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Question Bank No: 2

1. $\overset{\lim}{x - - > 1}$ $\frac{x^{2} - 2 x + 1}{| x^{2} - 1 |}$ =

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

2. $\overset{\lim}{x - - >}$ $0^{{(a}^{x} - b^{x}) / x} =$

a)		0
b)		1
c)		log a - log b
d)		log a/ log b

3. $\overset{\lim}{x - - >}$ $0^{\frac{(2 + x) \sin (2 + x) - 2 \sin 2}{x} =}$

a)		sin 2
b)		cos 2
c)		1
d)		2 cos 2 + sin 2

4. If f(x) = $\frac{3 x + \tan^{2} + x}{x}$ is continuous at x = 0, then f (0) =

a)		1
b)		2
c)		4
d)		3

5. If $| f (x) |$ is continuous at x = a, then f(x)

a)		is continuous at x = a
b)		is continuous at x = - a
c)		is continuous at x = $\sqrt{a}$
d)		is not continuous at x = a

6. If x is measured in degree, then (d/dx) (cos x) =

a)		- Sin x
b)		- [ $\frac{180}{π}$ ] sin x
c)		( $π / 180)$ sin x
d)		-( $π / 180)$ sin x

7. (d/dx) [ log (sec x - tan x ] =

a)		- sec x
b)		sec x + tan x
c)		sec x
d)		sec x - tan x

8. If x = a $\cos^{3}$ $θ$ and y = a $\sin^{3}$ $θ$ , then 1+ (dy/ ${dx)}^{2}$ =

a)		$\tan^{2}$ $θ$
b)		$\cot^{2}$ $θ$
c)		$\sec^{2}$ $θ$
d)		${cosec}^{2}$ $θ$

9. If xy = x+y, then (dy/dx) =

a)		xy/(1-x)
b)		(1+y)/(1-x)
c)		y/(1-xy)
d)		-1/( ${x - 1)}^{2}$

10. If x = ${at}^{2}$ , y = 2 at, then $d^{2}$ y/ ${dx}^{2}$ =

a)		-1/ ${(t}^{2}$ )
b)		-1 ${(2 at}^{3}$ )
c)		1/ ${(t}^{2}$ )
d)		Zero

11. The value of $x_{1}$ for which $| x |$ is continuous but not differentiables, is

a)		-1
b)		1
c)		0
d)		10

12. ( $d^{20} / {dx}^{20}$ ) ( 2 cos x.cos 3x) =

a)		$2^{20}$ (cos 2x + $2^{20}$ cos 4x)
b)		$2^{20}$ (cos 2x + $2^{20}$ sin 2x)
c)		-6 sin x sin 3 x
d)		6 sin x sin 3x

13. If the rate of change in the circumference of a circle is 0.3/sec, then the rate of change in the area of the circle when the radius is 5 cm, is

a)		1.5 sq.cm/sec
b)		0. 5 sq.cm/sec
c)		5 sq.cm/sec
d)		3 sq.cm/sec

14. A particle moves along a straight line according to the law s = $e^{t}$ ( Sin t - cos t) The acceleration at any time is

a)		$e^{t}$ (cos t+ Sin t)
b)		$e^{t}$ (cos t -Sin t)
c)		$2 e^{t}$ (cos t- Sin t)
d)		${2 e}^{t}$ (cos t+ Sin t)

15. If y = $x^{3}$ - ${ax}^{2}$ + 48 x + 7 is an increasing function for all real values of x, then a lies in

a)		(-14,14)
b)		(-12,12)
c)		( -16,16)
d)		(-21,21)

16. The value of 'a' for which the function f(x) = a sin x + (1/3) has an extremum ar x = $π / 3$ is

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

17. Rolle's theorem is not applicable for the function f(x) = $| x |$ in the interval [-1,1] because

a)		f(1) does not exist
b)		f(-1) does not exist
c)		f(x) is discountinuous at x = 0
d)		f(0) does not exist

18. $\int$ $\frac{2 dx}{{(e}^{x} + e^{- x})^{2}}$ dx =

a)		$\frac{{- e}^{- x}}{(e^{x} + e^{- x})}$
b)		$\frac{- 1}{e^{x} + e^{- x}}$
c)		$\frac{1}{{(e}^{x} + {1)}^{2}}$
d)		$\frac{1}{{(e}^{x} - e^{- x})}$

