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Question Bank No: 1

1. If f(x) = ${(x + 1)}^{\cot x}$ is continuous at x = 0, then f(0) is

a)		0
b)		$\frac{1}{e}$
c)		e
d)		None of these

2. The function f(x) = x( $\sqrt{x} - \sqrt{x + 1}$ ) is

a)		Continuous but not differentiable at x = 0
b)		differentiable at x = 0
c)		differentiable but not continuous at x = 0
d)		not differentiable at x = 0

3. If x + 4 $| y |$ = 6y, then y as a function of x is

a)		Continuous at x = 0
b)		derivable at x = 0
c)		$\frac{dy}{dx}$ = $\frac{1}{2}$ , for each x
d)		None of these

4. The function f(x) = $| x - 1 |$ $e^{x}$ is differntiable if x belongs to

a)		R
b)		R $- {1}$
c)		R $- {- 1}$
d)		R $- {0}$

5. If f(x) = a $| \sin x |$ +b $e^{| x |}$ +c ${| x |}^{3}$ and if f(x) is differentiable at x = 0, then

a)		a = b = c = 0
b)		a = b = 0, c $\in$ R
c)		b = c = 0, a $\in$ R
d)		a = c = 0, b $\in$ R

6. The function f(x) = $e^{- | x |}$ is

a)		Continuous everywhere but not differentiable at x = 0
b)		continuous and differentiable everywhere
c)		not continuous at x = 0
d)		None of these

7. If f(x) = $\frac{2 - {(256 - 7 x)}^{1 / 8}}{{(5 x + 32)}^{1 / 5} - 2}$ , (x $\neq$ 0) then for f to be continuous everywhere f(0) is equal to

a)		-1
b)		1
c)		16
d)		None of these

8. The function f(x) = $\frac{4 - x^{2}}{4 x - x^{2}}$ is

a)		Discontinous at only one point
b)		discontinuous at two points
c)		discontinuous at three points
d)		None of these

9. If f(x) = ${\begin{matrix} \sin x, & x \neq n π, n \in Z \\ 2, & otherwise \end{matrix}$ and ${\begin{matrix} x^{2} + 1, & x \neq 0, 2 \\ 4, & x = 0 \\ 5, & x = 2 \end{matrix}$ then $\lim_{x \to 0}$ g (f(x)) is

a)		5
b)		4
c)		2
d)		-5

10. $\lim_{x \to 0}$ $(\frac{\sin x - x}{x}) (\sin \frac{1}{x})$

a)		does not exist
b)		is equal to 0
c)		is equal to 1
d)		exists and different from 0 and 1

11. $\lim_{x \to 0}$ ${\frac{1 + \tan x}{1 + \sin x}}^{cosec x}$ is

a)		e
b)		$e^{- 1}$
c)		1
d)		None of these

12. $\lim_{x \to a^{-}}$ $(\frac{{| x |}^{3}}{a} - {[\frac{x}{a}]}^{3})$ , (a > 0 and $[x]$ denotes the greatest integer less than or equal to x) is

a)		$a^{2}$ $-$ 3
b)		$a^{2}$ $-$ 1
c)		$a^{2}$
d)		None of these

13. $\lim_{x \to 0}$ $\frac{x \cos x + - \log (1 + x)}{x^{2}}$ is

a)		$\frac{1}{2}$
b)		0
c)		1
d)		None of these

14. If f(x) = $\frac{\sin (e^{x - 2} - 1)}{\log (x - 1)}$ , then $\lim_{x \to 2}$ f(x) is

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

15. $\lim_{x \to 1}$ $\frac{x + x^{2} + ........ + x^{n} - n}{x - 1}$ is

a)		n
b)		$\frac{n + 1}{2}$
c)		$\frac{n (n + 1)}{2}$
d)		$\frac{n (n - 1)}{2}$

16. $\lim_{x \to 0}$ ${(\frac{a^{x} + b^{x} + c^{x}}{3})}^{1 / x}$ is

a)		abc
b)		$\sqrt{abc}$
c)		(abc)1/3
d)		None of these

17. $\lim_{x \to \infty}$ ${(\frac{x + 1}{x + 2})}^{2 x + 1}$ is

a)		e
b)		$e^{- 2}$
c)		$e^{- 1}$
d)		1

18. $\lim_{n \to \infty}$ $[\frac{1}{1 - n^{4}} + \frac{8}{1 - n^{4}} + ........ + \frac{n^{3}}{1 - n^{4}}]$ is

a)		$\frac{1}{4}$
b)		- $\frac{1}{4}$
c)		$\frac{1}{2}$
d)		None of these

19. $\lim_{x \to 0}$ $\frac{10^{x} - 2^{x} - 5^{x} + 1}{x \tan x}$ is

a)		log 2
b)		$\frac{\log 2}{\log 5}$
c)		(log 2) (log 5)
d)		log 10

20. $\lim_{x \to \frac{π}{4}}$ $\frac{\cos x - \sin x}{(\frac{π}{4} - x) (\cos x + \sin x)}$ is

a)		0
b)		1
c)		-1
d)		None of these

21. If f(x) $= {\begin{matrix} sinx \\ 2 \end{matrix}, \begin{matrix} x \neq n π \\ otherwise \end{matrix}$ = n is an integer g(x) $= {\begin{matrix} x^{2} + 1 \\ 4 \\ 5 \end{matrix} \begin{matrix} \begin{matrix} x \neq 0, 2 \\ x = 0 \end{matrix} \\ x = 2 \end{matrix}$ then $\begin{matrix} Lt \\ x \to 0 \end{matrix}$ g(f(x) is

