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CBSE Class 12 Mathematics Solved Question Paper 2011

Long Answer Type

21.

Using elementary transformations, find the inverse of the matrx

$(\begin{matrix} 1 & 3 & - 2 \\ - 3 & 0 & - 1 \\ 2 & 1 & 0 \end{matrix})$

22.
Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

Let the rectangle of length l and breadth b be inscribed in circle of radius a.

Then, the diagonal of the rectangle passes through the centre and is of length 2a cm.

Now, by applying the Pythagoras Theorem, we have:

( 2a )² = l² + b²

$\Rightarrow b2= 4 a2-l2$

$\Rightarrow b = \sqrt{4 a^{2} - l^{2}} ∴ Area of rectangle, A = l \sqrt{4 a^{2} - l^{2}} ∴ \frac{dA}{dl} = \sqrt{4 a^{2} - l^{2}} + l \frac{1}{2 \sqrt{4 a^{2} - l^{2}}} (- 2 l) = \sqrt{4 a^{2} - l^{2}} - \frac{l^{2}}{\sqrt{4 a^{2} - l^{2}}} = \frac{4 a^{2} - 2 l^{2}}{\sqrt{4 a^{2} - l^{2}}} \frac{{dA}^{2}}{{dl}^{2}} = \frac{\sqrt{4 a^{2} - l^{2}} (- 4 l) - (4 a^{2} - 2 l^{2}) \frac{(- 2 l)}{2 \sqrt{4 a^{2} - l^{2}}}}{(4 a^{2} - l^{2})} = \frac{(4 a^{2} - l^{2}) (- 4 l) + l (4 a^{2} - 2 l^{2})}{{(4 a^{2} - l^{2})}^{\frac{3}{2}}} = \frac{- 12 a^{2} l + 2 l^{3}}{{(4 a^{2} - l^{2})}^{\frac{3}{2}}} = \frac{- 2 l (6 a^{2} - l^{2})}{{(4 a^{2} - l^{2})}^{\frac{3}{2}}}$

$Now, \frac{dA}{d l} = 0 gives 4 a^{2} = 2 l^{2} \Rightarrow l = \sqrt{2 a} \Rightarrow b = \sqrt{4 a^{2} - 2 a^{2}} = \sqrt{2 a^{2}} = \sqrt{2} a When l = \sqrt{2} a, \frac{{dA}^{2}}{d l^{2}} = \frac{- 2 (\sqrt{2} a) (6 a^{2} - 2 a^{2})}{2 \sqrt{2} a^{3}} = \frac{- 8 \sqrt{2} a^{3}}{2 \sqrt{2} a^{3}} = - 4 < 0$

$∴$ Thus, frrom the second derivative test, when l = $\sqrt{2} a$ , the area of the rectangle is maximum.

Since l = b = $\sqrt{2} a$ , the rectangle is square.

Hence, of all the rectangles inscribed in the given circle, the square has the maximum area.

23.

Evaluate: $\int \frac{5 x + 3}{\sqrt{x^{2} + 4 x + 10}} dx$

24.

Evaluate: $\int \frac{2 x}{\sqrt{(x^{2} + 1) (x^{2} + 3)}} dx$

25.

Solve the following differential equation:

e^x tan y dx + ( 1 - e^x) sec² y dy = 0

26.

Solve the following differential equation:

$\cos^{2} x \frac{dy}{dx} + y = \tan x$

27.

Find a unit vector perpendicular to each of the vector $\vec{a} + \vec{b} and \vec{a} - \vec{b}$ , where

$\vec{a} = 3 \hat{i} + 2 \hat{j} + 2 \hat{k} and \vec{b} = \hat{i} + 2 \hat{j} - 2 \hat{k} .$

28.

Find the angle between the following pair of lines:

$\frac{- x + 2}{- 2} = \frac{y - 1}{7} = \frac{z + 3}{- 3} and \frac{x + 2}{- 1} = \frac{2 y - 8}{4} = \frac{z - 5}{4}$

And check whether the lines are parallel or perpendicular.

29.

Probabilities of solving problem independently by A and B are $\frac{1}{2}$ and $\frac{1}{3}$ respectively. If both try to solve the problem independently, find the probability that

(i) the problem is solved

(ii) exactly one of them solves the problem.