 4
    Question 7
    CBSEENMA11013120
    Wired Faculty App

    Evaluate  space space space space space space space space space space space space space space stack space lim with straight x rightwards arrow 3 below fraction numerator 3 straight x squared minus straight x minus 10 over denominator straight x squared minus 4 end fraction.
    Solution

    Evaluating the funtion at x =2, it is of the form <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>.
    Therefore,  space space space space space space space space space space space space space space space limit as straight x rightwards arrow 2 of fraction numerator 3 straight x squared minus straight x minus 10 over denominator straight x to the power of 2 straight x end exponent minus 4 end fraction equals limit as straight x rightwards arrow 2 of fraction numerator 3 straight x squared minus 6 straight x plus 5 straight x plus 10 over denominator straight x squared minus 2 squared end fraction equals limit as straight z rightwards arrow 2 of fraction numerator left parenthesis straight x minus 2 right parenthesis left parenthesis 3 straight x plus 5 right parenthesis over denominator left parenthesis straight x minus 2 right parenthesis left parenthesis straight x plus 2 right parenthesis end fraction equals limit as straight x rightwards arrow 2 of fraction numerator 3 straight x plus 5 over denominator straight x plus 2 end fraction
                                                            
                                                                                         space space space space space space space space space open square brackets because space straight x space not equal to 2 rightwards double arrow straight x minus 2 not identical to 0 close square brackets
                    space space space space space space space space equals space space fraction numerator 2 left parenthesis 2 minus 2 right parenthesis over denominator 2 plus 2 end fraction equals fraction numerator 2 left parenthesis 0 right parenthesis over denominator 4 end fraction equals 0

    Question 8
    CBSEENMA11013122
    Wired Faculty App

    Evaluate   space space space space space space space space space space limit as straight x rightwards arrow 2 of fraction numerator 3 straight x squared minus straight x minus 10 over denominator straight x squared minus 4 end fraction
    Solution

    Evaluating the function at x =2, it is of the form 0 over 0.
    Therefore,  <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
                                                                                              <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
                  space space space space space space space space space space space equals fraction numerator 3 left parenthesis 2 right parenthesis plus 5 over denominator 2 plus 2 end fraction equals 11 over 4
                                                                                                                                                                                                                                                                                                                                                                                                                                        

    Question 9
    CBSEENMA11013125
    Wired Faculty App

    Evaluate     space space space space space space space space space space limit as straight x rightwards arrow 1 of fraction numerator ax squared plus bx plus straight c over denominator cx squared plus bx plus straight a end fraction comma space straight a plus straight b plus straight c space not equal to 0.

                       
    Solution
    space space space space space space space space space space space space space space space space space space space space space space space space space space space limit as straight x rightwards arrow 1 of fraction numerator ax squared plus bc plus straight c over denominator cx squared plus bx plus straight a end fraction equals fraction numerator limit as straight x rightwards arrow 1 of ax squared plus bx plus straight c over denominator limit as straight x rightwards arrow 1 of cx squared plus bx plus straight a end fraction equals fraction numerator straight a left parenthesis 1 right parenthesis squared plus straight b left parenthesis 1 right parenthesis plus straight c over denominator straight c left parenthesis 1 right parenthesis squared plus straight b left parenthesis 1 right parenthesis plus straight a end fraction equals fraction numerator straight a plus straight b plus straight c over denominator straight c plus straight b plus straight a end fraction equals 1
                      
                                                                                               space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space space straight a plus straight b plus straight c not equal to 0 close square brackets
                                                                                                                                                                                                                                                                                                                                                                                                                  
    Question 10
    CBSEENMA11013128
    Wired Faculty App

    Evaluate space space space space space space space space space space space space space space space space space space space limit as straight x rightwards arrow square root of 2 of fraction numerator square root of 3 plus 2 straight x end root minus open parentheses square root of 2 space plus 1 close parentheses over denominator straight x squared minus 2 end fraction
    Solution

    Evaluating  the function at <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>  it si of space 0 over 0 form.
    Therefore, space limit as space space space space space space space space space space space space straight x rightwards arrow square root of 2 of fraction numerator square root of 3 plus 2 straight x end root minus open parentheses square root of 2 plus 1 close parentheses over denominator straight x squared minus 2 end fraction space equals limit as straight x rightwards arrow square root of 2 of fraction numerator square root of 3 plus 2 straight x end root space minus open parentheses square root of 2 plus 1 close parentheses over denominator straight x squared minus 2 end fraction straight x fraction numerator square root of 3 plus 2 straight x end root plus open parentheses square root of 2 plus 1 close parentheses over denominator square root of 3 plus 2 straight x end root plus left parenthesis square root of 2 plus 1 right parenthesis end fraction   
     
        <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
     
    space space space space space space space space space space space equals space space space fraction numerator 2 over denominator left parenthesis square root of 2 plus square root of 2 right parenthesis end root space left parenthesis square root of 3 plus 2 square root of 2 end root plus square root of 2 plus 1 right parenthesis end fraction
          <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
                                                           
                                              space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open parentheses because space square root of 3 plus 2 square root of 2 end root equals square root of left parenthesis square root of 2 plus 1 right parenthesis squared end root equals square root of 2 plus 1 close parentheses
                
                        space space space space space space space space space space space space space space space space space space equals space fraction numerator 1 over denominator 2 square root of 2 left parenthesis square root of 2 plus 1 right parenthesis end fraction
                      
                    
             
                     
                

    Tips: -

            

    Question 11
    CBSEENMA11013130
    Wired Faculty App

    Evaluate  space space space space space limit as straight x rightwards arrow 0 of fraction numerator square root of 1 plus straight x squared end root minus square root of 1 plus straight x end root over denominator square root of 1 plus straight x cubed end root space minus square root of 1 plus straight x end root end fraction
    Solution

    It is of space space space space 0 over 0 space form.
     space space therefore space space space space limit as straight x rightwards arrow 0 of fraction numerator square root of 1 plus straight x squared end root minus square root of 1 plus straight x end root over denominator square root of 1 plus straight x cubed end root minus square root of 1 plus straight x end root end fraction equals limit as straight x rightwards arrow 0 of fraction numerator square root of 1 plus straight x squared end root minus square root of 1 plus straight x end root over denominator square root of 1 plus straight x squared end root minus square root of 1 plus straight x end root end fraction straight x fraction numerator square root of 1 plus straight x squared end root plus square root of 1 plus straight x end root over denominator square root of 1 plus straight x squared end root plus square root of 1 plus straight z end root end fraction straight x fraction numerator square root of 1 plus straight x cubed end root minus square root of 1 plus straight x end root over denominator square root of 1 plus straight x cubed end root minus square root of 1 plus straight x end root end fraction
       
        space space space space space space space space space equals space limit as straight x rightwards arrow 0 of fraction numerator open square brackets left parenthesis square root of 1 plus straight x squared end root right parenthesis squared minus left parenthesis square root of 1 plus straight x end root right parenthesis squared close square brackets over denominator left square bracket left parenthesis square root of 1 plus straight x cubed end root right parenthesis minus left parenthesis square root of 1 plus straight x end root right parenthesis squared right square bracket end fraction straight x fraction numerator square root of 1 plus straight x cubed end root plus square root of 1 plus straight x end root over denominator square root of 1 plus straight x squared end root plus square root of 1 plus straight x end root end fraction
         space space space space space space space equals space space limit as straight x rightwards arrow 0 of fraction numerator 1 over denominator left parenthesis straight x plus 1 right parenthesis end fraction straight x fraction numerator square root of 1 plus straight x cubed end root plus square root of 1 plus straight x end root over denominator square root of 1 plus straight x squared end root plus square root of 1 plus straight x end root end fraction equals fraction numerator 1 over denominator 0 plus 1 end fraction straight x fraction numerator square root of 1 plus 0 end root plus square root of 1 plus 0 end root over denominator square root of 1 plus 0 end root plus square root of 1 plus 0 end root end fraction
        space space space space space space space space space space space space equals space 1 space cross times space 2 over 2 equals 1

    Question 12
    CBSEENMA11013133
    Wired Faculty App

    Evaluate space space space space space space space space space space space space limit as straight x rightwards arrow square root of 2 of fraction numerator straight x to the power of 9 minus 3 straight x to the power of 8 plus straight x to the power of 6 minus 4 straight x squared minus 16 straight x plus 84 over denominator straight x to the power of 5 minus 3 straight x to the power of 4 minus 4 straight x plus 12 end fraction
    Solution

    Since at space straight x space equals space square root of 2 comma end root numerator and denominator becomes zero.
    ∴ it  is of 0 over 0  form.
    ∴ By factor theorem, space space left parenthesis straight x minus square root of 2 right parenthesis is a factor of both numerator and denominator.
    rightwards double arrow space space straight x space plus square root of 2 is also a factor of both numerator and denominator because irrational roots occur in pairs.
    So, x2 -2 is a factor of both numerator and denominator.
    By long division method, dividing numerator and denominator by x2 -2.
    x9 - 3x8 + x6 - 9x4 - 4x2 - 16x + 84 = (x2 -2) (x7 - 3x6 + 2x5 -5x4 + 4x3 -19x2 +8x - 42)  and  x5-3x4 -4x + 12 = (x2 -2) (x3 - 3x2 + 2x -6)

    ∴  space space space limit as straight x rightwards arrow square root of 2 of fraction numerator straight x to the power of 9 minus 3 straight x to the power of 8 plus straight x to the power of 6 minus 9 straight x to the power of 4 minus 4 straight x squared minus 16 straight x plus 84 over denominator straight x to the power of 5 minus 3 straight x to the power of 4 minus 4 straight x plus 12 end fraction

    space space space space space space equals space space space stack space lim with straight x rightwards arrow square root of 2 below fraction numerator left parenthesis straight x squared minus 2 right parenthesis left parenthesis straight x to the power of 7 minus 3 straight x to the power of 6 plus 2 straight x to the power of 5 minus 5 straight x to the power of 4 plus 4 straight x cubed minus 19 straight x squared plus 8 straight x minus 42 right parenthesis over denominator left parenthesis straight x squared minus 2 right parenthesis left parenthesis straight x cubed minus 3 straight x squared plus 2 straight x minus 6 right parenthesis end fraction

       space space space space space space space space space equals space space fraction numerator left parenthesis square root of 2 right parenthesis end root to the power of 7 minus 3 left parenthesis square root of 2 right parenthesis to the power of 6 plus 2 left parenthesis square root of 2 right parenthesis end root to the power of 5 minus 5 left parenthesis square root of 2 right parenthesis to the power of 4 plus 4 left parenthesis square root of 2 right parenthesis end root cubed minus 19 left parenthesis square root of 2 right parenthesis end root squared plus 8 left parenthesis square root of 2 right parenthesis end root minus 42 over denominator left parenthesis square root of 2 right parenthesis cubed minus 3 left parenthesis square root of 2 right parenthesis squared plus 2 left parenthesis square root of 2 right parenthesis minus 6 end fraction
    space space space space space space space space space equals space fraction numerator 8 square root of 2 minus 24 plus 8 square root of 2 minus 20 plus 8 square root of 2 minus 38 plus 8 square root of 2 minus 42 over denominator 2 square root of 2 minus 6 plus 2 square root of 2 minus 6 end fraction space space space space       space space space space open square brackets because space straight x space not equal to square root of 2 space rightwards double arrow space straight x squared minus 2 not equal to 0 close square brackets

            <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

    Question 13
    CBSEENMA11013135
    Wired Faculty App

    Evaluate space space space space space space space space limit as straight x rightwards arrow 1 of open parentheses fraction numerator 1 over denominator straight x minus 1 end fraction minus fraction numerator 3 left parenthesis straight x minus 2 right parenthesis over denominator straight x cubed minus 3 straight x squared plus 2 end fraction close parentheses
    Solution
    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

      space space space space space space space space space space space equals space limit as straight x rightwards arrow 1 of fraction numerator left parenthesis straight x minus 1 right parenthesis left parenthesis straight x minus 4 right parenthesis over denominator left parenthesis straight x minus 1 right parenthesis left parenthesis straight x squared minus 2 straight x minus 2 right parenthesis end fraction equals limit as straight x rightwards arrow 1 of fraction numerator straight x minus 4 over denominator straight x squared minus 2 straight x minus 2 end fraction equals fraction numerator 1 minus 4 over denominator 1 minus 2 minus 2 end fraction equals fraction numerator negative 3 over denominator negative 3 end fraction equals 1
    Question 14
    CBSEENMA11013138
    Wired Faculty App

    Evaluate space space space space space space limit as straight x rightwards arrow 0 of straight f left parenthesis straight x right parenthesis comma space where space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left columnspacing 1.4ex end attributes row cell fraction numerator open vertical bar straight x close vertical bar over denominator straight x end fraction comma end cell cell straight x not equal to 0 end cell row cell 0 space comma end cell cell straight x equals 0 end cell end table close
    Solution

    L.H.L    =  space space space space space limit as straight x rightwards arrow 0 to the power of minus of space straight f left parenthesis straight x right parenthesis space equals space limit as straight x rightwards arrow 0 to the power of minus of fraction numerator open vertical bar straight x close vertical bar over denominator straight x end fraction                                                           [∵ here x < 0]

          space space space space space space space space space space space equals space limit as straight x rightwards arrow 0 to the power of minus of straight f left parenthesis straight x right parenthesis space equals space limit as straight x rightwards arrow 0 to the power of minus of left parenthesis negative 1 right parenthesis equals negative 1 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left square bracket because space for space straight x space less than 0 comma space open vertical bar x close vertical bar equals negative x right square bracket

    R.H.L.     space space space space equals space limit as straight x rightwards arrow 0 to the power of plus space of straight f left parenthesis straight x right parenthesis space equals limit as straight x rightwards arrow 0 to the power of plus of fraction numerator open curly brackets straight x close curly brackets over denominator straight x end fraction space space space space space space space space space space space space space space space space space space space space                                          [∵  here x > 0]

          
                  space space space space space equals space limit as straight x rightwards arrow 0 to the power of plus of straight x over straight x                                                                 [ ∵ for x >0, space space space space space open vertical bar straight x close vertical bar equals straight x]

               space space space space space space space space space space space space space equals space space limit as straight x rightwards arrow 0 to the power of plus of left parenthesis 1 right parenthesis equals 1
    Thus,               L.H.L  space space not equal to    R.H.L.
    ∴  space space space limit as straight x rightwards arrow 0 of straight f left parenthesis straight x right parenthesis space does not exist.

