Untitled Document

Question Bank No: 1

1. Which of the following is equal to ${\int f}_{0}^{2 η} (sinx) dx$

a)		2 $\int_{0}^{π}$ f $(sinx) dx$
b)		${\int f}_{π}^{π} (sinx) dx$
c)		2 $\int_{0}^{π}$ f $(sinx) dx$
d)		None of the above

2. $\int_{0}^{π / 4}$ $\tan^{4}$ x dx is

a)		$π / 4 + 1$
b)		$π / 4$ - 1/3
c)		$π / 4$ - 2/3
d)		1/3- $π / 4$

3. $\int_{0}^{π}$ dx/9 $\cos^{2}$ x + 4 $\sin^{2}$ x is

a)		$π$ /3
b)		$π / 36$
c)		$π$ /16
d)		None.

4. $\int_{- π / 4}^{π}$ $\cos^{4}$ x $\cos^{2}$ x dx =

a)		2 $π$
b)		$π / 4$
c)		$π / 8$
d)		None.

5. $\int_{- π / 4}^{π / 4}$ $\sin^{3}$ x $\cos^{2}$ x dx =

a)		$\sqrt{2}$
b)		2 $\sqrt{2}$
c)		1/2 $\sqrt{2}$
d)		0

6. $\int_{0}^{π}$ $\sin^{5}$ x dx

a)		8/15
b)		$-_{15}^{16}$
c)		8 $π / 15$
d)		0

7. $\int_{0}^{π}$ $\cos^{7}$ x dx =

a)		32/35
b)		16/35
c)		16 $π / 35$
d)		0

8. $\int_{0}^{π / 2}$ $\frac{\log sinx}{\log \sin x + \log \cos x}$ dx =

a)		0
b)		$π / 2$
c)		$π / 4$
d)		$π$

9. $\int_{0}^{1}$ ${x (1 - x)}^{n}$ dx =

a)		1/(n-1) (n-2)
b)		1/n+1+1/n+2
c)		1/(n+1) (n+2)
d)		(n+1) (n+2)

10. $\int_{0}^{π / 2}$ $a^{2}$ $\cos^{2}$ x + $b^{2}$ $\sin^{2}$ x dx =

a)		$a^{2}$ + $b^{2}$
b)		( $a^{2}$ + $b^{2}$ ) $π$
c)		( $a^{2}$ + $b^{2}$ ) $π$ /4
d)		ab $π$

11. $\int_{0}^{4}$ $\frac{\sqrt{x}}{\sqrt{x} + \sqrt{4 - x}}$ dx=

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

12. If $1_{n}$ = $\int_{0}^{π / 2}$ ${Sin}^{n}$ x dx then $\frac{1 n}{1 n - 2}$ =

a)		1/n
b)		n/n-2
c)		n/n-1
d)		n-1/n

13. $\int_{0}^{π}$ xf (sin x) dx and $\int_{0}^{π}$ f (sin x) dx are in the ratio

a)		$π : 3$
b)		$π : 2$
c)		2: $π$
d)		$π : 1$

14. $\int$ $e^{3 x}$ cos 2x dx=

a)		$e^{3 x}$ /13 (3 cos 2x + 2 sin 2x) + c
b)		$e^{3 x}$ /13 (3 cos 2x - 2 sin 2x) + c
c)		$e^{3 x}$ $\sqrt{13}$ (3 cos 2 x + 2 sin 2 x + c)
d)		$e^{3 x}$ (3 cos 2x - 2 sin 2x) + c

15. If a is such that $\int_{0}^{a}$ x dx $\leq$ a + 4, then

a)		0 $\leq$ a $\leq$ 4
b)		-2 $\leq$ a $\leq$ 0
c)		a $\leq$ -2 or a $\geq$ 4
d)		-2 $\leq$ a $\leq$ 4

16. The value of the integral $\int$ $\sin^{- 1}$ x dx is

a)		x $\sin^{- 1}$ x
b)		x $\sin^{- 1}$ x - $\sqrt{(1 - x) 2}$
c)		x $\sin^{- 1}$ x + $\sqrt{(1 - x) 2}$
d)		None of these

