Bihar Board 12th Maths Objective Questions and Answers

## Bihar Board 12th Maths Objective Answers Chapter 1 Relations and Functions

Question 1.

If R = {(x, y) : x, y ∈ I, x^{2} + y^{2} ≤ 4} is a relation in I, then domain of R is

(a) {0, 1, 2}

(b) {-2, -1, 0}

(c) {-2, -1, 0, 2}

(d) None of these

Answer:

(c) {-2, -1, 0, 2}

Question 2.

If A = {1, 2, 3}, B = {1, 4, 6, 9} and R is relation from A to B defined by ‘or is greater thany’. The range of R is

(a) {1, 4, 6, 9}

(b) {4, 6, 9}

(c) {1}

(d) None of these

Answer:

(c) {1}

Question 3.

The relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)} on set A = {1, 2, 3} is

(a) Reflexive but not symmetric

(b) Reflexive but not transitive

(c) Symmetric and transitive

(d) Neither symmetric nor transitive

Answer:

(a) Reflexive but not symmetric

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Question 4.

Let P = {(x, y) | x^{2} + y^{2} = 1, xy ∈ R}. Then P is

(a) Reflexive

(b) Symmetric

(c) Transitive

(d) Anti-symmetric

Answer:

(b) Symmetric

Question 5.

Let S be the set of all real numbers. Then, the relation R = {(a, b) : 1 + ab > 0} on S is

(a) Reflexive and symmetric but not transitive

(b) Reflexive and transitive but not symmetric

(c) Symmetric and transitive but not reflexive

(d) Reflexive, transitive and symmetric

Answer:

(a) Reflexive and symmetric but not transitive

Question 6.

Let R be a relation on the set N be defined by {(x, y) | x, y ∈ N, 2x + y = 41}. Then, R is

(a) Reflexive

(b) Symmetric

(c) Transitive

(d) None of these

Answer:

(d) None of these

Question 7.

Let R be the relation on the set of all real numbers defined by aRb iff |a – b| ≤ 1. Then, R is

(a) Reflexive and symmetric

(b) Symmetric only

(c) Transitive only

(d) Anti-symmetric only

Answer:

(a) Reflexive and symmetric

Question 8.

If R and R’ are symmetric relations (not disjoint) on a set A, then the relation R ∩ R’ is

(a) Reflexive

(b) Symmetric

(c) Transitive

(d) None of these

Answer:

(b) Symmetric

Question 9.

Let A = {1, 2, 3, 4} and R be a relation in A given by R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1), (3, 1). Then, R is

(a) Reflexive

(b) Transitive

(c) An equivalence relation

(d) None of these

Answer:

(a) Reflexive

Question 10.

Which one of the following relations on R is an equivalence relation?

(a) aR_{1}b ⇔ |a| = |b|

(b) aR_{2}b ⇔ a ≥ b

(c) aR_{3}b ⇔ a divides b

(d) aR_{4}b ⇔ a < b

Answer:

(a) aR_{1}b ⇔ |a| = |b|

Question 11.

Let R be a relation on the set N of natural numbers denoted by nRm ⇔ n is a factor of m (i.e. n | m). Then, R is

(a) Reflexive and symmetric

(b) Transitive and symmetric

(c) Equivalence

(d) Reflexive, transitive but not symmetric

Answer:

(d) Reflexive, transitive but not symmetric

Question 12.

Let S = {1, 2, 3, 4, 5} and let A = S × S. Define the relation R on A as follows:

(a, b) R (c, d) iff ad = cb. Then, R is

(a) reflexive only

(b) Symmetric only

(c) Transitive only

(d) Equivalence relation

Answer:

(d) Equivalence relation

Question 13.

Let R be the relation “is congruent to” on the set of all triangles in a plane is

(a) reflexive

(b) symmetric

(c) symmetric and reflexive

(d) equivalence

Answer:

(d) equivalence

Question 14.

Total number of equivalence relations defined in the set S = {a, b, c} is

(a) 5

(b) 3!

(c) 23

(d) 33

Answer:

(a) 5

Question 15.