    Question 15
    CBSEENMA11013141
    Wired Faculty App

    Suppose  that 


    space space space space space space space space space space space straight f left parenthesis straight x right parenthesis space open curly brackets table row cell table row cell table row cell straight a plus bx comma space straight x space less than 1 end cell end table end cell end table end cell row cell table row cell table row cell 4 comma space straight x space equals space 1 end cell row cell straight b minus ax comma space straight x space greater than 1 end cell end table end cell end table end cell end table close

    and if <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>, what are possible values of a and b?
    Solution

    L.H.L. =  space space space space space space space space space limit as straight x rightwards arrow 1 to the power of minus of straight f left parenthesis straight x right parenthesis space equals limit as straight x rightwards arrow 1 to the power of minus of left parenthesis straight a plus bx right parenthesis                                                         [∵  here x <1]
    Put x = 1-4, h>0 and hrightwards arrow0
    ∴                    L.H.L       = space space space space space space space space space space space stack space lim with h rightwards arrow 0 below left square bracket a plus b left parenthesis 1 minus h right parenthesis right square bracket space equals space a plus b left parenthesis 1 minus 0 right parenthesis equals a plus b

                             
                           R.H.L.     =      space space space space limit as straight x rightwards arrow 1 to the power of plus of straight f left parenthesis straight x right parenthesis space equals space limit as straight x rightwards arrow 1 to the power of plus of left parenthesis straight b minus ax right parenthesis                                [∵ here x > 1]
    Put x = 1-h, h>0 and hrightwards arrow0
    ∴      R.H.L     =     space space space space space space space limit as straight h rightwards arrow 0 of left square bracket straight b minus straight a left parenthesis 1 plus straight h right parenthesis right square bracket equals straight b minus straight a left parenthesis 1 plus 0 right parenthesis equals straight b minus straight a
    Also,                 f(1) = 4
    Now,    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    rightwards double arrow   
             space space space space space space limit as straight x rightwards arrow 1 to the power of minus of straight f left parenthesis straight x right parenthesis space equals straight f left parenthesis 1 right parenthesis space and space limit as straight x rightwards arrow 1 to the power of plus of straight f left parenthesis straight x right parenthesis space equals space straight f left parenthesis 1 right parenthesis
    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>         a + b = 4 adn b - a = 4
    Adding these equations, we get 2b = 8 rightwards double arrow b = 4
    ∴                         a = 4 - b = 4 - 4 = 0
    Hence,                 a = 0, b = 4

    Question 16
    CBSEENMA11013143
    Wired Faculty App

    Let f be a function defined by space space space space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left columnspacing 1.4ex end attributes row cell fraction numerator straight x over denominator open vertical bar straight x close vertical bar plus 2 straight x squared end fraction end cell cell straight x not equal to 0 end cell row cell 0 comma end cell cell straight x equals 0 end cell end table close   


    Evaluate space space limit as straight x rightwards arrow 0 to the power of asterisk times of straight f left parenthesis straight x right parenthesis comma  what can you say about space space space space space space space space space space space space limit as straight x rightwards arrow 0 of straight f left parenthesis straight x right parenthesis?
    Solution
    space space space space space space space space space space space space limit as space space space space straight x rightwards arrow 0 to the power of minus of straight f left parenthesis straight x right parenthesis space equals space limit as straight x rightwards arrow 0 to the power of minus of fraction numerator straight x over denominator open vertical bar straight x close vertical bar plus 2 straight x squared end fraction                                          [∵ here x <0]

                <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>                                                   [∵ x < 0 rightwards double arrowopen vertical bar x close vertical bar space equals space minus x]
        <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Also,      space space space space space space space space space space space space space space space space space limit as straight x rightwards arrow 0 to the power of plus of straight f left parenthesis straight x right parenthesis space equals space limit as straight x rightwards arrow 0 to the power of plus of fraction numerator straight x over denominator open vertical bar straight x close vertical bar plus 2 straight x squared end fraction                               [∵ here x >0]

            space space space space space space equals space limit as straight x rightwards arrow 0 to the power of plus of fraction numerator straight x over denominator straight x plus 2 straight x squared end fraction                                                         [∵ x >0 space space space space space space space space space space space space rightwards double arrow open vertical bar straight x equals straight x close vertical bar]
         space space space space space space space space space space space space space space equals space limit as straight x rightwards arrow 0 to the power of plus of fraction numerator straight x over denominator straight x left parenthesis 1 plus 2 straight x right parenthesis end fraction equals limit as straight x rightwards arrow 0 to the power of plus of fraction numerator 1 over denominator 1 plus 2 straight x end fraction equals fraction numerator 1 over denominator 1 plus 2 left parenthesis 0 right parenthesis end fraction equals 1
    Clearly,    space space space space space space space space space space space space space limit as straight x rightwards arrow 0 to the power of minus of straight f left parenthesis straight x right parenthesis space space not equal to space space limit as straight x rightwards arrow 0 to the power of plus of straight f left parenthesis straight x right parenthesis

    Hence,    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> does not exist.
    Question 17
    CBSEENMA11013145
    Wired Faculty App

    Evaluate space space space space space space limit as straight x rightwards arrow 0 of fraction numerator sin space ax over denominator sin space bs end fraction comma space straight a comma space straight b space not equal to 0.
    Solution
    space space space space space space space space space space space space space limit as straight x rightwards arrow 0 of fraction numerator sin space ax over denominator sin space bx end fraction equals limit as straight x rightwards arrow 0 of open parentheses fraction numerator sin space ax over denominator ax end fraction. fraction numerator bx over denominator sin space bx end fraction. straight a over straight b close parentheses equals straight a over straight b. limit as ax rightwards arrow 0 of fraction numerator sin space ax over denominator ax end fraction. fraction numerator straight a over denominator limit as bx rightwards arrow of space begin display style fraction numerator sin space bx over denominator bx end fraction end style end fraction
                                                                                         [∵ xrightwards arrow0rightwards double arrow ax rightwards arrow0 and bxrightwards arrow0]
    space space space space space space space space space space space equals space straight a over straight b minus 1 minus 1 over 1 equals straight a over straight b

    Sponsor Area

    Question 18
    CBSEENMA11013148
    Wired Faculty App

    Evaluate space space space space space limit as straight x rightwards arrow 0 of fraction numerator sin space ax over denominator bx end fraction.
    Solution
    space space space space space space space space space space limit as straight x rightwards arrow 0 of fraction numerator sin space ax over denominator bx end fraction equals 1 over straight b limit as straight x rightwards arrow 0 of fraction numerator sin space ax over denominator ax end fraction xa equals straight a over straight b semicolon fraction numerator sin space ax over denominator ax end fraction                                   [∵ xrightwards arrow0rightwards double arrowaxrightwards arrow0]

       space space space space space space space space space space space space space space equals straight a over straight b.1 equals straight a over straight b
    Question 19
    CBSEENMA11013149
    Wired Faculty App

    Evaluate space space space space space space space limit as straight x rightwards arrow 0 of fraction numerator sin space 5 straight x over denominator straight x end fraction
    Solution
    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
      
                                                                                                                                                                  
                                                                                 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space straight x rightwards arrow 0 rightwards double arrow 5 straight x rightwards arrow 0 space and space limit as straight x rightwards arrow 0 of fraction numerator sin space straight x space over denominator straight x end fraction equals 1 close square brackets
    Question 20
    CBSEENMA11013150
    Wired Faculty App

    Evaluate space space space space space space space space space space space space space space space space space limit as straight x rightwards arrow 0 of fraction numerator 1 minus cos space straight x over denominator straight x end fraction.
    Solution
    space space space space space space space space space space space limit as straight x rightwards arrow 0 of fraction numerator 1 minus cos space straight x over denominator straight x end fraction equals limit as straight x rightwards arrow 0 of fraction numerator 2 sin squared straight x divided by 2 over denominator straight x end fraction equals limit as straight x rightwards arrow 0 of fraction numerator sin begin display style straight x over 2 end style. sin begin display style straight x over 2 end style over denominator begin display style straight x over 2 end style end fraction
           space space space space space space space space space space space space space space space equals space limit as straight x rightwards arrow 0 of fraction numerator sinx divided by 2 over denominator straight x divided by 2 end fraction straight x limit as straight x rightwards arrow 0 of sin straight x over 2 equals 1 straight x 0 equals 0
    Question 21
    CBSEENMA11013152
    Wired Faculty App

    Evaluate space space limit as straight x rightwards arrow straight x over 4 of fraction numerator cos space straight x space minus space sin space straight x over denominator straight x minus straight pi divided by 4 end fraction.
    Solution
    space space space space space space space space space space space limit as straight x rightwards arrow straight pi over 4 of fraction numerator cos space space straight x space minus space sin space straight x over denominator straight x space minus space straight pi divided by 4 end fraction
    Put  space space straight x space equals straight pi over 4 plus straight h comma space straight x rightwards arrow straight pi over 4 rightwards double arrow straight h rightwards arrow 0              
                        

    ∴           space space space space space space space space space space space space space space space space space space space space limit as straight x rightwards arrow straight pi over 4 of fraction numerator cos space straight x space minus space sin space straight x over denominator straight x minus begin display style straight pi over 4 end style end fraction equals limit as straight h rightwards arrow 0 of fraction numerator cos open parentheses begin display style straight pi over 4 end style plus straight h close parentheses minus sin open parentheses begin display style straight pi over 4 end style plus straight h close parentheses over denominator begin display style straight pi over 4 end style plus straight h minus begin display style straight pi over 4 end style end fraction

            space space space space space space space space space space space space equals space limit as straight h rightwards arrow 0 of fraction numerator open parentheses cos straight pi over 4. cos space straight h minus sin straight pi over 4. sin space straight h close parentheses minus open parentheses sin straight pi over 4. cos space straight h minus sin straight pi over 4. sin space straight h close parentheses over denominator straight h end fraction

            space space space space space space space space space space space space space space space equals space space space limit as straight h rightwards arrow 0 of fraction numerator begin display style fraction numerator 1 over denominator square root of 2 end fraction end style cos space straight h space space minus begin display style fraction numerator 1 over denominator square root of 2 end fraction end style sin space straight h space minus space fraction numerator 1 over denominator square root of 2 end fraction cos space straight h space space minus fraction numerator 1 over denominator square root of 2 end fraction sin space straight h over denominator straight h end fraction

             <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

               space space space space space space space space equals space space fraction numerator negative 2 over denominator square root of 2 end fraction straight x fraction numerator square root of 2 over denominator square root of 2 end fraction equals negative square root of 2
         

              

    Question 22
    CBSEENMA11013153
    Wired Faculty App

    Evaluate space space space space space limit as straight x rightwards arrow straight pi of fraction numerator sin left parenthesis straight n minus straight pi right parenthesis over denominator straight pi left parenthesis straight pi minus straight x right parenthesis end fraction.
    Solution
    space space space space space limit as straight x rightwards arrow straight pi space of fraction numerator sin left parenthesis straight pi minus straight x right parenthesis over denominator straight pi left parenthesis straight pi minus straight x right parenthesis end fraction
    space space space space space space space space Put space straight x space equals straight pi plus straight h comma space straight x rightwards arrow straight pi rightwards double arrow straight h rightwards arrow 0

    ∴  <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

                space space space space space equals space 1 over straight pi straight x 1 equals 1 over straight pi

    Question 23
    CBSEENMA11013155
    Wired Faculty App

    Evaluate space space space space space space space space space space space space limit as straight x rightwards arrow 0 of fraction numerator sin space space ax plus bx over denominator ax plus sin space bx end fraction comma space straight a comma space space straight b comma space straight a plus straight b not equal to 0
    Solution
    space space space space space space space space space space space space space space space space space space space space space space limit as straight x rightwards arrow 0 of fraction numerator sin space ax plus bx over denominator ax plus sin space bx end fraction equals limit as straight x rightwards arrow 0 of fraction numerator begin display style fraction numerator sin space ax space plus space space bx over denominator straight x end fraction end style over denominator begin display style fraction numerator ax plus sin space space bx over denominator straight x end fraction end style end fraction equals fraction numerator begin display style fraction numerator sin space ax over denominator straight x end fraction plus straight b end style over denominator straight a plus begin display style fraction numerator sin space bx over denominator straight x end fraction end style end fraction

        space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space limit as straight x rightwards arrow 0 of fraction numerator begin display style fraction numerator straight a space sin space ax over denominator ax end fraction plus straight b end style over denominator straight a plus begin display style fraction numerator straight b space sin space bx over denominator bx end fraction end style end fraction equals fraction numerator straight a. limit as ax rightwards arrow 0 of begin display style fraction numerator siin space ax over denominator ax end fraction end style plus straight b over denominator straight a plus straight b. limit as bx rightwards arrow 0 of begin display style fraction numerator sin space bx over denominator bx end fraction end style end fraction equals fraction numerator straight a plus straight b over denominator straight a plus straight b end fraction equals 1
    Question 24
    CBSEENMA11013156
    Wired Faculty App

