CBSE Class 12 Mathematics Relations and Functions VBQs Set 03

Assertion and Reason Questions

Choose the correct option :
(a) Both (A) and (B) are true and R is the correct explanation A.
(b) Both (A) and (R) are true but R is not correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.

Question. Assertion (R) : The function f(x) = | x | is not one-one.
Reason (R) : The negative real number are not the images of any real numbers.
(a) (a)
(b) (b)
(c) (c)
(d) (d)
Answer: (c) A is true but R is false.

Question. Assertion (A) : A function y = f(x) is defined by \( x^2 – \cos^{-1} y = \pi \), then domain of f(x) is R.
Reason (R) : \( \cos^{-1} y \in [0, \pi] \).
(a) (a)
(b) (b)
(c) (c)
(d) (d)
Answer: (d) A is false but R is true.

Question. Assertion (A) : If f(x) is odd function and g(x) is even function, then f(x) + g(x) is neither even nor odd.
Reason (R) : Odd function is symmetrical in opposite quadrants and even function is symmetrical about the y-axis.
(a) (a)
(b) (b)
(c) (c)
(d) (d)
Answer: (b) Both (A) and (R) are true but R is not correct explanation of A.

Question. Assertion (A) : Every even function y = f(x) is not one-one, \( \forall x \in D_f \).
Reason (R) : Even function is symmetrical about the y-axis.
(a) (a)
(b) (b)
(c) (c)
(d) (d)
Answer: (a) Both (A) and (B) are true and R is the correct explanation A.

Question. Assertion (A) : The function \( f(x) = x^2 – x + 1 \), \( x \ge \frac{1}{2} \) and \( g(x) = \frac{1}{2} + \sqrt{x - \frac{3}{4}} \) then the number of solutions of the equation f(x) = g(x) is two.
Reason (R) : f(x) and g(x) are mutually inversion.
(a) (a)
(b) (b)
(c) (c)
(d) (d)
Answer: (d) A is false but R is true.

Question. Assertion (A) : f(x) = sin x + cos ax is a periodic function.
Reason (R) : a is rational number.
(a) (a)
(b) (b)
(c) (c)
(d) (d)
Answer: (a) Both (A) and (B) are true and R is the correct explanation A.

Question. Assertion (A) : The least period of the function, f (x) = cos (cos x) + cos (sin x) + sin 4x is \( \pi \).
Reason (R) : \( \therefore f (x + \pi) = f (x) \).
(a) (a)
(b) (b)
(c) (c)
(d) (d)
Answer: (d) A is false but R is true.

Question. Assertion (A) : If f (x + y) + f (x – y) = 2f(x) · f (y) \( \forall x, y \in R \) and f (0) ≠ 0, then f(x) is an even function.
Reason (R) : If f (– x) = f (x), then f (x) is an even function.
(a) (a)
(b) (b)
(c) (c)
(d) (d)
Answer: (b) Both (A) and (R) are true but R is not correct explanation of A.

Question. Assertion (A) : The equation \( x^4 = (\lambda x – 1)^2 \) has atmost two real solutions (is \( \lambda > 0 \)).
Reason (R) : Curves \( f(x) = x^4 \) and \( g(x) = (\lambda x – 1)^2 \) has atmost two points.
(a) (a)
(b) (b)
(c) (c)
(d) (d)
Answer: (d) A is false but R is true.

Question. Assertion (A) : The domains of \( f (x) = \sqrt{\cos(\sin x)} \) and \( g(x) = \sqrt{\sin(\cos x)} \) are same.
Reason (R) : –1 ≤ cos (sin x) ≤ 1 and – 1 ≤ sin (cos x) ≤ 1
(a) (a)
(b) (b)
(c) (c)
(d) (d)
Answer: (d) A is false but R is true.

Question. Assertion (A) : If \( f(x) = x^5 – 16x + 2 \), then f(x) = 0 has only one root in the interval [–1, 1].
Reason (R) : f (– 1) and f (1) are of opposite sign.
(a) (a)
(b) (b)
(c) (c)
(d) (d)
Answer: (b) Both (A) and (R) are true but R is not correct explanation of A.

Question. Assertion (A) : The domain of the function \( f (x) = \sin^{-1} x + \cos^{-1} x + \tan^{-1} x \) is [–1, 1].
Reason (R) : \( \sin^{-1} x \) and \( \cos^{-1} x \) is defined in | x| ≤ 1 and \( \tan^{-1} x \) defined for all x.
(a) (a)
(b) (b)
(c) (c)
(d) (d)
Answer: (a) Both (A) and (B) are true and R is the correct explanation A.

Question. Assertion (A) : The period of f(x) = sin 3x cos [3x] – cos 3x sin [3x] is \( \frac{1}{3} \) where [ ] denotes the greatest integer function ≤ x.
Reason (R) : The period of {x} is 1, where {x} denotes the fractional part function of x.
(a) (a)
(b) (b)
(c) (c)
(d) (d)
Answer: (a) Both (A) and (B) are true and R is the correct explanation A.

Question. Assertion (A) : The relation R given by R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} on a set A = {1, 2, 3} is not symmetric.
Reason : For symmetric relation \( R = R^{-1} \).
(a) (a)
(b) (b)
(c) (c)
(d) (d)
Answer: (a) Both (A) and (B) are true and R is the correct explanation A.

