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

17.
If $x = a (θ - \sin θ), y = (1 + \cos θ), find \frac{d^{2} y}{{dx}^{2}}$

$x = a (θ - \sin θ), y = a (1 + \cos θ) Differentiating x and y w . r . t . θ, \frac{dx}{dθ} = a (1 - \cos θ) . . . . . . . . . (i) \frac{dy}{dθ} = - a \sin θ . . . . . . . . . . (ii) Dividing (2) by (1), \frac{(\frac{dy}{dθ})}{(\frac{dx}{dθ})} = \frac{- a \sin θ}{a (1 - \cos θ)}$

$\Rightarrow \frac{dy}{dx} = \frac{- sinθ}{1 - \cos θ} \Rightarrow \frac{dy}{dx} = \frac{- 2 \sin \frac{θ}{2} \cos \frac{θ}{2}}{2 \sin^{2} \frac{θ}{2}} \Rightarrow \frac{dy}{dx} = \frac{- \cos \frac{θ}{2}}{\sin \frac{θ}{2}} \Rightarrow \frac{dy}{dx} = - \cot \frac{θ}{2}$

Differentiating w.r.t. x,

$\frac{d}{dx} (\frac{dy}{dx}) = \frac{d}{dθ} (\frac{dy}{dx}) x \frac{dθ}{dx} \Rightarrow \frac{d^{2} y}{{dx}^{2}} = \frac{d}{dθ} (\frac{dy}{dx}) x \frac{dθ}{dx} \Rightarrow \frac{d^{2} y}{{dx}^{2}} = \frac{d}{dθ} (- \cot \frac{θ}{2}) x \frac{dθ}{dx} . . . . [From equation (iii)] \frac{d^{2} y}{{dx}^{2}} = - (- {cosec}^{2} \frac{θ}{2} x \frac{1}{2}) x \frac{dθ}{dx} = \frac{1}{2} {cosec}^{2} \frac{θ}{2} x \frac{1}{(\frac{dx}{dθ})}$

$= \frac{1}{2} {cosec}^{2} \frac{θ}{2} x \frac{1}{a (1 - \cos θ)} . . . . . . . . [From equation (i)] = \frac{{cosec}^{2} \frac{θ}{2}}{2 a (1 - \cos θ)} = \frac{{cosec}^{2} \frac{θ}{2}}{2 a (2 \sin^{2} \frac{θ}{2})} = \frac{1}{4 a} x {cosec}^{4} \frac{θ}{2}$

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?