The relation R is defined on the set of natural numbers as {(a, b) : a = 2b}. Then, R^{-1} is given by

(a) {(2, 1), (4, 2), (6, 3),….}

(b) {(1, 2), (2, 4), (3, 6), ……..}

(c) R^{-1} is not defiend

(d) None of these

Answer:

(b) {(1, 2), (2, 4), (3, 6), ……..}

Question 16.

The domain of the function \(f(x)=\frac{1}{\sqrt{\{\sin x\}+\{\sin (\pi+x)\}}}\) where {.} denotes fractional part, is

(a) [0, π]

(b) (2n + 1) π/2, n ∈ Z

(c) (0, π)

(d) None of these

Answer:

(d) None of these

Question 17.

Range of \(f(x)=\sqrt{(1-\cos x) \sqrt{(1-\cos x) \sqrt{(1-\cos x) \ldots \ldots \infty}}}\)

(a) [0, 1]

(b) (0, 1)

(c) [0, 2]

(d) (0, 2)

Answer:

(c) [0, 2]

Question 18.

Consider the function y = f(x) satisfying the condition \(f\left(x+\frac{1}{x}\right)=x^{2}+\frac{1}{x^{2}}(x \neq 0)\). Then which of the

following is not true?

(a) Domain of f(x) is (-∞, -2] ∪ [2, ∞)

(b) f(x) is an even function

(c) Range of f(x) is [2, ∞)

(d) None of these

Answer:

(d) None of these

Question 19.

The function f : R → R defined by f(x) = 6x + 6 |x| is

(a) One-one and onto

(b) Many-one and onto

(c) One-one and into

(d) Many-one and into

Answer:

(c) One-one and into

Question 20.

f : N → N where f(x) = x – (-1)^{x}, then f is

(a) One-one and into

(b) Many-one and into

(c) One-one and onto

(d) Many-one and onto

Answer:

(c) One-one and onto

Question 21.

A function f from the set of natural numbers to integers is defined by

(a) one-one but not onto

(b) onto but not one-one

(c) one-one and onto both

(d) neither one-one nor onto

Answer:

(c) one-one and onto both

Question 22.

Let X = {-1, 0, 1}, Y = {0, 2} and a function f : X → Y defiend by y = 2x^{4}, is

(a) one-one onto

(b) one-one into

(c) many-one onto

(d) many-one into

Answer:

(c) many-one onto

Question 23.

Let f : R → R be a function defined by \(f(x)=\frac{e^{|x|}-e^{-x}}{e^{x}+e^{-x}}\) then f(x) is

(a) one-one onto

(b) one-one but not onto

(c) onto but not one-one

(d) None of these

Answer:

(d) None of these

Question 24.

Let g(x) = x^{2} – 4x – 5, then

(a) g is one-one on R

(b) g is not one-one on R

(c) g is bijective on R

(d) None of these

Answer:

(b) g is not one-one on R

Question 25.

Let A = R – {3}, B = R – {1}. Let f : A → B be defined by \(f(x)=\frac{x-2}{x-3}\). Then,

(a) f is bijective

(b) f is one-one but not onto

(c) f is onto but not one-one

(d) None of these

Answer:

(a) f is bijective

Question 26.

The mapping f : N → N is given by f(n) = 1 + n^{2}, n ∈ N when N is the set of natural numbers is

(a) one-one and onto

(b) onto but not one-one

(c) one-one but not onto

(d) neither one-one nor onto

Answer:

(c) one-one but not onto

Question 27.

The function f : R → R given by f(x) = x^{3} – 1 is

(a) a one-one function

(b) an onto function

(c) a bijection

(d) neither one-one nor onto

Answer:

(c) a bijection

Question 28.

Let f : [0, ∞) → [0, 2] be defined by \(f(x)=\frac{2 x}{1+x}\), then f is

(a) one-one but not onto

(b) onto but not one-one

(c) both one-one and onto

(d) neither one-one nor onto

Answer:

(a) one-one but not onto

Question 29.

If N be the set of all-natural numbers, consider f : N → N such that f(x) = 2x, ∀ x ∈ N, then f is

(a) one-one onto

(b) one-one into

(c) many-one onto

(d) None of these

Answer:

(b) one-one into

Question 30.

