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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.

$f (x) = \{\begin{cases} a \sin \frac{π}{2} (x + 1), x \leq 0 \\ \frac{\tan x - \sin x}{x^{3}}, x > 0 \end{cases} The given function f is defined for all x \in R . It is known that a function f is continuous at x = 0, if \lim_{x \to 0^{-}} f (x) = \lim_{x \to 0^{+}} f (x) = f (0) \lim_{x \to 0^{-}} f (x) = \lim_{x \to 0} [a \sin \frac{π}{2} (x + 1)] = a \sin \frac{π}{2} = a (1) = a \lim_{x \to 0^{+}} f (x) = \lim_{x \to 0} \frac{\tan x - \sin x}{x^{3}} = \lim_{x \to 0} \frac{\frac{\sin x}{\cos x} - \sin x}{x^{3}}$

$= \lim_{x \to 0} \frac{\sin x (1 - \cos x)}{x^{3}} = \lim_{x \to 0} \frac{\sin x . 2 \sin^{2} \frac{x}{2}}{x^{3} \cos x} = 2 \lim_{x \to 0} \frac{1}{\cos x} x \lim_{x \to 0} \frac{\sin x}{x} x \lim_{x \to 0} {[\frac{\sin \frac{x}{2}}{x}]}^{2} = 2 x 1 x 1 x \frac{1}{4} x \lim_{\frac{x}{2} \to 0} {[\frac{\sin \frac{x}{2}}{\frac{x}{2}}]}^{2} = 2 x 1 x 1 x \frac{1}{4} x 1 = \frac{1}{2} Now, f (0) = a \sin \frac{π}{2} (0 + 1) = a \sin \frac{π}{2} = a x 1 = a Since f is continuous at x = 0, a = \frac{1}{2}$

16.

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

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?