Continuity and Differentiability

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Question 1

limit as straight n rightwards arrow infinity of space open parentheses fraction numerator left parenthesis straight n plus 1 right parenthesis left parenthesis straight n plus 2 right parenthesis....3 straight n over denominator straight n to the power of 2 straight n end exponent end fraction close parentheses to the power of 1 divided by straight n end exponent is equal to
  • 18/e4

  • 27/e2

  • 9/e2

  • 3 log 3-2

Solution

B.

27/e2

Let space straight I space equals space limit as straight n stack rightwards arrow infinity with space on top of open parentheses fraction numerator left parenthesis straight n plus 1 right parenthesis. left parenthesis straight n plus 2 right parenthesis..... left parenthesis 3 straight n right parenthesis over denominator straight n to the power of 2 straight n end exponent end fraction close parentheses to the power of 1 over straight n end exponent
space equals space space limit as straight n stack rightwards arrow infinity with space on top of open parentheses fraction numerator left parenthesis straight n plus 1 right parenthesis. left parenthesis straight n plus 2 right parenthesis..... left parenthesis straight n plus 2 straight n right parenthesis over denominator straight n to the power of 2 straight n end exponent end fraction close parentheses to the power of 1 over straight n end exponent

equals space limit as straight n stack rightwards arrow infinity with space on top of open square brackets open parentheses fraction numerator straight n plus 1 over denominator straight n end fraction close parentheses plus open parentheses fraction numerator straight n plus 2 over denominator straight n end fraction close parentheses.... open parentheses fraction numerator straight n plus 2 straight n over denominator straight n end fraction close parentheses close square brackets to the power of 1 over straight n end exponent
taking space log space on space both space sides space comma we space get
log space straight I space equals space limit as straight n stack rightwards arrow infinity with space on top of open square brackets log space open curly brackets open parentheses 1 plus 1 over straight n close parentheses open parentheses 1 plus 2 over straight n close parentheses.... open parentheses 1 plus fraction numerator 2 straight n over denominator straight n end fraction close parentheses close curly brackets close square brackets
rightwards double arrow space log space straight I space equals space limit as straight n stack rightwards arrow infinity with space on top of 1 over straight n open square brackets log open parentheses 1 plus 1 over straight n close parentheses plus space log space open parentheses 1 plus 2 over straight n close parentheses space plus..... plus space log space open parentheses 1 plus fraction numerator 2 straight n over denominator straight n end fraction close parentheses close square brackets
rightwards double arrow space log space straight I space equals limit as straight n stack rightwards arrow infinity with space on top of 1 over straight n sum from straight r space equals 1 to 2 straight n of space log space open parentheses 1 plus straight r over straight n close parentheses
rightwards double arrow space log space straight I space equals space integral subscript 0 superscript 2 space log space left parenthesis 1 plus straight x right parenthesis space dx
rightwards double arrow space log space straight I space equals space open square brackets log space space left parenthesis 1 space plus straight x right parenthesis. straight x space minus space integral fraction numerator 1 over denominator 1 plus straight x end fraction. straight x space dx close square brackets subscript 0 superscript 2
rightwards double arrow space log space space straight I space equals space left square bracket log space left parenthesis 1 space plus straight x right parenthesis. straight x right square bracket subscript 0 superscript 2 space minus integral subscript 0 superscript 2 open parentheses 1 minus fraction numerator 1 over denominator 1 plus straight x end fraction close parentheses dx
rightwards double arrow space log space space straight I space equals space 2. space log space 3 space minus space left square bracket straight x minus log space vertical line 1 plus straight x vertical line subscript 0 superscript 2
rightwards double arrow space log space space straight I space equals space 3. space log space 3 space minus 2
rightwards double arrow space log space space straight I space equals space space log space 27 space minus 2
therefore comma space straight l space equals space straight e to the power of log space 27 minus 2 end exponent space equals space 27. straight e to the power of negative 2 end exponent space equals space 27 over straight e squared
Question 2

fraction numerator straight d squared straight x over denominator dy squared end fraction equal to
  • open parentheses fraction numerator straight d squared straight x over denominator dy squared end fraction close parentheses to the power of negative 1 end exponent
  • negative open parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction close parentheses to the power of negative 1 end exponent open parentheses dy over dx close parentheses to the power of negative 3 end exponent
  • open parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction close parentheses open parentheses dy over dx close parentheses to the power of negative 2 end exponent
  • negative open parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction close parentheses open parentheses dy over dx close parentheses to the power of negative 3 end exponent