    Evaluate space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space limit as straight x rightwards arrow 0 of left parenthesis cosec space straight x minus cot space straight x right parenthesis
    Solution
    space space space space space space space space space space space space space space space space space space space space space limit as straight x rightwards arrow 0 of left parenthesis cosec space straight x space minus space cot space straight x right parenthesis space equals space limit as straight x rightwards arrow 0 of open parentheses fraction numerator 1 over denominator siin space straight x end fraction minus fraction numerator cos space straight x over denominator siin space straight x end fraction close parentheses equals limit as straight x rightwards arrow 0 of fraction numerator 1 minus cos space straight x over denominator sin space straight x end fraction equals limit as straight x rightwards arrow 0 of fraction numerator 2 sin squared begin display style straight x over 2 end style over denominator sin space straight x end fraction

          space space space space space space space space space space space space space space space space equals space 2 limit as straight x rightwards arrow 0 of fraction numerator sin begin display style straight x over 2 end style. sin begin display style straight x over 2 end style over denominator sin space straight x end fraction equals limit as straight x rightwards arrow of fraction numerator begin display style fraction numerator sinx divided by 2 over denominator straight x divided by 2 end fraction end style over denominator begin display style fraction numerator sin space straight x over denominator straight x end fraction end style end fraction. sin straight x over 2

                space space space space space space space space space space space space space equals space fraction numerator limit as begin display style straight x over 2 end style rightwards arrow 0 of begin display style fraction numerator sinx divided by 2 over denominator straight x divided by 2 end fraction end style over denominator limit as straight x rightwards arrow 0 of begin display style fraction numerator sin space straight x over denominator straight x end fraction end style end fraction. limit as straight x rightwards arrow 0 of sin straight x over 2 equals 1 over 1.0 space equals 0
    Question 25
    CBSEENMA11013158
    Wired Faculty App

    Find the derivative of x2-2 at x = 10.

    Solution

    Let  f(x)  = x2-2
    Now, f (10) denote the derivative of x2-2 at x = 10.

    ∴    f' (10) space space space space space space space space space space space space space space space space space space space space space space equals limit as straight h rightwards arrow 0 of fraction numerator straight f left parenthesis 10 plus straight h right parenthesis minus straight f left parenthesis 10 right parenthesis over denominator straight h end fraction equals limit as straight h rightwards arrow 0 of fraction numerator left square bracket 10 plus straight h right square bracket squared minus 2 right square bracket minus left parenthesis left parenthesis 10 right parenthesis squared minus 2 right square bracket over denominator straight h end fraction

                   space space space space space space space space space space space space space space space equals space limit as straight h rightwards arrow 0 of fraction numerator 100 plus straight h squared minus 20 straight h minus 2 minus 100 plus 2 over denominator straight h end fraction equals limit as straight h rightwards arrow 0 of fraction numerator straight h squared plus 20 straight h over denominator straight h end fraction
                          space space space space space space space equals space space space limit as straight h rightwards arrow 0 of left parenthesis straight h plus 20 right parenthesis equals 0 plus 20
    Hence, f(10)=20

    Question 26
    CBSEENMA11013159
    Wired Faculty App

    Find the derivative of f (x) = 7 at x = 3

    Solution

    f(3) space space space space space space space space space space space space space space space equals space limit as h rightwards arrow 0 of fraction numerator f left parenthesis 3 plus h right parenthesis minus left parenthesis 3 right parenthesis over denominator h end fraction equals limit as h rightwards arrow 0 of fraction numerator 7 minus 7 over denominator h end fraction equals limit as h rightwards arrow 0 of 0 equals 0

    ∴              f(3) = 0

    Question 27
    CBSEENMA11013163
    Wired Faculty App

    Find the derivative of space fraction numerator 1 over denominator square root of 3 straight x plus 1 end root end fraction at x =2.
    Solution
    space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight f left parenthesis 2 right parenthesis space equals space space limit as straight h rightwards arrow 0 of fraction numerator straight f left parenthesis 2 plus straight h right parenthesis minus straight f left parenthesis 2 right parenthesis over denominator straight h end fraction equals limit as straight h rightwards arrow 0 of fraction numerator begin display style fraction numerator 1 over denominator square root of 3 left parenthesis 2 plus straight h right parenthesis plus 1 end root end fraction minus fraction numerator 1 over denominator square root of 3 left parenthesis 2 right parenthesis end root plus 1 end fraction end style over denominator straight h end fraction
                                  space space space space space space space space space equals space limit as straight h rightwards arrow 0 of fraction numerator begin display style fraction numerator 1 over denominator square root of 7 plus 3 straight h end root end fraction end style minus begin display style fraction numerator 1 over denominator square root of 2 end fraction end style over denominator straight h end fraction equals limit as straight h rightwards arrow 0 of 1 over straight h open square brackets fraction numerator square root of 7 minus square root of 7 plus 3 straight h end root over denominator square root of 7 space space square root of 7 plus 3 straight h end root end fraction close square brackets

                                  <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
                      space space space space space space space space space space space space space space space space space space space space space space space space equals space limit as straight h rightwards arrow 0 of 1 fourth. fraction numerator 7 minus left parenthesis 7 plus 3 straight h right parenthesis over denominator square root of 7 square root of 7 plus 3 straight h end root left parenthesis square root of 7 plus square root of 7 plus 34 right parenthesis end root end fraction

                               space space space space space space space space space space space space space space space equals space limit as straight h rightwards arrow 0 of fraction numerator negative 3 over denominator square root of 7 plus square root of 7 plus 3 straight h end root left parenthesis square root of 7 plus square root of 7 plus 3 straight h right parenthesis end root end fraction equals fraction numerator negative 3 over denominator square root of 7 square root of 7 left parenthesis square root of 7 plus square root of 7 right parenthesis end fraction equals fraction numerator negative 3 over denominator 14 square root of 7 end fraction

                                


    Question 28
    CBSEENMA11013165
    Wired Faculty App

    Find the derivative of the following functions from first principle. 

    (x-1) (x-2)

    Solution

    Let   f(x) = (x-1)(x-2) = x2 -3x + 2.

    ∴      space space space space space space space space space space space space space space straight f space left parenthesis straight x right parenthesis space equals space limit as straight h rightwards arrow 0 of fraction numerator straight f left parenthesis straight x plus straight h right parenthesis minus straight f left parenthesis straight x right parenthesis over denominator straight h end fraction
            <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
         space space space space space space space space space space space space space space space space space space equals space limit as straight h rightwards arrow 0 of fraction numerator straight x squared plus 2 hx plus straight h squared minus negative 3 straight x minus 3 straight h plus 2 minus straight x squared plus 3 straight x minus 2 over denominator straight h end fraction
              <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
                        = 0 + 2x-3 = 2x -3

    Question 29
    CBSEENMA11013167
    Wired Faculty App

    Find the derivative of the following functions from first principle. 

    space space space space space fraction numerator straight x plus 1 over denominator straight x minus 1 end fraction

    Solution
    space space space space straight f left parenthesis straight x right parenthesis space fraction numerator straight x plus 1 over denominator straight x minus 1 end fraction

    space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space therefore space space space space space space space space straight f left parenthesis straight x right parenthesis space equals space limit as straight h rightwards arrow 0 of fraction numerator straight f left parenthesis straight x plus straight h right parenthesis minus straight f left parenthesis straight x right parenthesis over denominator straight h end fraction equals limit as straight h rightwards arrow 0 of fraction numerator begin display style fraction numerator straight x plus straight h plus 1 over denominator straight x plus straight h minus 1 end fraction minus fraction numerator straight x plus 1 over denominator straight x minus 1 end fraction end style over denominator straight h end fraction

                   space space space space space space space space space space space space space space space space space space space space space space equals space limit as straight h rightwards arrow 0 of 1 over straight h open square brackets fraction numerator left parenthesis straight x plus 1 plus straight h right parenthesis left parenthesis straight x minus 1 right parenthesis minus left parenthesis straight x minus 1 plus straight h right parenthesis left parenthesis straight x plus 1 right parenthesis over denominator left parenthesis straight x plus straight h minus 1 right parenthesis left parenthesis straight x minus 1 right parenthesis end fraction close square brackets


                 space space space space space space space space space space space space space space space space space space space space space space space space space equals space limit as straight h rightwards arrow 0 of 1 over straight h open square brackets fraction numerator straight x squared minus 1 plus hx minus straight h minus left parenthesis straight x squared minus 1 right parenthesis minus straight h left parenthesis straight x plus 1 right parenthesis over denominator left parenthesis straight x plus straight h minus 1 right parenthesis left parenthesis straight x minus 1 right parenthesis end fraction close square brackets
                 space space space space space space space space space space space space space space space space space space space space space space space space space equals space limit as straight h rightwards arrow 0 of 1 over straight h open square brackets fraction numerator straight x squared minus 1 plus hx minus straight h minus straight x squared plus 1 minus hx minus straight h over denominator left parenthesis straight x plus straight h minus 1 right parenthesis left parenthesis straight x minus 1 right parenthesis end fraction close square brackets


                          <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>


                        

    Question 31
    CBSEENMA11013171
    Wired Faculty App

    Let f(x) space space space space space equals space open curly brackets table attributes columnalign left end attributes row cell 1 minus straight x squared comma space straight x less or equal than 0 end cell row cell 1 plus straight x squared comma space straight x space greater than 0 end cell end table close find Lf'(0)  and Rf'(0)

    Is f derivable at x = 0?
    Solution

    Here, f(0) = 1-02 = 1  
    space space space space space space space space space space space space space Lf apostrophe left parenthesis 0 right parenthesis space equals space limit as straight h rightwards arrow 0 to the power of minus of fraction numerator straight f left parenthesis 0 plus straight h right parenthesis minus straight f left parenthesis 0 right parenthesis over denominator straight h end fraction equals limit as straight h rightwards arrow 0 to the power of minus of fraction numerator straight f left parenthesis straight h right parenthesis minus 1 over denominator straight h end fraction                           (Note that here h < 0)
              space space space space space space space space space space space space equals space space limit as straight h rightwards arrow 0 to the power of minus of fraction numerator 1 minus straight h squared minus 1 over denominator straight h end fraction equals limit as straight h rightwards arrow 0 to the power of minus of left parenthesis negative straight h right parenthesis space equals space 0
    space space space space space space space space space space space space space Rf apostrophe left parenthesis 0 right parenthesis space equals space limit as straight h rightwards arrow 0 to the power of plus of fraction numerator straight f left parenthesis 0 plus straight h right parenthesis minus straight f left parenthesis 0 right parenthesis over denominator straight h end fraction equals limit as straight h rightwards arrow 0 to the power of plus of fraction numerator straight f left parenthesis straight h right parenthesis minus 1 over denominator straight h end fraction                          (Note that here h > 0)
                        <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Clearly, Lf'(0) = 0 = Rf'(0) rightwards double arrowf is derivable at x =0

    Question 32
    CBSEENMA11013172
    Wired Faculty App

    Find the derivative of the function f(x) = 99x at x= 100.

    Solution
    space space space space space space straight f left parenthesis 100 right parenthesis space space equals space limit as straight h rightwards arrow 0 of fraction numerator straight f left parenthesis 100 plus straight h right parenthesis minus straight f left parenthesis 100 right parenthesis over denominator straight h end fraction equals limit as straight h rightwards arrow 0 of fraction numerator 99 left parenthesis 100 plus straight h right parenthesis minus 99 left parenthesis 100 right parenthesis over denominator straight h end fraction

        space space space space space space space equals space limit as straight h rightwards arrow 0 of fraction numerator 99 straight h over denominator straight h end fraction equals limit as straight h rightwards arrow 0 of 99 equals 99

     

    ∴ derivative of function of f(x) = 99x at x = 100 is 99.

