CBSE Class 12 Mathematics — Long Answer Questions

(a) A relation R is defined on Z, the set of integers, as R = {(x, y) : |x - y| is divisible by a prime number 'p', x, y in Z}. Check whether R is an equivalence relation or not. OR (b) A function $f : \mathbb{R} - \{3/5\} \to \mathbb{R} - \{3/5\}$ is defined as $f(x) = \frac{3x + 2}{5x - 3}$. Show that f is one-one and onto.

(a) If $A = \begin{bmatrix}0 & 2 & 1\\ -2 & -1 & -2\\ 1 & -1 & 0\end{bmatrix}$, find $A^{-1}$ and use it to solve the following system of equations: $-2y + z = 7$, $2x - y - z = 8$, $x - 2y = 10$. OR (b) If $\begin{bmatrix}3 & -1 & \sin 3x\\ -7 & 4 & \cos 2x\\ -11 & 7 & 2\end{bmatrix}$ is a singular matrix, then find all values of x, where $x \in [0, \pi/2]$.

If $x = \cos t, y = \cos mt$, prove that $(1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} + m^2 y = 0$.

Check whether the lines given by $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$ and $\frac{x - 4}{5} = \frac{y - 1}{2} = z$ are parallel or not. If parallel, find the distance between them, otherwise find their point of intersection, if the lines are intersecting.