List of top Mathematics Questions

Find the shortest distance between the lines:
\( \frac{x+1}{7} = \frac{y+1}{-6} = \frac{z+1}{1} \) and
\( \frac{x-3}{1} = \frac{y-5}{-2} = \frac{z-7}{1} \).

Solve the differential equation \( ydx - (x + 2y^2) dy = 0 \).

There are three groups of children having 3 girls and one boy, 2 girls and 2 boys, one girl and 3 boys respectively. One child is selected at random from each group. Find the probability that the three selected children have one girl and 2 boys.

Find the area of the region enclosed by the parabola \( y^2 = 4ax \) and its latus rectum.

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

Find the area of a parallelogram whose adjacent sides are given by vectors \( \vec{a} = 3\hat{i} + \hat{j} + 2\hat{k} \) and \( \vec{b} = \hat{i} + 2\hat{j} - 2\hat{k} \).

Show that the given function \( f(x) = \cos x \) is increasing in \( (\pi, 2\pi) \).

Find the angle between the pair of lines \( \frac{x+3}{3} = \frac{y-1}{5} = \frac{z+3}{4} \) and \( \frac{x+1}{1} = \frac{y-4}{1} = \frac{z-5}{2} \).

The volume of a cube is increasing at the rate of 9 cm\(^3\)/s. If the length of its edge is 10 cm, then its surface area is increasing with which rate?

The order of the differential equation \( \frac{d^3y}{dx^3} + \frac{d^2y}{dx^2} + y \cos x = 0 \) is :

The value of \( \tan^{-1}\sqrt{3} - \sec^{-1}(-2) \) is :

If P(A) = 0.3, P(B) = 0.4, then find P(A/B) if A and B are independent events.

Find the order and degree of the differential equation \( \frac{d^2y}{dx^2} + \frac{dy}{dx} + y \cdot \sin x = 0 \).

If \( A = \begin{bmatrix} \cos\alpha & -\sin\alpha
\sin\alpha & \cos\alpha \end{bmatrix} \) and \( A + A' = I \), then find the value of \( \alpha \).

The value of \( \int_1^{\sqrt{3}} \frac{dx}{1 + x^2} \) will be :

If three vectors \( \vec{a} \), \( \vec{b} \) and \( \vec{c} \) satisfying the condition \( \vec{a} + \vec{b} + \vec{c} = \vec{0} \). If \( |\vec{a}| = 3 \), \( |\vec{b}| = 4 \) and \( |\vec{c}| = 2 \), then find the value of \( \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} \).

The radius of an air bubble is increasing at the rate of \( \frac{1}{2} \) cm/s. At what rate is the volume of the bubble increasing while the radius is 1 cm?

There are 4 white and 2 black balls in a bag and in another bag 3 white and 5 black balls. Find the probability of getting both black balls if a ball is drawn from each bag.

If the Cartesian equation of a line is \( \frac{x-5}{3} = \frac{y+4}{7} = \frac{z-6}{2} \), then find its equation in vector form.

Prove that (4, 4, 2), (3, 5, 2) and (-1, -1, 2) are vertices of a right angle triangle.

Find the degree of the differential equation \( xy \frac{d^2y}{dx^2} + x \left(\frac{dy}{dx}\right)^2 - y\left(\frac{dy}{dx}\right) = 2 \)

Prove that the function f(x) = |x| is continuous at x = 0.

The value of \( \int_{0}^{\pi/2} \frac{dx}{1 + \sqrt{\tan x}} \) will be

Let $$ \alpha = \frac{1}{\sin 60^\circ \sin 61^\circ} + \frac{1}{\sin 62^\circ \sin 63^\circ} + \cdots + \frac{1}{\sin 118^\circ \sin 119^\circ}. $$ Then the value of $$ \left( \frac{\csc 1^\circ}{\alpha} \right)^2 $$ is \rule{1cm}{0.15mm}.