Let A = {x : -1 ≤ x ≤ 1} and f : A → A is a function defined by f(x) = x |x| then f is

(a) a bijection

(b) injection but not surjection

(c) surjection but not injection

(d) neither injection nor surjection

Answer:

(a) a bijection

Question 31.

Let f : R → R be a function defined by f(x) = x^{3} + 4, then f is

(a) injective

(b) surjective

(c) bijective

(d) none of these

Answer:

(c) bijective

Question 32.

The function f : A → B defined by f(x) = 4x + 7, x ∈ R is

(a) one-one

(b) Many-one

(c) Odd

(d) Even

Answer:

(a) one-one

Question 33.

The smallest integer function f(x) = [x] is

(a) One-one

(b) Many-one

(c) Both (a) & (b)

(d) None of these

Answer:

(b) Many-one

Question 34.

The function f : R → R defined by f(x) = 3 – 4x is

(a) Onto

(b) Not onto

(c) None one-one

(d) None of these

Answer:

(a) Onto

Question 35.

The number of bijective functions from set A to itself when A contains 106 elements is

(a) 106

(b) (106)^{2}

(c) 106!

(d) 2^{106}

Answer:

(c) 106!

Question 36.

If f(x) = (ax^{2} + b)^{3}, then the function g such that f(g(x)) = g(f(x)) is given by

(a) \(g(x)=\left(\frac{b-x^{1 / 3}}{a}\right)\)

(b) \(g(x)=\frac{1}{\left(a x^{2}+b\right)^{3}}\)

(c) \(g(x)=\left(a x^{2}+b\right)^{1 / 3}\)

(d) \(g(x)=\left(\frac{x^{1 / 3}-b}{a}\right)^{1 / 2}\)

Answer:

(d) \(g(x)=\left(\frac{x^{1 / 3}-b}{a}\right)^{1 / 2}\)

Question 37.

If f : R → R, g : R → R and h : R → R is such that f(x) = x^{2}, g(x) = tanx and h(x) = logx, then the value of [ho(gof)](x), if x = \(\frac{\sqrt{\pi}}{2}\) will be

(a) 0

(b) 1

(c) -1

(d) 10

Answer:

(a) 0

Question 38.

If f : R → R and g : R → R defined by f(x) = 2x + 3 and g(x) = x^{2} + 7, then the value of x for which f(g(x)) = 25 is

(a) ±1

(b) ±2

(c) ±3

(d) ±4

Answer:

(b) ±2

Question 39.

Let the functions f, g, h are defined from R to R such that

ho(fog)(x) is defined by

(a) x

(b) x^{2}

(c) 0

(d) none of these

Answer:

(b) x^{2}

Question 40.

(a) 1

(b) -1

(c) √3

(d) 0

Answer:

(b) -1

Question 41.

Let f : N → R : f(x) = \(\frac{(2 x-1)}{2}\) and g : Q → R : g(x) = x + 2 be two functions. Then, (gof) (\(\frac{3}{2}\)) is

(a) 3

(b) 1

(c) \(\frac{7}{2}\)

(d) None of these

Answer:

(a) 3

Question 42.

Let \(f(x)=\frac{x-1}{x+1}\), then f(f(x)) is

(a) \(\frac{1}{x}\)

(b) \(-\frac{1}{x}\)

(c) \(\frac{1}{x+1}\)

(d) \(\frac{1}{x-1}\)

Answer:

(b) \(-\frac{1}{x}\)

Question 43.

If f(x) = \(1-\frac{1}{x}\), then f(f(\(\frac{1}{x}\)))

(a) \(\frac{1}{x}\)

(b) \(\frac{1}{1+x}\)

(c) \(\frac{x}{x-1}\)

(d) \(\frac{1}{x-1}\)

Answer:

(c) \(\frac{x}{x-1}\)

Question 44.

If f : R → R, g : R → R and h : R → R are such that f(x) = x^{2}, g(x) = tan x and h(x) = log x, then the value of (go(foh)) (x), if x = 1 will be

(a) 0

(b) 1

(c) -1

(d) π

Answer:

(a) 0

Question 45.

