JEE Main 2024 8 April Shift 1 Question Paper (Available)- Download Solution PDF with Answer Key

Let \(A = \begin{bmatrix} 2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b \end{bmatrix}\). If \(A^3 = 4A^2 - A - 21I\), where I is the identity matrix of order \(3 \times 3\), then \(2a + 3b\) is equal to:

• (1) -10
• (2) -13
• (3) -9
• (4) -12

The value of \(k \in \mathbb{N}\) for which the integral \[ I_n = \int_0^1 (1 - x^k)^n \, dx, \, n \in \mathbb{N}, \] satisfies \(147 \, I_{20} = 148 \, I_{21}\) is:

• (1) 10
• (2) 8
• (3) 14
• (4) 7

Let \(f(x) = 4\cos^3 x + 3\sqrt{3} \cos^2 x - 10\). The number of points of local maxima of \(f\) in interval \((0, 2\pi)\) is:

• (1) 1
• (2) 2
• (3) 3
• (4) 4

The set of all α, for which the vectors \(\vec{a} = \alpha \hat{ti} + 6\hat{j} - 3\hat{k}\) and \(\vec{b} = \hat{ti} - 2\hat{j} - 2\alpha t\hat{k}\) are inclined at an obtuse angle for all \(t \in \mathbb{R}\) is:

• (1) [0,1)
• (2) (-2,0]
• (3) \(\left[-\frac{4}{3}, 0\right]\)
• (4) \(\left[-\frac{4}{3}, 1\right]\)

Let the circles \(C_1 : (x - \alpha)^2 + (y - \beta)^2 = r_1^2\) and \(C_2 : (x - 8)^2 + \left( y - \frac{15}{2} \right)^2 = r_2^2\) touch each other externally at the point \((6, 6)\). If the point \((6, 6)\) divides the line segment joining the centres of the circles \(C_1\) and \(C_2\) internally in the ratio \(2 : 1\), then \((\alpha + \beta) + 4\left(r_1^2 + r_2^2\right)\) equals _____.

• (1) 110
• (2) 130
• (3) 125
• (4) 145

Let \( f(x) \) be a positive function such that the area bounded by \( y = f(x) \), \( y = 0 \), from \( x = 0 \) to \( x = a>0 \) is \[ \int_0^a f(x) \, dx = e^{-a} + 4a^2 + a - 1. \] Then the differential equation, whose general solution is \[ y = c_1 f(x) + c_2, \] where \( c_1 \) and \( c_2 \) are arbitrary constants, is:

• (1) \((8e^x - 1)\frac{d^2y}{dx^2} + \frac{dy}{dx} = 0\)
• (2) \((8e^x + 1)\frac{d^2y}{dx^2} - \frac{dy}{dx} = 0\)
• (3) \((8e^x + 1)\frac{d^2y}{dx^2} + \frac{dy}{dx} = 0\)
• (4) \((8e^x - 1)\frac{d^2y}{dx^2} - \frac{dy}{dx} = 0\)

Let H: \(\frac{-x^2}{a^2} + \frac{y^2}{b^2} = 1\) be the hyperbola, whose eccentricity is \(\sqrt{3}\) and the length of the latus rectum is \(4\sqrt{3}\). Suppose the point \((\alpha, 6)\), \(\alpha>0\) lies on H. If \(\beta\) is the product of the focal distances of the point \((\alpha, 6)\), then \(\alpha^2 + \beta\) is equal to:

• (1) 170
• (2) 171
• (3) 169
• (4) 172

The number of critical points of the function \(f(x) = (x - 2)^{2/3}(2x + 1)\) is:

• (1) 2
• (2) 0
• (3) 1
• (4) 3

Let \(y = y(x)\) be the solution of the differential equation \((1 + y^2)e^{\tan x} dx + \cos^2 x (1 + e^{2\tan x}) dy = 0\), \(y(0) = 1\). Then \(y\left(\frac{\pi}{4}\right)\) is equal to:

• (1) \(\frac{2}{e}\)
• (2) \(\frac{1}{e^2}\)
• (3) \(\frac{1}{e}\)
• (4) \(\frac{2}{e^2}\)

Let the sum of two positive integers be 24. If the probability, that their product is not less than \(\frac{3}{4}\) times their greatest positive product, is \(\frac{m}{n}\), where \(\gcd(m, n) = 1\), then \(n - m\) equals :

• (1) 9
• (2) 11
• (3) 8
• (4) 10

For the function \(f(x) = (\cos x) - x + 1, x \in \mathbb{R}\), find the correct relationship between the following two statements
(S1) \(f(x) = 0\) for only one value of x is \([0, \pi]\).
(S2) \(f(x)\) is decreasing in \(\left[0, \frac{\pi}{2}\right]\) and increasing in \(\left[\frac{\pi}{2}, \pi\right]\).

• (1) Both (S1) and (S2) are correct
• (2) Only (S1) is correct
• (3) Both (S1) and (S2) are incorrect
• (4) Only (S2) is correct

Let \(\text{P}(x, y, z)\) be a point in the first octant, whose projection in the xy-plane is the point \(\text{Q}\). Let \(\text{OP} = \gamma\); the angle between \(\text{OQ}\) and the positive x-axis be \(\theta\); and the angle between \(\text{OP}\) and the positive z-axis be \(\phi\), where \(\text{O}\) is the origin. Then the distance of \(\text{P}\) from the x-axis is:

• (1) \(\gamma \sqrt{1 - \sin^2 \phi \cos^2 \theta}\)
• (2) \(\gamma \sqrt{1 + \cos^2 \theta \sin^2 \phi}\)
• (3) \(\gamma \sqrt{1 - \sin^2 \theta \cos^2 \phi}\)
• (4) \(\gamma \sqrt{1 + \cos^2 \phi \sin^2 \theta}\)

Let \(z\) be a complex number such that \(|z + 2| = 1\) and \(\text{Im}\left(\frac{z+1}{z+2}\right) = \frac{1}{5}\). Then the value of \(|\text{Re}(z+2)|\) is:

• (1) \(\frac{\sqrt{6}}{5}\)
• (2) \(\frac{1+\sqrt{6}}{5}\)
• (3) \(\frac{24}{5}\)
• (4) \(\frac{2\sqrt{6}}{5}\)

The sum of all the solutions of the equation \[(8)^{2x} - 16 \cdot (8)^x + 48 = 0\]is:

• (1) \(1 + \log_6(8)\)
• (2) \(\log_6(6)\)
• (3) \(1 + \log_6(6)\)
• (4) \(\log_6(4)\)

The equations of two sides AB and AC of a triangle ABC are \(4x + y = 14\) and \(3x - 2y = 5\), respectively. The point \(\left(2, -\frac{4}{3}\right)\) divides the third side BC internally in the ratio 2 : 1. The equation of the side BC is:

• (1) x – 6y – 10 = 0
• (2) x – 3y – 6 = 0
• (3) x + 3y + 2 = 0
• (4) x + 6y + 6 = 0

If \(\sin x = -\frac{3}{5}\), where \(\pi<x<\frac{3\pi}{2}\), then \(80(\tan^2 x - \cos x)\) is equal to:

• (1) 109
• (2) 108
• (3) 18
• (4) 19

Let \([t]\) be the greatest integer less than or equal to t. Let A be the set of all prime factors of 2310 and \(f: A \to \mathbb{Z}\) be the function \(f(x) = \left[ \log_2 \left( x^2 + \left[ \frac{x^3}{5} \right] \right) \right]\). The number of one-to-one functions from A to the range of f is:

• (1) 20
• (2) 120
• (3) 25
• (4) 24

If the set \(R = {(a, b) : a + 5b = 42, a, b \in \mathbb{N}}\) has \(m\) elements and \(\sum_{n=1}^m (1 + i^n) = x + iy\), where \(i = \sqrt{-1}\), then the value of \(m + x + y\) is:

• (1) 8
• (2) 12
• (3) 4
• (4) 5

If the shortest distance between the lines.
L1: \(\vec{r} = (2 + \lambda)\hat{i} + (1 - 3\lambda)\hat{j} + (3 + 4\lambda)\hat{k}\), \(\lambda \in \mathbb{R}\).
L2: \(\vec{r} = 2(1 + \mu)\hat{i} + 3(1 + \mu)\hat{j} + (5 + \mu)\hat{k}\), \(\mu \in \mathbb{R}\) is \(\frac{m}{\sqrt{n}}\), where gcd(m, n) = 1, then the value of m + n equals.

