List of top Mathematics Questions asked in PSEB XII

Evaluate \[ \int \frac{3x + 2}{(x^2 + 1)(x - 2)} \, dx \]

A laboratory blood test is 99% effective in diagnosing a certain disease when it is in fact present. However, the test also yields a false positive result for 0.5% of the healthy persons tested (i.e., if a healthy person is tested, then with probability 0.005, the test will imply he has the disease). If 0.1% of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive?

Prove that the function \( f : \mathbb{R} \to \mathbb{R} \), \( f(x) = \frac{5x + 3}{4} \), is one-one and onto.

A problem is given to 3 students whose chances of solving it are \( \frac{1}{3} \), \( \frac{1}{5} \), and \( \frac{1}{6} \). What is the probability that:
(i) Exactly one of them solves the problem?

(ii) The problem is solved?

Find the value of \( p \) if the vectors \( \vec{p} \hat{i} - 8 \hat{j} + 5 \hat{k} \) and \( 5 \hat{i} + 2 \hat{j} - 3 \hat{k} \) are perpendicular to each other.

Find \( \frac{dy}{dx} \), if \( y = \tan^{-1} \left( \frac{2x}{1 - x^2} \right) \)

Evaluate \( \int \frac{\sin^{-1}(x)}{1 + x^2} \, dx \)

Evaluate \( \int \frac{2x - 3}{x^2 + 1} \, dx \)

Find \( \frac{dy}{dx} \), if \( x^3 + 2x^2 y - 3xy^2 + y^3 = 100 \)

Find the area of the triangle whose sides are given by vectors: \[ \vec{a} = 2\hat{i} + 3\hat{j} - \hat{k}, \quad \vec{b} = \hat{i} - \hat{j} + 3\hat{k} \]

Using integration, find the area bounded by the circle \( x^2 + y^2 = 16 \) in the first quadrant.

Find the critical points of the function \( f(x) = 2x^3 - 3x^2 - 12x + 15 \):

If the order of the matrix \( A \) is \( 3 \times 2 \), then the order of matrix \( A^T \) is:

Principal value of \( \sin^{-1}(1) \) is:

Evaluate \( \int_{\pi/6}^{\pi/3} \frac{\sqrt{\cos x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx \):

Evaluate \( \int_0^1 \frac{dx}{1 + x^2} \):

Find the order of the differential equation \( \frac{d^2y}{dx^2} - 2\frac{dy}{dx} + 3y = 0 \):

The radius of a circle is increasing at the rate of 2 m/s. The rate of change of its circumference is:

If \( A \) is a square matrix of order \( 2 \times 2 \) and \( |A| = 5 \), then \( | \text{Adj}(A) | \) is:

If \( y = \log x \), then \( \frac{dy}{dx} \) is:

Vector equation of the line \( \frac{x - 5}{4} = \frac{y - 3}{5} = \frac{z + 3}{-8} \) is:

Objective function of a linear programming problem is:

Range of the function \( \cos^{-1} \):

Relation \( R = \{(x, y): x<y^2 \text{ where } x, y \in \mathbb{R}\} \) is:

If \( \vec{a} \) is a non-zero vector then \( |\vec{a} \times \vec{a}| \) is equal to: