Class 12 Mathematics 2016 Board Exam

If $x\in \mathbb{N}$ and $\left|\begin{array}{cc}x+3& -2\\ -3x& 2x\end{array}\right|=8$, then find the value of $x$.

Use elementary column operation $C_2\to C_2+2C_1$ in the following matrix equation :

$$\left[\begin{array}{cc}2& 1\\ 2& 0\end{array}\right]=\left[\begin{array}{cc}3& 1\\ 2& 0\end{array}\right]\left[\begin{array}{cc}1& 0\\ -1& 1\end{array}\right]$$

Write the number of all possible matrices of order $2\times 2$ with each entry $1$, $2$ or $3$.

Write the position vector of the point which divides the join of points with position vectors $3\vec{a}-2\vec{b}$ and $2\vec{a}+3\vec{b}$ in the ratio $2:1$.

Write the number of vectors of unit length perpendicular to both the vectors $\vec{a}=2\hat{i}+\hat{j}+2\hat{k}$ and $\vec{b}=\hat{j}+\hat{k}$.

Find the vector equation of the plane with intercepts $3$, $-4$ and $2$ on $x$, $y$ and $z$-axis respectively.

Find the coordinates of the point where the line through the points $A(3,4,1)$ and $B(5,1,6)$ crosses the $XZ$ plane. Also find the angle which this line makes with the $XZ$ plane.

The two adjacent sides of a parallelogram are $2\hat{i}-4\hat{j}-5\hat{k}$ and $2\hat{i}+2\hat{j}+3\hat{k}$. Find the two unit vectors parallel to its diagonals. Using the diagonal vectors, find the area of the parallelogram.

In a game, a man wins ₹ $5$ for getting a number greater than $4$ and loses ₹ $1$ otherwise, when a fair die is thrown. The man decided to throw a die thrice but to quit as and when he gets a number greater than $4$. Find the expected value of the amount he wins/loses.

OR

A bag contains $4$ balls. Two balls are drawn at random (without replacement) and are found to be white. What is the probability that all balls in the bag are white?

Differentiate $x^{\sin x}+(\sin x{)}^{\cos x}$ with respect to $x$.

OR

If $y=2\cos (\log x)+3\sin (\log x)$, prove that $x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+y=0$.