List of top Mathematics & Statistics Questions

Prove the Section Formula for internal division in vectors.

Prove that \( \displaystyle \int \sqrt{a^2 - x^2} \, dx = \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\sin^{-1}\left(\frac{x}{a}\right) + C \).

The population of a town grows at a rate proportional to its size. If it grows from 40,000 to 60,000 in 40 years, what will it be in another 20 years?

A die is tossed twice. If a “success” is getting a number greater than 4, find the probability distribution of the number of successes.

Evaluate \( \displaystyle \int \frac{x+2}{2x^2 + 6x + 5} \, dx \) using the substitution or partial fraction method.

Write the converse, inverse, and contrapositive of: "If voltage increases, then current decreases".

Find the principal value of \( \sin^{-1}\left(-\frac{1}{2}\right) \).

Differentiate \( \cos^{-1} \left( \frac{1 - x^2}{1 + x^2} \right) \) with respect to \( x \).

Find the area of a triangle with vertices \( A(1,2,3), B(2,3,4), \) and \( C(3,4,5) \).

Write the dual of the statement \( p \land [\neg q \lor (p \land q) \lor \neg r] \).

If the mean and variance of a binomial distribution are 18 and 12 respectively, then \( n \) is equal to .......

Find the adjoint of the matrix \( \left[ \begin{matrix} 2 & -2 \\ 4 & 3 \end{matrix} \right] \).

Write the order of the differential equation \[ \sqrt{1 + \left( \frac{dy}{dx} \right)^2} = \left( \frac{d^2y}{dx^2} \right)^{\frac{3}{2}}. \]

The value of \( \int x^x(1 + \log x) \, dx \) is equal to ...........

If \(|\vec{a}| = 5\), \(|\vec{b}| = 13\), and \(|\vec{a} \times \vec{b}| = 25\), then \(|\vec{a} \cdot \vec{b}|\) is equal to ........

Write the condition for the function \( f(x) \), to be strictly increasing, for all \( x \in \mathbb{R} \).

The area bounded by the line \( y = x \), X-axis and the lines \( x = -1 \) and \( x = 4 \) is equal to .......... (in square units).

Let \( f(1) = 3 \), \( f'(1) = -\frac{1}{3} \), \( g(1) = -4 \), and \( g'(1) = -\frac{8}{3} \). The derivative of \( \sqrt{[f(x)]^2 + [g(x)]^2} \) w.r.t. \( x \) at \( x = 1 \) is ........

Divide the number 20 into two parts such that the sum of their squares is minimum.

Find x, y, z if
\[ \begin{bmatrix} 5 & 1 & 0 \\ 1 & 1 & 1 \end{bmatrix} \begin{bmatrix} 0 & 1 & -2 \\ 1 & -2 & 3 \\ -1 & 1 & 1 \end{bmatrix} \begin{bmatrix} x-1 \\ y+1 \\ 2z \end{bmatrix} = \begin{bmatrix} 2 \\ 1 \end{bmatrix}. \]

If a body cools from 80°C to 50°C at room temperature of 25°C in 30 minutes, find the temperature of the body after 1 hour.

Prove that: \( 2 \tan^{-1} \left( \frac{1}{3} \right) + \cos^{-1} \left( \frac{3}{5} \right) = \frac{\pi}{2} \).

Express the following switching circuit in the symbolic form of logic. Construct the switching table and interpret it. \includegraphics[width=0.5\linewidth]{01.jpeg}

Find the coordinates of the points of intersection of the lines represented by \( x^2 - y^2 - 2x + 1 = 0 \).

The displacement of a particle at time \( t \) is given by \( s = 2t^3 - 5t^2 + 4t - 3 \). Find the velocity and displacement at the time when the acceleration is \( 14 \, \text{ft/sec}^2 \).