    Question 33
    CBSEENMA11013174
    Wired Faculty App

    Let       space space space straight f left parenthesis straight x right parenthesis space equals space open curly brackets table attributes columnalign left end attributes row cell straight x plus straight a comma space straight x less or equal than 1 end cell row cell ax squared plus 1 comma space straight x greater than 1 end cell end table close

    find a so that f may be derivable at x =1.
    Solution

    Here, f(1) = 1+a
    space space space space space space space space space space space Lf apostrophe left parenthesis 1 right parenthesis space equals space limit as straight h rightwards arrow 0 to the power of minus of fraction numerator straight f left parenthesis 1 plus straight h right parenthesis minus straight f left parenthesis 1 right parenthesis over denominator straight h end fraction equals limit as straight h rightwards arrow 0 to the power of minus of fraction numerator left parenthesis 1 plus straight h right parenthesis plus straight a minus left parenthesis 1 plus straight a right parenthesis over denominator straight h end fraction
        <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>                                                            (here  h < 0 rightwards double arrow1+h < 1)

    space space space space space space Rf apostrophe left parenthesis 1 right parenthesis space space equals space limit as straight h rightwards arrow 0 to the power of plus of fraction numerator straight f left parenthesis 1 plus straight h right parenthesis minus straight f left parenthesis 1 right parenthesis over denominator straight h end fraction                                                    (here h >0 rightwards double arrow1 + h > 1)

               space space space space space space equals space limit as straight h rightwards arrow 0 to the power of plus of fraction numerator straight a left parenthesis 1 plus straight h right parenthesis squared plus 1 minus left parenthesis 1 plus straight a right parenthesis over denominator straight h end fraction
                 
        space space space space space space space space space equals space limit as straight h rightwards arrow 0 to the power of plus of fraction numerator ah squared plus 2 ah over denominator straight h end fraction equals limit as straight h rightwards arrow 0 to the power of plus of left parenthesis ah plus 2 straight a right parenthesis equals 0 plus 2 straight a equals 2 straight a
    Since f is derivable at x =1

    ∴    Lf'(1) = Rf'(1)
    rightwards double arrow       1 = 2a              rightwards double arrow  space space space space straight a space equals space 1 half

        
              

                                     
    Question 34
    CBSEENMA11013176
    Wired Faculty App

    If f is derivable at x=a, evaluate space space space space space space space space space stack space lim with straight x rightwards arrow straight a below fraction numerator xf left parenthesis straight a right parenthesis minus af left parenthesis straight x right parenthesis over denominator straight x minus straight a end fraction.
    Solution

    Since  f is derivable at x =a

    ∴       space space space space space space space space space space space space space space straight f apostrophe left parenthesis straight a right parenthesis space space space space equals space limit as straight h rightwards arrow straight a of fraction numerator straight f left parenthesis straight a plus straight h right parenthesis minus straight f left parenthesis straight a right parenthesis over denominator straight h end fraction
    Now,    space space space space space space space space space space space space limit as straight x rightwards arrow 0 of fraction numerator xf left parenthesis straight a right parenthesis minus af left parenthesis straight x right parenthesis over denominator straight x minus straight a end fraction equals limit as straight h rightwards arrow 0 of fraction numerator left parenthesis straight a plus straight h right parenthesis space straight f left parenthesis straight a right parenthesis space minus af left parenthesis straight a plus straight h right parenthesis over denominator straight a plus straight h minus straight a end fraction

                                                                                     [By putting x =a +h as xrightwards arrowa, h rightwards arrow0]


    space space space space space space space space space space space space space space space space space equals space space space limit as straight h rightwards arrow 0 of fraction numerator af left parenthesis straight a right parenthesis plus hf left parenthesis straight a right parenthesis minus af left parenthesis straight a plus straight h right parenthesis over denominator straight h end fraction equals limit as straight h rightwards arrow 0 of fraction numerator straight a left square bracket straight f left parenthesis straight a right parenthesis minus straight f left parenthesis straight a plus straight h right parenthesis plus right square bracket plus hf left parenthesis straight a right parenthesis over denominator straight h end fraction

        space space space space space space space space space space space space space equals negative straight a space limit as straight h rightwards arrow 0 of open square brackets fraction numerator straight f left parenthesis straight a plus straight h right parenthesis minus straight f left parenthesis straight a right parenthesis over denominator straight h end fraction plus straight f left parenthesis straight a right parenthesis close square brackets
         space space space space space space space space space space space space space space equals space minus straight a space limit as straight h rightwards arrow 0 of fraction numerator straight f left parenthesis straight a plus straight h right parenthesis minus straight f left parenthesis straight a right parenthesis over denominator straight h end fraction plus limit as straight h rightwards arrow 0 of straight f left parenthesis straight a right parenthesis

                     = -af'(a) + f(a)                                                                           [By using (1)]
                      =  f(a)-af'(a)                                                             
    Question 35
    CBSEENMA11013180
    Wired Faculty App

    For the function :

    space space space space space space space space straight f left parenthesis straight x right parenthesis space equals straight x to the power of 100 over 100 plus straight x to the power of 99 over 99 plus.... plus fraction numerator straight x squared 2 over denominator 2 end fraction plus straight x plus 1  Prove that f(1) = 100 f'(0)
    Solution

    Here,    space space space space space space space space space space space space space space space space space space space space space space space space space space straight f space left parenthesis straight x right parenthesis space equals space straight x to the power of 100 over 100 plus straight x to the power of 99 over 99 plus..... plus straight x squared over 2 plus straight x plus 1
    rightwards double arrow                        <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
                         
              space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals straight d over dx open parentheses straight x to the power of 100 over 100 close parentheses plus straight d over dx open parentheses straight x to the power of 99 over 99 close parentheses plus..... plus straight d over dx open parentheses straight x squared over 2 close parentheses plus straight d over dx plus left parenthesis straight x right parenthesis plus straight d over dx plus left parenthesis 1 right parenthesis
                           
                               space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 over 100 straight x 100 straight x to the power of 99 plus 1 over 99 space straight d over dx straight x to the power of 99 plus..... plus 1 half space straight d over dx straight x squared plus straight d over dx straight x to the power of 1 plus 0

               space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 over 100 straight x 100 straight x to the power of 99 plus 1 over 99 straight x 99 straight x to the power of 98 plus...... plus 1 half straight x 2 straight x to the power of 1 plus 1 straight x to the power of 0
    rightwards double arrow      space space space space space space space space space space space space space space space space space space space space space space space space space space space straight f left parenthesis x right parenthesis space equals space straight x to the power of 99 space end exponent plus straight x to the power of 98 plus...... plus space straight x to the power of 1 plus 1
    L.H.S.  = f(1) = (1)99 + (1)98 + ....... +(1)1 + 1 = 1+1+ ....+1+1 = 100
    R.H.S. = [100f'(0) = 100(0)99 +(0)98 + ..... + (0)1 +1] = 100

    ∴ L.H.S =R.H.S.
    Hence,   f'(1) = 100 f'(0)


                   
    Question 36
    CBSEENMA11013182
    Wired Faculty App

    For some constants a and b, find the derivatives of the following functions :

    (x-a)(x-b)

    Solution

    Let       space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight y space equals space left parenthesis straight x minus straight a right parenthesis left parenthesis straight x minus straight b right parenthesis equals straight x squared space minus ax minus 0 bx plus ab
    rightwards double arrow                             y = x2 - (a+b)x + ab

    ∴     space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space dx over dx equals straight d over dx straight x squared minus straight d over dx left parenthesis straight a plus straight b right parenthesis straight x plus straight d over dx left parenthesis ab right parenthesis equals 2 straight x minus left parenthesis straight a plus straight b right parenthesis straight d over dx straight x plus 0
                                     = 2x-(a+b).1=2x-(a+b)

    Question 37
    CBSEENMA11013183
    Wired Faculty App

    For some constants a and b, find the derivatives of the following functions :

    (ax2 +b)2 

    Solution

    y = (ax2 +b)2 

    rightwards double arrow    space space space space space space space space space space straight y space equals space space straight a squared straight x to the power of 4 plus 2 abx squared plus straight b squared


    rightwards double arrow          space space space space space space space space space space space space space space space space space dy over dx equals straight d over dx left parenthesis straight a squared straight x to the power of 4 right parenthesis plus straight d over dx left parenthesis 2 ab space straight x squared right parenthesis plus straight d over dx left parenthesis straight b squared right parenthesis equals straight a squared straight d over dx straight x to the power of 4 plus 2 ab straight d over dx straight x squared plus 0

                 space space space space space space space space space space space space space space equals space space straight a squared straight x space 4 straight x cubed plus 2 abx 2 straight x equals 4 straight a squared straight x cubed plus 4 abx
    Question 38
    CBSEENMA11013185
    Wired Faculty App

    For some constants a and b, find the derivatives of the following functions :


    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

    Solution

    Let  y = <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    rightwards double arrow    space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space dy over dx equals straight d over dx open parentheses fraction numerator straight x minus over denominator straight x minus straight b end fraction close parentheses equals fraction numerator left parenthesis straight x minus straight b right parenthesis begin display style straight d over dx end style left parenthesis straight x minus straight a right parenthesis minus left parenthesis straight x minus straight a right parenthesis begin display style straight d over dx end style left parenthesis straight x minus straight b right parenthesis over denominator left parenthesis straight x minus straight b right parenthesis squared end fraction

                                                               space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space straight d over dx open parentheses straight u over straight v close parentheses equals fraction numerator straight v begin display style straight d over dx end style straight u minus straight u begin display style straight d over dx end style straight v over denominator straight v squared end fraction close square brackets  
       space space space space space space space space space space space space equals space fraction numerator left parenthesis straight x minus straight b right parenthesis left parenthesis 1 minus 0 right parenthesis minus left parenthesis straight x minus straight a right parenthesis left parenthesis 1 minus 0 right parenthesis over denominator left parenthesis straight x minus straight b right parenthesis squared end fraction space equals space fraction numerator straight x minus straight b minus straight x plus straight a over denominator left parenthesis straight x minus straight b right parenthesis squared end fraction equals fraction numerator straight a minus straight b over denominator left parenthesis straight x minus straight b right parenthesis squared end fraction

    Question 39
    CBSEENMA11013187
    Wired Faculty App

    Find the derivative of the following :

    (5x3+3x+1)(x-1)

    Solution

    Let        y = (5x3 +3x -1) (x-1)
    differentiating both sides w.r.t. x, we  get
    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
           space space space space space space space space space space space space space space equals space left parenthesis 5 straight x cubed plus 3 straight x minus 1 right parenthesis straight d over dx left parenthesis straight x minus 1 right parenthesis plus left parenthesis straight x minus 1 right parenthesis straight d over dx left parenthesis 5 straight x cubed plus 3 straight x minus 1 right parenthesis
                     =(5x3 +3x -1) (1-0)+(x-1)(5x2 +3 -0) 
                     = 5x3 +3x -1+15x3 +3x -3
                     = 20x3 -15x2 + 6x -4

    Sponsor Area

    Question 40
    CBSEENMA11013189
    Wired Faculty App

    Find the derivative of the following :

    x-4 (3 - 4x-5)

    Solution

    Let y = x-4 (3 - 4x-5)
    rightwards double arrow y = 3 x-4 -4x-9 
    Differentiating both sides w.r.t. x,  we get
     space space space space space space space space space space space space space space space space space space space dy over dx equals straight d over dx open parentheses 3 straight x to the power of negative 4 space end exponent minus 4 straight x to the power of negative 9 end exponent close parentheses equals straight d over dx open parentheses 3 straight x to the power of negative 4 end exponent close parentheses minus straight d over dx left parenthesis 4 straight x to the power of negative 9 end exponent right parenthesis
        space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 3 straight d over dx left parenthesis straight x to the power of negative 4 end exponent right parenthesis minus 4 straight d over dx left parenthesis straight x to the power of negative 9 end exponent right parenthesis equals 3 left parenthesis negative 4 right parenthesis straight x to the power of negative 5 end exponent minus 4 left parenthesis negative 9 right parenthesis straight x to the power of negative 10 end exponent
               <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

    Question 41
    CBSEENMA11013192
    Wired Faculty App

    Find the derivative of the following :


    space space space space space space space space fraction numerator 2 over denominator straight x plus 1 end fraction minus fraction numerator straight x squared over denominator 3 straight x minus 1 end fraction

    Solution

    Let               space space space space space space space space space space space space space straight y space equals space fraction numerator 2 over denominator straight x plus 1 end fraction minus fraction numerator straight x squared over denominator 3 straight x minus 1 end fraction
    Differentiating both sides w.r.t. x, we get

    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

       space space space space space space space space space space space space space space equals space 2 space straight d over dx left parenthesis straight x plus 1 right parenthesis to the power of negative 1 end exponent minus fraction numerator left parenthesis 3 straight x minus 1 right parenthesis begin display style straight d over dx end style straight x squared minus straight x squared space begin display style straight d over dx end style left parenthesis 3 straight x minus 1 right parenthesis over denominator left parenthesis 3 straight x minus 1 right parenthesis squared end fraction
         space space space space space space space space space space space space space space space space space space equals space fraction numerator negative 2 over denominator left parenthesis straight x plus 1 right parenthesis squared end fraction minus fraction numerator 6 straight x squared minus 2 straight x minus 3 straight x squared over denominator left parenthesis 3 straight x minus 1 right parenthesis end fraction equals fraction numerator negative 2 over denominator left parenthesis straight x plus 1 right parenthesis squared end fraction minus fraction numerator 3 straight x squared minus 2 straight x over denominator left parenthesis 3 straight x minus 1 right parenthesis end fraction
      

    Question 42
    CBSEENMA11013194
    Wired Faculty App

    Find the derivative of the following :



    space space space space space space space fraction numerator 1 plus begin display style 1 over straight x end style over denominator 1 minus begin display style 1 over straight x end style end fraction

    Solution

    Let space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight y space equals space space fraction numerator 1 plus begin display style 1 over straight x end style over denominator 1 minus begin display style 1 over straight x end style end fraction equals fraction numerator begin display style fraction numerator straight x plus 1 over denominator straight x end fraction end style over denominator begin display style fraction numerator straight x minus 1 over denominator straight x end fraction end style end fraction equals fraction numerator straight x plus 1 over denominator straight x end fraction straight x fraction numerator straight x over denominator straight x minus 1 end fraction equals fraction numerator straight x plus 1 over denominator straight x minus 1 end fraction
    Differentiating both sides w.r.t.   x, we  get
        <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

                  space space space space space space space space space space space space space space space space space space space space space space space space equals space space space fraction numerator left parenthesis straight x minus 1 right parenthesis left parenthesis 1 plus 0 right parenthesis minus left parenthesis straight x plus 1 right parenthesis left parenthesis 1 minus 0 right parenthesis over denominator left parenthesis straight x minus 1 right parenthesis squared end fraction equals fraction numerator straight x minus 1 minus straight x minus 1 over denominator left parenthesis straight x minus 1 right parenthesis squared end fraction equals fraction numerator negative 2 over denominator left parenthesis straight x minus 1 right parenthesis squared end fraction

    Question 43
    CBSEENMA11013196
    Wired Faculty App

    Find the derivatives of the following :

