List of top Mathematics Questions

Solve the following linear programming problem by the graphical method, under the following constraints: \[ x + 3y \leq 60, \quad x + y \geq 10, \quad x \leq y, \quad x \geq 0, \quad y \geq 0. \] Find the minimum and maximum values of \[ Z = 3x + 9y. \]

Evaluate: \[ \int_0^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x + \cos x}} \, dx. \]

Evaluate: \[ \int_0^\pi \frac{x \sin x}{1 + \cos^2 x} \, dx. \]

Solve the following system of equations by matrix method: \[ 3x - 2y + 3z = 8, \quad 2x + y - z = 1, \quad 4x - 3y + 2z = 4. \]

If the normal of the curve \[ \frac{x^{2/3}}{a^{2/3}} + \frac{y^{2/3}}{a^{2/3}} = 1 \] makes an angle \( \theta \) with the x-axis, prove that the equation of the normal is \[ y \cos \theta - x \sin \theta = a \cos 2\theta. \]

If the coordinates of mid-points of the sides of a triangle are \( (1, 5, -1), (0, 4, -2) \) and \( (2, 3, 4) \), then find the coordinates of its vertices.

Find the differential coefficient of \( \tan^{-1}\left( \frac{2x}{1-x^2} \right) \) \text{with respect to} \( x \).

Find the shortest distance between the lines \[ \vec{r} = (i + 2j + k) + \lambda(i - j + k) \] and \[ \vec{r} = (2i - j - k) + \mu(2i + j + 2k) \]

If two dice are thrown together, then find the probability of getting at least one 6.

Evaluate: \[ \int \frac{\sec^2(2x)}{(\cot x - \tan x)^2} \, dx. \]

Find the equation of normal at the point \( (1, 1) \) of the curve \( x^{2/3} + y^{2/3} = 2 \).

Find the equation of the plane passing through the points \((-2i + 6j - 6k), (-3i + 10j - 9k), (-5i - 6j - 6k)\).

Find the value of \( \int_a^b x^2 \, dx \) with the help of definite integral as the limit of a sum.

If \( P(A) = \frac{1}{2} \), \( P(B) = \frac{1}{3} \), and \( P(A \cup B) = \frac{2}{3} \), prove that the events \( A \) and \( B \) are independent.

Solve the differential equation \( (x - y) \, dy - (x + y) \, dx = 0 \).

Find the area of the triangle whose two sides are represented by the vectors \[ \vec{a} = 3\hat{i} - \hat{j} + 5\hat{k}, \quad \vec{b} = \hat{i} + 2\hat{j} - \hat{k}. \]

Find the coordinates of the point which divides the line joining the points \( (2, -5, 1) \) and \( (1, 4, -6) \) internally in the ratio 2 : 3.

Prove: \[ \left| \begin{matrix} 1+\alpha & 1 & 1 \\ 1+\beta & 1 & 1 \\ 1 & 1 & 1+\gamma \\ \end{matrix} \right| = abc \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + 1 \right) \]

If \( \vec{a} = 2\hat{i} + 2\hat{j} + 3\hat{k} \), \( \vec{b} = -\hat{i} + 2\hat{j} + \hat{k} \), and \( \vec{c} = 3\hat{i} + \hat{j} \) are such that \( \vec{a} + \lambda \vec{b} \) is perpendicular to \( \vec{c} \), then find the value of \( \lambda \).

If \[ A = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix}, \] prove that \[ A^3 = \begin{bmatrix} \cos 3\theta & \sin 3\theta \\ -\sin 3\theta & \cos 3\theta \end{bmatrix}. \]

If \( f : \mathbb{R} \to \mathbb{R} \), where \( f(x) = \sin x \) and \( g : \mathbb{R} \to \mathbb{R} \), where \( g(x) = x^2 \), then find the range of \( f(x) \) and \( g(x) \).

Two integers among 1 to 11 are selected at random. If their sum is even, then find the probability that both integers are odd.

Find the differential equation of the family of curves \[ y = a \sin(x + b), \] where \( a \) and \( b \) are arbitrary constants.

The angle between the vectors \[ \mathbf{A} = 2\hat{i} + \hat{j} + 3\hat{k} \quad \text{and} \quad \mathbf{B} = 3\hat{i} - 2\hat{j} + \hat{k} \] will be