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

22. $\begin{matrix} Lt \\ x \to \propto \end{matrix} {(\frac{x + 6}{x + 1})}^{x + 4} equals$

a)		0
b)		1
c)		$e^{4}$
d)		$e^{5}$

23. $\begin{matrix} Lt \\ x \to 0 \end{matrix} [\frac{x}{\tan^{- 1} 2 x}] equals$

a)		0
b)		$\frac{1}{2}$
c)		1
d)		$\propto$

24. $\begin{matrix} Lt \\ x \to 0 \end{matrix} \frac{{\sin x}^{0}}{x} is$

a)		1
b)		$π$
c)		$\frac{π}{180}$
d)		$\frac{180}{π}$

25. f (x) $\begin{matrix} = & 1 + x & x > 0 \\ = & x & x < 0 \end{matrix}$ then limit f f(x) as x tends to zero is

a)		1
b)		0
c)		$\frac{1}{2}$
d)		non existent Since limit does not exist

26. $\begin{matrix} Lt \\ x \to 0 \end{matrix} \frac{e x^{2} - cosx}{x^{2}} is equal to$

a)		$\frac{3}{2}$
b)		$\frac{1}{2}$
c)		-1
d)		none of these

27. If f(x) = $\begin{matrix} x \sin \frac{1}{x} \\ = 0 \end{matrix}$ $, \begin{matrix} x \neq 0 \\ x = 0 \end{matrix}$ then $\begin{matrix} Lt f (x) \\ x \to 0 \end{matrix}$ equals

a)		1
b)		0
c)		-1
d)		none of these

28. $\begin{matrix} Lt \\ x \to 0 \end{matrix} \frac{{(1 - x)}^{n}}{x} is$

a)		Ln
b)		Ln-1
c)		-1
d)		n

29. $\begin{matrix} Lt \\ x \to 0 \end{matrix} \frac{\log \cos x}{x} is equal to$

a)		0
b)		$\propto$
c)		1
d)		none of these

30. $\begin{matrix} Lt \\ x \to 0 \end{matrix} \frac{a^{x} - b^{x}}{x} is$

a)		log ab
b)		log $(\frac{a}{b})$
c)		$\frac{loga}{logb}$
d)		$\frac{logb}{loga}$

31. $\begin{matrix} Lt \\ x \to x \end{matrix} \frac{logx - 1}{x - e} is equal to$

a)		1
b)		$\frac{1}{2}$
c)		$\frac{1}{e}$
d)		0

32. $\begin{matrix} Lt \\ x \to 2 \end{matrix} + [\frac{{[x]}^{3}}{3} - {[\frac{x}{3}]}^{3}] is equal to$

a)		0
b)		$\frac{64}{27}$
c)		$\frac{8}{3}$
d)		none of these

33. If $\begin{matrix} Lt \\ x \to 5 \end{matrix} \frac{x^{k} - 5^{k}}{x - 5} = 500 then positive value of K is$

a)		3
b)		4
c)		5
d)		6

34. $\begin{matrix} Lt \\ x \to 9 \end{matrix} \frac{x^{\frac{3}{2}} - 27}{x - 9} =$

a)		$\frac{3}{2}$
b)		$\frac{9}{2}$
c)		$\frac{2}{3}$
d)		$\frac{1}{3}$

35. If $\begin{matrix} Lt \\ x \to 0 \end{matrix} \frac{Sin Px}{\tan 4 x} = 4 then the value of P is$

a)		6
b)		9
c)		12
d)		4

36. If ${}^{n}{C 2}$ = ${}^{n}{C 3}$ , then the value of ${}^{n}{C 4}$ is

a)		2
b)		3
c)		5
d)		4

37. The value of n, when ${}^{n}{P 2}$ = 20

a)		3
b)		4
c)		6
d)		5

38. There are four letter boxes in a post office. In how many ways can a man post 8 distinct letters

a)		8 x 8
b)		$8^{4}$
c)		$4^{8}$
d)		${}^{8}{P 4}$

39. The total number of combinations of n different things taken 1, 2, 3, --- n at a time is

a)		$2^{n} + 1$
b)		$2^{n + 1}$
c)		$2^{n} - 1$
d)		$2^{n - 1}$

40. If C (10, 4) + C (10, 5) = C (11, r), then r is equal to

a)		5
b)		4
c)		3
d)		6

41. The number of ways in which four men and four women can be seated at round table so that no two women may be together is

a)		576
b)		48
c)		16
d)		144

42. P (10, r) = 2 P(9, r), r is equal to

a)		2
b)		4
c)		5
d)		6

43. The possible outcomes when a coin is tossed five times

a)		$2^{5}$ ?
b)		$5^{2}$
c)		10
d)		5/2

44. The number of ways of selecting 4 players out of 5 is

a)		5
b)		5!
c)		C (5, 4)
d)		30

45. 66 games were played in a tournament where each players one against the rest. The number of players are

a)		33
b)		12
c)		13
d)		11

46. The number of different four digit numbers that can be formed with the digits 2, 3, 4, 5, 7 using each digit only once is

a)		4!
b)		4 (4!)
c)		5!
d)		5(7!)

47. A polygon has 44 diagonals. The number of its sides are

a)		10
b)		11
c)		12
d)		13

48. Six identical coins are arranged in a row. The number of ways in which the number of tails is equal to number of heads is

a)		20
b)		120
c)		9
d)		40

49. The number of ways in which 11 different things can be divided into two groups containing 6 and 5 things respectively is

a)		P (11, 5)
b)		P (11, 6)
c)		P (11, 2)
d)		C (11, 5)

50. The number of ways in which p + q things can be divided into two groups containing p and q things respectively is

a)		C (p + q, q)
b)		P (p + q, p)
c)		P (p + q, q)
d)		C (p + q, p - q)