    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

    Solution
    space space space space space space space space space space space straight y space equals space fraction numerator px to the power of 2 space end exponent plus qx plus straight r over denominator ax plus straight b end fraction
    Differentiating both sides w.r.t.  x, we get
    space space space space space space space space space space space space space space space space space space dy over dx equals straight d over dx open parentheses fraction numerator px squared plus qx plus straight r over denominator ax plus straight b end fraction close parentheses
     space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator left parenthesis ax plus straight b right parenthesis begin display style straight d over dx end style left parenthesis px squared plus qx plus straight r right parenthesis space minus left parenthesis px squared plus qx plus straight r right parenthesis begin display style straight d over dx end style left parenthesis ax plus straight b right parenthesis space over denominator left parenthesis ax plus straight b right parenthesis squared end fraction

              space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space space fraction numerator left parenthesis ax plus straight b right parenthesis left parenthesis 2px plus straight q right parenthesis minus left parenthesis px squared plus qx plus straight r right parenthesis left parenthesis straight a plus 0 right parenthesis over denominator left parenthesis ax plus straight b right parenthesis squared end fraction

            space space space space space space space space space space space space space space space space space space space space space space space space space equals space space space space space fraction numerator 2 ap space straight x squared aqx plus 2 bpx plus bq minus apx squared minus aqx minus ar over denominator left parenthesis ax plus straight b right parenthesis squared end fraction

                   <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
     

    Question 44
    CBSEENMA11013197
    Wired Faculty App

    Find the derivatives of the following :

    (ax+b)n (cx+ d)n 

    Solution

    Let         y  =     (ax+b)n (cx+ d)n 
    differentiating both  sides  w.r.t.   x,   we  get
    space space space space space space space space space dy over dx equals straight d over dx left square bracket left parenthesis ax plus straight b right parenthesis to the power of straight n space left parenthesis cx plus space straight d right parenthesis to the power of straight n right square bracket
      space space space space space space space space space space space space space space space space space equals space space space space left parenthesis ax space plus straight b right parenthesis to the power of straight n straight n left parenthesis cx plus straight d right parenthesis to the power of straight n minus 1 space end exponent straight c space plus space left parenthesis cx plus straight d right parenthesis to the power of straight n space straight d over dx left parenthesis ax plus straight b right parenthesis to the power of straight n
          space space space space space space space equals space space space space left parenthesis ax plus straight b right parenthesis to the power of straight n straight n left parenthesis cx plus straight d right parenthesis to the power of straight n minus 1 end exponent straight c plus left parenthesis cx plus straight d right parenthesis to the power of straight n space straight n space left parenthesis ax plus straight b right parenthesis to the power of straight n minus 1 end exponent a
        <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

          space space space space space space space space space space space space equals space straight n open parentheses ax plus straight b close parentheses to the power of straight n minus 1 end exponent open parentheses cx plus straight d close parentheses to the power of straight n minus 1 end exponent open parentheses 2 ax plus bc plus ad close parentheses

    Question 45
    CBSEENMA11013200
    Wired Faculty App

    Derivative of xn.

    Solution

    Let          f(x) = xn
                           space space space space space space space space space space space space space space space space space space space space space space space space space straight f left parenthesis straight x right parenthesis space equals space limit as straight h rightwards arrow 0 of fraction numerator straight f left parenthesis straight x plus straight h right parenthesis minus straight f left parenthesis straight x right parenthesis over denominator straight h end fraction equals limit as straight h rightwards arrow 0 of fraction numerator left parenthesis straight x plus straight h right parenthesis to the power of straight n minus left parenthesis straight x right parenthesis to the power of straight n over denominator straight h end fraction
     
                  space space space space space space space space space space space space space space space equals space space limit as straight h rightwards arrow 0 of fraction numerator open square brackets straight x open parentheses 1 plus begin display style straight h over straight x end style close parentheses close square brackets to the power of straight n minus straight x to the power of straight n over denominator straight h end fraction equals limit as straight h rightwards arrow 0 of fraction numerator straight x to the power of straight n open parentheses 1 plus begin display style straight h over straight x end style close parentheses to the power of straight n minus straight x to the power of straight n over denominator straight h end fraction
                 <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> 

                         space space space space space space space space space space equals space space space straight x to the power of straight n space space limit as straight h rightwards arrow 0 of fraction numerator open parentheses 1 plus begin display style nh over straight x end style plus begin display style fraction numerator straight n left parenthesis straight n minus 1 right parenthesis over denominator 2 factorial end fraction end style plus begin display style h squared over x squared end style plus.... close parentheses minus 1 over denominator straight h end fraction

                           space space space space space space space space space equals space straight x to the power of straight n limit as straight h rightwards arrow 0 of fraction numerator straight h open square brackets begin display style straight n over straight x end style plus begin display style fraction numerator straight n left parenthesis straight n minus 1 right parenthesis over denominator 21 end fraction end style begin display style straight h over straight x squared end style plus..... close square brackets over denominator straight h end fraction

            space space space space space space space space space space space space space space space space equals space space straight x to the power of straight n space limit as straight h rightwards arrow 0 of open square brackets straight n over straight x plus fraction numerator straight n left parenthesis straight n minus 1 right parenthesis over denominator 2 factorial end fraction space straight h over straight x squared plus.... close square brackets equals straight x to the power of straight n. end exponent straight n over straight x equals nx to the power of straight n minus 1 end exponent

           Hence,   space space space space straight d over dx left parenthesis straight x to the power of straight n right parenthesis space equals space nx to the power of straight n minus 1 end exponent

                  

    Question 46
    CBSEENMA11013202
    Wired Faculty App

    Derivative of (ax + b)n, n being a rational number.
    Solution

    Let           f(x) = (ax+b)n

     space space space space space space space space space space space space straight f left parenthesis straight x right parenthesis space equals space limit as straight h rightwards arrow 0 of fraction numerator straight f left parenthesis straight x plus straight h right parenthesis minus straight f left parenthesis straight x right parenthesis over denominator straight h end fraction equals limit as straight h rightwards arrow 0 of fraction numerator open square brackets straight a open parentheses straight x plus straight h close parentheses plus straight b close square brackets to the power of straight n minus left parenthesis ax plus straight b right parenthesis to the power of straight n over denominator straight h end fraction

         space space space space space space space space space space space space space space space space space equals space limit as straight h rightwards arrow 0 of fraction numerator left square bracket ax plus straight b plus ha right square bracket to the power of straight n minus left parenthesis ax plus straight b right parenthesis to the power of straight n over denominator straight h end fraction
         space space space space space space space space space space space equals space limit as straight h rightwards arrow 0 of left parenthesis ax plus straight b right parenthesis to the power of straight n fraction numerator open curly brackets open parentheses 1 plus begin display style fraction numerator ah over denominator ax plus straight b end fraction end style close parentheses to the power of straight n minus 1 close curly brackets over denominator straight h end fraction 

        space space space space space space space space space space space space space space space space space space space space space space equals space left parenthesis a x plus b right parenthesis to the power of n. limit as h rightwards arrow 0 of 1 over h open curly brackets open parentheses 1 plus fraction numerator n h a over denominator a x plus b end fraction plus fraction numerator n left parenthesis n minus 1 right parenthesis over denominator left enclose bottom enclose 2 end enclose end fraction. fraction numerator h squared a squared over denominator left parenthesis a x plus b right parenthesis squared end fraction plus..... close parentheses minus 1 close curly brackets

             space space space space space space space space space space space space space space space space space space space space space space equals space left parenthesis ax plus straight b right parenthesis to the power of straight n. limit as straight h rightwards arrow 0 of open curly brackets fraction numerator nha over denominator ax plus straight b end fraction plus fraction numerator straight n left parenthesis straight n minus 1 right parenthesis over denominator bottom enclose left enclose 2 end enclose end fraction. fraction numerator straight h squared straight a squared over denominator left parenthesis ax plus straight b right parenthesis squared end fraction plus...... close curly brackets

          space space space space space space space space space space space space equals space left parenthesis ax plus straight b right parenthesis to the power of straight n. limit as straight h rightwards arrow 0 of 1 over straight h open curly brackets fraction numerator na over denominator ax plus straight b end fraction plus fraction numerator straight n left parenthesis straight n minus 1 right parenthesis over denominator bottom enclose left enclose 2 end enclose end fraction. fraction numerator ha squared over denominator left parenthesis ax plus straight b right parenthesis squared end fraction plus...... close curly brackets
         space space space space space space space space equals space left parenthesis ax plus straight b right parenthesis to the power of straight n open parentheses fraction numerator na over denominator ax plus straight b end fraction plus 0 close parentheses equals na left parenthesis ax plus straight b right parenthesis to the power of straight n minus 1 end exponent
    Hence,  space space space space space space straight d over dx left parenthesis ax plus straight b right parenthesis to the power of straight n equals space na left parenthesis ax plus straight b right parenthesis to the power of straight n minus 1 end exponent
    Corollary :  Put a = 1m, b = 0

    space space space space space space straight d over dx left parenthesis straight x to the power of straight n right parenthesis space equals space nx to the power of straight n minus 1 end exponent


             

    Question 47
    CBSEENMA11013203
    Wired Faculty App

    Derivative of sin x :

    Solution

    Let       space space space space space space straight f left parenthesis straight x right parenthesis space equals space space sin space straight x comma space Therefore comma space straight f left parenthesis straight x right parenthesis space space equals space limit as straight h rightwards arrow 0 of fraction numerator straight f left parenthesis straight x plus straight h right parenthesis minus straight f left parenthesis straight x right parenthesis over denominator straight h end fraction
                 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space limit as straight h rightwards arrow 0 of fraction numerator sin left parenthesis straight x plus straight h right parenthesis minus sinx over denominator straight h end fraction equals limit as straight h rightwards arrow 0 of fraction numerator 2 cos open parentheses begin display style fraction numerator straight x plus straight h plus straight x over denominator 2 end fraction end style close parentheses sin open parentheses begin display style fraction numerator straight x plus straight h minus straight x over denominator 2 end fraction end style close parentheses over denominator straight h end fraction
    space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space limit as straight h rightwards arrow 0 of fraction numerator 2 cos open parentheses straight x straight h over 2 close parentheses sin space begin display style straight h over 2 end style over denominator straight h end fraction equals space limit as straight h rightwards arrow 0 space cos space open parentheses straight x straight h over 2 close parentheses. limit as straight h over 2 rightwards arrow 0 of of fraction numerator sin begin display style straight h over 2 end style over denominator begin display style straight h over 2 end style end fraction
                                                                                    
                           = cos (x+0).1 = cos x                                     space space space space space space space space space space open square brackets because limit as straight x rightwards arrow 0 of fraction numerator sin space straight x over denominator straight x end fraction equals 1 close square brackets
      So,                space space space space space space space space space space space space space space space space space space space space space space space space straight d over dx left parenthesis sinx right parenthesis space equals space xos space straight x space for space all space straight x element of straight Ro,    

    Question 48
    CBSEENMA11013205
    Wired Faculty App

    Find the derivatives of the following functions :

    x100 

    Solution

    LKet  y = x100 

    ∴   space space space space space space space space space space space dy over dx equals 100 straight x to the power of 100 minus 1 end exponent equals 100 straight x to the power of 99           space space space space space space space space space space space space space space space open square brackets because straight d over dx left parenthesis straight x to the power of straight n right parenthesis equals nx to the power of straight n minus 1 end exponent close square brackets

    Question 49
    CBSEENMA11013207
    Wired Faculty App

    Find the derivatives of the following functions :

    6x100 - x55 + x

    Solution

    Let     y  = 6x100 - x55 + x

    ∴     <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
      space space space space space space space space space space equals space 6 straight d over dx left parenthesis straight x to the power of 100 right parenthesis minus straight d over dx left parenthesis straight x to the power of 55 right parenthesis plus straight d over dx straight x to the power of 1     <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

            space space equals 6 straight x 100 straight x to the power of 100 minus 1 end exponent minus 55 straight x to the power of 55 minus 1 end exponent plus 1 straight x to the power of 1 minus 1 end exponent      <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
       <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

    Question 50
    CBSEENMA11013209
    Wired Faculty App

    Find the derivatives of the following functions :

    space space space fraction numerator 1 over denominator square root of straight x end fraction

    Solution

    Let        space space space space space space space space straight y space equals space fraction numerator 1 over denominator square root of straight x end fraction equals straight x to the power of negative 1 half end exponent

    ∴       space space space space space space space space space dy over dx equals straight d over dx left parenthesis straight x to the power of negative 1 half end exponent right parenthesis space equals space minus 1 half straight x to the power of negative 1 half minus 1 end exponent         space space space space space space open square brackets because straight d over dx left parenthesis straight x to the power of straight n right parenthesis equals space nx to the power of straight n minus 1 end exponent close square brackets

           

                    space space space space space space space equals space space minus 1 half straight x to the power of negative 3 over 2 end exponent
    Question 51
    CBSEENMA11013210
    Wired Faculty App

    Find the derivative of space space space straight y space equals space 10 to the power of 10 plus 10 over straight x plus straight x over 10 plus 10 straight x plus straight x to the power of 10
    Solution

    Given space space space space space space space space space space space space space space space space space space straight y space equals 10 to the power of 10 plus 10 over straight x plus straight x over 10 plus 10 straight x plus straight x to the power of 10



    Question 52
    CBSEENMA11013212
    Wired Faculty App

    Find the derivative of (2x+3)(5x2-7x+1)
    Solution

    Let             f(x) =   (2x+3)(5x2-7x+1)

            <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
                                                                                                                              (Byn Product rule)

                                 = (2x+3) (10x-7)+(5x2 -7x+1)(2)
                                  