19. $\int \frac{x^{3}}{x + 1}$ dx =

a)		$\frac{x^{3}}{3}$ - $\frac{x^{2}}{2}$ + c
b)		$\frac{x^{3}}{3}$ - $\frac{x^{2}}{2}$ + x-log (x+1)+ C
c)		$\frac{x^{3}}{3}$ + $\frac{x^{2}}{2}$ + +x + log (x+1) + C
d)		$\frac{x^{4}}{2 (x + 1)^{2}}$ + C

20. $\int \frac{e^{x} (1 + x)}{{Cos}^{2} (x e^{x})}$ dx =

a)		2 log cos (x $e^{x})$ + C
b)		sec (x $x^{x})$ + C
c)		tan (x $e^{x})$ + C
d)		tan ( x+ $e^{x})$ + C

21. $\int a^{x} e^{x}$ dx =

a)		$a^{x} e^{x}$ + C
b)		$[a^{x} e^{x} / \log a]$ + C
c)		[ ${ae)}^{x} / (x + 1)]$ + C
d)		$[a^{x} e^{x} / (1 + \log a)]$ + C

22. $\int \frac{4 x}{(x^{2} + 1) (x^{2} + 3)}$ dx

a)		log[( $x^{2} + 1) / (x^{2} + 3)$ ] + C
b)		log[( $x^{2} + 3) / (x^{2} + 1)$ ] + C
c)		tan -1 x + (1/ $\sqrt{3}$ )+ C
d)		2log[( $x^{2} + 1) / (x^{2} + 3)$ ] + C

23. $\int \frac{{Sin}^{2} x}{{Cos}^{4} x}$ dx =

a)		(1/3) $\tan^{2}$ x + C
b)		(1/2) $\tan^{2}$ x + C
c)		(1/3) $\tan^{3}$ x + C
d)		3 sin 2x - 4 cos 4x + C

24. $\int_{- 1 / 2}^{1 / 2}$ $\frac{dx}{(1 - x^{2})^{1 / 2}}$ =

a)		$π / 6$
b)		$π / 4$
c)		$π / 2$
d)		2 $π / 3$

25. $\int_{0}^{π / 2}$ $\frac{{Sin}^{n} θ}{\sin^{n} θ + \cos^{n} θ}$ d $θ$

a)		1
b)		0
c)		$π / 2$
d)		$π / 4$

26. The area ( in square units) enclosed by the curve $x^{2}$ y = 36, the x - axis and the lines
x = 6 and x = 9 is

a)		2
b)		1
c)		4
d)		None

27. The degree of the differential equation [ 5 + (dy/ ${dx}^{2}]^{5 / 3}$ = $x^{5}$ ( $d^{2} y / {dx}^{2}]$ is

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

28. The general solution of x ( 1+ $y^{2})^{1 / 2}$ dx + y ( 1+ $y^{2})^{1 / 2}$ dy = 0 is

a)		$\sin^{- 1}$ x + $\sin^{- 1}$ y = C
b)		$x^{2}$ + $y^{2}$ = ${(1 + x}^{2})^{1 / 2}$ + ${{(1 + y}^{2})}^{1 / 2}$ + C
c)		${(1 + x}^{2})^{1 / 2}$ + ${{(1 + y}^{2})}^{1 / 2}$ + C
d)		$\tan^{- 1}$ x - $\tan^{- 1}$ y = C

29. The general solution of (x+1) dy/dx + 1 = ${2 e}^{- y}$ is

a)		$e^{y}$ = 2x + C
b)		$e^{- y}$ = 2x + C
c)		$e^{y} (x + 1)$ = 2x + C
d)		$e^{y}$ (x+1) =C

30. The general solution of dy/dx+ y cot x = cosec x is

a)		x+y sin x = C
b)		x+ y cos x = C
c)		y = x (sinx + cos x) + C
d)		y sin x = x + C

