List of top Mathematics Questions asked in PSEB XII

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

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

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

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

The rate of change of the area of a circle with respect to its radius \( r \) at \( r = 6 \, \text{cm} \) is:

Maximum value of \( Z = 3x + y \) for the constraints \( x + y \leq 4, x \geq 0, y \geq 0 \) is:

\( \frac{d}{dx} \left( \cos^{-1}x \right) \) is:

If \( AB = C \) where \( B \) and \( C \) are of order \( 3 \times 5 \), then the order of \( A \) is:

The value of \( \cos^{-1} \left( \cos \frac{2\pi}{3} \right) \) is equal to:

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

\( \int_0^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx \) equals:

If \( A \) is an invertible matrix, then \( \det(A^{-1}) \) is:

If \( A = \text{diag}(d_1, d_2, \ldots, d_n) \), then \( |A| \) is:

If \( y = e^x \), then \( \frac{d^2y}{dx^2} \) is:

If \( A \) is a non-singular matrix of order 3 and \( |A| = 2 \), then \( |\text{adj}(A)| \) equals:

Let \( A \) be a square matrix of order 3x3. Then \( |2A| \) is equal to:

Find the shortest distance between the following pairs of lines: \[ \mathbf{r_1} = \hat{i} - 4 \hat{j} + 5 \hat{k} + \mu(5 \hat{i} + 9 \hat{j} + \hat{k}), \quad \mathbf{r_2} = 2 \hat{i} + 8 \hat{j} - 6 \hat{k} + \lambda(3 \hat{i} - 2 \hat{j} + \hat{k}) \]

The value of \( \int_0^{\pi/2} \sin^2 x \cos^3 x \, dx \) is:

Find the angle between the lines: \[ \mathbf{r_1} = 3 \hat{i} + 8 \hat{j} + 3 \hat{k} + \mu (\hat{i} + 2 \hat{j} - \hat{k}), \quad \mathbf{r_2} = -3 \hat{i} + 9 \hat{j} - \hat{k} + \lambda (5 \hat{i} + 3 \hat{j} + 4 \hat{k}) \]

If \( x = \frac{1 - t^2}{1 + t^2} \), \( y = \frac{2t}{1 + t^2} \), then prove that \( \frac{dy}{dx} + \frac{x}{y} = 0 \).

Evaluate \[ \int \frac{x^2}{x^4 + 1} \, dx \]

Find the height of the right circular cone of maximum volume, which is inscribed in a sphere of radius 12 cm.

Solve the following system of linear equations by matrix method: \[ 5x + y - 3z = -10, \quad 3x - 2y + z = -3, \quad x + 3y + z = 4 \]

Using determinants, find the value of \( k \) if the area of the triangle formed by the points \( (-3, 6) \), \( (-4, 4) \), and \( (k, -2) \) is 12 sq. units.

If \( y = x^{\sin x} + (\sin x)^x \), then find \( \frac{dy}{dx} \).