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Question Bank No: 3

1. If y = $x^{\sin x}$ then dy/dx =

a)		$\frac{Sin x}{x}$ (log sin x + Cos x)
b)		y [ $\frac{Sin x}{x}$ + log x Cosx]
c)		y (x sin x + sin x log x)
d)		0

2. Derivative of $\sin^{2}$ x.w.r.t $x^{2}$ is

a)		1
b)		2 sin x cos x
c)		$\frac{\sin 2 x}{x}$
d)		$\frac{\sin 2 x}{2 x}$

3. The area bounded by the curve $x^{2}$ y = $x^{2}$ - 4 the x- axix and the lines x = 2 and x= 4 in sq. units is

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

4. The area bounded by the parabola $y^{2}$ = 4 ax and its latus rectum in sq. unit is

a)		( ${4 a}^{2} / 3)$
b)		( ${8 a}^{2} / 3)$
c)		(8a/3)
d)		0

5. The area of the region lying above the x - axis and below the graph of y = $x^{2}$ + 4x - 3 in sq. unit is

a)		(4/3)
b)		(2/3)
c)		(7/3)
d)		1

6. The area of the curve ( $x^{2}$ /4) + ( $y^{2}$ /9) = 1 in sq.units is

a)		6 $π$
b)		4 $π$
c)		3 $π$
d)		7 $π$

7. The order and the degree of the differential equation $\frac{d^{2} y}{{dx}^{2}}$ + $\sqrt{x + [\frac{dy}{dx}]^{3}}$ = 0 is

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

8. The differential equation of the family of all circle passing through the origin and having the centers on the x-axis is =

a)		2xy dy/dx + ( $x^{2}$ + $y^{2}$ ) = 0
b)		2xy dy/dx + ( $x^{2}$ - $y^{2}$ ) = 0
c)		2xy dx/dy - ( $x^{2}$ - $y^{2}$ ) = 0
d)		None of these

9. The differential equation of the family of curves y = ${Ae}^{3 x}$ + ${Be}^{5 x}$ , where A and B are constants is

a)		$\frac{d^{2} y}{{dx}^{2}}$ - 8 $\frac{dy}{dx}$ - 15y = 0
b)		$\frac{d^{2} y}{{dx}^{2}}$ - 8 $\frac{dy}{dx}$ + 15y = 0
c)		$\frac{d^{2} y}{{dx}^{2}}$ + 8 $\frac{dy}{dx}$ - 15y = 0
d)		None of these

10. The solution of $\sqrt{(1 - x^{2})}$ dy + $\sqrt{(1 - y^{2})}$ dx = 0 is

a)		c + $\sin^{- 1}$ x + $\cos^{- 1}$ y
b)		C = $\sin^{- 1}$ x + $\cos^{- 1}$ y
c)		c= $\tan^{- 1}$ x + $\tan^{- 1}$ y
d)		None

11. The solution of the equation ( $x^{3}$
- ${3 xy}^{2})$
dx = ( $y^{3}$
- ${3 x}^{2} y)$
dy is:

a)		$c^{2}$ ( $x^{2} - y^{2})^{2}$ = $y^{2} + x^{2}$
b)		$c^{2}$ ( $x^{2} + y^{2})^{2}$ = $y^{2} - x^{2}$
c)		$c^{2}$ ( $x^{2} + y^{2})^{2}$ = $y^{2} + x^{2}$
d)		None of these

12. The equation : $\frac{dy}{dx}$ = $\frac{y}{x}$ + tan $[\frac{y}{x}]$ is

a)		Variable separable
b)		Linear
c)		Homogeneous
d)		None of these

13. The curve satisfying cos y dy + cos x sin y dx = 0 , given y = ( $π / 2$ ) when x = ( $π$ /2), is

a)		log $\| \sin y \|$ = Sin x + 1
b)		log $\| C o s y \|$ = cos x + 1 = 0
c)		log $[Cos y]$ = sin y + 1
d)		None of these

14. The domain of definition of function y = ${3 e}^{{\sqrt{x}}^{2} - 1}$ log(x-1) is,

a)		(1, $\propto$ )
b)		[1, $\propto$ )
c)		Set of all real different from {1}
d)		(- $\propto$ ,-1) $\cup$ (1, $\propto$ )

15. Let [x] denote the greatest integer $\leq$ x. The domain of function f(x) = $\sqrt{\frac{4 - x^{2}}{[x] + 2}}$ is

a)		(- $\propto$ ,-2) $\cup$ [-1,2]
b)		[0,2]
c)		[-1,2]
d)		(0,2)

16. If f(x) = Cos (log x ) then f(x) f(y) - $\frac{1}{2}$ (f( $\frac{x}{y}$ )+f(xy)) has the value

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

17. If f(x+ $\frac{1}{x}$ ) = $x^{2} + \frac{1}{x^{2}}$ , x $\neq$ 0 then f(x) is equal to

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

18. The domain of y = $\frac{1}{\sqrt{| x |} - x}$ is

a)		[0, $\propto$ ]
b)		(- $\propto$ , 0)
c)		( $\propto$ , 0)
d)		[1, $\propto$ )

19. The range of f(x) = $\frac{x + 2}{| x + 2 |}$ , x $\neq$ -2 is

a)		{ -1,0,1 }
b)		{-1,1}
c)		{-1}
d)		{1}

20. The domain of the function f(x) = $\sqrt{\log_{10}} x^{2}$ is

a)		x $\geq$ 0
b)		$\| x \|$ $\geq$ 1
c)		$\| x \|$ $\leq$ 1
d)		$\| x \|$ $\geq$ 4

21. If f(x) = $\frac{\sin [x]}{[x]}$ , [x] $\neq$ 0 then $\overset{Lim}{x \to 0}$ f(x) equals

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

22. $\overset{Lim}{x \to 0}$ $\frac{a^{x} - 1}{\sqrt{1 + x} - 1}$ is,

a)		${2 \log}_{e}^{a}$
b)		$\frac{1}{2}$ $\log_{e}^{a}$
c)		a $\log_{e}^{2}$
d)		None of these

23. $\underset{x \to 0}{Lim}$ ( $\frac{1 + \tan x}{1 - \tan x}$ $)^{1 / x}$ is

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

24. $\overset{Lim}{x \to 0}$ $\frac{27^{x} 3^{x} - 9^{x} + 1}{1 - Cos x}$ is,

a)		4 ${(\log 3)}^{2}$
b)		${(\log 3)}^{2}$
c)		0
d)		log 3

25. The function f(x) = $\frac{| x |}{x^{2} + 2 x}$ , x $\neq$ 0 and f(0) is not continuous at x = 0 because,

a)		$\overset{Lim}{x \to 0}$ f(x) $\neq$ f(0)
b)		$\overset{Lim}{x \to 0}$ + f(x) does not exist
c)		$\overset{Lim}{x \to 0}$ - f(x) does not exist
d)		$\overset{Lim}{x \to 0}$ f(x) does not exist

26. $\frac{Lim}{x \to π / 2}$ [Sin x] is equal to

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

27. $\overset{Lim}{n \to \propto}$ $\frac{\sum_{r = 1}^{n} r^{2}}{n^{3}}$ is equal to

a)		$\frac{1}{2}$
b)		$\frac{1}{3}$
c)		$\frac{1}{4}$
d)		$\frac{- 1}{2}$

28. $\overset{Lim}{n \to \propto}$ [ $\frac{1}{{1 - n}^{2}}$ + $\frac{2}{{1 - n}^{2}}$ + $\frac{3}{{1 - n}^{2}}$ ........+ $\frac{n}{{1 - n}^{2}}$ ] = is equal to

