List of top Questions asked in PSEB XII

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:

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

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

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?

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

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

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

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

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

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?

Objective function of a L.P.P is a function to be optimized.

\( \int_1^1 (x^{19} + x^{21}) \, dx = 0 \)

\( \int dx = x^2 + C \)

The function \( y = \sin x \) is increasing in \( \left[ 0, \frac{\pi}{2} \right] \).

\( \int e^x(f(x) + f'(x)) \, dx \) equals: