Untitled Document

Question Bank No: 1

1. If $x^{2}$ + $y^{2}$ = $r^{2}$ and k = $\frac{1}{r}$ , then k is

a)		$\frac{\| y " \|}{\sqrt{1 + {y'}^{2}}}$
b)		$\frac{\| y " \|}{\sqrt{{(1 + {y'}^{2})}^{3}}}$
c)		$\frac{2 \| y " \|}{\sqrt{1 + {y'}^{2}}}$
d)		$\frac{\| y " \|}{\sqrt{{(1 + {y'}^{2})}^{3}}}$

2. If y = $\sin^{- 1} (\sqrt{x - ax} - \sqrt{a - ax})$ , then $\frac{dy}{dx}$ is

a)		$\frac{1}{2 \sqrt{x} \sqrt{1 - x}}$
b)		$\frac{1}{\sin \sqrt{a - a x}}$
c)		sin $\sqrt{x} . \sin \sqrt{a}$
d)		zero

3. If x = $\sqrt{\frac{1 - t^{2}}{1 + t^{2}}}$ and y = $\frac{\sqrt{1 + t^{2}} - \sqrt{1 - t^{2}}}{\sqrt{1 + t^{2}} + \sqrt{1 - t^{2}}}$ then the value of $\frac{d^{2} y}{d x^{2}}$ at t = 0 is given by

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

4. If $y^{\frac{1}{m}}$ = x + $\sqrt{1 + x^{2}}$ , then (1 + $x^{2}$ ) $y_{2}$ + x $y_{1}$ is equal to

a)		$m^{2}$ y
b)		m $y^{2}$
c)		$m^{2}$ $y^{2}$
d)		None of these

5. If F(x) = $\frac{1}{x^{2}} \int_{4}^{x}$ (4 $t^{2} - 2 F^{'} (t)$ ) dt, then $F^{'}$ (4) is

a)		$\frac{32}{9}$
b)		$\frac{64}{9}$
c)		$\frac{64}{3}$
d)		None of these

6. If y (x + $\sqrt{1 + x^{2})^{m}}$ , then (1 + $x^{2}$ ) $y_{2}$ + x $y_{1} -$ $m^{2}$ y =

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

7. If x = t + $\frac{1}{t}$ , y = t $- \frac{1}{t}$ , then $\frac{d^{2} y}{d x^{2}}$ is

a)		$-$ 4t ${(t^{2} - 1)}^{- 2}$
b)		- 4 $t^{3}$ ${(t^{2} - 1)}^{- 3}$
c)		${(t^{2} + 1) (t^{2} - 1)}^{- 1}$
d)		- 4 $t^{2}$ ${(t^{2} - 1)}^{- 2}$

8. If x = a (cos t+log tan $\frac{t}{2}$ ), y = a sin t, then $\frac{dy}{dt}$ is

a)		tan t
b)		cos t
c)		sec t
d)		cosec t

9. If g is inverse of f and $f^{'}$ (x) = $\frac{1}{2 + x^{n}},$ then $g^{'}$ (x) is equal to

a)		2 + $x^{n}$
b)		2 + ${(f (x))}^{n}$
c)		2 + ${(g) x}^{n}$
d)		None of these

10. If y = $\cos^{- 1}$ $(\frac{3 cosx + 5 sinx}{\sqrt{34}})$ $\frac{dy}{dx}$ , is

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

11. If u = $e^{\sin - 1} x$ and $υ$ = $\log_{e}$ x, then $\frac{du}{dv}$

a)		xu
b)		$\frac{u}{\sqrt{1 - x^{2}}}$
c)		$\frac{xu}{\sqrt{1 - x^{2}}}$
d)		None of these

12. If y = $e^{x}$ + sin x, then $\frac{d^{2} x}{{dy}^{2}}$ is

a)		${(- \sin x + e^{x})}^{- 1}$
b)		$\frac{\sin x - e^{x}}{{(\cos x + e^{x})}^{2}}$
c)		$\frac{\sin x - e^{x}}{{(\cos x + e^{x})}^{3}}$
d)		$\frac{\sin x + e^{x}}{{(\cos x + e^{x})}^{3}}$

13. If y = $\sin^{- 1}$ x satisfies (1 $-$ $x^{2}$ ) $y_{2}$ = f(x) $y_{1}$ , then f(x) is

a)		x
b)		1
c)		y
d)		None of these

14. If f(x) = $| x - 2 |$ and g(x) = f(f(x)), then for 2 $<$ x $<$ 4, $g^{'}$ (x) is

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

15. If f(x) = sin x, g(x) = $x^{2}$ , h(x) = $\log_{e}$ x and F(x) = (hogof)(x), then $f^{″}$ (x) is

a)		2 ${cosec}^{3}$ x
b)		2 cot $x^{2} -$ 4 $x^{2}$ ${cosec}^{2}$ $x^{2}$
c)		2x cot $x^{2}$
d)		$-$ 2 ${cosec}^{2}$ x

16. If y = xk, k $\in$ R, then the value of k so that y is differentiable (n – 1) times at x = 0 but is not differentiable at x = 0 is

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

17. If f(x) = $\frac{\tan x + \sec x - 1}{\tan x - \sec x + 1}$ , then $f^{'}$ (x) is

a)		sec x(tanx $-$ sec x)
b)		sec x (sec x $-$ tan x)
c)		sec x (sec x + tan x)
d)		None of these

18. If y = ${cosec}^{- 1}$ $(\frac{x + 1}{x - 1})$ + $\cos^{- 1}$ $(\frac{x - 1}{x + 1})$ , then $\frac{dy}{dx}$ is

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

19. If y = $\sin^{- 1}$ $(\frac{\sqrt{1 - x} + \sqrt{1 - x}}{2})$ , then $\frac{dy}{dx}$ is

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

20. If y = $\sin^{- 1}$ x + $\sin^{- 1}$ $\sqrt{1 - x^{2}}$ , then $\frac{dy}{dx}$ is

a)		0
b)		$\frac{2}{\sqrt{1 - x^{2}}}$
c)		$\frac{π}{2}$
d)		None of these