CBSE Class 12 Mathematics Relations And Functions Assignment Set 04

Question. Let f : (0,1) → R be defined by f(x) = b - x / 1 - bx , where b is a constant such that 0 < b <1.Then:
a. f is not invertible on (0, 1)
b. f ≠ f⁻¹ one (0, 1) and f'(b) = 1/f '(0)
c. f = f⁻¹ on (0, 1) and 1/f '(0)
d. f⁻¹ − is differentiable on 1/(0, 1)
Answer : A

Question. For every integer n, let n a and n b be real numbers. Let function f : R→ R be given by

If f is continuous, then which of the following hold(s) for all n?
a. a_n−1 − b_n−1 = 0
b. a_n −b_n = 1
c. a_n - b_n+1 = 1
d. a_n−1 b_n = − 1
Answer : B, D

Question. For every pair of continuous function f , g : [0,1]→ R such that max { f (x) : x∈[0,1]} = max{g(x) : x∈[0,1]}. The correct statement: (s) is (are)
a. [ f (c)]]² + 3 f (c) = [g(c)]² + 3g(c) for some c ∈ [0,1]
b. [ f (c)]]² + f (c) = [g(c)]² + 3g(c) for some c ∈ [0,1]
c. [ f (c])² + 3 f (c) = [g(c)]² + g(c) for some c ∈ [0,1]
d. [ f (c)]_² = [g(c)]² for some c ∈ [0,1]
Answer : A, D

Question. Let : ( - π/2, π/2) be given byf(x)=[log(sec x+tan x)]3.
Then:
a. f(x) is an odd function
b. f(x) is a one-one function
c. f(x) is an onto function
d. f(x) is an even function
Answer : A, B, C

Question. Let L be the set of all straight lines in the Euclidean plane.
Two lines l₁ and l₂ are said to be related by the relation R if l₁ is parallel to l₂. Then the relation R is:
a. Reflexive
c. Symmetric
b. Transitive
d. Equivalence
Answer : ALL

Assertion and Reason

Note: Read the Assertion (A) and Reason (R) carefully to mark the correct option out of the options given below:
a. If both assertion and reason are true and the reason is the correct explanation of the assertion.
b. If both assertion and reason are true but reason is not the correct explanation of the assertion.
c. If assertion is true but reason is false.
d. If the assertion and reason both are false.
e. If assertion is false but reason is true.

Question. Let F(x) be an indefinite integral of sin² x: 
Assertion: The function F(x) satisfies F(x + π) = F (x) for all real x.
Reason: sin² (x + π) = sin² x for all real x.
Answer : D

Question. Let f(x) = 2 + cos x for all real x:
Assertion: For each real t, there exists a point c in [t,t +π] such that f’(c) = 0.
Reason: f’(t) = f (t + 2π) for each real t.
Answer : B

Question. Assertion: The curve y = - x²/2 + x + 1 is symmetric with respect to the line x = 1.
Reason: A parabola is symmetric about its axis.
Answer : A

Question. Consider the following relations. R = {(x,y)| x,y}are real numbers and  x= xy for some rational number w} S = {(m/n. p/q)}m,n,p,q are integer such that n.q ≠ 0 and qm = p_n}
Assertion: S is an equivalence relation but R is not an equivalence relation.
Reason: R and S both are symmetric.
Answer : C

Question. Let R be a relation on the set N of natural numbers defined by n Rm ⇔ n is a factor of m (i.e., n | m) :
Assertion: R is not an equivalence relation
Reason: R is not symmetric
Answer : A

Question. Let A = {1,2,3}and B = {3,8}?
Assertion: (A ∪ B) × (A∩B) = {(1,3), (2,3), (3,3)(8,3)}
Reason: (A × B) ∩ (B × A) = {(3,3)}
Answer : B

Question. Assertion: f:R→R is a function defined by f (x) = 5x + 3. If 1, g = f−1, then g(x) = x - 3 / 5
Reason: If f : A→B is a bijection and g : B → A is the inverse of f, then fog is the identity function on A.
Answer : C

Question. Let X and Y be two sets:
Assertion: X ∩ (Y ∩ X ) ' = φ
Reason: If X∪Y has m elements and X ∩ Y has n elements then symmetric difference X Y has m − n elements
Answer : B

Question. Let f be a function defined by f(x) = (x −1)² + 1, (x ≥ 1)
Assertion: The set 1 {x : f (x) = f^-1 (x)} = {1,2}
Reason: f is a bijection and f^-1(x) = 1 + √x - 1,  x ≥ 1
Answer : A

Question. Consider the following relation R on the set of real square matrices of order 3. R {(A,B) : A = P^-1 BP for some invertible matrix P}
Assertion: R is an equivalence relation.
Reason: For any two invertible 3 × 3 matrices M and N
Answer : B

Question. Let f(x) = sin x + cos x, g(x) = sin x / 1 - cos x ?
Assertion: f is neither an odd function nor an even function.
Reason: g is an odd function.
Answer : B

