Question

The value of p and q for which the function f(x) = $open curly brackets table attributes columnalign left end attributes row cell table attributes columnalign left columnspacing 1.4ex end attributes row cell fraction numerator sin space left parenthesis straight p space plus 1 right parenthesis space straight x space plus space sin space straight x over denominator straight x end fraction comma end cell cell straight x less than 0 end cell row cell straight q comma end cell cell straight x space equals space 0 space is space continuous space for space all space straight x space in space straight R comma space are colon end cell end table fraction numerator square root of straight x space plus straight x squared end root minus square root of straight x over denominator straight x to the power of 3 divided by 2 end exponent end fraction space comma space straight x greater than space 0 space end cell row space end table close$

p = 1/2. q = -3/2

p = 5/2, q = 1/2

p = - 3/2, q = 1/2

p = 1/2, q = 3/2

Solution

p = - 3/2, q = 1/2

f(0) = q
$straight f left parenthesis 0 to the power of plus right parenthesis space equals stack space lim with straight x space rightwards arrow 0 to the power of plus below space fraction numerator left parenthesis 1 space plus straight x right parenthesis to the power of 1 divided by 2 end exponent minus 1 space over denominator straight x end fraction space equals space stack space lim with straight x space rightwards arrow 0 to the power of minus below space fraction numerator left parenthesis 1 space plus begin display style 1 half end style straight x...... negative 1 space over denominator straight x end fraction space equals space 1 half straight f left parenthesis 0 to the power of minus right parenthesis space equals space stack space lim with straight x space rightwards arrow 0 to the power of minus below fraction numerator sin space left parenthesis straight p plus 1 right parenthesis space straight x space plus space sin space straight x over denominator straight x end fraction straight f left parenthesis 0 to the power of minus right parenthesis space space equals space stack space lim with straight x space rightwards arrow 0 to the power of minus below space fraction numerator left parenthesis cos space left parenthesis straight p space plus 1 right parenthesis space straight x right parenthesis space left parenthesis straight p plus 1 right parenthesis space space plus space left parenthesis cos space straight x right parenthesis over denominator 1 end fraction space equals space left parenthesis straight p space plus 1 right parenthesis space space plus 1 space equals space straight p space plus 2 therefore space straight p space plus 2 space equals space straight q space equals space 1 half rightwards double arrow space straight p space equals space minus 3 divided by 2 comma space straight q space equals space 1 divided by 2$

Some More Questions From Relations and Functions Chapter

Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive.

Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

Give an example of a relation which is

(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive.

Let L be the set of all lines in a plane and R be the relation in L defined as R = {(L₁, L₂) : L₁ is perpendicular to L₂}. Show that R is symmetric but neither reflexive nor transitive.

Determine whether each of the following relations are reflexive, symmetric and transitive :

(i) Relation R in the set A = {1, 2, 3,....., 13, 14} defined as

R = {(x, y) : 3 x – y = 0}

(ii) Relation R in the set N of natural numbers defined as R = {(x, y) : y = x + 5 and x < 4} (iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = {(x,y) : y is divisible by x} (iv) Relation R in the set Z of all integers defined as R = {(x,y) : x – y is an integer}

(v) Relation R in the set A of human beings in a town at a particular time given by
(a)    R = {(x, y) : x and y work at the same place}
(b)    R = {(x,y) : x and y live in the same locality}
(c)    R = {(x, y) : x is exactly 7 cm taller than y}
(d)    R = {(x, y) : x is wife of y}
(e)    R = {(x,y) : x is father of y}

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Relations And Functions

Some More Questions From Relations and Functions Chapter

Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive.

Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

Give an example of a relation which is

(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive.

Let L be the set of all lines in a plane and R be the relation in L defined as R = {(L₁, L₂) : L₁ is perpendicular to L₂}. Show that R is symmetric but neither reflexive nor transitive.

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive.

Show that the relation R in R defined as R = {(a, b) : a ≤ b}, is reflexive and transitive but not symmetric.

Check whether the relation R in R defined by R = {(a,b) : a ≤ b³} is refleive, symmetric or transitive.

If R and R’ arc reflexive relations on a set then so are R ∪ R’ and R ∩ R’.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

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Relations And Functions

The value of p and q for which the function f(x) = p = 1/2. q = -3/2 p = 5/2, q = 1/2 p = - 3/2, q = 1/2 p = 1/2, q = 3/2

Some More Questions From Relations and Functions Chapter

Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive.

Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

Give an example of a relation which is (i) Symmetric but neither reflexive nor transitive.(ii) Transitive but neither reflexive nor symmetric.(iii) Reflexive and symmetric but not transitive.(iv) Reflexive and transitive but not symmetric.(v) Symmetric and transitive but not reflexive.

Let L be the set of all lines in a plane and R be the relation in L defined as R = {(L1, L2) : L1 is perpendicular to L2}. Show that R is symmetric but neither reflexive nor transitive.

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive.

Show that the relation R in R defined as R = {(a, b) : a ≤ b}, is reflexive and transitive but not symmetric.

Check whether the relation R in R defined by R = {(a,b) : a ≤ b3} is refleive, symmetric or transitive.

If R and R’ arc reflexive relations on a set then so are R ∪ R’ and R ∩ R’.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Give an example of a relation which is

(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive.

Let L be the set of all lines in a plane and R be the relation in L defined as R = {(L₁, L₂) : L₁ is perpendicular to L₂}. Show that R is symmetric but neither reflexive nor transitive.

Check whether the relation R in R defined by R = {(a,b) : a ≤ b³} is refleive, symmetric or transitive.