Dashboard Overview

Q1. If the rank of matrix $$\begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & k \\ 3 & 6 & 9 \end{pmatrix}$$ is 2, then $k$ equals:

Reason: Rank-based parameter questions appear in the first 5 questions every single year.

Q2. If $\lambda_1, \lambda_2, \lambda_3$ are eigenvalues of a $3\times3$ matrix with trace = 7 and determinant = 6, and two eigenvalues are 1 and 2, then the third eigenvalue is:

Reason: Eigenvalue + trace logic was repeated in 2022, 2023, and 2024.

Q3. If $f(x) = x^3 - 3x + a$ is continuous and differentiable everywhere, and has a stationary point at $x = 1$, then $a$ equals:

Reason: Stationary condition-based polynomials appear regularly.

Q4. Evaluate: $$\lim_{x \to 0} \frac{\sin(3x) - 3x}{x^3}$$

Reason: L’Hospital cubic expansion asked every year.

Q5. The area bounded between the curves $y = x^2$ and $x = y^2$ is:

Reason: Exact same symmetric curve logic seen in 2023.

Q6. Solve the differential equation: $$\frac{dy}{dx} = \frac{x+y}{x}$$

Reason: Homogeneous substitution is almost guaranteed.

Q7. If two numbers are chosen from 1 to 8 without replacement, the probability that the minimum is less than 4 is:

Reason: "Minimum less than k" is a repeated probability structure.

Q8. If vectors $\mathbf{a}$ and $\mathbf{b}$ are unit vectors with an angle of 60°, then $|\mathbf{a} \times \mathbf{b}|$ equals:

Reason: Direct vector magnitude formula is recurring.

Q9. If $f(x)=x^2+2bx+4$ has a minimum value of 3, then $b$ equals:

Reason: Quadratic minimum relation is repeated.

Q10. Evaluate: $$\int_0^\pi x \sin x \, dx$$

Reason: Definite integral by parts appears yearly.

Q31. Let $A$ be a symmetric matrix and $B$ a skew-symmetric matrix. Then which of the following are skew-symmetric?

Reason: Matrix symmetry properties logic appears every year in Category II.

Q32. If $f(x) = \lim_{n\to\infty} \frac{x^{2n} - 1}{x^{2n} + 1}$, then:

Reason: Limit and function value definition pattern was repeated in 2023.

Q41. The RMS speed of an ideal gas does not change if:

Reason: Gas law properties and RMS speed are recurring basics.

Q42. The magnetic field at the centre of a circular coil of radius $R$ carrying current $I$ is:

Reason: Coil centre magnetic field formulas repeated across 3 years.

Q43. In an RC discharge transient circuit, the total heat energy produced in the resistor equals:

Reason: Identical concept asked in the 2023 paper.

Q44. If the Brewster angle for a medium is 45°, the refractive index of the medium is:

Reason: Polarization and Brewster's law are frequent syllabus checks.

Q45. The initial activity of a radioactive sample of mass $m$ and molar mass $M$ with decay constant $\lambda$ is:

Reason: Radioactive rate equations are high-yield syllabus points.

Q71. The energy of the 2nd excited state of a hydrogen-like $Li^{2+}$ ion is:

Reason: Bohr hydrogenic energy transition states recur in chemistry section.

Q72. A solution's pH changes from 1 to 2. The dilution factor of the volume is:

Reason: Logarithmic concentration checks for pH appear consistently.

Q73. For the reaction $\text{CO(g)} + \frac{1}{2}\text{O}_2\text{(g)} \to \text{CO}_2\text{(g)}$, the ratio $K_p/K_c$ equals:

Reason: Gas phase equilibrium constants relationships are high-probability points.

Q91. For maximum power transfer from a linear active network, the load resistance $R_L$ equals:

Reason: Maximum power transfer theorem is a very high probability numeric concept.

Q92. If the frequency of a transformer is doubled while maintaining constant voltage, the core flux:

Reason: Faraday's voltage and frequency ratio calculations recur in transformer theory.