• (1) 384
• (2) 387
• (3) 377
• (4) 390

Let \(I(x) = \int \frac{6}{\sin^2 x (1 - \cot x)^2} dx\). If \(I(0) = 3\), then \(I\left(\frac{\pi}{12}\right)\) is equal to:

• (1) \(\sqrt{3}\)
• (2) \(3\sqrt{3}\)
• (3) \(6\sqrt{3}\)
• (4) \(2\sqrt{3}\)

If the orthocentre of the triangle formed by the lines 2x + 3y – 1 = 0, x + 2y – 1 = 0 and ax + by – 1 = 0, is the centroid of another triangle, whose circumecentre and orthocentre respectively are (3, 4) and (–6, –8), then the value of |a– b| is_____.

If the range of \(f(\theta) = \frac{\sin^4\theta + 3\cos^2\theta}{\sin^4\theta + \cos^2\theta}, \, \theta \in \mathbb{R}\) is \([\alpha, \beta]\), then the sum of the infinite G.P., whose first term is \(64\) and the common ratio is \(\frac{\alpha}{\beta}\), is equal to ____.

Let \(\vec{a} = 9\hat{i} - 13\hat{j} + 25\hat{k}\), \(\vec{b} = 3\hat{i} + 7\hat{j} - 13\hat{k}\), and \(\vec{c} = 17\hat{i} - 2\hat{j} + \hat{k}\) be three given vectors. If \(\vec{r}\) is a vector such that \(\vec{r} \times \vec{a} = (\vec{b} + \vec{c}) \times \vec{a}\) and \(\vec{r} \cdot (\vec{b} - \vec{c}) = 0\), then \(\frac{|593\vec{r} + 67\vec{a}|^2}{(593)^2}\) is equal to _______.

Let the positive integers be written in the form :

If the \(k^\text{th}\) row contains exactly \(k\) numbers for every natural number \(k\), then the row in which the number \(5310\) will be, is _____

The number of 3-digit numbers, formed using the digits 2, 3, 4, 5 and 7, when the repetition of digits is not allowed, and which are not divisible by 3, is equal to ______.

Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables \(X\) and \(Y\) respectively denote the number of blue and yellow balls. If \(\bar{X}\) and \(\bar{Y}\) are the means of \(X\) and \(Y\) respectively, then \(7\bar{X} + 4\bar{Y}\) is equal to _____.

Let \(\alpha = \sum_{r=0}^n (4r^2 + 2r + 1) \binom{n}{r}\) and \(\beta = \left( \sum_{r=0}^n \frac{\binom{n}{r}}{r+1} \right) + \frac{1}{n+1}\). If \(140 < \frac{2\alpha}{\beta} < 281\), then the value of \(n\) is _____.

Let \(A = \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix}\). If the sum of the diagonal elements of \(A^{13}\) is \(3^n\), then \(n\) is equal to _____.

Let the area of the region enclosed by the curve \(y = \min\{\sin x, \cos x\}\) and the x-axis between \(x = -\pi\) to \(x = \pi\) be \(A\). Then \(A^2\) is equal to _____.

The value of \[ \lim_{x \to 0} 2 \left( \frac{1 - \cos x \sqrt{\cos 2x} \, \sqrt[3]{\cos 3x} \ldots \sqrt[10]{\cos 10x}}{x^2} \right) \] is _____.

A stationary particle breaks into two parts of masses \(m_A\) and \(m_B\), which move with velocities \(v_A\) and \(v_B\), respectively. The ratio of their kinetic energies (\(K_B : K_A\)) is:

• (1) \(v_B : v_A\)
• (2) \(m_B : m_A\)
• (3) \(m_B v_B : m_A v_A\)
• (4) \(1 : 1\)

Two charged conducting spheres of radii a and b are connected to each other by a conducting wire. The ratio of charges of the two spheres respectively is:

• (1) \(\sqrt{ab}\)
• (2) ab
• (3) \(\frac{a}{b}\)
• (4) \(\frac{b}{a}\)

Two planets A and B having masses \(m_1\) and \(m_2\) move around the sun in circular orbits of \(r_1\) and \(r_2\) radii, respectively. If the angular momentum of A is \(L\) and that of B is \(3L\), the ratio of time periods \(\frac{T_A}{T_B}\) is:

