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Extra questions on relations and functions class 12



In this page we have extra questions on relations and functions class 12. Hope you like them and do not forget to like , social share and comment at the end of the page.

Multiple Choice Questions

Question 1
Let R be a relation on the set N of natural numbers defined by nRm if n divides m. Then R is
(a) Equivalence
(b) Reflexive and symmetric
(c) Transitive and symmetric
(d) Reflexive, transitive but not symmetric

Question 2
Let P ={1,2,4,5}. The total number of distincts relations that can be defined on P is
(a) 16
(b) $2^{16}$
(c) 32
(d) 256

Question 3
The function $f:[0,\infty)$ -> R given by $f(x)= \frac {x}{x+1}$ is
(a) one -one and onto
(b) one-one but not onto
(c) onto but not one-one
(d) Neither one-one nor onto

Question 4
Let f : R -> R be defined by $f(x) = 2x - 1. Then $f^{–1}(x)$ is given by
(a) $\frac {x-2}{2}$
(b) $\frac {x+1}{2}$
(c) $\frac {x-1}{2}$
(d) $\frac {2x-2}{3}$

Question 5
The function f:R-> R given by $f(x)= \frac {x^2 -8}{x^2+2}$ is
(a) one -one and onto
(b) one-one but not onto
(c) onto but not one-one
(d) Neither one-one nor onto

Question 6
let A={1,2,3} Which of the following relation defined on A is a reflexive relation?
(a) {(1, 1), (1, 2), (2, 1), (2, 2)}
(b) {(1, 2), (2, 1)}
(c) {(1, 1), (2, 2), (3, 3)}
(d) {(1, 2), (2, 3), (3, 1)}

Question 7
Which of the following is an onto function f:R -> R?
(a) $f(x) = x^2$
(b) $f(x) = 2x + 1$
(c) $f(x) = |x|$
(d) $f(x) = x^2 + 1$

Question 8
For real numbers x and y, define xRy if and only if $x – y + \sqrt 2$ is an irrational number. Then the relation R is
(a) symmetric
(b) Reflexive
(c) Transitive
(d) None of these

Question 9
$f(x) =x^2 +1$ and $g(x) = sin x$, then the value of $g \circ f$ is
(a) $sin^2x + 1$
(b) $sin (x^2 +1)$
(c) $x sin x + 1$
(d) None of these

Question 10
let f(x)=[x] and g(x) =|x|, the the value of $(g \circ f) (\frac {-5}{3}) - (f \circ g)(\frac {-5}{3}) $ is
(a) 1/2
(b) 1
(c) -1
(d) -2

Fill in the blanks

Question 11
(i) Let the relation R be defined in N by aRb if 2a + 3b = 30. Then R = _____
(ii) if $f(x)=x^2 +2$ and $g(x)= 1 - \frac{1}{1-x}$, the $f \circ g$ is _____
(iii) The inverse of the function $f(x)= \frac {x}{x+2}$ is _______

Long Answer type

Question 12
Let f : W -> W be defined as
$f(n)=\begin{cases} & \text{ n-1 if } n \ is \ odd \\ & \text{ n+1 if } n \ is \ even \end{cases}$
Then show that f is invertible.Also, find the inverse of f.

Question 13
Show that f : [– 1, 1] -> R , given by $f(x) =\frac {x}{x+2}$ is one-one.
Find the inverse of the function f : [– 1, 1] -> Range of f

Question 14
Show that the relation R in the set N X N defined by (a,b)R(c,d) if $a^2+d^2 =b^2+ c^2$ for all a,b,c,d in N is an equivalence relation

Question 15
Functions f , g : R -> R are defined, respectively, by $f(x) = x^2 + 3x + 1$,g(x) = 2x – 3, find
(i) $f \circ g$
(ii) $g \circ f$
(iii)$f \circ f$
(iv) $g \circ g$

Question 16
Show that the relation R in the set A X A defined by (a,b)R(c,d) if $a+d =b+c$ for all a,b,c,d in A is an equivalence relation. Here A={1,2,3...10}

Question 17
If $f(x) =\frac {4x+3}{6x-4}$ and $x \ne \frac {-2}{3}$ , show that fof(x) = x for all x $x \ne \frac {-2}{3}$ . Also, find the inverse of f

Question 18
Let A= R -{-1} and * be the binary operation on A defined by
a*b = a+b+ab
(i) Show that * is commutative and Associative
(ii) Find the identity element for * on A
(iii) Prove that every element of A is invertible

Question 19
Prove that the function f:[0, $\infty$) -> R given by $f(x) = 9x^2+ 6x – 5$ is not invertible. Modify the codomain of the function f to make it invertible, and hence find $f^{–1}$

Question 20
Check whether the relation R in the set R of real numbers, defined by R = {(a, b) : 1 + ab > 0}, is reflexive, symmetric or transitive
Question 21
Show that the relation R on the set Z of integers, given by R = {(a, b) : 2 divides (a – b)} is an equivalence relation
Question 22
Prove that the relation R in the set A = {1, 2, 3, 4, 5, 6, 7} given by R = {(a, b) : |a – b| is even} is an equivalence relation
Question 23
Show by examples that the relation R in , defined by
R = {(a, b) : $a \le b^3$} is neither reflexive nor transitive.
Question 24
Let R be a relation on the set A of ordered pairs of positive integers defined by (x, y) R (u, v) if and only if xv = yu. Show that R is an equivalence relation

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