17. Area bounded by the curve y = 2x - $x^{2}$ and the line y = - x is given by

a)		2 $\sqrt{2}$ - 1
b)		(2 $\sqrt{2}$ + 1)/7
c)		2
d)		2 $\sqrt{2}$

18. Let $A_{1}$ be the area of the parabola $y^{2}$ = 4ax lying between vertex and latus rectum and $A_{2}$ be the area between latus rectum and double ordinate x = 2a. Then $A_{1}$ / $A_{2}$

a)		2 $\sqrt{2}$ - 1
b)		(2 $\sqrt{2}$ + 1)/7
c)		(2 $\sqrt{2}$ - 1)/7
d)		$\sqrt{2}$

19. solution of sin x cos y dy + cos x sin y dx is

a)		Cos x + y = k
b)		Sin x + y = k
c)		sin x sin y = k
d)		sin x - sin y = k

20. Solution of ${xy}^{2}$ dy/dx = $x^{3}$ + $y^{3} is$

a)		$x^{3}$ = c $e^{y 3 / x 3}$
b)		$x^{3}$ $y^{3}$ = log x+c
c)		xy = c
d)		x/y = c

21. Solution of $\sec^{2}$ x tan y dx + $\sec^{2}$ y tan x dy = 0

a)		tan x tan y = k
b)		tan x + sec x = tan y + sec y + k
c)		tan x sec y + sec y tan x + k
d)		tan x + sec x = tan y - sec y +k

22. If f (x) = ${\frac{x^{2} - 9}{\underset{2 x + k otherwise}{x - 3}}}$ if x $\neq$ 3, is continuous at x = 3 then k =

a)		3
b)		0
c)		-6
d)		$\frac{1}{6}$

23. $\frac{d}{dx}$ $x^{x}$ is equal to

a)		log x
b)		log ex
c)		$x^{x}$ log x
d)		$x^{x}$ log ex

24. If the displacement s of a particle at time t is given by $s^{2}$ = ${at}^{2}$ + 2bt + c, then acceleration varies as

a)		${1 / s}^{2}$
b)		1/s
c)		${1 / s}^{3}$
d)		$s^{3}$

25. If PQ and PR are the two sides of a triangle, then the angle between them which gives maximum area of the triangle is

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

26. The function y = a(1-cos x ) is maximum when s is eaual to

a)		$π$
b)		$π / 2$
c)		- $π / 2$
d)		$- π / 6$

27. $\int$ $\frac{\sin x}{\sin (x - a)}$ dx is equal to

a)		x cos a + sin a log sin(x-a) + C
b)		(x-a)cos x + log sin(x-a) + C
c)		Sin (x-a) + sin x +C
d)		Cos (x-a) + cos x + C

28. $\int$ $13^{x}$ dx is

a)		$\frac{13^{x}}{\log 13}$ + C
b)		$13^{x + 1}$ + C
c)		14 x + C
d)		$14^{x + 1}$ + C

29. $\int_{0}^{π / 2}$ sin 2x log tan x dx is equal to

a)		$π$
b)		$π / 2$
c)		1
d)		None of these

30. $\int_{0}^{π / 2}$ x sin x dx is equal to

a)		$π / 4$
b)		$π / 2$
c)		$π$
d)		1

31. $\lim_{x \to 0}$ $\frac{1 - \cos mx}{1 - \cos nx}$ =

a)		m/n
b)		$m^{2}$ / $n^{2}$
c)		0
d)		$n^{2}$ / $m^{2}$

32. Which of the following is not true ?

a)		A polynomial function is always continuous
b)		A continuous function is always differentiable
c)		a differentiable function is always continuous
d)		$e^{x}$ is continuous for all x

