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CBSE Class 12 Mathematics Solved Question Paper 2008

Short Answer Type

If f(x) = x + 7 and g(x) = x – 7, x $\in$ R, find (fog) (7)

Evaluate: $\sin [\frac{π}{3} - \sin^{- 1} (\frac{1}{2})]$

Find the value of x and y if: $2 [\begin{matrix} 1 & 3 \\ 0 & x \end{matrix}] + [\begin{matrix} y & 0 \\ 1 & 2 \end{matrix}] = [\begin{matrix} 5 & 6 \\ 1 & 8 \end{matrix}]$

Evaluate: $[\begin{matrix} a + ib & c + id \\ - c + id & a - ib \end{matrix}]$

Find the co-factor of a12 in the following:

$|\begin{matrix} 2 & - 3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & - 7 \end{matrix}|$

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

7.
Evaluate: $\int_{0}^{4} \frac{dx}{1 + x^{2}} dx$

$\int_{0}^{4} \frac{dx}{1 + x^{2}} dx Let X = tanθ \Rightarrow θ = \tan^{- 1} X dx = \sec^{2} θ dθ When X = 0, θ = \tan^{- 1} (0) = 0 When X = 1, θ = \tan^{- 1} 1 = \frac{π}{4} ∴ \int_{0}^{4} \frac{dx}{1 + x^{2}} = \int_{0}^{\frac{π}{4}} \frac{\sec^{2} θ}{1 + \tan^{2} θ} dθ = \int_{0}^{\frac{π}{4}} \frac{\sec^{2} θ}{\sec^{2} θ} dθ = \int_{0}^{\frac{π}{4}} dθ = {[θ]}_{0}^{\frac{π}{4}} = [\frac{π}{4} - 0] = \frac{π}{4}$

Find a unit vector in the direction of $\vec{a} = 3 \hat{i} - 2 \hat{j} + 6 k$

Find the angle between the vectors $\vec{a} = \hat{i} - \hat{j} + \hat{k} and \vec{b} = \hat{1} + \hat{j} - \hat{k}$

10.

For what value of $λ$ are the vectors $\vec{a} = 2 \hat{i} + λ \hat{j} + \hat{k} and \vec{b} = \hat{i} - 2 \hat{j} - 3 \hat{k}$ are perpendicular to each other?