List of top Mathematics Questions

In $\triangle ABC$, if $AB = 6\sqrt{3}$ cm, $AC = 12$ cm, and $BC = 6$ cm, then $\angle B$ will be:

If \( \triangle ABC \) is an isosceles triangle with \( AB = AC \), then:

If \( m = 5 \) and \( n = m + 7 \), then the value of \( \sqrt{m^2 + n^2} \) will be:

For any two positive integers ‘a’ and ‘b’:

The quotient of a non-zero rational and irrational number will be:

The equivalent rational numbers of any rational number will be:

If $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ are two linear equations, and if $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$, then:

Prove that the function \[ f(x) = \begin{cases} x^3 - 3 & \text{if } x \leq 2 \\ x^2 + 1 & \text{if } x>2 \end{cases} \] is continuous at \( x = 2 \).

Solve the differential equation: \[ \tan^{-1}(y - x) \, dy = (1 + y^2) \, dx. \]

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

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 \[ Z = 3x + 9y. \]

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 \[ y \cos \theta - x \sin \theta = a \cos 2\theta. \]

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

Find the inverse of the matrix \[ A = \begin{pmatrix} 2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{pmatrix} \] by elementary transformations.

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

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

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

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

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

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

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