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CBSE Class 12 Mathematics Solved Question Paper 2011

Short Answer Type

11.

Write the intercept cut off by the plane 2x + y – z = 5 on x-axis.

Long Answer Type

12.

Prove the following:

$\cot^{- 1} [\frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}}] = \frac{x}{2}, x \in (0, \frac{π}{4})$

13.

Find the value of $\tan^{- 1} (\frac{x}{y}) - \tan^{- 1} (\frac{x - y}{x + y})$

14.

Using properties of determinants, prove that

$|\begin{matrix} - a^{2} & ab & ac \\ ba & - b^{2} & bc \\ ca & cb & - c^{2} \end{matrix}| = 4 a^{2} b^{2} c^{2}$

15.

Find the value of ‘a’ for which the function f defined as

$f (x) = \{\begin{cases} a \sin \frac{π}{2} (x + 1), x \leq 0 \\ \frac{\tan x - \sin x}{x^{3}}, x > 0 \end{cases}$

is continuous at x = 0.

16.
Differentiate $X^{x \cos x} + \frac{x^{2} + 1}{x^{2} - 1} w . r . t . x$

$Let y = x^{x \cos x} and z = \frac{x^{2} + 1}{x^{2} - 1} Consider y = x^{x \cos x}$

Taking log on both sides,

log y = log ( x^{x co}^{s x} )

log y = x cos x log x

Differentiating with respect to x,

$\frac{1}{y} \frac{dy}{dx} = (x \cos x) \frac{1}{x} + \log x \frac{d}{dx} (x \cos x) \frac{1}{y} \frac{dy}{dx} = \cos x + \log x (\cos x - x \sin x) \frac{dy}{dx} = y [\cos x + \log x (\cos x - x \sin x)] \frac{dy}{dx} = x^{x \cos x} [\cos x + \log x (\cos x - x \sin x)] . . . . . . . (i) Consider z = \frac{x^{2} + 1}{x^{2} - 1}$

Differentiating with respect to x,

$\frac{dz}{dx} = \frac{(x^{2} - 1) . \frac{d}{dx} (x^{2} + 1) - (x^{2} + 1) . \frac{d}{dx} (x^{2} - 1)}{{(x^{2} + 1)}^{2}} = \frac{(x^{2} - 1) (2 x) - (x^{2} + 1) (2 x)}{{(x^{2} + 1)}^{2}} = \frac{2 x^{3} - 2 x - 2 x^{3} - 2 x}{{(x^{2} + 1)}^{2}} = \frac{- 4 x}{{(x^{2} + 1)}^{2}} . . . . . . . . . . . . (ii)$

Adding (i) and (ii)

$\frac{d}{dx} \{x^{x \cos x} + \frac{x^{2} + 1}{x^{2} - 1}\} = \frac{dy}{dx} + \frac{dz}{dx} = x^{x \cos x} [\cos x + \log x (\cos x - x \sin x)] - \frac{4 x}{{(x^{2} - 1)}^{2}}$

17.

If $x = a (θ - \sin θ), y = (1 + \cos θ), find \frac{d^{2} y}{{dx}^{2}}$

18.

Sand is pouring from a pipe at the rate of 12 cm3/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of t heradius of the base. How fast is the sand cone increasing when the height is 4 cm?