If f(x) = \(\frac{3 x+2}{5 x-3}\) then (fof)(x) is

(a) x

(b) -x

(c) f(x)

(d) -f(x)

Answer:

(a) x

Question 46.

If f(x) = (ax^{2} – b)^{3}, then the function g such that f{g(x)} = g{f(x)} is given by

(a) \(g(x)=\left(\frac{b-x^{1 / 3}}{a}\right)^{1 / 2}\)

(b) \(g(x)=\frac{1}{\left(a x^{2}+b\right)^{3}}\)

(c) \(g(x)=\left(a x^{2}+b\right)^{1 / 3}\)

(d) \(g(x)=\left(\frac{x^{1 / 3}+b}{a}\right)^{1 / 2}\)

Answer:

(d) \(g(x)=\left(\frac{x^{1 / 3}+b}{a}\right)^{1 / 2}\)

Question 47.

If f : [1, ∞) → [2, ∞) is given by f(x) = x + \(\frac{1}{x}\), then f^{-1} equals to

(a) \(\frac{x+\sqrt{x^{2}-4}}{2}\)

(b) \(\frac{x}{1+x^{2}}\)

(c) \(\frac{x-\sqrt{x^{2}-4}}{2}\)

(d) \(1+\sqrt{x^{2}-4}\)

Answer:

(a) \(\frac{x+\sqrt{x^{2}-4}}{2}\)

Question 48.

Let f(x) = x^{2} – x + 1, x ≥ \(\frac{1}{2}\), then the solution of the equation f(x) = f^{-1}(x) is

(a) x = 1

(b) x = 2

(c) x = \(\frac{1}{2}\)

(d) None of these

Answer:

(a) x = 1

Question 49.

Which one of the following function is not invertible?

(a) f : R → R, f(x) = 3x + 1

(b) f : R → [0, ∞), f(x) = x^{2}

(c) f : R^{+} → R^{+}, f(x) = \(\frac{1}{x^{3}}\)

(d) None of these

Answer:

(d) None of these

Question 50.

The inverse of the function \(y=\frac{10^{x}-10^{-x}}{10^{x}+10^{-x}}\) is

(a) \(\log _{10}(2-x)\)

(b) \(\frac{1}{2} \log _{10}\left(\frac{1+x}{1-x}\right)\)

(c) \(\frac{1}{2} \log _{10}(2 x-1)\)

(d) \(\frac{1}{4} \log \left(\frac{2 x}{2-x}\right)\)

Answer:

(b) \(\frac{1}{2} \log _{10}\left(\frac{1+x}{1-x}\right)\)

Question 51.

If f : R → R defind by f(x) = \(\frac{2 x-7}{4}\) is an invertible function, then find f^{-1}.

(a) \(\frac{4 x+5}{2}\)

(b) \(\frac{4 x+7}{2}\)

(c) \(\frac{3 x+2}{2}\)

(d) \(\frac{9 x+3}{5}\)

Answer:

(b) \(\frac{4 x+7}{2}\)

Question 52.

Consider the function f in A = R – {\(\frac{2}{3}\)} defiend as \(f(x)=\frac{4 x+3}{6 x-4}\). Find f^{-1}.

(a) \(\frac{3+4 x}{6 x-4}\)

(b) \(\frac{6 x-4}{3+4 x}\)

(c) \(\frac{3-4 x}{6 x-4}\)

(d) \(\frac{9+2 x}{6 x-4}\)

Answer:

(a) \(\frac{3+4 x}{6 x-4}\)

Question 53.

If f is an invertible function defined as f(x) = \(\frac{3 x-4}{5}\), then f^{-1}(x) is

(a) 5x + 3

(b) 5x + 4

(c) \(\frac{5 x+4}{3}\)

(d) \(\frac{3 x+2}{3}\)

Answer:

(c) \(\frac{5 x+4}{3}\)

Question 54.

If f : R → R defined by f(x) = \(\frac{3 x+5}{2}\) is an invertible function, then find f^{-1}.