Question. Assertion: A function f:R→R satisfied the equation f (x) – f(y) = x – y ∀ x, y ∈ R and f(3) = 2, then f(xy) = xy − 1
Reason: f (x) = f (1/ x) ∀ x ∈ R, x ≠ 0, and f(2) = 7/3 if
f(x) = x² + x + 1 / x² - x + 1
Answer : B

Question. Assertion: Let A{2, 3, 7, 9}and B = {4, 9, 49, 81} f:A → B is a function defined as f(x) = x₂. Then is a bijection from A to B.
Reason: A function f from a set A to a set B is a bijection if f(A) = B and f(x₁) ≠ f(x₂) if x₁ ≠ x₂ for all x₁, x₂ ∈A and n(A) = n(B).
Answer : A

Comprehension Based

Paragraph-I

Let f be a function satisfying f(x) = a^x / a^x + √a = ga (x) (a > 0)

Question. Let f(x) = g9(x), then the value of

is: (where [.] denotes the greatest integer function)
a. 995
b. 996
c. 997
d. 998
Answer : C

Question. Let 4 f (x) = g (x), then

a. zero
b. even
c. odd
d. none of these
Answer : B

Question. The value of g5 (x) + g5 (1–x) is:
a. 1
b. 5
c. 10
d. none of these
Answer : A

Question. The value of

a. 0
b. 2n–1
c. 2n
d. none of these
Answer : B

Paragraph-II

Let F(x) = f (x) + g(x),G(x) = f (x) − g(x) and H (x) = f(x)/g)(x),
where f(x) = 1 − 2sin² x and g(x) = cos 2x, ∀f : R → [−1,1] and g : R →[−1,1].

Question. Domain and range of H (x) are respectively:
a. R and {1}
b. R and {0, 1}
c. R ∼ {(2n + 1)π/4},and{1} n ∈  I
d. R  ∼ {(2n + 1)π/2}, and {0,1},
Answer : C

Question. If F: R → [–2, 2], then:
a. F (x) is one – one function
b. F (x) is onto function
c. F (x) is into function
d. none of the above
Answer : D

Question. Which statement is correct?
a. period of f(x), g(x) and F(x) makes are AP with common difference π/3
b. period of f(x), g(x) and F(x) are same and is equal to 2π
c. sum of periods of f(x), g(x) and F(x) is 3π
d. sum of periods of f(x), g(x) and F(x) is 6π
Answer : C

Question. Which statement is correct?
a. the domain of G(x) and H(x) are same
b. the rang of G(x) and H(x) are same
c. the union of domain of G(x) and H(x) are all real
d. the union of domain of G(x) and H(x) are rational numbers
Answer : C

Question. If the solutions of F (x) – G (x) = 0 are x₁, x₂, x₃ ,…x_n where x∈ [0, 5π], then:
a. x₁, x₂, x₃, …x_n are in AP with common difference π/4
b. the number of solution of F (x) – G (x) = 0 is 10, ∀ x ∈ [0, 5π].
c. the sum of all solutions of F(x) −G(x) = 0,∀ x ∈ [0,5π ] is 25π
d. (b) and (c) are correct
Answer : D

Match the Column

Question. Let the functions defined in Column I have domain (–π/2, π/2) and range (–∞,∞)?
    Column I             Column II
(A) 1 + 2x          1. onto but not one-one
(B) tan x            2. one-one but not onto
                          3. one-one and onto
                          4. neither one-one nor onto
a. A→ 2; B→ 3
b. A→ 2; B→ 4
c. A→ 1; B→ 3
d. A→ 4; B→ 1
Answer : A

Question. Let f(x) = x² - 6x + 5 / x² - 5x + 6 ?
              Column I                                 Column II

(A) If –1< x <1, then f(x) satisfies        1. 0 < f (x) < 1
(B) If 1 < x < 2, then f(x) satisfies        2. f (x) < 0
(C) If 3 < x < 5, then f(x) satisfies        3. f (x) > 0
(D) If x > 5, then f(x) satisfies              4. f (x) < 1
a. A→3; B→1; C→2; D→2
b.A→1; B→2; C→2; D→1
c. A→1; B→3; C→2; D→4
d. A→4; B→1; C→3; D→2
Answer : B

Question. Let f₁ : R→R, f₂ :[0,∞] → R, f₃ : R → Rand f₄ : R → [0,∞) be

     Column I                      Column II
(A) f₄ is               1. onto but not one-one
(B) f₃ is               2. neither continuous nor one-one
(C) f₂ o f1 is        3. differentiable but not one-one
(D) f₂ is               4. continuous and one-one
a. A→3; B→1; C→4; D→2
b. A→1; B→3; C→4; D→2
c. A→3; B→1; C→2; D→4
d. A→1; B→3; C→2; D→4
Answer : D

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