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JEE Mathematics Solved Question Paper 2012

Multiple Choice Questions

11.

If y = $5^{\tan (x)}$ , then $\frac{dy}{dx}$ at $x = \frac{π}{4}$ is equal to

$5 \log (5)$
$10 \log (5)$
0
${(\log (5))}^{2}$

12.

If $y = \sin^{- 1} (x) and z = \cos^{- 1} (\sqrt{1 - x^{2}})$ , then $\frac{dy}{dz}$ is equal to

$\frac{x}{\sqrt{1 - x^{2}}}$
$\frac{1}{2}$
$\frac{- x}{\sqrt{1 - x^{2}}}$
1

13.

If u = 2(t - sin(t)) and v = 2 (1 - cos(t)), then $\frac{dv}{du}$ at t = $\frac{2 π}{3}$ is equal to

$\sqrt{3}$
$- \sqrt{3}$
$2 \sqrt{3}$
$\frac{1}{\sqrt{3}}$

14.

If $f (x) = \log [e^{x} {(\frac{3 - x}{3 + x})}^{\frac{1}{3}}]$ , then f'(1) is equal to

$\frac{3}{4}$
$\frac{2}{3}$
$\frac{1}{3}$
$\frac{1}{2}$

15.

If y = ${(\log (x))}^{2}$ , then $\frac{dy}{dx}$ at x = e is equal to

2
$\frac{e}{2}$
e
$\frac{2}{e}$

16.

If y^x = 2^x, then $\frac{dy}{dx}$ is equal to

$\frac{y}{x} \log (\frac{2}{y})$
$\frac{x}{y} \log (\frac{2}{y})$
$\frac{y}{x} \log (\frac{y}{2})$
$\frac{x}{y} \log (\frac{y}{2})$

17.
If x² + 2xy + 2y² = 1, then $\frac{dy}{dx}$ at the point where y = 1 is equal to

1

2

- 1

0

$Given, x^{2} + 2 xy + 2 y^{2} = 1 . . . (i) Put y = 1, x^{2} + 2 x (1) + 2 {(1)}^{2} = 1 \Rightarrow x^{2} + 2 x + 1 = 0 \Rightarrow {(x + 1)}^{2} = 0 \Rightarrow x = - 1 On differentiating Eq . (i) w . r . t . x, we get 2 x + 2 x \frac{dy}{dx} + 2 y + 4 y \frac{dy}{dx} = 0 \Rightarrow 2 \frac{dy}{dx} (x + 2 y) = - 2 (y + x) \Rightarrow \frac{dy}{dx} = \frac{- (y + x)}{x + 2 y} When x = - 1, y = 1 \frac{dy}{dx} = - \frac{(1 - 1)}{- 1 + 2} = 0$