List of practice Questions

Chandan, Deepak and Elvish were partners in a firm sharing profits and losses in the ratio of 1 : 2 : 2. Their Balance Sheet as at 31st March, 2024 stood as follows:

Balance Sheet of Chandan, Deepak and Elvish as at 31^st March, 2024

Liabilities	Amount (₹)	Assets	Amount (₹)
Capitals:		Fixed Assets	27,00,000
Chandan	7,00,000	Stock	3,00,000
Deepak	5,00,000	Debtors	2,00,000
Elvish	3,00,000	Cash	1,00,000
General Reserve	4,50,000
Creditors	13,50,000
Total	33,00,000	Total	33,00,000

Chandan retired from the firm on 1st April, 2024 on the following terms:
[(i)] Fixed assets were to be depreciated by 10%.
[(ii)] Debtors of ₹ 30,000 were to be written off as bad debts.
[(iii)] Goodwill of the firm was valued at ₹ 6,00,000 and the retiring partner’s share is adjusted through the capital accounts of the remaining partners.
[(iv)] Chandan was paid through cash brought in by Deepak and Elvish in such a way so as to make their capitals proportionate to their new profit sharing ratio. Prepare Revaluation Account and Partners’ Capital Accounts.

“Planning is the widely accepted strategy for judicious use of resources.” Explain the statement.

Give a schematic representation of oogenesis in human females.

Match List-I with List-II:

\[ \begin{array}{|l|l|} \hline \textbf{List-I (Equation)} & \textbf{List-II (Roots)} \\ \hline (A)\; (z + 1)^{2n} + (z - 1)^{2n} = 0 & (I)\; \cos\!\left(\tfrac{\pi}{5}\right) \pm i\sin\!\left(\tfrac{\pi}{5}\right), -1, \cos\!\left(\tfrac{3\pi}{5}\right) \pm i\sin\!\left(\tfrac{3\pi}{5}\right) \\ \hline (B)\; z^5 + z^4 + z^3 + z^2 + z + 1 = 0 & (II)\; \text{purely imaginary number} \\ \hline (C)\; (z-1)^5 + z^5 = 0 & (III)\; -1,\; \pm\frac{1}{2} \pm i\frac{\sqrt{3}}{2} \\ \hline (D)\; z^5 + 1 = 0 & (IV)\; \tfrac{1}{2}\left[1+i\cot\!\left(\tfrac{p\pi}{10}\right)\right], \; p = 1,3,5,7,9 \\ \hline \end{array} \]
Choose the correct answer from the options given below:

Which of the following statements are correct?
If \[ A = \begin{pmatrix} p & q \\ 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & q \\ 0 & 1 \end{pmatrix}, \] then

(A) \[ B^n = \begin{pmatrix} 1 & nq \\ 0 & 1 \end{pmatrix} \]

(B) \[ A^n = \begin{pmatrix} p^n & q\frac{p^n-1}{p-1} \\ 0 & 1 \end{pmatrix}, \; \text{if } p \neq 1 \]

(C) \[ AB = \begin{pmatrix} p & pq+q \\ 0 & 1 \end{pmatrix} \]

(D) \[ B^{n-1} = \begin{pmatrix} 1 & (n+1)q \\ 0 & 1 \end{pmatrix} \]

(E) \[ AB^n = \begin{pmatrix} p & (np+1)q \\ 0 & 1 \end{pmatrix} \]

Choose the correct answer from the options given below:

If the matrix \[ A = \begin{pmatrix} 2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{pmatrix} \] satisfies the matrix equation:

If the rank of matrix \[ \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 0 & 7 & \lambda \end{pmatrix} \] is 2, then the value of \(\lambda\) is:

