Untitled Document

Question Bank No: 1

1. In order that a relation R defined in a non-empty set A is an equivalence relation, it is sufficient to show that R is

a)		reflexive
b)		symmetric
c)		transitive
d)		reflexive, symmetric and transitive

2. If R $\underline{\subset}$ A $\times$ B and S $\underline{\subset}$ B $\times$ C be two relations, then ${(SOR)}^{- 1}$ is equal to

a)		$S^{- 1} o R^{- 1}$
b)		$R^{- 1}$ ${oS}^{- 1}$
c)		SoR
d)		None of these

3. The void relation in a set A is

a)		reflexive
b)		symmetric and transitive
c)		reflexive and transitive
d)		reflexive and symmetric

4. Let R be a reflexive relation defined in a finite set having n elements and let there be k ordered pairs in R. Then

a)		k $\geq$ n
b)		k $\leq$ n
c)		k = n
d)		None of these

5. Let A be the set of all children in the world and R be a relation in A defined by xRy if x and y have same sex. Then R is

a)		not reflexive
b)		not symmetric
c)		not transitive
d)		an equivalence relation

6. If A is the set of even natural numbers less than 8 and B is the set of all prime numbers less than 7, then the number of relations from A to B is

a)		$2^{9}$
b)		$9^{2}$
c)		$3^{2}$
d)		$2^{9}$ – 1

7. The relation R = {(1, 1), (2, 2), (3, 3) ont he set {1, 2, 3} is

a)		Symmetric only
b)		reflexive only
c)		an equivalence relation
d)		transitive only

8. Let A = {2, 3, 4, 5, ...., 16, 17, 18}. Let = be the equivalence relation on A $\times$ A, the cartesian product of A and A, defined by (a, b) $\approx$ (c, d) if ad = bc. Then the number of ordered pairs in the equivalence class of (3, 2) is

a)		4
b)		5
c)		6
d)		7

9. Let A = {a, b, c} and B = {1, 2}. Consider a relation R defined from set A to set. Then R is equal to a subset of

a)		A
b)		B
c)		A $\times$ B
d)		B $\times$ A

10. A relation from P to Q is

a)		A universal set of P $\times$ Q
b)		P $\times$ Q
c)		an equivalent set of P $\times$ Q
d)		a subset of P $\times$ Q

11. If R $\underline{\subset}$ A $\times$ Band S $\underline{\subset}$ B $\times$ C be two relations, then ${(SOR)}^{- 1}$ is equal to

a)		$S^{- 1} o R^{- 1}$
b)		RoS
c)		$R^{- 1}$ ${oS}^{- 1}$
d)		None of these

12. Which one of the following relations in Z is an equivalence relation?

a)		a $R_{1}$ b $\Leftrightarrow$ a $\leq$ b
b)		a $R_{2}$ b $\Leftrightarrow$ a/b
c)		a $R_{3}$ b $\Leftrightarrow$ a=b
d)		a $R_{4}$ b $\Leftrightarrow$ a > b

13. Let R be an equivalence relation in a finite set A having n elements. The number of ordered pairs in R is

a)		less than n
b)		less than or equal to n
c)		greater than n
d)		greater than or equal to

14. Let R be a relation defined in the set of natural numbers N as R = {(x, y) : x, y $\in$ N 2x + y = 41}. Which of the following is true?

a)		R is reflexive
b)		R is symmetric
c)		R is transitive
d)		At least one is false

15. Which of the following relations in R is an equivalence relation?

a)		x $R_{1}$ y $\Leftrightarrow$ $\| x \|$ = $\| y \|$
b)		x $R_{2}$ y $\Leftrightarrow$ x $\geq$ y
c)		x $R_{3}$ y $\Leftrightarrow$ x/y
d)		x $R_{4}$ y $\Leftrightarrow$ x < y

16. Let S be a non-empty set. In P(S), let R be a relation defined as ARB $\Rightarrow$ A $\cap$ B $\neq$ $φ$ . The relation R is

a)		reflexive
b)		symmetric
c)		transitive
d)		None of these

17. If R be an anti-symmetric relation in a set A such that (a, b), (b, a) $\in$ R, then

a)		a $\geq$ b
b)		a $\leq$ b
c)		a = b
d)		None of these

18. Let A = {1, 2, 3}, B = {1, 3, 5}. If relation R from A to B is given by {(1, 3), (2, 5), (3, 3)}, then $R^{- 1}$ is

a)		{(3, 3), (3, 1), (5, 3)}
b)		{(1, 3), (2, 5), (5, 3)}
c)		{(1, 3), (5, 2)}
d)		None of these