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

29. $\overset{Lim}{n \to \propto}$ ( $\sqrt{x + \sqrt{x + \sqrt{x}}}$ - $\sqrt{x}$ ) is equal to

a)		0
b)		$\frac{1}{4}$
c)		$\frac{1}{2}$
d)		-1/2

30. $\frac{d}{dx}$ sin ${(x}^{0}) =$

a)		Cos ${(x}^{0})$
b)		$\frac{π}{180}$ Cos ${(x}^{0})$
c)		$\frac{180}{π}$ Cos ${(x}^{0})$
d)		-Sin $x^{0}$

31. $\frac{d^{{([x]}^{3})}}{dx}$ =

a)		3 ${\| x \|}^{2}$
b)		-3 ${\| x \|}^{2}$
c)		3x $\| x \|$
d)		${3 x}^{2}$

32. If f(x) is an even function, then f'(x) is

a)		neither even nor odd
b)		an odd function
c)		0
d)		none of these

33. If y = $\sec^{- 1}$ ( $\frac{x + 1}{x - 1})$ + $\sec^{- 1}$ ( $\frac{x - 1}{x - - + 1})$ then dy/dx =

a)		1
b)		0
c)		$\frac{x - 1}{x + 1}$
d)		$\frac{x + 1}{x - 1}$

34. If $x^{y}$ = $e^{x - y}$ then dy/dx is,

a)		$\frac{1}{{(1 + \log x)}^{2}}$
b)		$\frac{y}{{(1 + \log x)}^{2}}$
c)		$\frac{\log x}{{(1 + \log x)}^{2}}$
d)		None of these

35. Which of the following is not true.

a)		$\frac{dy}{dx}$ = $\frac{1}{dy / dx}$
b)		$\frac{d^{2} y}{{dx}^{2}}$ = $\frac{1}{\frac{d^{2} x}{{dy}^{2}}}$
c)		( $\frac{dy}{dx})^{2}$ = $\frac{1}{{(dx / dy)}^{2}}$
d)		$\frac{dy}{dx}$ = $\frac{dy}{d \emptyset}$ . $\frac{d \emptyset}{dx}$

36. If y = ${Cos}^{- 1}$ ( $\frac{4 Cos x - 3 \sin x}{5}$ ) then dy/dx =

a)		0
b)		1
c)		a Constant
d)		x

37. The function f(x) = $\frac{x}{{1 + x}^{2}}$ decreases in the interval,

a)		(-1,1)
b)		[0, $\propto$ ]
c)		(- $\propto$ , $- 1$ )
d)		None of these

38. The normal to the curve y = x+ Sin x Cos x at x = $π$ /2 is

a)		x = 2
b)		x = $π$
c)		x+ $π$ = 0
d)		2x = $π$

39. The minimum value of $x^{x}$ is attained at x =

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

40. What is the derivative of $\frac{Sec x + \tan x}{\sec x - \tan x}$

a)		2 Sec x ( sec x + tan x)
b)		2 ${Sec}^{2} x$ ( sec x + tan x $)^{2}$
c)		2 sec x ( ${\sec x + \tan x)}^{2}$
d)		Sec x( ${\sec x + \tan)}^{2}$

41. Which one of the following is correct for the graph of y = $| x |$

a)		It lies only in the first quadrant of the xy - plane
b)		t lies only in the first and third quadrant of the xy- plane
c)		It lies only in the first and second quadrant of the xy - plane
d)		It lies only in the third and fourth quadrant of the xy - plane

42. f and g are continuous function on [o,a], satisfying f(a-x) = f(x) and g(x)+g(a-x) = 2 then, $\int_{0}^{a}$ f(x) g(x) dx = .............

a)		$\int_{0}^{a}$ g(x) dx
b)		$\int_{0}^{a}$ f(x) dx
c)		f(a).g(a)
d)		0

43. f(x) = $\int_{0}^{x}$ t smt dt , then f ' (x) =

a)		x Cos x +sin x
b)		x sin x
c)		x Cos x
d)		None of these