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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.

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

The given differential equation is:

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

$\Rightarrow$ e^x tan y dx = - ( 1 - e^x) sec² y dy

$\Rightarrow$ e^x tan y dx = ( e^x - 1 ) sec² y dy

$\Rightarrow \frac{e^{x}}{e^{x} - 1} dx = \frac{\sec^{2} y}{\tan y} dy$

On integrating on both sides, we get

$\int \frac{e^{x}}{e^{x} - 1} dx = \int \frac{\sec^{2} y}{\tan y} dy . . . . . . . . . . (i) Let I_{1} = \int \frac{\sec^{2} y}{\tan y} dy Put \tan y = t \Rightarrow \sec^{2} y dy = dt ∴ \int \frac{\sec^{2} y}{\tan y} dy = \int \frac{dt}{t} = \log |t| = \log \tan y . . . . . . . . . (ii) Let I_{2 =} \int \frac{e^{x}}{e^{x} - 1} dx$

Put e^x - 1 = u

$∴$ e^x dx = du

$\int \frac{e^{x}}{e^{x} - 1} dx = \int \frac{du}{u}$

= log u

= log ( e^x - 1 ) ............(iii)

From (i), (ii), and (iii), we get

log tan y = log ( e^x - 1 ) + log C

$\Rightarrow$ log tan y = log C ( e^x - 1 )

$\Rightarrow$ tan y = C ( e^x - 1 )

The solution of the given differential equation is tan y = C ( e^x - 1 ).

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.