Solution

D.

negative open parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction close parentheses open parentheses dy over dx close parentheses to the power of negative 3 end exponent dy over dx space equals space fraction numerator 1 over denominator begin display style dx over dy end style end fraction

fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space straight d over dx open parentheses fraction numerator 1 over denominator dx divided by dy end fraction close parentheses space
equals space fraction numerator begin display style straight d end style over denominator begin display style dx end style end fraction open parentheses fraction numerator begin display style 1 end style over denominator begin display style dx divided by dy end style end fraction close parentheses. fraction numerator begin display style straight d end style over denominator begin display style dx end style end fraction
equals space minus space space fraction numerator begin display style 1 end style over denominator begin display style open parentheses dx over dy close parentheses squared end style end fraction. fraction numerator begin display style fraction numerator straight d squared straight x over denominator dy squared end fraction end style over denominator begin display style dx over dy end style end fraction
space equals space fraction numerator begin display style negative fraction numerator straight d squared straight x over denominator dy squared end fraction end style over denominator begin display style open parentheses dx over dy close parentheses cubed end style end fraction
equals negative open parentheses fraction numerator begin display style straight d squared straight x end style over denominator begin display style dy squared end style end fraction close parentheses open parentheses fraction numerator begin display style dy end style over denominator begin display style dx end style end fraction close parentheses cubed
Question 3

limit as straight x space rightwards arrow 2 of space open parentheses fraction numerator square root of 1 minus cos space open curly brackets 2 left parenthesis straight x minus 2 right parenthesis close curly brackets end root over denominator straight x minus 2 end fraction close parentheses
  • does not exist

  • equal square root of 2

  • equal negative square root of 2

  • equal fraction numerator 1 over denominator square root of 2 end fraction

Solution

A.

does not exist

stack lim space with straight x space rightwards arrow 2 below square root of 2 space fraction numerator vertical line sin space left parenthesis straight x minus 2 right parenthesis vertical line over denominator left parenthesis straight x minus 2 right parenthesis end fraction
therefore does not exist
Question 4

A function y = f(x) has a second order derivative f″(x) = 6(x – 1). If its graph passes through the point (2, 1) and at that point the tangent to the graph is y = 3x – 5, then the function is

  • (x-1)2

  • (x-1)3

  • (x+1)3

  • (x+1)2

Solution

B.

(x-1)3

f''(x) = 6(x - 1) f'(x) = 3(x - 1)2 + c ........ (i)
At the point (2, 1) the tangent to graph is y = 3x - 5 Slope of tangent = 3
∴ f'(2) = 3(2 - 1)2 + c = 3
3 + c = 3
⇒ c = 0
∴ From equation (i) f'(x) = 3(x - 1)2
f'(x) = 3(x - 1)2 f(x) = (x - 1)3 + k ...... (ii)
Since graph passes through (2, 1)
∴ 1 = (2 - 1)2 + k k = 0
∴ Equation of function is f(x) = (x - 1)3

Question 5

A spherical iron ball 10 cm in radius is coated with a layer of ice of uniform thickness than melts at a rate of 50 cm3 /min. When the thickness of ice is 5 cm, then the rate at which the thickness of ice decreases, is

  • fraction numerator 1 over denominator 36 space straight pi end fraction space cm divided by min
  • fraction numerator 1 over denominator 18 space straight pi end fraction space cm divided by min
  • fraction numerator 1 over denominator 54 space straight pi end fraction space cm divided by min
  • fraction numerator 5 over denominator 6 space straight pi end fraction space cm divided by min

Solution

B.

fraction numerator 1 over denominator 18 space straight pi end fraction space cm divided by min dv over dt space equals space 50
4 πr squared space dr over dt space equals space 50
rightwards double arrow space dr over dt space equals space fraction numerator 50 over denominator 4 straight pi space left parenthesis 15 right parenthesis squared end fraction space where space straight r space equals space 15
space equals space fraction numerator 1 over denominator 16 space straight pi end fraction