(a) \(\frac{2 x-5}{3}\)

(b) \(\frac{x-5}{3}\)

(c) \(\frac{5 x-2}{3}\)

(d) \(\frac{x-2}{3}\)

Answer:

(a) \(\frac{2 x-5}{3}\)

Question 55.

Let f : R → R, g : R → R be two functions such that f(x) = 2x – 3, g(x) = x^{3} + 5. The function (fog)^{-1} (x) is equal to

(a) \(\left(\frac{x+7}{2}\right)^{1 / 3}\)

(b) \(\left(x-\frac{7}{2}\right)^{1 / 3}\)

(c) \(\left(\frac{x-2}{7}\right)^{1 / 3}\)

(d) \(\left(\frac{x-7}{2}\right)^{1 / 3}\)

Answer:

(d) \(\left(\frac{x-7}{2}\right)^{1 / 3}\)

Question 56.

Let * be a binary operation on set of integers I, defined by a * b = a + b – 3, then find the value of 3 * 4.

(a) 2

(b) 4

(c) 7

(d) 6

Answer:

(c) 7

Question 57.

If * is a binary operation on set of integers I defined by a * b = 3a + 4b – 2, then find the value of 4 * 5.

(a) 35

(b) 30

(c) 25

(d) 29

Answer:

(b) 30

Question 58.

Let * be the binary operation on N given by a * b = HCF (a, b) where, a, b ∈ N. Find the value of 22 * 4.

(a) 1

(b) 2

(c) 3

(d) 4

Answer:

(b) 2

Question 59.

Consider the binary operation * on Q defind by a * b = a + 12b + ab for a, b ∈ Q. Find 2 * \(\frac{1}{3}\)

(a) \(\frac{20}{3}\)

(b) 4

(c) 18

(d) \(\frac{16}{3}\)

Answer:

(a) \(\frac{20}{3}\)

Question 60.

Consider a binary operation * on the set {1, 2, 3, 4, 5} given by the following multiplication table. Compute (2 * 3) * (4 * 5).

(a) 1

(b) 2

(c) 3

(d) 4

Answer:

(a) 1

Question 61.

If the binary operation * is defind on the set Q+ of all positive rational numbers by a * b = \(\frac{a b}{4}\). Then, \(3 *\left(\frac{1}{5} * \frac{1}{2}\right)\) is equal to

(a) \(\frac{3}{160}\)

(b) \(\frac{5}{160}\)

(c) \(\frac{3}{10}\)

(d) \(\frac{3}{40}\)

Answer:

(a) \(\frac{3}{160}\)

Question 62.

The number of binary operations that can be defined on a set of 2 elements is

(a) 8

(b) 4

(c) 16

(d) 64

Answer:

(c) 16

Question 63.

Let * be a binary operation on Q, defined by a * b = \(\frac{3 a b}{5}\) is

(a) Commutative

(b) Associative

(c) Both (a) and (b)

(d) None of these

Answer:

(c) Both (a) and (b)

Question 64.

(a) 0

(b) 1

(c) 2

(d) 3

Answer:

(a) 0

Question 65.

Let * be a binary operation on set Q of rational numbers defined as a * b = \(\frac{a b}{5}\). Write the identity for *.

(a) 5

(b) 3

(c) 1

(d) 6

Answer:

(a) 5

Question 66.

For binary operation * defind on R – {1} such that a * b = \(\frac{a}{b+1}\) is

(a) not associative

(b) not commutative

(c) commutative

(d) both (a) and (b)

Answer:

(d) both (a) and (b)

Question 67.

The binary operation * defind on set R, given by a * b = \(\frac{a+b}{2}\) for all a,b ∈ R is

(a) commutative

(b) associative

(c) Both (a) and (b)

(d) None of these

Answer:

(a) commutative

Question 68.

Let A = N × N and * be the binary operation on A defined by (a, b) * (c, d) = (a + c, b + d). Then * is

(a) commutative

(b) associative

(c) Both (a) and (b)

(d) None of these

Answer:

(c) Both (a) and (b)

Question 69.

Find the identity element in the set I^{+} of all positive integers defined by a * b = a + b for all a, b ∈ I^{+}.

(a) 1

(b) 2

(c) 3

(d) 0

Answer:

(d) 0

Question 70.

