List of practice Questions

If for some $ \alpha, \beta $; $ \alpha \leq \beta $, $ \alpha + \beta = 8 $ and \[ \sec^2(\tan^{-1} \alpha) + \csc^2(\cot^{-1} \beta) = 36, \] then $ \alpha^2 + \beta $ is:

If the system of equations \[ 2x - y + z = 4, \] \[ 5x + \lambda y + 3z = 12, \] \[ 100x - 47y + \mu z = 212, \] has infinitely many solutions, then $ \mu - 2\lambda $ is equal to:

The product of all the rational roots of the equation \[ (x^2 - 9x + 11)^2 - (x - 4)(x - 5) = 3, \] is equal to:

If $ \lim_{x \to 0} \left( \frac{\tan x}{x} \right)^{\frac{1}{x^2}} = p $, then $ 96 \log_e p $ is equal to _______

The area of the region \[ \{ (x, y) : x^2 + 4x + 2 \leq y \leq |x| + 2 \} \] is equal to:

A 3 m long wire of radius 3 mm shows an extension of 0.1 mm when loaded vertically by a mass of 50 kg in an experiment to determine Young's modulus. The value of Young's modulus of the wire as per this experiment is $ P \times 10^{11} \, \text{N/m}^2 $, where the value of $ P $ is: (Take $ g = 3\pi \, \text{m/s}^2 $)

Consider the region \[ R = \left\{ (x, y): x \leq y \leq 9 - \frac{11}{3} x^2, x \geq 0 \right\}. \] The area of the largest rectangle of sides parallel to the coordinate axes and inscribed in $ R $ is:

Let the product of the focal distances of the point $$ \left( \sqrt{3}, \frac{1}{2} \right) $$ on the ellipse $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad (a > b), $$ be $ \frac{7}{4} $. Then the absolute difference of the eccentricities of two such ellipses is:

Let I be the identity matrix of order 3 × 3 and for the matrix $ A = \begin{pmatrix} \lambda & 2 & 3 \\ 4 & 5 & 6 \\ 7 & -1 & 2 \end{pmatrix} $, $ |A| = -1 $. Let B be the inverse of the matrix $ \text{adj}(A \cdot \text{adj}(A^2)) $. Then $ |(\lambda B + I)| $ is equal to _______

Let $ y = y(x) $ be the solution of the differential equation \[ \left( xy - 5x^2 \sqrt{1 + x^2} \right) dx + (1 + x^2) dy = 0, \quad y(0) = 0. \] Then $ y(\sqrt{3}) $ is equal to:

Let in a $ \triangle ABC $, the length of the side AC is 6, the vertex B is $ (1, 2, 3) $ and the vertices A, C lie on the line \[ \frac{x - 6}{3} = \frac{y - 7}{2} = \frac{z - 7}{-2}. \] Then the area (in sq. units) of $ \triangle ABC $ is:

Evaluate the limit: \[ \lim_{x \to 0} \csc{x} \left( \sqrt{2 \cos^2{x} + 3 \cos{x}} - \sqrt{\cos^2{x} + \sin{x} + 4} \right) \] is equal to:

The sum $ 1 + \frac{1 + 3}{2!} + \frac{1 + 3 + 5}{3!} + \frac{1 + 3 + 5 + 7}{4!} + ... $ upto $ \infty $ terms, is equal to

The distance of the point (7, 10, 11) from the line $ \frac{x - 4}{1} = \frac{y - 4}{0} = \frac{z - 2}{3} $ along the line $ \frac{x - 7}{2} = \frac{y - 10}{3} = \frac{z - 11}{6} $ is

Let $ f(x) = \frac{2^{x+2} + 16}{2^{2x+1} + 2^{x+4} + 32} $. Then the value of \[ 8 \left( f\left( \frac{1}{15} \right) + f\left( \frac{2}{15} \right) + \dots + f\left( \frac{59}{15} \right) \right) \] is equal to:

Let C be the circle of minimum area enclosing the ellipse E: $ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $ with eccentricity $ \frac{1}{2} $ and foci $ (\pm 2, 0) $. Let PQR be a variable triangle, whose vertex P is on the circle C and the side QR of length 29 is parallel to the major axis and contains the point of intersection of E with the negative y-axis. Then the maximum area of the triangle PQR is:

If the probability that the random variable X takes the value x is given by $ P(X = x) = k(x + 1)3^{-x} $, $ x = 0, 1, 2, 3, ... $, where k is a constant, then $ P(X \ge 3) $ is equal to

For a nucleus of mass number $ A $ and radius $ R $, the mass density of the nucleus can be represented as:

The amplitude and phase of a wave that is formed by the superposition of two harmonic travelling waves, $ y_1 (x, t) = 4 \sin(kx - \omega t) $ and $ y_2 (x, t) = 2 \sin(kx - \omega t + \frac{2\pi}{3}) $, are: (Take the angular frequency of initial waves same as $ \omega $)

Let $ S_n = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \dots $ up to $ n $ terms. If the sum of the first six terms of an A.P. with first term $ -p $ and common difference $ p $ is $ \sqrt{2026 S_{2025}} $, then the absolute difference between the 20th and 15th terms of the A.P. is:

Let $ f : \mathbb{R} - \{0\} \to \mathbb{R} $ be a function such that \[ f(x) - 6f\left(\frac{1}{x}\right) = \frac{35}{3x} - \frac{5}{2}. \] If $ \lim_{x \to 0} \left( \frac{1}{\alpha x} + f(x) \right) = \beta $, then $ \alpha, \beta \in \mathbb{R} $, and $ \alpha + 2\beta $ is equal to: