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

$\int \frac{5 x + 3}{\sqrt{x^{2} + 4 x + 10}} dx Now, 5 x + 3 = A \frac{d}{dx} (x^{2} + 4 x + 10) + B \Rightarrow 5 x + 3 = A (2 x + 4) + B \Rightarrow 5 x + 3 = 2 A x + 4 A + B \Rightarrow 2 A = 5 and 4 A + B = 3 \Rightarrow A = \frac{5}{2} Thus, 4 (\frac{5}{2}) + B = 3 \Rightarrow 10 + B = 3 \Rightarrow B = 3 - 10 = - 7$

On substituting the values of A and B, we get

$\int \frac{(5 x + 3)}{\sqrt{x^{2} + 4 x + 10}} dx = \int \frac{[\frac{5}{2} \frac{d}{dx} (x^{2} + 4 x + 10) - 7]}{\sqrt{x^{2} + 4 x + 10}} dx = \int [\frac{\frac{5}{2} (2 x + 4) - 7}{\sqrt{x^{2} + 4 x + 10}}] dx = \frac{5}{2} \int \frac{2 x + 4}{\sqrt{x^{2} + 4 x + 10}} dx - 7 \int \frac{dx}{\sqrt{x^{2} + 4 x + 10}} = \frac{5}{2} I_{1} - 7 I_{2} . . . . . . . . . . . (i)$

$I_{2} = \int \frac{2 x + 4}{\sqrt{x^{2} + 4 x + 10}} dx Put x^{2} + 4 x + 10 = z^{2} (2 x + 4) dx = 2 z dz Thus, I_{1} = \int \frac{2 z}{z} dz = 2 z = 2 \sqrt{x^{2} + 4 x + 10} + C_{1} I_{2} = \int \frac{dx}{\sqrt{x^{2} + 4 x + 10}} = \int \frac{dx}{\sqrt{x^{2} + 4 x + 4 + 6}} = \int \frac{dx}{\sqrt{(x + 2)^{2} + {(\sqrt{6})}^{2}}} = \log |(x + 2) + \sqrt{x^{2} + 4 x + 10}| + C_{2}$

Substituting I₁and I₂ in ( i ), we get

$∴ \int \frac{5 x + 3}{\sqrt{x^{2} + 4 x + 10}} = \frac{5}{2} (2 \sqrt{x^{2} + 4 x + 10} + C_{1}) - 7 [\log |(x + 2) + \sqrt{x^{2} + 4 x + 10}| + C_{2}] = 5 \sqrt{x^{2} + 4 x + 10} - 7 [\log |(x + 2) + \sqrt{x^{2} + 4 x + 10}|] + \frac{5}{2} C_{1} - 7 C_{2} = 5 \sqrt{x^{2} + 4 x + 10} - 7 [\log |(x + 2) + \sqrt{x^{2} + 4 x + 10}|] + C, where C = \frac{5}{2} C_{1} - 7 C_{2}$

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.