List of top Mathematics Questions asked in PSEB XII

The matrix

\[ A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \]

is:

If

\[ A = \begin{bmatrix} 3 & 2 \\ 4 & 7 \end{bmatrix}, \quad f(x) = x^2 + 2x - 3 \]

then find \( f(A) \).

Express

\[ \begin{bmatrix} 5 \\ 3 \end{bmatrix} \]

as the sum of a symmetric matrix and a skew-symmetric matrix.

If

\[ X = \begin{bmatrix} 3 & 4 \\ 2 & -1 \end{bmatrix}, \quad 2X - Y = \begin{bmatrix} 5 & 10 \\ 3 & -5 \end{bmatrix} \]

then find the matrix \( Y \).

Solve: \[ x \log x \frac{dy}{dx} + y = \frac{2}{x} \log x \]

Solve: \[ x^2 \, dy - (3x^2 + xy + y^2) \, dx = 0 \quad \text{given that} \quad y = 1 \, \text{when} \, x = 1 \]

Solve the following linear programming problem graphically: Maximise and minimise \( Z = 4x + 2y - 7 \) subject to the constraints: \[ x + 3y \leq 60, \quad x + y \geq 10, \quad x - y \leq 0, \quad x \geq 0, \quad y \geq 0 \]

If

\[ A = \begin{bmatrix} 2 & 1 & 3 \\ 4 & 1 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & -1 \\ 0 & 2 \\ 5 & 0 \end{bmatrix} \]

then verify that \( (AB)^T = B^T A^T \).

The volume of a spherical balloon is increasing at the rate of 25 c.c./s. Find the rate of change of its surface area at the instant when its radius is 5 cm.

Direction ratios of the straight line \[ \vec{r} = 4\hat{i} - 7\hat{j} + 5\hat{k} + s(9\hat{i} - 2\hat{j} + 5\hat{k}) \text{ are:} ........... \]

If \( y = \tan x \), then at \( x = \frac{\pi}{4} \), \[ \frac{dy}{dx} = \quad ......... \]

If \( P(A) = \frac{1}{2}, P(B) = \frac{3}{8} \) and \( P(A \cap B) = \frac{1}{5} \), then \( P(A \cup B) \) is equal to:

A wire of length 28m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and circle is minimized?

Evaluating \( \int \frac{dx}{x^3 + 1} \):

Prove that \[ 2 \tan^{-1} \left( \frac{1}{3} \right) + \tan^{-1} \left( \frac{1}{7} \right) = \frac{\pi}{4} \]

Evaluate \[ \int \frac{dx}{x^2 - 8x + 17} \]

If \( y = (\tan^{-1} x)^2 \), prove that \( (x^2 + 1)^2 y_2 + 2x (x^2 + 1) y_1 = 2 \).

Differentiate \( x \sin x + (\sin x)^x \) with respect to \( x \).

Solve the L.P.P graphically
Maximize \( Z = 7x + 8y \) subject to constraints \[ x + y \geq 50, \quad x + 2y \leq 100, \quad x - y \geq 10, \quad x \geq 0, \quad y \geq 0 \]

Evaluate \( \int_{-1}^{1} \log \left( \frac{2 + x}{2 - x} \right) \, dx \)

Find the intervals in which the function \( f(x) = 2x^3 - 15x^2 + 36x + 1 \) is strictly increasing or decreasing?

Evaluate \( \int e^x \left( \cot x + \log \sin x \right) \, dx \)

\( \tan^{-1} \sqrt{3} - \sec^{-1}(-2) \) is equal to \( \frac{\pi}{3} \).