Match List-I with List-II

\[ \begin{array}{|l|l|} \hline \textbf{List-I} & \textbf{List-II} \\ \hline (A) \; \lim_{x \to 0} \frac{\ln(1+x)}{\sin x} & (I) \; 1 \\ (B) \; \lim_{x \to \infty} 2x \tan\left(\tfrac{1}{x}\right) & (II) \; 0 \\ (C) \; \lim_{x \to \infty} \frac{x^2}{e^x} & (III) \; 2 \\ (D) \; \lim_{x \to 1} x^{\tfrac{1}{x-1}} & (IV) \; e \\ \hline \end{array} \]
Choose the correct answer from the options given below:

Match List-I with List-II

\[ \begin{array}{|l|l|} \hline \textbf{List-I (Functions)} & \textbf{List-II (Concavity and Convexity)} \\ \hline (A) \; f(x) = e^{-x^2} & (I) \; \text{Concave downward in } (-\infty, -1) \\ (B) \; f(x) = (1+x^2)e^{-x} & (II) \; \text{Concave upward in } (-\infty, 1) \\ (C) \; f(x) = 3x^4+4x^3-6x^2+12x+12 & (III) \; \text{Concave downward in } \left(-\tfrac{1}{\sqrt{2}}, \tfrac{1}{\sqrt{2}}\right) \\ (D) \; f(x) = (x+1)^{1/3} & (IV) \; \text{Concave upward in } (-\infty, -1) \\ \hline \end{array} \]
Choose the correct answer from the options given below:

Match List-I with List-II

\[ \begin{array}{|l|l|} \hline \textbf{List-I} & \textbf{List-II} \\ \hline (A) \; 9x^2 - 12xy + 4y^2 - 74x - 98y + 324 = 0 & (I) \; \text{Hyperbola} \\ (B) \; 12x^2 + 7xy - 12y^2 + 10x + 55y - 125 = 0 & (II) \; \text{A pair of straight lines} \\ (C) \; x^2 + 3xy + 2y^2 + x + y = 0 & (III) \; \text{Ellipse} \\ (D) \; 5x^2 + y^2 - 30x + 1 = 0 & (IV) \; \text{Parabola} \\ \hline \end{array} \]
Choose the correct answer from the options given below:

Match List-I with List-II

\[ \begin{array}{|l|l|} \hline \textbf{List-I} & \textbf{List-II} \\ \hline (A) \; \dfrac{y^2}{36} - \dfrac{x^2}{16} = 1 & (I) \; \text{Eccentricity is } 2\sqrt{2} \\ (B) \; 7x^2 + 12xy - 2y^2 - 2x + 4y - 7 = 0 & (II) \; \text{Eccentricity is } \tfrac{3}{2} \\ (C) \; 7x^2 - y^2 = 224 & (III) \; \text{Eccentricity is } \tfrac{\sqrt{13}}{3} \\ (D) \; \dfrac{x^2}{16} - \dfrac{y^2}{20} = \dfrac{1}{9} & (IV) \; \text{Asymptotes are } y = \pm \tfrac{3}{2}x \\ \hline \end{array} \]

Match List-I with List-II

\[ \begin{array}{|l|l|} \hline \textbf{List-I} & \textbf{List-II} \\ \hline (A) \; P \text{ and } Q \text{ are two perpendicular forces, acting at a point} & (I) \; R = |P - Q| \\ (B) \; P \text{ and } Q \text{ are equal, forces acting at a point at an angle } \alpha & (II) \; R = P + Q \\ (C) \; P \text{ and } Q \text{ are acting at a point in same direction} & (III) \; R = 2P \cos(\tfrac{\alpha}{2}) \\ (D) \; P \text{ and } Q \text{ are acting at a point in opposite direction} & (IV) \; R = \sqrt{P^2 + Q^2} \\ \hline \end{array} \]
Choose the correct answer from the options given below:

The integral equation corresponding to the boundary value problem \[ \frac{d^2y}{dx^2} + \lambda y(x) = 0; \quad y(0) = 0; \quad y(1) = 0 \] is

where \[ k(x,t) = \begin{cases} t(1-x), & \text{if } t < x \\ x(1-t), & \text{if } t > x \end{cases} \]