                                 = 20x2 -14x + 30x -21 + 10x2 -14x + 2
                                 = 30x2 + 2x - 19

    Question 53
    CBSEENMA11013220
    Wired Faculty App

    Differentiate the following functions w.r.t. x


    space space space space space space space space space space space space space space space space space space space space space space space space space fraction numerator straight x minus 1 over denominator 3 straight x cubed plus 2 end fraction comma space straight x space not equal to space cube root of fraction numerator negative 2 over denominator 3 end fraction end root

    Solution

    Let     space space space space space space space space space space space space space space space space space space space space space straight f left parenthesis straight x right parenthesis space equals space fraction numerator straight x minus 1 over denominator 3 straight x cubed plus 2 end fraction
     space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight f left parenthesis straight x right parenthesis space space equals space fraction numerator left parenthesis 3 straight x cubed plus 2 right parenthesis begin display style straight d over dx end style left parenthesis straight x minus 1 right parenthesis minus left parenthesis straight x minus 1 right parenthesis begin display style straight d over dx end style left parenthesis 3 straight x cubed plus 2 right parenthesis over denominator open parentheses 3 straight x cubed plus 2 close parentheses squared end fraction        (By Quotient Rule)
              space space space space space space space space space space space space space space space space space space space equals space space space space fraction numerator left parenthesis 3 straight x cubed plus 2 right parenthesis left parenthesis 1 minus 0 right parenthesis minus left parenthesis straight x minus 1 right parenthesis left parenthesis 9 straight x squared plus 0 right parenthesis over denominator left parenthesis 3 straight x cubed plus 2 right parenthesis squared end fraction equals fraction numerator 3 straight x cubed plus 2 minus 9 straight x cubed plus 9 straight x squared over denominator left parenthesis 3 straight x cubed plus 2 right parenthesis squared end fraction


                    space space space space space space space space space space space space space equals space space fraction numerator negative 6 straight x cubed plus 9 straight x squared plus 2 over denominator left parenthesis 3 straight x cubed plus 2 right parenthesis squared end fraction

    Question 54
    CBSEENMA11013224
    Wired Faculty App

    Solve   space space space space space space space space space space limit as straight z rightwards arrow 1 of fraction numerator straight z to the power of begin display style 1 third end style end exponent minus 1 over denominator straight z to the power of begin display style 1 over 6 end style end exponent minus 1 end fraction
    Solution
    space space space space space space space space space space space space space space limit as straight z rightwards arrow 1 of fraction numerator straight z to the power of begin display style 1 third end style end exponent minus 1 over denominator straight z to the power of begin display style 1 over 6 end style minus 1 end exponent end fraction equals limit as straight z rightwards arrow 1 of fraction numerator open parentheses straight z begin display style fraction numerator 1 third minus 1 over denominator straight z minus 1 end fraction end style close parentheses over denominator open parentheses straight z fraction numerator 1 over 6 minus 1 over denominator straight z minus 1 end fraction close parentheses end fraction equals fraction numerator limit as straight z rightwards arrow 1 of begin display style fraction numerator straight z to the power of begin display style 1 third end style end exponent minus 1 over denominator straight z minus 1 end fraction end style over denominator limit as straight z rightwards arrow 1 of fraction numerator straight z to the power of begin display style 1 over 6 end style end exponent minus 1 over denominator straight z minus 1 end fraction end fraction equals fraction numerator begin display style 1 third end style left parenthesis 1 right parenthesis to the power of begin display style 1 third end style minus 1 end exponent over denominator begin display style 1 over 6 end style left parenthesis 1 right parenthesis to the power of begin display style 1 over 6 end style minus 1 end exponent end fraction equals 1 third straight x 6 over 1 equals 2
                                                                                          (By dividing Num and den by z -1)
    Question 55
    CBSEENMA11013226
    Wired Faculty App

    If <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>,  find n.
    Solution

    Now,   space space space space space space space space space space space limit as straight x rightwards arrow 2 of fraction numerator straight x to the power of straight n minus 2 to the power of straight n over denominator straight x minus 2 end fraction equals 80
    rightwards double arrow   space space space space space space space space space space space space space space space space straight n.2 to the power of straight n minus 1 space equals space 80
    rightwards double arrow          space space space space space space space straight n.2 to the power of straight n minus 1 end exponent equals 80
    rightwards double arrow            space space space space straight n.2 to the power of straight n minus 1 end exponent equals 5.16
    rightwards double arrow        <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    rightwards double arrow           space space space space space straight n space equals space 5

    Question 56
    CBSEENMA11013228
    Wired Faculty App

    Evaluate  <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Solution

    It is of 0 over 0 form.

    ∴         space space space space space space space space space space space space space space space space space space space space space space space space space limit as straight x rightwards arrow 1 of fraction numerator left parenthesis straight x plus straight x squared plus straight x cubed plus.... plus straight x to the power of straight n right parenthesis minus straight n over denominator straight x minus 1 end fraction equals limit as straight x rightwards arrow 1 of fraction numerator left parenthesis straight x minus 1 right parenthesis plus left parenthesis straight x squared minus 1 right parenthesis plus left parenthesis straight x cubed minus 1 right parenthesis plus... plus left parenthesis straight x to the power of straight n minus 1 right parenthesis over denominator straight x minus 1 end fraction

        space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space limit as x rightwards arrow 1 of open square brackets 1 plus fraction numerator x squared minus 1 over denominator x minus 1 end fraction plus fraction numerator x cubed minus 1 over denominator x minus 1 end fraction plus.... plus fraction numerator x to the power of n minus 1 over denominator x minus 1 end fraction close square brackets
                      space space space space space space space space space space space space space space space space space space space space space equals space limit as straight x rightwards arrow 1 of 1 plus limit as straight x rightwards arrow of fraction numerator straight x squared minus 1 over denominator straight x minus 1 end fraction plus limit as straight x rightwards arrow 1 of fraction numerator straight x cubed minus 1 over denominator straight x minus 1 end fraction plus...... plus limit as straight x rightwards arrow 1 of fraction numerator straight x to the power of straight n minus 1 over denominator straight x minus 1 end fraction
                                       = 1 + 2(1)2 + 3(1)2 + ......+ n(1)n-1 = 1 + 2 + 3 + ..... +n
                                          
                                space space space space space space space space space equals space sum subscript straight n fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction
     


    Question 57
    CBSEENMA11013231
    Wired Faculty App

    Differentiate the following functions w.r.t. x


    space space space space space space space space space space space space fraction numerator left parenthesis straight x plus 2 right parenthesis left parenthesis 1 minus 3 straight x right parenthesis over denominator 2 straight x plus 1 end fraction semicolon space space straight x space not equal to space fraction numerator negative 1 over denominator 2 end fraction

    Solution

    Let                 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight f left parenthesis straight x right parenthesis space space equals space space space fraction numerator left parenthesis straight x plus 2 right parenthesis left parenthesis 1 minus 3 straight x right parenthesis over denominator 2 straight x plus 1 end fraction semicolon space straight x space not equal to fraction numerator negative 1 over denominator 2 end fraction space space space or space straight f left parenthesis straight x right parenthesis space space equals space fraction numerator negative 3 straight x squared minus 5 straight x plus 2 over denominator 2 straight x plus 1 end fraction

    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
                                                                                                                        (By Quotient Rule)

           space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator left parenthesis 2 straight x plus 1 right parenthesis left parenthesis negative 6 straight x minus 5 right parenthesis minus left parenthesis negative 3 straight x squared minus 5 straight x plus 2 right parenthesis left parenthesis 2 plus 0 right parenthesis over denominator left parenthesis 2 straight x plus 1 right parenthesis squared end fraction
            space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space space space fraction numerator negative 12 straight x squared minus 10 straight x minus 6 straight x minus 5 plus 6 straight x squared plus 10 straight x minus 4 over denominator left parenthesis 2 straight x plus 1 right parenthesis squared end fraction equals fraction numerator negative 6 straight x squared minus 6 straight x minus 9 over denominator left parenthesis 2 straight x plus 1 right parenthesis squared end fraction

    Question 58
    CBSEENMA11013232
    Wired Faculty App

    Find the derivatives of the following :



    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

    Solution
    space space space space space space space space space space space space space 3 over 2 square root of straight x minus fraction numerator 1 over denominator 2 square root of straight x end fraction plus 1 over straight x to the power of 3 square root of 2 end exponent
    Question 59
    CBSEENMA11013234
    Wired Faculty App

    Find the derivatives of the following :



    space space space space space space space space space space fraction numerator 3 straight x plus 2 over denominator left parenthesis straight x plus 5 right parenthesis left parenthesis 2 straight x plus 1 right parenthesis plus 3 end fraction

    Solution
    space space space space space space space space space space space space space space minus space fraction numerator 2 left parenthesis 3 straight x squared plus 4 straight x minus 1 right parenthesis over denominator left parenthesis 2 straight x squared plus 11 straight x plus 8 right parenthesis squared end fraction
    Question 60
    CBSEENMA11013235
    Wired Faculty App

    Find the derivatives of the following :


    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

    Solution
    space space space space space space space space space space space space space space space space space space space space space space 1 minus 1 over straight x squared
    Question 61
    CBSEENMA11013236
    Wired Faculty App

    Find the derivatives of the following :


    space space space space space space space space space space space space space space space space space space space fraction numerator ax squared plus bx plus straight c over denominator px squared plus qx plus straight r end fraction

    Solution
    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Question 62
    CBSEENMA11013238
    Wired Faculty App

    Find  space space space space space space space space space space space space space space space space space space space space dy over dx space space if space left parenthesis straight i right parenthesis space straight y space space equals space fraction numerator 1 minus square root of straight x over denominator 1 plus square root of straight x end fraction
    Solution
    space space space space space space space space space space space space space space space fraction numerator negative 1 over denominator square root of straight x left parenthesis 1 plus square root of straight x right parenthesis squared end fraction
    Question 63
    CBSEENMA11013239
    Wired Faculty App

    Find  space space space space space space space space space space space space space space space space space space space space space space space space space space space space dy over dx space space if space left parenthesis straight i right parenthesis space straight y space space equals space fraction numerator straight x squared minus straight x plus 1 over denominator straight x squared plus straight x plus 1 end fraction
    Solution
    space space space space space space space space space space space space space space space space space space fraction numerator 2 left parenthesis straight x squared minus 1 right parenthesis over denominator left parenthesis straight x squared plus straight x plus 1 right parenthesis squared end fraction
    Question 64
    CBSEENMA11013242
    Wired Faculty App

    Derivative of tan x :
    Solution

    Let       f(x) =, tan x,  Therefore,    space space space space space space space space space space space space space space space space straight f left parenthesis straight x right parenthesis space equals space limit as straight h rightwards arrow 0 of fraction numerator straight f left parenthesis straight x plus straight h right parenthesis minus straight f left parenthesis straight x right parenthesis over denominator straight h end fraction
    space space space space space space space space space space space space space space space equals space space limit as straight h rightwards arrow 0 of fraction numerator tan left parenthesis straight x plus straight h right parenthesis minus tanx over denominator straight h end fraction equals limit as straight h rightwards arrow 0 of fraction numerator begin display style fraction numerator sin left parenthesis straight x plus straight h right parenthesis over denominator cos left parenthesis straight x plus straight h right parenthesis end fraction end style minus begin display style fraction numerator sin space straight x over denominator cos space straight x end fraction end style over denominator straight h end fraction

     space space space space space space space space space space space space space space space space space space space space space space equals space space limit as straight h rightwards arrow 0 of fraction numerator sin left parenthesis straight x plus straight h right parenthesis cps space straight x minus space sin space straight x space cos space left parenthesis straight x plus straight h right parenthesis over denominator cos left parenthesis straight x plus straight h right parenthesis. cos space straight x. straight h end fraction equals limit as straight h rightwards arrow 0 of fraction numerator sin left parenthesis straight x plus straight h minus straight x right parenthesis over denominator straight h space cos space straight x. space cos space left parenthesis straight x plus straight h right parenthesis end fraction

            space space space space space space space space space space space space space equals space space space limit as straight h rightwards arrow 0 of fraction numerator sin space space straight h over denominator straight h end fraction. space limit as straight h rightwards arrow 0 of fraction numerator 1 over denominator cos space straight x. space cos space left parenthesis straight x plus straight h right parenthesis end fraction               

    space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space space space 1. fraction numerator 1 over denominator cos space straight x space. space cos space straight x end fraction equals space fraction numerator 1 over denominator cos squared straight x end fraction equals space space space sec squared straight x space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space space limit as straight x rightwards arrow 0 of fraction numerator sin space straight x over denominator straight x end fraction equals 1 close square brackets   

    So,  space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight d over dx left parenthesis tan space straight x right parenthesis space equals space sec squared straight x space for space all space straight x space element of space straight R space open parentheses straight R space minus space odd space multiples space of space straight pi over 2 close parentheses  

    Question 65
    CBSEENMA11013243
    Wired Faculty App

    Derivative of cot x :

    Solution
    Question 66
    CBSEENMA11013245
    Wired Faculty App

    Derivative of sec x :  

    Solution
    Question 67
    CBSEENMA11013246
    Wired Faculty App

    Derivative of cosec x :  

    Solution
    Question 68
    CBSEENMA11013248
    Wired Faculty App

    Find the derivatives of the following functions from first principle :

    sin (x+1)

    Solution
    Lef f(x)  = sin (x+1)


    Question 69
    CBSEENMA11013250
    Wired Faculty App

    Find the derivatives of the following functions from first principle :

    space space space space space space space space space space space space space space open parentheses straight x minus straight pi over 8 close parentheses