19. Two points A and B in a plane are related if OA = OB, where O is a fixed point. This relation is

a)		reflexive but not symmetric
b)		symmetric but not transitive
c)		an equivalent relation
d)		None of these

20. Let R be a reflexive relation in a finite set having n elements and let there be m ordered pairs in R. Then

a)		m $\geq$ n
b)		m $\leq$ n
c)		m = n
d)		None of these

21. The domain of the function f(x)=x $\frac{1 + 2 {(x + 4)}^{- 0.5}}{2 - {(x + 4)}^{0.5}}$ + 5 ${(x + 4)}^{0.5}$ is

a)		R
b)		(-4, 4)
c)		(0, $\infty$ )
d)		(-4, 0) $\cup$ (0, $\infty$ )

22. If f(x) = $x^{2}$ + 4x + 1, then

a)		f(x) = f (-x) for all x
b)		f(x) $>$ 1 for all x $\geq$ 0
c)		f(x) $\geq$ 1 for all x $\geq$ 0
d)		f(x) $\geq$ 1 for all x $\geq$ 0 or x $\leq$ -4

23. No. of onto functions defined from the set {1, 2, 3, ......, 10} to the set {1, 2} is

a)		1024
b)		$10^{10}$
c)		1022
d)		None of these

24. If f : A $\to$ B is a bijection, then

a)		o(A) $\leq$ o(B)
b)		o(B) $\leq$ o(A)
c)		o(A) $<$ o(B)
d)		None of these

25. A mapping f : X $\to$ Y is one-one if

a)		$x_{1}$ = $x_{2}$ $\Rightarrow$ f( $x_{1}$ ) = f( $x_{2}$ )
b)		f $x_{1}$ = f $x_{2}$ $\Rightarrow$ $x_{1}$ = $x_{2}$
c)		f $x_{1}$ $\neq$ f $x_{2}$ for all $x_{1}$ , $x_{2}$ $\in$ X
d)		None of these

26. If f : R $\to$ r and g : R $\to$ R are defined by f(x) = 2x + 3 and g(x) = $x^{2}$ + 7, then the values of x such that g $(f (x))$ = 8 are

a)		1, 2
b)		-1, 2
c)		-1, -2
d)		1, -2

27. Let f(x) = $x^{2}$ and g(x) = $\sqrt{x}$ , then

a)		(gof) (-2) = 2
b)		(fog) (2) = 4
c)		(gof) (2) = 4
d)		(fog) (3) = 6

28. The fundamental period of the function f(x) = 2 cos $\frac{1}{3}$ (x - $π$ ) is

a)		6 $π$
b)		4 $π$
c)		2 $π$
d)		$π$

29. If f(x) = $\frac{x}{x - 1}$ , then $\frac{f (a)}{f (a + 1)}$ is equal to

a)		f(-a)
b)		f(1/a)
c)		f( $a^{2}$ )
d)		f $(- \frac{a}{a - 1})$

30. Mapping f: R $\to$ R which is defined as f(x) = cos x, x $\in$ R is

a)		Neither one-one nor onto
b)		one-one
c)		onto
d)		one-one and onto

31. If f(x) = $\frac{x}{x - 1}$ , x $\in$ R $-$ {1}, then $f^{- 1}$ (x) is equal to

a)		$\frac{x}{x - 1}$
b)		$\frac{x - 1}{x}$
c)		$\frac{1 - x}{x}$
d)		$\frac{x}{1 - x}$

32. Let f(x) = $x^{2}$ $-$ x + 1, x $\geq$ 1/2, then the solution of the equation $f^{- 1}$ (x) = f(x) is

a)		x = 1
b)		x = 2
c)		x = 1/2
d)		None of these

33. If a function f : $[2, \infty)$ $\to B$ B defined by f(x) = $x^{2} -$ 4x + 5 is a bijection, then B is equal to

a)		R
b)		$[1, \infty)$
c)		$[4, \infty)$
d)		$[5, \infty)$

34. The inverse of the function f(x) = $\frac{10^{x} - 10^{- x}}{10^{x} + 10^{- x}} +$ 1 is

a)		$f^{- 1}$ (x) = $\log_{10} \frac{x}{2 - x}$
b)		$f^{- 1}$ (x) = $\frac{1}{2}$ $\log_{10} \frac{x}{2 - x}$
c)		$f^{- 1}$ (x) = $\frac{1}{2}$ $\log_{10} \frac{x}{1 - x}$
d)		None of these