Match List-I with List-II

\[ \begin{array}{|l|l|} \hline \textbf{List-I (Curve)} & \textbf{List-II (Orthogonal trajectory)} \\ \hline (A) \; xy = c & (I) \; \tfrac{y^2}{2} + x^2 = c \\ (B) \; e^x + e^{-y} = c & (II) \; y(y^2 + 3x^2) = c \\ (C) \; y^2 = cx & (III) \; y^2 - x^2 = 2c \\ (D) \; x^2 - y^2 = cx & (IV) \; e^y - e^{-x} = c \\ \hline \end{array} \]
Choose the correct answer from the options given below:

Match List-I with List-II

\[ \begin{array}{|l|l|} \hline \textbf{List-I} & \textbf{List-II} \\ \hline (A) \; \text{Classius Clapeyron equation} & (I) \; PV^\gamma = \text{constant} \\ (B) \; \text{Gibbs Function} & (II) \; U + PV \\ (C) \; \text{Enthalpy} & (III) \; U - TS + PV \\ (D) \; \text{Adiabatic change in Perfect Gas} & (IV) \; \dfrac{dP}{dT} = \dfrac{L}{T (V_2 - V_1)} \\ \hline \end{array} \]
Choose the correct answer from the options given below:

Match List-I with List-II

\[ \begin{array}{|l|l|} \hline \textbf{List-I} & \textbf{List-II} \\ \hline (A) \; \text{Circular Fringes} & (I) \; \text{Nicol prism} \\ (B) \; \text{Straight parallel and equidistant interference pattern} & (II) \; \text{Newton's Ring experiment} \\ (C) \; \text{Polarizer} & (III) \; \text{Interference in wedge-shaped film} \\ (D) \; \text{E-ray and O-ray travel with same speed} & (IV) \; \text{Optic axis} \\ \hline \end{array} \]
Choose the correct answer from the options given below:

Match List-I with List-II

\[ \begin{array}{|l|l|} \hline \textbf{List-I} & \textbf{List-II} \\ \hline (A) \; \text{Force} & (I) \; \text{Torque} \\ (B) \; \text{Distance covered} & (II) \; \text{Angle described} \\ (C) \; \text{Mass} & (III) \; \text{Moment of inertia} \\ (D) \; \text{Linear velocity} & (IV) \; \text{Angular velocity} \\ \hline \end{array} \]
Choose the correct answer from the options given below:

Match List-I with List-II

\[ \begin{array}{|l|l|} \hline \textbf{List-I} & \textbf{List-II} \\ \hline (A) \; \text{Zener diode} & (I) \; \text{Negative resistance region} \\ (B) \; \text{Tunnel diode} & (II) \; \text{Voltage regulator} \\ (C) \; \text{Rectifier} & (III) \; \text{Pulsating d.c.} \\ (D) \; \text{Light emitting diode} & (IV) \; \text{Gallium Arsenide phosphide (GaAsP)} \\ \hline \end{array} \]
Choose the correct answer from the options given below:

Match List-I with List-II

\[ \begin{array}{|l|l|} \hline \textbf{List-I} & \textbf{List-II} \\ \hline (A) \; \text{Displacement current } (J_d) & (I) \; \frac{\epsilon_0}{2} \int E^2 d\tau \\ (B) \; \text{Poynting vector} & (II) \; \nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0} \\ (C) \; \text{Energy stored in electric field } (\vec{E}) & (III) \; \frac{1}{\mu_0}(\vec{E} \times \vec{B}) \\ (D) \; \text{Gauss's Law} & (IV) \; \epsilon_0 \frac{\partial \vec{E}}{\partial t} \\ \hline \end{array} \]
Choose the correct answer from the options given below:

The general solution of differential equation \( \frac{d^2y}{dx^2} + 9y = \cos(3x) \) is:

The particular integral of differential equation \( \frac{d^2y}{dx^2} + 2\frac{dy}{dx} + y = e^{-x}\log x \) is:

Which one of the following statement is not correct?