    Solution
    Let      <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Question 70
    CBSEENMA11013252
    Wired Faculty App

    Find the derivatives of the following functions :

    5 sin x - 6 cos x +7
    Solution

    Let y = 5 sin x - 6 cos x +7
    Differntiating both sides w.r.t. x, we get.
    space space space space space space space space space space space space space space space dy over dx equals space straight d over dx left parenthesis 5 space sin space straight x space minus space 6 space cos space straight x space plus space 7 right parenthesis space equals space 5 space straight d over dx left parenthesis sin space straight x right parenthesis space minus space 6 straight d over dx cos space straight x space plus space straight d over dx left parenthesis 7 right parenthesis
                   = 5cos x - 6(-sin x)+0 - 5 cos x + 6 sin x

    Question 71
    CBSEENMA11013254
    Wired Faculty App

    Find the derivatives of the following functions :

    5 sec x + 4 cos x
    Solution

    Let y = 5 sec x + 4 cos x
    Diffentiating  both sides w.r.t.  x, we get
    space space space space space space space space space space space space space space space space space space space space space space space space space space space dy over dx equals straight d over dx left parenthesis 5 space sec space straight x space plus space 4 space cos space straight x right parenthesis space equals space 5 straight d over straight x left parenthesis sec space straight x right parenthesis space plus space 4 straight d over dx left parenthesis cos space straight x right parenthesis

                           = 5 (sec x tan x) + (-sin x) = 5 sec x tan x -4 sin x

    Question 72
    CBSEENMA11013255
    Wired Faculty App

    Find the derivatives of the following functions :

    sin x cos x
    Solution

    Let y sin x cos x 
    Differentiating both sides w.r.t.  x,  we get

    space space space space space space space space space space space space space space space space space space space space space space space space dy over dx space equals space straight d over dx left parenthesis sin space straight x space cos space straight x space right parenthesis space equals space sin space straight x space straight d over dx left parenthesis cos space straight x right parenthesis space plus space cos space straight x straight d over dx left parenthesis sin space straight x right parenthesis
    space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space sin space straight x left parenthesis negative sin space straight x right parenthesis space plus space cos space straight x left parenthesis cos space straight x right parenthesis space equals space minus space sin squared straight x plus cos squared straight x space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space cos squared straight x minus sin squared straight x equals cos 2 straight x

    Question 73
    CBSEENMA11013256
    Wired Faculty App

    Find the derivatives of the following functions :

    2 tan x - 7 sec x
    Solution

    Let y = 2tan x-7 sec x
    Differentiating both sides w.r.t.  x, we get
    space space space space space space space space space space space space space space space space space space space space space space space space space space dy over dx equals straight d over dx left parenthesis 2 space tan space straight x space minus space 7 space sec space straight x space right parenthesis space equals space straight d over dx left parenthesis 2 tan space straight x right parenthesis space minus straight d over dx left parenthesis 7 sec space straight x right parenthesis

              space space space space space space space space space space space space space space space space space space space space equals space 2 straight d over dx tan space straight x space minus 7 straight d over dx sec space straight x space equals space 2 space sex squared straight x space minus space 7 space sec space straight x space tan space straight x

    Question 74
    CBSEENMA11013260
    Wired Faculty App

    Find the derivatives of the following functions :

    space space space space space space space space fraction numerator sec space straight x space minus 1 over denominator sec space straight x space plus 1 end fraction

    Solution

    Let    space space space space space space space space space space space space space space space straight y space equals space fraction numerator sec space straight x space minus 1 over denominator sec space straight x plus 1 end fraction
    Differentiating both sides w.r.t.   s,  we get
    space space space space space space space space space space space space space dy over dx equals straight d over dx open parentheses fraction numerator sec space straight x minus 1 over denominator sec space straight x space plus 1 end fraction close parentheses equals fraction numerator left parenthesis sec space straight x plus 1 right parenthesis begin display style straight d over dx end style left parenthesis sec space straight x minus 1 right parenthesis space minus space left parenthesis sec space straight x minus 1 right parenthesis begin display style straight d over dx end style left parenthesis sec space straight x plus 1 right parenthesis over denominator left parenthesis sec space straight x space plus 1 right parenthesis squared end fraction

      space space space space space space space space space space space space space space space space equals space fraction numerator left parenthesis sec space straight x plus 1 right parenthesis left parenthesis sec space straight x space tan space straight x minus 0 right parenthesis minus left parenthesis sec space straight x minus 1 right parenthesis left parenthesis sec space straight x space tan space straight x plus 0 right parenthesis over denominator left parenthesis sec space straight x plus 1 right parenthesis squared end fraction
    space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator sec squared straight x space tan space straight x space plus sec space straight x space tan space straight x minus sec squared space xtan space straight x space plus space sec space straight x space tan space straight x over denominator left parenthesis sec space straight x space plus 1 right parenthesis squared end fraction equals fraction numerator 2 space sec space straight x space tan space straight x over denominator left parenthesis sec space straight x space plus 1 right parenthesis squared end fraction

    Question 75
    CBSEENMA11013262
    Wired Faculty App

    Find the derivatives of the following functions :


    space space space space fraction numerator sin left parenthesis straight x plus straight a right parenthesis over denominator cos space straight x end fraction

    Solution

    Let y <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Differentiating both sides w.r.t.   x,  we  get

    space space space space space space space space space space space dy over dx equals straight d over dx left square bracket cos space straight a space tan space straight x space plus space sin space straight a right square bracket space equals space cos space straight a space straight d over dx tan space straight x space plus straight d over dx sin space straight a
              = cos a sec2x +0 = cos a sec2x

    Question 76
    CBSEENMA11013264
    Wired Faculty App

    Find the derivatives of the following functions :


    (ax2+sin x) (p + q cos x)

    Solution

    Let y = (ax2+sin x) (p + q cos x)
    Differentiating both sides w.r.t. x, we get

    space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space dy over dx equals straight d over dx left square bracket ax squared space plus sin space straight x right square bracket space straight d over dx left parenthesis straight p plus straight q space cos space straight x right parenthesis space plus space left parenthesis straight p space plus space straight q space cos space straight x right parenthesis space straight d over dx space left parenthesis ax squared plus sinx right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space left parenthesis ax squared plus sin space straight x right parenthesis space left square bracket 0 plus straight q left parenthesis negative sin space straight x right parenthesis right square bracket plus left parenthesis straight p plus straight q space cos space straight x 0 left square bracket 2 ax space plus cos space straight x right square bracket space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space minus straight q space sin space straight x left parenthesis ax squared plus sin space straight x right parenthesis space plus space left parenthesis space straight p plus straight w plus cos space straight x right parenthesis left parenthesis 2 ax space plus cos space straight x right parenthesis

    Question 77
    CBSEENMA11013268
    Wired Faculty App

    Find the derivatives of the following functions :


    "<pre  

    Solution

    Let     space space space space space space space space space space space space space space space straight y space equals space fraction numerator straight x squared cos open parentheses begin display style straight pi over 4 end style close parentheses over denominator sin space straight x end fraction space equals space space cos open parentheses straight pi over 4 close parentheses straight x squared cosec space straight x
    Differentiating both sides w.r.t.  x,  we get

    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
               <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

      space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space straight x space cos open parentheses straight pi over 4 close parentheses cosec space straight x space left square bracket 2 minus straight x space cot space straight x right square bracket

    Question 78
    CBSEENMA11013272
    Wired Faculty App

    Find the derivatives of the following functions :

    (x + sec x)(x - tan x)

    Solution

    Let     y  =  (x + sec x)(x - tan x)
    Differentiating bioth  sides w.r.t.    x, we  get

    space space space space space space space space space space space space space space space space space space space space space space space space space space space space dy over dx equals space space left parenthesis straight x space plus space sec space straight x right parenthesis space straight d over dx left parenthesis straight x space minus space tan space straight x right parenthesis plus left parenthesis straight x space minus space tan space straight x right parenthesis straight d over dx left parenthesis straight x plus sec space straight x right parenthesis
               space space space space space space space space space space space space space space space space space space space equals space left parenthesis straight x plus sec right parenthesis space left parenthesis 1 plus space sec squared space straight x right parenthesis plus left parenthesis straight x minus tan space straight x right parenthesis space left parenthesis 1 plus sec space straight x space tan space straight x right parenthesis

    Question 79
    CBSEENMA11013274
    Wired Faculty App

    Find the derivatives of the following functions :

    space space space space space space space space space space space space space space space space space space fraction numerator straight x over denominator 1 plus tan space straight x end fraction

    Solution

    Let     space space space space space space space space space space space space fraction numerator straight x over denominator 1 plus tan space straight x end fraction
    Differentiating both sides w.r.t. x, we get
    space space space space space space space space space space space space space space space space space space space space space space space dy over dx equals fraction numerator left parenthesis 1 plus tan space straight x right parenthesis begin display style straight d over dx end style straight x minus straight x begin display style straight d over dx end style left parenthesis 1 plus tan space straight x right parenthesis over denominator blank end fraction equals space fraction numerator left parenthesis 1 plus tan space straight x right parenthesis space 1 minus straight x left parenthesis 0 plus sec squared straight x right parenthesis over denominator left parenthesis 1 plus tan space straight x right parenthesis squared end fraction
                      space space space space space space space space space space space space space space space space space equals space space fraction numerator 1 plus tan space straight x space minus space straight x space sec squared straight x over denominator left parenthesis 1 plus tan space straight x right parenthesis squared end fraction

    Sponsor Area

    Question 80
    CBSEENMA11013592
    Wired Faculty App

    Evaluate the following limits :


    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Solution

    (i) 12         (ii) 6  space space space space space space space space space space space space space space space space space open parentheses iii close parentheses space space space straight pi space space space space space space space space space space space space space space left parenthesis iv right parenthesis space space space space space minus 1 fourth

    Question 81
    CBSEENMA11013596
    Wired Faculty App

    Evaluate the following limits :

    space space space space space space space space space space space space space space space space space space left parenthesis straight i right parenthesis space space limit as straight x rightwards arrow 1 of fraction numerator straight x to the power of 15 minus 1 over denominator straight x to the power of 10 minus 1 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space left parenthesis ii right parenthesis space limit as straight x rightwards arrow 0 of fraction numerator left parenthesis straight x plus 1 right parenthesis to the power of 5 minus 1 over denominator straight x end fraction
    Solution
    space space space space space space left parenthesis straight i right parenthesis space 3 over 2         (ii) 5
    Question 84
    CBSEENMA11013604
    Wired Faculty App

    Evaluate  space space space limit as straight x rightwards arrow straight a of fraction numerator straight x cubed minus straight a cubed over denominator straight x squared minus straight a squared end fraction semicolon space space straight a not equal to 0.
    Solution
    space space space space space space space 3 over 2 straight a
    Question 85
    CBSEENMA11013605
    Wired Faculty App

    Evaluate  <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Solution

    2

    Question 86
    CBSEENMA11013606
    Wired Faculty App

    Evaluate   space space space space space space space space space space space space space space space space space space space space space space left parenthesis straight i right parenthesis space stack space lim with straight x rightwards arrow 0 below fraction numerator tan space straight x space minus space sin space straight x over denominator sin cubed straight x end fraction space space space space space space space space space space space space space left parenthesis ii right parenthesis space space limit as straight x rightwards arrow 0 space of fraction numerator sinx minus 2 sin 3 straight x minus sin 5 straight x over denominator straight x end fraction
    Solution
    space space space space space space space space left parenthesis straight i right parenthesis space 1 half     (ii) 0
    Question 87
    CBSEENMA11013607
    Wired Faculty App

    Evaluate  space space space space space space space space space space space limit as straight x rightwards arrow 0 of fraction numerator 2 sin space straight x space minus sin space 2 straight x over denominator straight x cubed end fraction
    Solution

    1

    Question 88
    CBSEENMA11013608
    Wired Faculty App

    Evaluate : space space space space space space space space space limit as straight x rightwards arrow straight n divided by 2 of left parenthesis sec space straight x space minus space tan space straight x space right parenthesis
    Solution

    0

    Question 89
    CBSEENMA11013609
    Wired Faculty App

    Evaluate : space space space space space space space space space space limit as straight X rightwards arrow 0 equals straight N divided by 6 of fraction numerator square root of 3 sin space straight x space minus space cos space straight x over denominator straight x space minus space straight pi divided by 6 end fraction
    Solution

    2

    Question 90
    CBSEENMA11013610
    Wired Faculty App

    Evaluate :   space space space space space space space space space space space space limit as straight x rightwards arrow straight pi divided by 6 of fraction numerator cot squared straight x minus 3 over denominator cosec space straight x space minus 2 end fraction
    Solution

    4

    Question 91
    CBSEENMA11013611
    Wired Faculty App

    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

    Solution

    1

    Question 92
    CBSEENMA11013612
    Wired Faculty App

    Find k if stack space space space space space space lim with space space space space space space space space space straight x rightwards arrow 0 below space straight x space cosec space kx.
    Solution
    space space space space straight k space equals plus-or-minus 1
    Question 93
    CBSEENMA11013613
    Wired Faculty App

    Evaluate  space space space space space space space space space space limit as straight x rightwards arrow 0 to the power of minus of fraction numerator straight x over denominator open vertical bar straight x close vertical bar plus straight x squared end fraction
    Solution

    -1

    Question 94
    CBSEENMA11013614
    Wired Faculty App

    Evaluate  <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Solution
    space space space space 35 over 18 straight a to the power of 17 divided by 30 end exponent
    Question 95
    CBSEENMA11013615
    Wired Faculty App

    Find the derivatives of followings functions from first principle.