31. $\underset{X \to 3}{Lt}$ $\frac{x^{2} - 9}{| x - 3 |}$ =

a)		0
b)		3
c)		$\infty$
d)		does not exist

32. If f (x) = $\frac{\log (1 + ax) - \log (1 - bx)}{x}$ for x $\neq$ 0 and f (0) = k and f (x) is continuous at x = 0, then k =

a)		a+b
b)		a-b
c)		a
d)		b

33. Lt 2 $\frac{(1 - Cos x)}{x^{2}}$ is

a)		1
b)		1/2
c)		1/4
d)		0

34. ${Lt}_{x - - > 0}^{X - - > 0}$ $\frac{\log Cos x}{x^{2}}$ equals

a)		0
b)		1
c)		2
d)		3

35. ${Lt}_{n - - > \infty}^{X - - > 0}$ $\frac{1}{{1 - n}^{2}}$ + $\frac{2}{1 - n^{2}}$ + .......... $\frac{n}{{1 - n}^{2}}$ equals

a)		0
b)		-1/2
c)		1/2
d)		None

36. If f (x) $\frac{1 - Cos Kx;}{x \sin x}$ ; x $\neq$ 0 and f(0) = 1/2 and if f(x) is continuous at x = 0 then K equals

a)		0
b)		$\pm$ 1
c)		2
d)		3

37. The function f(x) = 3x-5 for x <3 = x+1 for x>3
= k for x = 3 is continuous at x = 3 if k equals

a)		1
b)		2
c)		3
d)		4

38. d/dx ( $\sin^{- 1}$ x + ${Cos}^{- 1}$ x) =

a)		1
b)		$π / 2$
c)		0
d)		2

39. If y = $X^{x^{x ...... infinity}}$ thendy/dx =

a)		$\frac{y^{2}}{1 - y \log x}$
b)		$\frac{y^{2}}{x (1 - y \log x)}$
c)		$\frac{y^{2}}{\log x - y}$
d)		0

40. If f(x) = log (x + $\sqrt{x^{2} + 1)}$ then $f^{1}$ (x) equals

a)		$\sqrt{x^{2} + 1}$
b)		$\frac{x}{{\sqrt{x}}^{2} + 1}$
c)		$\frac{1}{\sqrt{x^{2} + 1}}$
d)		0

41. If y = $\tan^{- 1}$ $[\frac{\sqrt{{1 + x}^{2}} - 1}{x}]$ then $y^{1}$ (0) =

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

42. If $x^{y}$ = $e^{x - y}$ then dy/dx =

a)		$\frac{y}{{(1 + logx)}^{2}}$
b)		$\frac{x}{{(1 + logx)}^{2}}$
c)		$\frac{xy}{{(1 + logx)}^{2}}$
d)		None

43. If y = $\tan^{- 1}$ $[\frac{\sin x + Cos x}{Cos x - Sin x}]$ then dy/dx =

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

44. If $x^{2}$ + $y^{2}$ + 2 gx + 2 fy + c = 0 then dy/dx =

a)		$\frac{g + x}{f + y}$
b)		$\frac{x + C}{y - f}$
c)		$\frac{xy}{fg}$
d)		None

45. If y = a sin mx + b cos mx then $d^{2}$ y/ ${dx}^{2}$ =

a)		$m^{2}$ y
b)		${- m}^{2}$ y
c)		my
d)		-my

46. If y = $\log_{10^{x}}$ then dy/dx =

a)		$\frac{1}{x}$
b)		0
c)		$\frac{l o g 10}{x}$
d)		$\frac{1}{xlog 10}$

47. The point on the curvey y = 12x- $x^{3}$ , then tangent at which are parallel to x axis are

a)		( 2,16) and (-2,-16)
b)		(2,16) and ( -2,16)
c)		( 1,4) and ( -1,-4)
d)		None

48. f(x) = $x^{3}$ - ${6 x}^{2}$ + 9x + 8 then f(x) is decreasing in

a)		(1,3)
b)		(2,3)
c)		(0,1)
d)		R

49. The function of f(x) = ${2 x}^{3}$ - ${3 x}^{2}$ - 12x + 4 has a

a)		minima 1
b)		maxima 1
c)		maxima 11
d)		minima 2

50. If x is in degree measure then d/dx (sin x)

a)		Cos x
b)		180/ $π$ Cox x
c)		$π /$ 180 Cos x
d)		cos x