• (1) \(\left(\frac{r_2}{r_1}\right)^{\frac{3}{2}}\)
• (2) \(\left(\frac{r_1}{r_2}\right)^3\)
• (3) \(\frac{1}{27} \left(\frac{m_2}{m_1}\right)^3\)
• (4) \(27 \left(\frac{m_1}{m_2}\right)^3\)

Average force exerted on a non-reflecting surface at normal incidence is \(2.4\times10^{-4}\ N\). If \(360\ W/cm^2\) is the light energy flux during span of 1 hour 30 minutes, Then the area of the surface is:

• (1) \(20\ m^2\)
• (2) \(0.2\ m^2\)
• (3) \(0.1\ m^2\)
• (4) \(0.02\ m^2\)

In an expression \(a \times 10^b\):

• (1) \(a\) is the order of magnitude for \(b \leq 5\)
• (2) \(b\) is the order of magnitude for \(a \leq 5\)
• (3) \(b\) is the order of magnitude for \(5<a \leq 10\)
• (4) \(b\) is the order of magnitude for \(a \geq 5\)

A clock has a 75 cm long second hand and a 60 cm long minute hand, respectively. In 30 minutes duration, the tip of the second hand will travel \(x\) distance more than the tip of the minute hand. The value of \(x\) in meters is nearly (Take \(\pi = 3.14\)):

• (1) 139.4
• (2) 140.5
• (3) 220.0
• (4) 118.9

The output Y of following circuit for given inputs is :

• (1) \(A \cdot B (A + B)\)
• (2) \(A \cdot B\)
• (3) 0
• (4) \(\overline{A} \cdot B\)

A player caught a cricket ball of mass 150 g moving at a speed of 20 m/s. If the catching process is completed in 0.1 s, the magnitude of force exerted by the ball on the hand of the player is:

• (1) 150 N
• (2) 3 N
• (3) 30 N
• (4) 300 N

Three bodies A, B and C have equal kinetic energies and their masses are 400 g, 1.2 kg and 1.6 kg respectively. The ratio of their linear momenta is :

• (1) \(1 : \sqrt{3} : 2\)
• (2) \(1 : \sqrt{3} : \sqrt{2}\)
• (3) \(\sqrt{2} : \sqrt{3} : 1\)
• (4) \(\sqrt{3} : \sqrt{2} : 1\)

Young's modulus is determined by the equation given by:\[ Y = \frac{49000 \, M}{\ell} \, \text{dyne/cm}^2 \] where \(M\) is the mass and \(\ell\) is the extension of the wire used in the experiment. Now, the error in Young's modulus (\(Y\)) is estimated by taking data from the \(M-\ell\) plot on graph paper. The smallest scale divisions are \(5 \, \text{g}\) and \(0.02 \, \text{cm}\) along the load axis and extension axis, respectively. If the value of \(M\) and \(\ell\) are \(500 \, \text{g}\) and \(2 \, \text{cm}\), respectively, then the percentage error of \(Y\) is:

• (1) 0.2 %
• (2) 0.02 %
• (3) 2 %
• (4) 0.5 %

Paramagnetic substances:
A. align themselves along the directions of external magnetic field.
B. attract strongly towards external magnetic field.
C. has susceptibility little more than zero.
D. move from a region of strong magnetic field to weak magnetic field.
Choose the most appropriate answer from the options given below:

• (1) A, B, C, D
• (2) B, D Only
• (3) A, B, C Only
• (4) A, C Only

A mixture of one mole of a monoatomic gas and one mole of a diatomic gas (rigid) are kept at room temperature (\(27^\circ \text{C}\)). The ratio of specific heat of gases at constant volume respectively is:

• (1) \(\frac{7}{5}\)
• (2) \(\frac{3}{2}\)
• (3) \(\frac{3}{5}\)
• (4) \(\frac{5}{3}\)

A LCR circuit is at resonance for a capacitor C, inductance L and resistance R. Now the value of resistance is halved keeping all other parameters same. The current amplitude at resonance will be now:

• (1) Zero
• (2) double
• (3) same
• (4) halved

The diameter of a sphere is measured using a vernier caliper whose 9 divisions of main scale are equal to 10 divisions of vernier scale. The shortest division on the main scale is equal to l mm. The main scale reading is 2 cm and second division of vernier scale coincides with a division on main scale. If mass of the sphere is 8.635 g, the density of the sphere is:

• (1) \(2.5 \, \text{g/cm}^3\)
• (2) \(1.7 \, \text{g/cm}^3\)
• (3) \(2.2 \, \text{g/cm}^3\)
• (4) \(2.0 \, \text{g/cm}^3\)

Critical angle of incidence for a pair of optical media is \(45^\circ\).
The refractive indices of the first and second media are in the ratio:

• (1) \(\sqrt{2} : 1\)
• (2) \(1 : 2\)
• (3) \(1 : \sqrt{2}\)
• (4) \(2 : 1\)

A proton and an electron are associated with the same de-Broglie wavelength. The ratio of their kinetic energies is:
Given: \(h = 6.63 \times 10^{-34} \, \text{Js}\), \(m_e = 9.0 \times 10^{-31} \, \text{kg}\), and \(m_p = 1836 \times m_e\)

• (1) \(1 : 1836\)
• (2) \(1 : \frac{1}{1836}\)
• (3) \(1 : \frac{1}{\sqrt{1836}}\)
• (4) \(1 : \sqrt{1836}\)

Correct Bernoulli's equation is (symbols have their usual meaning):

• (1) \(P + mgh + \frac{1}{2} mv^2 = \text{constant}\)
• (2) \(P + \rho gh + \frac{1}{2} \rho v^2 = \text{constant}\)
• (3) \(P + \rho gh + \rho v^2 = \text{constant}\)
• (4) \(P + \frac{1}{2} \rho gh + \frac{1}{2} \rho v^2 = \text{constant}\)

The binding energy of a certain nucleus is \(18 \times 10^8 \, \text{J}\). How much is the difference between the total mass of all the nucleons and the nuclear mass of the given nucleus?

• (1) \(0.2 \, \mu \text{g}\)
• (2) \(20 \, \mu \text{g}\)
• (3) \(2 \, \mu \text{g}\)
• (4) \(10 \, \mu \text{g}\)

In the given circuit, the terminal potential difference of the cell is :

• (1) 2 V
• (2) 4 V
• (3) 1.5 V
• (4) 3 V

Two different adiabatic paths for the same gas intersect two isothermal curves as shown in the \(P-V\) diagram. The relation between the ratio \(\frac{V_a}{V_d}\) and the ratio \(\frac{V_b}{V_c}\) is:

• (1) \(\frac{V_a}{V_d} = \left(\frac{V_b}{V_c}\right)^{-1}\)
• (2) \(\frac{V_a}{V_d} \neq \frac{V_b}{V_c}\)
• (3) \(\frac{V_a}{V_d} = \frac{V_b}{V_c}\)
• (4) \(\frac{V_a}{V_d} = \left(\frac{V_b}{V_c}\right)^2\)

A closed and an open organ pipe have same lengths. If the ratio of frequencies of their seventh overtones is \(\left(\frac{a-1}{a}\right)\), then the value of \(a\) is ______.

An electric field, \(\vec{E} = \frac{2\hat{i} + 6\hat{j} + 8\hat{k}}{\sqrt{6}}\) passes through the surface of \(4 \, \text{m}^2\) area having unit vector \(\hat{n} = \frac{2\hat{i} + \hat{j} + \hat{k}}{\sqrt{6}}\). The electric flux for that surface is _________ \( \text{V m} \).

Three vectors \(\overrightarrow{OP}\), \(\overrightarrow{OQ}\), and \(\overrightarrow{OR}\) each of magnitude \(A\) are acting as shown in figure. The resultant of the three vectors is \(A \sqrt{x}\). The value of \(x\) is ______.

A square loop PQRS having 10 turns, area \(3.6 \times 10^{-3} \, \text{m}^2\), and resistance \(100 \, \Omega\) is slowly and uniformly being pulled out of a uniform magnetic field of magnitude \(B = 0.5 \, \text{T}\) as shown. Work done in pulling the loop out of the field in \(1.0 \, \text{s}\) is ______ \( \times 10^{-6} \, \text{J} \).

