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CBSE Class 12 Mathematics Solved Question Paper 2010

Long Answer Type

11.

Find all points of discontinuity of f, where f is defined as following:

$f (x) = \{\begin{cases} \begin{matrix} |x| + 3, x \leq - 3 \\ - 2 x, - 3 < x < 3 \end{matrix} \\ 6 x + 2, x \geq 3 \end{cases}$

12.

$Find \frac{dy}{dx}, if y = {(cosx)}^{x} + {(sinx)}^{\frac{1}{x}}$

13.

Prove the following:

$\tan^{- 1} \sqrt{x} = \frac{1}{2} \cos^{- 1} (\frac{1 - x}{1 + x}), x \in (0, 1)$

14.

Prove the following:

$\cos^{- 1} (\frac{12}{13}) + \sin^{- 1} (\frac{3}{5}) = \sin^{- 1} (\frac{56}{65})$

15.

Let $*$ be a binary operation on Q defined by $a * b = \frac{3 ab}{5}$
Show that $*$ is commutative as well as associative. Also find its identity element, if it exists.

16.

Using elementary row operations, find the inverse of the following matrix:

$[\begin{matrix} 2 & 5 \\ 1 & 3 \end{matrix}]$

17.

Find the equations of the normals to the curve y = x³ + 2x + 6 which are parallel to the line x + 14y + 4 = 0.

18.
Using properties of determinants show the following:
$|\begin{matrix} {(b + c)}^{2} & ab & ca \\ ab & {(a + c)}^{2} & bc \\ ac & bc & {(a + b)}^{2} \end{matrix}| = 2 abc (a + b + c)^{3}$

Consider,

$∆ = |\begin{matrix} (b + c)^{2} & ab & ac \\ ab & (a + c)^{2} & bc \\ ac & bc & (a + b)^{2} \end{matrix}| By performing R_{1} \to a R_{1}, R_{2} \to b R_{2}, R_{3} \to c R_{3} and dividing the determinant by abc, we get ∆ = \frac{1}{abc} |\begin{matrix} a (b + c)^{2} & a^{2} b & a^{2} c \\ {ab}^{2} & b (a + c)^{2} & b^{2} c \\ {ac}^{2} & {bc}^{2} & c (a + b)^{2} \end{matrix}|$

Now, taking a, b, c common from C₁, C₂, and C₃

$∆ = \frac{abc}{abc} |\begin{matrix} (b + c)^{2} & a^{2} & a^{2} \\ b^{2} & (a + c)^{2} & b^{2} \\ c^{2} & c^{2} & (a + b)^{2} \end{matrix}| \Rightarrow ∆ = |\begin{matrix} (b + c)^{2} & a^{2} & a^{2} \\ b^{2} & (c + a)^{2} & b^{2} \\ c^{2} & c^{2} & (a + b)^{2} \end{matrix}| Applying C_{1} \to C_{1} - C_{2}, C_{2} \to C_{2} - C_{3} \Rightarrow ∆ = |\begin{matrix} (b + c)^{2} - a^{2} & 0 & a^{2} \\ b^{2} - (c + a)^{2} & (c + a)^{2} - b^{2} & b^{2} \\ 0 & c^{2} - (a + b)^{2} & (a + b)^{2} \end{matrix}| ∆ = {(a + b + c)}^{2} |\begin{matrix} b + c - a & 0 & a^{2} \\ b - c - a^{} & c + a - b & b^{2} \\ 0 & c - a - b & (a + b)^{2} \end{matrix}|$

$Applying R_{3} \to R_{3} - (R_{1} + R_{2}) ∆ = {(a + b + c)}^{2} |\begin{matrix} b + c - a & 0 & a^{2} \\ b - c - a & c + a - b & b^{2} \\ 2 a - 2 b & - 2 a & 2 ab \end{matrix}|$

$Applying C_{1} \to C_{1} + C_{2} ∆ = {(a + b + c)}^{2} |\begin{matrix} b + c - a & 0 & a^{2} \\ 0 & c + a - b & b^{2} \\ - 2 b & - 2 a & 2 ab \end{matrix}| Applying C_{3} \to C_{3} + {bC}_{2} ∆ = {(a + b + c)}^{2} |\begin{matrix} b + c - a & 0 & a^{2} \\ 0 & c + a - b & bc + ab \\ - 2 b & - 2 a & 0 \end{matrix}|$

$Applying C_{1} \to {aC}_{1} and C_{2} \to {bC}_{2} ∆ = \frac{{(a + b + c)}^{2}}{ab} |\begin{matrix} ab + ac - a^{2} & 0 & a^{2} \\ 0 & bc + ab - b^{2} & bc + ab \\ - 2 ab & - 2 ab & 0 \end{matrix}| Applying C_{1} \to C_{1} - C_{2} ∆ = \frac{{(a + b + c)}^{2}}{ab} |\begin{matrix} ab + ac - a^{2} & 0 & a^{2} \\ - bc - ab + b^{2} & bc + ab - b^{2} & bc + ab \\ 0 & - 2 ab & 0 \end{matrix}|$

Expanding along R₃

$= \frac{{(a + b + c)}^{2}}{ab} (2 ab ({ab}^{2} c + a^{2} b^{2} + {abc}^{2} + a^{2} bc - a^{2} bc - a^{3} b + a^{2} bc + a^{3} b - a^{2} b^{2})) = 2 (a + b + c)^{2} ({ab}^{2} c + {abc}^{2} + a^{2} bc) = 2 (a + b + c)^{3} abc = 2 abc (a + b + c)^{3} = R . H . S .$

19.

Find the values of x for which f(x) = [x(x - 2)]2 is an increasing function. Also, find the points on the curve where the tangent is parallel to x-axis.

20.

Show that the right circular cylinder, open at the top, and of given surface area and maximum volume is such that its height is equal to the radius of the base.