The price of the oranges in the market is dependent on the amount of oranges (in kgs) which can be represented as y = 3x + 5. Reema went to buy the oranges for a family function in her house. The total number of oranges she wants to buys is 5 ≤ x ≤ 10 according to her assumption of people coming to the party. Answer the following questions on the basis of the given information.

Question. How many ordered pairs can be represented for the equation y = 3x + 5 for 5 ≤ x ≤ 10 ?
(a) 4
(b) 5
(c) 6
(d) 7
Answer: (c) 6

Question. What is the domain of the given relation R = {(5, 20), (6, 23), (7, 26), (8, 29), (9, 32), (10, 35)} ?
(a) {– 5, – 4, 0, 1, 2}
(b) {0, 1, 2, 3, 4, 5}
(c) {4, 5, 6, 7, 8}
(d) {5, 6, 7, 8, 9, 10}
Answer: (d) {5, 6, 7, 8, 9, 10}

Question. How can range of the relation be represented for the relation R = {(5, 20), (6, 23), (7, 26), (8, 29), (9, 32), (10, 35)} ?
(a) {(x, y) | y = x + 3; 3 : 17 ≤ x ≤ 32}
(b) {(x, y) | y = x + 3 : 20 ≤ x ≤ 35}
(c) {(x, y) | y = x + 3 : 17 < x < 32}
(d) {(x, y) | y = x + 3 : 20 < x < 35}
Answer: (a) {(x, y) | y = x + 3; 3 : 17 ≤ x ≤ 32}

Question. What is co-domain for the given relation ?
(a) {(x, y) | y = x + 3 : 17 ≤ x ≤ 32}
(b) {(x, y) | y = x + 3 : 20 ≤ x ≤ 35}
(c) {(x, y) | y = x + 3 : 17 < x < 32}
(d) {(x, y) | y = x + 3 : 20 < x < 35}
Answer: (a) {(x, y) | y = x + 3 : 17 ≤ x ≤ 32}

Question. How many subsets are there for the given relation R = {(5, 20), (6, 23), (7, 26), (9, 29), (9, 32), (10, 35)} ?
(a) 16
(b) 32
(c) 64
(d) 128
Answer: (c) 64

Sherlin and Danju are playing Ludo at home during Covid-19. While rolling the dice, Sherlin’s sister Raji observed and noted the possible outcomes of the throw every time belongs to set {1, 2, 3, 4, 5, 6}. Let A be the set of players while B be the set of all possible outcomes.

Question. Let R : B → B be defined by R = {(x, y) : y is divisible by x} is :
(a) Reflexive and transitive but not symmetric
(b) Reflexive and symmetric and not transitive
(c) Not reflexive but symmetric and transitive
(d) Equivalence
Answer: (a) Reflexive and transitive but not symmetric

Question. Raji wants to know the number of functions from A to B. How many number of functions are possible ?
(a) \( 6^2 \)
(b) \( 2^6 \)
(c) 6!
(d) \( 2^{12} \)
Answer: (a) \( 6^2 \)

Question. Let R be a relation on B defined by R = {(1, 2), (2, 2), (1,3 ), (3, 4), (3, 1), (4, 3), (5, 5)}. Then R is :
(a) Symmetric
(b) Reflexive
(c) Transitive
(d) None of the options
Answer: (d) None of the options

Question. Raji wants to know the number of relations possible from A to B. How many numbers of relations are possible ?
(a) \( 6^2 \)
(b) \( 2^6 \)
(c) 6!
(d) \( 2^{12} \)
Answer: (d) \( 2^{12} \)

Question. Let R : B → B be defined by R = {(1, 1), (1, 2), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}, then R is :
(a) Symmetric
(b) Reflexive and Transitive
(c) Transitive and symmetric
(d) Equivalence
Answer: (b) Reflexive and Transitive

Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line y = x – 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.

Question. Let relation R be defined by R = {(L1, L2) : L1 || L2 where L1, L2 ∈ L} then R is .................. .
(a) Equivalence
(b) Only reflexive
(c) Not reflexive
(d) Symmetric but not transitive
Answer: (a) Equivalence

Question. Let R = {(L1, L2) : \( L_1 \perp L_2 \) where L1, L2 ∈ L} which of the following is true ?
(a) R is symmetric but neither reflexive nor transitive
(b) R is reflexive and transitive but not symmetric
(c) R is reflexive but neither symmetric nor transitive
(d) R is an equivalence relation
Answer: (a) R is symmetric but neither reflexive nor transitive

Question. The function f : R → R defined by f(x) = x – 4 is :
(a) Bijective
(b) Surjective but not injective
(c) Injective but not surjective
(d) Neither surjective nor injective
Answer: (a) Bijective

Question. Let f : R → R be defined by f(x) = x – 4. Then the range of f(x) is :
(a) R
(b) Z
(c) W
(d) Q
Answer: (a) R

Question. Let R = {(L1, L2) : L1 is parallel to L2 and L1 : y = x – 4} then which of the following can be taken as L2 ?
(a) 2x – 2y + 5 = 0
(b) 2x + y = 5
(c) 2x + 2y + 7 = 0
(d) x + y = 7
Answer: (a) 2x – 2y + 5 = 0