33. $\lim_{x \to 0}$ $\frac{a^{x} - b^{x}}{e^{x} - 1}$ is equal to

a)		log a/b
b)		log b/a
c)		log ab
d)		log a+b

34. The derivative of $x^{6}$ + $6^{x}$ with respect to x ix

a)		12 x
b)		x+4
c)		${6 x}^{5}$ + $6^{x}$ log 6
d)		${6 x}^{5}$ + x $6^{x - 1}$

35. If x = a $\cos^{4}$ $θ$ , y = a $\sin^{4}$ $θ, then \frac{dy}{dx}$ at $θ$ = $\frac{3 π}{4}$ is

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

36. If sin y + $e^{- x \cos y}$ = e, then $\frac{dy}{dx}$ at (1, $π$ ) is,

a)		sin Y
b)		- x cos y
c)		e
d)		None

37. If x = $\sin^{- 1}$ ${(3 t - 4 t}^{3}$ ) and y = $\cos^{- 1}$ [ $\frac{{1 - t}^{2}}{{1 + t}^{2}}$ ] then $\frac{dy}{dx}$ is equal to

a)		1/2
b)		2/5
c)		3/2
d)		2/3

38. The second derivative of a $\sin^{3}$ t with respect to a $\cos^{3}$ t at t = $\frac{π}{4}$ is

a)		$\frac{4 \sqrt{2}}{3 a}$
b)		2
c)		$\frac{1}{12 a}$
d)		0

39. The equation of the tangent to the curve (1 + $x^{2}$ ) y = 2-x where it crosses the x - axis is

a)		x + 5y = 2
b)		x-5y = 2
c)		5x - y = 2
d)		5x +y-2 = 0

40. The side of an equilateral triangle are increasing at the rate of 2 cms/sec. The rate at which the area increases, when the side is 10 cms is

a)		$\sqrt{3}$ sq.units/sec
b)		10 sq.units/sec
c)		10 $\sqrt{3}$ sq units sec.
d)		$\frac{10}{\sqrt{3}}$ sq.units.sec

41. $\int_{- 2}^{2}$ $| 1 - x^{2} |$ dx is equal to

a)		4
b)		2
c)		-2
d)		0

42. $\int$ $\frac{\sqrt{\tan x}}{\sin x \cos x}$ dx is equal to

a)		2 tan x
b)		$\sqrt{\cot x}$
c)		2 $\sqrt{\tan x}$
d)		$\tan^{2}$ x

43. $\int$ tanx dx =

a)		$\sec^{2}$ x
b)		$\sqrt{\cot x}$
c)		2 $\sqrt{\tan x}$
d)		log sec x

44. $\int$ $\frac{dx}{x^{2} + 4 x + 13}$ is equal to

a)		log( $x^{2}$ + 4x+13) + C
b)		1/3 $\tan^{- 1}$ [ $\frac{x + 2}{3}]$ + C
c)		log (2x+4) + C
d)		$\frac{1}{{(x}^{2} + 4 x + 13)} +$ C

45. $\int_{0}^{2}$ $\frac{d}{dx}$ [sin - 1[ $\frac{2 x}{1 + x^{2}}$ ] ] dx ix equal to

a)		0
b)		$π$
c)		$π / 2$
d)		$π / 4$

46. $\int_{0}^{π / 2}$ $\frac{\sin x}{\sin x + \cos x}$ dx evaluates to

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

47. The area bounded by the parabolas $y^{2}$ = 4 ax and $x^{2}$ = 4ay is

a)		$\frac{{8 a}^{2}}{3}$
b)		$\frac{{16 a}^{2}}{3}$
c)		$\frac{{32 a}^{2}}{3}$
d)		$\frac{{64 a}^{2}}{3}$

48. The area bounded by the curve y = sin x between the ordinates x = 0, x = $π$ and then the x - axis is

a)		2.sq.units
b)		4.sq.units
c)		1.sq.units
d)		3.sq.units

49. The degree of the differential equation $\frac{d^{2} y}{{dx}^{2}}$ +[ $\frac{dx}{dy}]^{3}$ + 6y = 0 is

a)		1
b)		3
c)		2
d)		5

50. The solution of the equation (2y - 1) dx - (2x + 3) dy = 0 is

a)		$\frac{2 x - 1}{2 y + 2}$ = C
b)		$\frac{2 x + 3}{2 y - 1}$ = C
c)		$\frac{2 x - 1}{2 y - 1}$ = C
d)		$\frac{2 y + 1}{2 x - 3}$ = C