    (i) sin 2x        (ii) cot (3x+1)           space space space space space space space space space space space space space space space left parenthesis iii right parenthesis space space space fraction numerator 1 over denominator square root of ax plus straight b end root end fraction
    Solution

    (i) 2 cos 2x       (ii) -3 cosec2 (3x + 1)       space space space space space space space space space left parenthesis iii right parenthesis space minus straight a over 2 left parenthesis ax plus straight b right parenthesis to the power of negative 3 divided by 2 end exponent

    Question 96
    CBSEENMA11013616
    Wired Faculty App

    Find the derivative of sin2x from first principle.
    Solution

    sin  2x

    Question 98
    CBSEENMA11013618
    Wired Faculty App

    Differentiate following functions w.r.t. x

    (i) x cosec x    (ii) (1-2 tan x) (5 +4 sin x) (iii) x4 tan x    (iv) x sin x + cos x
    Solution

    (I) (1 - x cot x) cosec x   (ii) 4 (1 - 2 tan x) cos x -2 (5 +4 sin x) sec2x  (iii) x3 (x xec2x + 4 tan x)

    Question 99
    CBSEENMA11013619
    Wired Faculty App

    Differentiate the following functions w.r.t. x.

    space space space space space space space space space space space space space space space space left parenthesis straight i right parenthesis space space space space fraction numerator left parenthesis straight x squared plus 1 right parenthesis left parenthesis straight x plus 3 right parenthesis over denominator straight x end fraction comma space straight x not equal to 0 space space space space space left parenthesis ii right parenthesis space space open parentheses straight x plus 1 over straight x close parentheses squared space space space space space space space space left parenthesis iii right parenthesis space space space fraction numerator 2 straight x plus 3 over denominator straight x squared minus 5 end fraction
    Solution
    space space space space space space space space space space space space space space space space space space space left parenthesis straight i right parenthesis space space space 2 straight x plus 3 minus 3 over straight x squared space space space space space space space space left parenthesis ii right parenthesis space space space 2 straight x minus 2 over straight x cubed space space space space space space space left parenthesis iii right parenthesis space space space fraction numerator negative 2 left parenthesis straight x squared plus 3 straight x plus 5 right parenthesis over denominator left parenthesis straight x squared minus 5 right parenthesis squared end fraction
    Question 100
    CBSEENMA11015471
    Wired Faculty App

    limit as straight x space rightwards arrow 0 of space fraction numerator sin begin display style left parenthesis end style begin display style straight pi end style begin display style space end style begin display style begin display style cos end style squared end style begin display style straight x end style begin display style right parenthesis end style over denominator straight x squared end fraction space is space equal space to

    • π/2

    • 1

    • -π

    • π
    Solution

    D.

    π

    limit as straight x space rightwards arrow 0 of space fraction numerator sin begin display style space end style begin display style left parenthesis end style begin display style straight pi end style begin display style space end style begin display style begin display style cos end style squared end style begin display style straight x end style begin display style right parenthesis end style over denominator straight x squared end fraction space equals space limit as straight x space rightwards arrow 0 of space fraction numerator sin begin display style space end style begin display style straight pi end style begin display style left parenthesis end style begin display style 1 end style begin display style minus end style begin display style begin display style sin end style squared end style begin display style straight x end style begin display style right parenthesis end style over denominator straight x squared end fraction space equals space limit as straight x space rightwards arrow 0 of space fraction numerator sin begin display style left parenthesis end style begin display style straight pi end style begin display style minus end style begin display style begin display style πsin end style squared end style begin display style straight x end style begin display style right parenthesis end style over denominator straight x squared end fraction equals limit as straight x space rightwards arrow 0 of space fraction numerator sin begin display style left parenthesis end style begin display style straight pi end style begin display style space end style begin display style begin display style sin end style squared end style begin display style space end style begin display style straight pi end style begin display style right parenthesis end style over denominator straight x squared end fraction space space space space space left square bracket because space sin space left parenthesis straight pi space minus space straight theta space equals space sin space straight theta right square bracket equals space limit as straight x space rightwards arrow 0 of space fraction numerator sin begin display style space end style begin display style begin display style πsin end style squared end style begin display style straight x end style over denominator straight pi space sin squared straight x end fraction space straight x space left parenthesis straight pi right parenthesis open parentheses fraction numerator begin display style sin squared end style begin display style straight x end style over denominator straight x squared end fraction close parentheses space equals space straight pi space open square brackets because space limit as straight theta rightwards arrow 0 of space fraction numerator sin begin display style space end style begin display style straight theta end style over denominator straight theta end fraction space equals 1 space close square brackets
    Question 101
    CBSEENMA11015492
    Wired Faculty App

    limit as straight x space rightwards arrow 0 of fraction numerator left parenthesis 1 minus cos space 2 straight x right parenthesis left parenthesis 3 plus cos space straight x right parenthesis over denominator straight x space tan space 4 straight x end fraction space is space equal space to space

    • -1/4

    • 1/2

    • 1

    • 2
    Solution

    D.

    2

    Let space l space equals space space limit as straight x rightwards arrow 0 of space fraction numerator left parenthesis 1 minus cos space 2 straight x right parenthesis over denominator straight x squared end fraction fraction numerator left parenthesis 3 plus cos space straight x right parenthesis over denominator 1 end fraction. fraction numerator straight x over denominator tan space 4 straight x end fraction equals space limit as straight x rightwards arrow 0 of space fraction numerator 2 space sin squared straight x over denominator straight x squared end fraction. fraction numerator 3 plus cosx over denominator 1 end fraction. fraction numerator straight x over denominator tan space 4 straight x end fraction equals space 2 space limit as straight x rightwards arrow 0 of open parentheses fraction numerator sin space straight x over denominator straight x end fraction close parentheses squared. space limit as straight x rightwards arrow 0 of left parenthesis 3 plus space cos space straight x right parenthesis. space limit as straight x rightwards arrow 0 of fraction numerator 4 straight x over denominator space 4 space tan space 4 straight x end fraction 2.4.1 fourth space equals space 2
    Question 102
    CBSEENMA11015531
    Wired Faculty App

    The equation of the tangent to the curve y = x +4/x2, that is parallel to the x-axis, is

    • y= 0

    • y= 1

    • y= 2 

    • y= 3
    Solution

    D.

    y= 3

    We have, 
    straight y space equals space straight x space plus space 4 over straight x squared
    On differentiating w.r.t x, we get
    dy over dx space equals space 1 space minus space 8 over straight x cubed
    since the tangent is parallel to X- axis, therefore
    dy/dx = 0
    ⇒ x3 = 8
    ⇒ x = 2 abd y =3

    Question 103
    CBSEENMA11015534
    Wired Faculty App

    Let cos (α + β) = 4/5 and let sin (α - β) = 5/13, where 0 ≤α,β ≤ π/4. Then tan 2α is equal to

    • 25/16

    • 56/33

    • 19/12

    • 20/7
    Solution

    B.

    56/33

    Cos (α + β) = 4/5 
    ⇒  (α + β) ∈ 1st quadrant
    and sin (α - β) = 5/13
    ⇒  (α - β) ∈ 1st quadrant
    ⇒ 2α =(α + β) +  (α - β)
    ∴space tan space 2 straight alpha space equals space fraction numerator tan space left parenthesis straight alpha space space plus space straight beta right parenthesis space plus space tan space left parenthesis straight alpha space minus straight beta right parenthesis over denominator 1 minus tan space left parenthesis straight alpha space plus straight beta right parenthesis space tan left parenthesis straight alpha minus straight beta right parenthesis end fraction space equals space fraction numerator begin display style 3 over 4 end style plus begin display style 5 over 12 end style over denominator 1 minus begin display style 3 over 4 end style. begin display style 5 over 12 end style end fraction space equals space 56 over 33

    Question 104
    CBSEENMA11015546
    Wired Faculty App

    stack lim space with straight x space rightwards arrow straight pi over 2 below space fraction numerator cot space straight x space minus cos space straight x over denominator left parenthesis straight pi minus 2 straight x right parenthesis cubed end fraction space equals

    • 1/4

    • 1/24

    • 1/16

    • 1/8
    Solution

    C.

    1/16

    limit as straight x rightwards arrow straight pi over 2 of space fraction numerator cot space straight x space left parenthesis 1 minus sin space straight x right parenthesis over denominator negative 8 space open parentheses straight x space minus begin display style straight pi over 2 end style close parentheses cubed end fraction space equals space limit as straight x rightwards arrow fraction numerator begin display style straight pi end style over denominator begin display style 2 end style end fraction of space fraction numerator tan begin display style space end style begin display style open parentheses straight pi over 2 minus straight x close parentheses end style over denominator 8 open parentheses begin display style straight pi over 2 end style minus straight x close parentheses end fraction space fraction numerator open parentheses 1 minus space cos space open parentheses begin display style straight pi over 2 end style minus straight x close parentheses close parentheses over denominator open parentheses begin display style straight pi over 2 minus straight x end style close parentheses end fraction space equals space 1 over 8.1.1 half space equals space 1 over 16
    Question 105
    CBSEENMA11015547
    Wired Faculty App

    If 5(tan2x – cos2x) = 2cos 2x + 9, then the value of cos4x is

    • -7/9

    • -3/5

    • 1/3

    • 2/9
    Solution

    A.

    -7/9

    5 space open square brackets fraction numerator 1 minus straight t over denominator straight t end fraction minus straight t close square brackets space equals space 2 left parenthesis 2 straight t minus 1 right parenthesis space plus space 9 left curly bracket space Let space cos squared space straight x space equals space straight t right curly bracket

    ⇒5(1 – t – t2) = t(4t + 7)
    ⇒ 9t2 + 12t – 5 = 0
    ⇒ 9t2 + 15t – 3t – 5 = 0
    ⇒ (3t – 1) (3t + 5) = 0
    ⇒ t = t/3 as t≠-5/3.
    cos2x = 2(1/3)-1 = -1/3
    cos space 4 straight x space equals space 2 space open parentheses negative 1 third close parentheses squared minus 1 space equals space minus 7 over 9

    Question 106
    CBSEENMA11015561
    Wired Faculty App

    The differential equation which represents the family of curves y=c1ec2xe, where c1 and c2 are arbitrary constants, is

    • y' =y2

    •  y″ = y′ y

    • yy″ = y′

    • yy″ = (y′)
    Solution

    D.

    yy″ = (y′)

    y c1ec2x = …..(i)
    y' = c1c2ec2x
    y' = c2y.....(from (i) ....(ii)
    y" = c2y' ....... (iii)
    from (ii) & (iii)
    fraction numerator straight y apostrophe over denominator straight y end fraction space equals space fraction numerator straight y " over denominator straight y apostrophe end fraction rightwards double arrow space yy " space equals space left parenthesis straight y apostrophe right parenthesis squared

    Question 107
    CBSEENMA11015572
    Wired Faculty App

    The solution of the differential equation dy over dx space equals space fraction numerator straight x plus straight y over denominator straight x end fraction   satisfying the condition y (1) = 1 is  

    • y = ln x + x

    • y = x ln x + x2

    •  y = xe(x−1)

    • y = x ln x + x
    Solution

    D.

    y = x ln x + x

    y = vx
    dy over dx space equals space straight v space plus space straight x dv over dx straight v space plus space straight x dv over dx space equals space 1 space plus space straight v rightwards double arrow space dv space equals space dx over straight x therefore space straight v space equals space log space straight x space plus space straight c rightwards double arrow space straight y over straight x space equals space log space straight x space space plus straight c
    Since, y (1) = 1, we have y = x log x + x

    Question 108
    CBSEENMA11015678
    Wired Faculty App

    If limit as straight x space rightwards arrow infinity of space open parentheses 1 plus straight a over straight x plus straight b over straight x squared close parentheses to the power of 2 straight x end exponent space equals space straight e squared then the values of a and b, are

    • straight a element of space straight R with equals below space straight b element of space straight R with equals below
    • straight a space equals 1 comma space straight b element of space straight R with equals below
    • straight a element of space straight R with equals below space comma space straight b space equals 2

    • a = 1 and b = 2
    Solution

    D.

    a = 1 and b = 2

    limit as straight x space rightwards arrow infinity of space open parentheses 1 plus straight a over straight x plus straight b over straight x squared close parentheses to the power of 2 straight x end exponent space equals space limit as straight x space rightwards arrow infinity of space open parentheses 1 plus fraction numerator begin display style straight a end style over denominator begin display style straight x end style end fraction plus fraction numerator begin display style straight b end style over denominator begin display style straight x squared end style end fraction close parentheses to the power of open parentheses fraction numerator 1 over denominator begin display style straight a over straight x plus straight b over straight x squared end style end fraction close parentheses space straight x space 2 straight x space straight x space open parentheses straight a over straight x plus straight b over straight x squared close parentheses end exponent space equals space straight e to the power of 2 straight a end exponent rightwards double arrow space straight a space equals 1 straight b element of straight R
    Question 109
    CBSEENMA11015694
    Wired Faculty App

    For each t ∈R, let [t] be the greatest integer less than or equal to t. Then

    limx→0+ x1x+2x+......+15x

    • does not exist (in R)

    • is equal to 0

    • is equal to 15

    • is equal to 120
    Solution

    D.

    is equal to 120

    limx→0+ x1x+2x +.......+15x ⇒Now 1x - 1 < 1x≤1x2x-1<2x≤2x15x-1 <15x≤15xlimx→0+ x∑   15x -15 < limx→0+ x1x + 2x + ..... + 15x≤limx→0+ x ∑  15 xlimx→0+ x ∑ 15-5x < L ≤ limx→0+∑ 15L 120

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