Let * be a binary operation on set Q – {1} defind by a * b = a + b – ab : a, b ∈ Q – {1}. Then * is

(a) Commutative

(b) Associative

(c) Both (a) and (b)

(d) None of these

Answer:

(c) Both (a) and (b)

Question 71.

The binary operation * defined on N by a * b = a + b + ab for all a, b ∈ N is

(a) commutative only

(b) associative only

(c) both commutative and associative

(d) none of these

Answer:

(c) both commutative and associative

Question 72.

The number of commutative binary operation that can be defined on a set of 2 elements is

(a) 8

(b) 6

(c) 4

(d) 2

Answer:

(d) 2

Question 73.

Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as aRb if a is congruent to b ∀ a, b ∈ T. Then R is

(a) reflexive but not transitive

(b) transitive but not symmetric

(c) equivalence

(d) None of these

Answer:

(c) equivalence

Question 74.

The maximum number of equivalence relations on the set A = {1, 2, 3} are

(a) 1

(b) 2

(c) 3

(d) 5

Answer:

(d) 5

Question 75.

Let us define a relation R in R as aRb if a ≥ b. Then R is

(a) an equivalence relation

(b) reflexive, transitive but not symmetric

(c) symmetric, transitive but not reflexive

(d) neither transitive nor reflexive but symmetric

Answer:

(b) reflexive, transitive but not symmetric

Question 76.

Let A = {1, 2, 3} and consider the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}. Then R is

(a) reflexive but not symmetric

(b) reflexive but not transitive

(c) symmetric and transitive

(d) neither symmetric, nor transitive

Answer:

(a) reflexive but not symmetric

Question 77.

The identity element for the binary operation * defined on Q – {0} as a * b = \(\frac{a b}{2}\) ∀ a, b ∈ Q – {0) is

(a) 1

(b) 0

(c) 2

(d) None of these

Answer:

(c) 2

Question 78.

Let A = {1, 2, 3, …. n} and B = {a, b}. Then the number of surjections from A into B is

(a) \(^{n} P_{2}\)

(b) 2^{n} – 2

(c) 2^{n} – 1

(d) none of these

Answer:

(b) 2^{n} – 2

Question 79.

Let f : R → R be defind by f(x) = \(\frac{1}{x}\) ∀ x ∈ R. Then f is

(a) one-one

(b) onto

(c) bijective

(d) f is not defined

Answer:

(d) f is not defined

Question 80.

Which of the following functions from Z into Z are bijective?

(a) f(x) = x^{3}

(b) f(x) = x + 2

(c) f(x) = 2x + 1

(d) f(x) = x^{2} + 1

Answer:

(b) f(x) = x + 2

Question 81.

Let f : R → R be the functions defined by f(x) = x^{3} + 5. Then f^{-1}(x) is

(a) \((x+5)^{\frac{1}{3}}\)

(b) \((x-5)^{\frac{1}{3}}\)

(c) \((5-x)^{\frac{1}{3}}\)

(d) 5 – x

Answer:

(b) \((x-5)^{\frac{1}{3}}\)

Question 82.

Let f : R – {\(\frac{3}{5}\)} → R be defined by f(x) = \(\frac{3 x+2}{5 x-3}\). Then

(a) f^{-1}(x) = f(x)

(b) f^{-1}(x) = -f(x)

(c) (fof) x = -x

(d) f^{-1}(x) = \(\frac{1}{19}\) f(x)

Answer:

(a) f^{-1}(x) = f(x)

Question 83.

Let f : [0, 1] → [0, 1] be defined by

(a) constant

(b) 1 + x

(c) x

(d) None of these

Answer:

(c) x

Question 84.

(a) 9

(b) 14

(c) 5

(d) None of these

Answer:

(a) 9

Question 85.

Let f : R → R be given by f(x) = tan x. Then f^{-1}(1) is

(a) \(\frac{\pi}{4}\)

(b) {nπ + \(\frac{\pi}{4}\); n ∈ Z}

(c) Does not exist

(d) None of these

Answer:

(b) {nπ + \(\frac{\pi}{4}\); n ∈ Z}