35. Let f : R $\to$ R be a function defined by f(x) = cos (2x + 7). This function is

a)		injective
b)		surjective
c)		bijective
d)		None of these

36. The function f(x) = cos x, x $\in$ R is

a)		an even function
b)		an odd function
c)		a power function
d)		None of these

37. Let A and B be two finite sets having m and n elements respectively. The total number of mappings from A to B is

a)		mn
b)		$2^{mn}$
c)		$n^{m}$
d)		$m^{n}$

38. If f(x) = 5x, x $\in$ R, then f(0), f(1), f(2), .... are in

a)		A. P
b)		G.P
c)		H.P
d)		None of these

39. The domain of the function f(x) = $| x |$ is

a)		{-3}
b)		R $-$ {-3}
c)		R $-$ {3}
d)		None of these

40. The range of the function f(x) = $\frac{x - 2}{2 - x}$ is

a)		R
b)		R - ${1}$
c)		${- 1}$
d)		R - {-1}

41. If f(x)= $\frac{2^{x} + 2^{- x}}{2}$ then f(x=y) F(x-y) is equal to

a)		$\frac{1}{2} [f (x + y) + f (x - y)]$
b)		$\frac{1}{2} [f (2 x) + f (2 y)]$
c)		$\frac{1}{2} [f (x + y) . f (x - y)]$
d)		none of these

42. The domain of the function f(x) = $\sqrt{\frac{1 - | x |}{| x | - 2}}$ is

a)		$(- \infty, - 1) \cup (1, \infty)$
b)		$(- \infty, - 2) \cup (2, \infty)$
c)		$(- 2, - 1) \cup [1, 2)$
d)		None of these

43. If f(x+2y, x-2y) = xy then f(x,y) is

a)		$\frac{x^{2} - y^{2}}{8}$
b)		$\frac{x^{2} - y^{2}}{4}$
c)		$\frac{x^{2} + y^{2}}{4}$
d)		$\frac{x^{2} - y^{2}}{2}$

44. If f(x)= ${(a - x^{n})}^{\frac{1}{n}}$ then f $(f (x))$ is

a)		$x^{\frac{1}{n}}$
b)		$x^{n}$
c)		$a - x$
d)		x

45. If f is a function such that f(0)=2, f(1)=3 and f (x+2)=2 f(x) $- f (x + 1)$ for every real x then f(5) is

a)		7
b)		13
c)		1
d)		5

46. The function f(x) = $\frac{x}{e^{x} - 1} + \frac{x}{2} + 1$ is

a)		an odd funtion
b)		an even function
c)		neither even or odd function
d)		a periodic function

47. The range of the function f(x) = $\frac{5}{3 - x^{2}}$ is

a)		$(- \infty, 0)$ $\cup [\frac{5}{3}, \infty)$
b)		$(- \infty, 0)$ $\cup (\frac{5}{3}, \infty)$
c)		$(- \infty, 0)$ $\cup (\frac{5}{3}, \infty]$
d)		none of these

48. If f(x).f(y)=f(x)+f(y) $- 2 \forall x, y \in R$ and if f(x) is not a constant function, then the value of f(1) is

a)		1
b)		2
c)		0
d)		-1

49. The domain of the function f(x)= $\sqrt{\log_{10} (\frac{3 - x}{x})}$ is

a)		(0, $\frac{3}{2}$ )
b)		(0,3)
c)		(- $\propto, \frac{3}{2}$ )
d)		$(0, \frac{3}{2}]$

50. If 3 f(x)+5 f( $\frac{1}{x}) = \frac{1}{x} - 3$ $\forall x (\neq 0) \in R, then f (x) =$

a)		$\frac{1}{14} (\frac{3}{x} + 5 x - 6)$
b)		$\frac{1}{14} (\frac{- 3}{x} + 5 x - 6)$
c)		$\frac{1}{14} (\frac{- 3}{x} + 5 x + 6)$
d)		$\frac{1}{14} (\frac{3}{x} - 5 x + 6)$