Relations and Functions Objective
- The function f : A → B defined by f(x) = 4x + 7, x ∈ R is
(a) one-one
(b) Many-one
(c) Odd
(d) Even
Ans (a)
- The smallest integer function f(x) = [x] is
(a) One-one
(b) Many-one
(c) Both (a) & (b)
(d) None of these
Ans (b)
- 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
Ans (a)
- The number of bijective functions from set A to itself when A contains 106 elements is
(a) 106
(b) (106)2
(c) 106!
(d) 2106
Ans (c)
- If f : R → R, g : R → R and h : R → R is such that f(x) = x2, g(x) = tanx and h(x) = logx, then the value of [ho(gof)](x), if x = √ π/2 will be
(a) 0
(b) 1
(c) -1
(d) 10
Ans (a)
- If f : R → R and g : R → R defined by f(x) = 2x + 3 and g(x) = x2+ 7, then the value of x for which f(g(x)) = 25 is
(a) ±1
(b) ±2
(c) ±3
(d) ±4
Ans (b)
- Let f : N → R : f(x) = (2x−1)/2 and g : Q → R : g(x) = x + 2 be two functions. Then, (gof) (3/2) is
(a) 3
(b) 1
(c) 7/2
(d) None of these
Ans (a)
- If f : R → R, g : R → R and h : R → R are such that f(x) = x2, 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
Ans (a)
- The number of binary operations that can be defined on a set of 2 elements is
(a) 8
(b) 4
(c) 16
(d) 64
Ans (c)
- Let * be a binary operation on Q, defined by a * b = 3ab/5 is
(a) Commutative
(b) Associative
(c) Both (a) and (b)
(d) None of these
Ans (c)
- Let * be a binary operation on set Q of rational numbers defined as a * b = ab/5. Write the identity for *.
(a) 5
(b) 3
(c) 1
(d) 6
Ans (a)
- For binary operation * defind on R – {1} such that a * b = a/b+1 is
(a) not associative
(b) not commutative
(c) commutative
(d) both (a) and (b)
Ans (d)
- The binary operation * defind on set R, given by a * b = a+b/2 for all a,b ∈ R is
(a) commutative
(b) associative
(c) Both (a) and (b)
(d) None of these
Ans (a)
- 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
Ans (c)
- 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
Ans (d)
- 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
Ans (c)
- 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
Ans (c)
- 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
Ans (d)
- 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 ∈ Then R is
(a) reflexive but not transitive
(b) transitive but not symmetric
(c) equivalence
(d) None of these
Ans (c)
- The maximum number of equivalence relations on the set A = {1, 2, 3} are
(a) 1
(b) 2
(c) 3
(d) 5
Ans (d)
- 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
Ans (b)
- 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
Ans (a)
- The identity element for the binary operation * defined on Q – {0} as a * b = ab/2 ∀ a, b ∈ Q – {0) is
(a) 1
(b) 0
(c) 2
(d) None of these
Ans (c)
- Let A = {1, 2, 3, …. n} and B = {a, b}. Then the number of surjections from A into B is
(a) nP2
(b) 2n – 2
(c) 2n – 1
(d) none of these
Ans (b)
- Let f : R → R be defind by f(x) = 1/x ∀ x ∈ Then f is
(a) one-one
(b) onto
(c) bijective
(d) f is not defined
Ans (d)
- Which of the following functions from Z into Z are bijective?
(a) f(x) = x3
(b) f(x) = x + 2
(c) f(x) = 2x + 1
(d) f(x) = x2 + 1
Ans (b)
- Let f : R → R be given by f(x) = tan x. Then f-1(1) is
(a) π/4
(b) {nπ + π/4; n ∈ Z}
(c) Does not exist
(d) None of these
Ans (b)
- 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
Ans (d)
- 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
Ans (d)
- 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
Ans (d)
- Total number of equivalence relations defined in the set S = {a, b, c} is
(a) 5
(b) 3!
(c) 23
(d) 33
Ans (a)
- The relation R is defined on the set of natural numbers as {(a, b) : a = 2b}. Then, R-1is 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
Ans (b)
- Let X = {-1, 0, 1}, Y = {0, 2} and a function f : X → Y defiend by y = 2x4, is
(a) one-one onto
(b) one-one into
(c) many-one onto
(d) many-one into
Ans (c)
- Let g(x) = x2– 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
Ans (b)
- The mapping f : N → N is given by f(n) = 1 + n2, 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
Ans (c)
- The function f : R → R given by f(x) = x3– 1 is
(a) a one-one function
(b) an onto function
(c) a bijection
(d) neither one-one nor onto
Ans (c)
- Let f : [0, ∞) → [0, 2] be defined by f(x)=2x/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
Ans (a)
- 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
Ans (b)
- 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
Ans (a)
- Let f : R → R be a function defined by f(x) = x3+ 4, then f is
(a) injective
(b) surjective
(c) bijective
(d) none of these
Ans (c)
- 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
Ans (c)
- 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
Ans (b)
- Let * be the binary operation on N given by a * b = HCF (a, b) where, a, b ∈ Find the value of 22 * 4.
(a) 1
(b) 2
(c) 3
(d) 4
Ans (b)