A liquid column of height \(0.04 \, \text{cm}\) balances excess pressure of a soap bubble of certain radius. If density of liquid is \(8 \times 10^3 \, \text{kg m}^{-3}\) and surface tension of soap solution is \(0.28 \, \text{N m}^{-1}\), then diameter of the soap bubble is _________ \(\text{cm}\).
\((\text{if } g = 10 \, \text{m s}^{-2})\)

A uniform thin metal plate of mass \(10 \, \text{kg}\) with dimensions is shown. The ratio of \(x\) and \(y\) coordinates of the center of mass of the plate is \(\frac{n}{9}\). The value of \(n\) is ______.

Resistance of a wire at \(0^\circ C\), \(100^\circ C\) and \(t^\circ C\) is found to be \(10 \, \Omega\), \(10.2 \, \Omega\) and \(10.95 \, \Omega\) respectively. The temperature \(t\) in Kelvin scale is ______.

A parallel beam of monochromatic light of wavelength \(600 \, \text{nm}\) passes through single slit of \(0.4 \, \text{mm}\) width. Angular divergence corresponding to second order minima would be ______ \( \times 10^{-3} \, \text{rad}\).

An electron with kinetic energy \(5 \, \text{eV}\) enters a region of uniform magnetic field of \(3 \, \mu\text{T}\) perpendicular to its direction. An electric field \(E\) is applied perpendicular to the direction of velocity and magnetic field. The value of \(E\), so that the electron moves along the same path, is ______ \( \text{NC}^{-1} \).
Given: mass of electron = \(9 \times 10^{-31} \, \text{kg}\), electric charge = \(1.6 \times 10^{-19} \, \text{C}\)

In an alpha particle scattering experiment, distance of closest approach for the \(\alpha\) particle is \(4.5 \times 10^{-14} \, \text{m}\). If the target nucleus has atomic number 80, then maximum velocity of \(\alpha\)-particle is ______ \( \times 10^5 \, \text{m/s} \) approximately.
\[ \left( \frac{1}{4\pi \epsilon_0} = 9 \times 10^9 \, \text{SI unit}, \, \text{mass of } \alpha \, \text{particle} = 6.72 \times 10^{-27} \, \text{kg} \right) \]

Combustion of glucose (\(C_6H_{12}O_6\)) produces \(CO_2\) and water. The amount of oxygen (in g) required for the complete combustion of \(900 \, \text{g}\) of glucose is:
\([ \text{Molar mass of glucose in g mol}^{-1} = 180 ]\)

• (1) 480
• (2) 960
• (3) 800
• (4) 32

Thiosulphate reacts differently with iodine and bromine in the reaction given below:  
\[ 2S_2O_3^{2-} + I_2 \rightarrow S_4O_6^{2-} + 2I^- \] \[ S_2O_3^{2-} + 5Br_2 + 5H_2O \rightarrow 2SO_4^{2-} + 4Br^- + 10H^+ \] 
Which of the following statements justifies the above dual behaviour of thiosulphate?

• (1) Bromine undergoes oxidation and iodine undergoes reduction by iodine in these reactions
• (2) Thiosulphate undergoes oxidation by bromine and reduction by iodine in these reactions
• (3) Bromine is a stronger oxidant than iodine
• (4) Bromine is a weaker oxidant than iodine

Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R:
Assertion A: The stability order of \(+1\) oxidation state of Ga, In, and Tl is \( \text{Ga}<\text{In}<\text{Tl} \).
Reason R: The inert pair effect stabilizes the lower oxidation state down the group.
In the light of the above statements, choose the correct answer from the options given below:

• (1) Both A and R are true and R is the correct explanation of A.
• (2) A is true but R is false.
• (3) Both A and R are true but R is NOT the correct explanation of A.
• (4) A is false but R is true.

Iron (III) catalyses the reaction between iodide and persulphate ions, in which
A. \(\text{Fe}^{3+}\) oxidises the iodide ion
B. \(\text{Fe}^{3+}\) oxidises the persulphate ion
C. \(\text{Fe}^{2+}\) reduces the iodide ion
D. \(\text{Fe}^{2+}\) reduces the persulphate ion
Choose the most appropriate answer from the options given below:

• (1) B and C only
• (2) B only
• (3) A only
• (4) A and D only

Which of the following are aromatic?

• (1) B and D only
• (2) A and C only
• (3) A and B only
• (4) C and D only

Among the following halogens \(F_2, \, Cl_2, \, Br_2, \, \text{and} \, I_2\)
Which can undergo disproportionation reaction?

• (1) Only \(I_2\)
• (2) \(Cl_2, \, Br_2, \, \text{and} \, I_2\)
• (3) \(F_2, \, Cl_2, \, \text{and} \, Br_2\)
• (4) \(F_2 \, \text{and} \, Cl_2\)

Which among the following compounds will undergo fastest \(S_N2\) reaction?

• (1)

• (2)

• (3)

• (4)

• (1) A-IV, B-III, C-I, D-II
• (2) A-II, B-IV, C-I, D-III
• (3) A-III, B-I, C-IV, D-II
• (4) A-III, B-IV, C-I, D-II

• (1) A-II, B-III, C-IV, D-I
• (2) A-III, B-II, C-I, D-IV
• (3) A-IV, B-III, C-II, D-I
• (4) A-II, B-I, C-IV, D-III

List-I (Name of the test)	List-II (Reaction sequence involved) [M is metal]
(A) Borax bead test	I
(B) Charcoal cavity test	II
(C) Cobalt nitrate test	III
(D) Flame test	IV

• (1) A-III, B-I, C-IV, D-II
• (2) A-III, B-II, C-IV, D-I
• (3) A-III, B-I, C-II, D-IV
• (4) A-III, B-IV, C-I, D-II

Number of complexes with even number of electrons in \(t_{2g}\) orbitals is: \[[\text{Fe(H}_2\text{O)}_6]^{2+}, \, [\text{Co(H}_2\text{O)}_6]^{2+}, \, [\text{Co(H}_2\text{O)}_6]^{3+}, \, [\text{Cu(H}_2\text{O)}_6]^{2+}, \, [\text{Cr(H}_2\text{O)}_6]^{2+}\]

• (1) 1
• (2) 3
• (3) 2
• (4) 5

For the given hypothetical reactions, the equilibrium constants are as follows:
\[ X \rightleftharpoons Y; \, K_1 = 1.0 \]
\[ Y \rightleftharpoons Z; \, K_2 = 2.0 \]
\[ Z \rightleftharpoons W; \, K_3 = 4.0 \]
The equilibrium constant for the reaction
\[ X \rightleftharpoons W\]
is:

• (1) 6.0
• (2) 12.0
• (3) 8.0
• (4) 7.0

An octahedral complex with the formula \(\text{CoCl}_3n\text{NH}_3\) upon reaction with excess of \(\text{AgNO}_3\) solution gives 2 moles of \(\text{AgCl}\). Consider the oxidation state of Co in the complex is \(x\). The value of "x + n" is ______.

• (1) 3
• (2) 6
• (3) 8
• (4) 5

In the given compound, the number of 2° carbon atom/s is _____.

• (1) Three
• (2) One
• (3) Two
• (4) Four

Identify the major products A and B respectively in the following set of reactions.

• (1)

• (2)

• (3)

• (4)

Identify the product (P) in the following reaction:

• (1)

• (2)

• (3)

• (4)

Given below are two statements:
Statement I:

IUPAC name of Compound A is 4-chloro-1, 3-dinitrobenzene.
Statement II:

IUPAC name of Compound B is 4-ethyl-2-methylaniline.
In the light of the above statements, choose the most appropriate answer from the options given below:

• (1) Both Statement I and Statement II are correct
• (2) Statement I is incorrect but Statement II is correct
• (3) Statement I is correct but Statement II is incorrect
• (4) Both Statement I and Statement II are incorrect

Given below are two statements:
Statement I: \(N(CH_3)_3\) and \(P(CH_3)_3\) can act as ligands to form transition metal complexes.
Statement II: As \(N\) and \(P\) are from the same group, the nature of bonding of \(N(CH_3)_3\) and \(P(CH_3)_3\) is always the same with transition metals.
In the light of the above statements, choose the most appropriate answer from the options given below:

• (1) Statement I is incorrect but Statement II is correct
• (2) Both Statement I and Statement II are correct
• (3) Statement I is correct but Statement II is incorrect
• (4) Both Statement I and Statement II are incorrect

• (1) Can be oxidized to a dicarboxylic acid with Br2 water
• (2) despite the presence of – CHO does not give Schiff's test
• (3) has 4-asymmetric carbon atom
• (4) will coexist in equilibrium with 2 other cyclic structure

• (1) A-III, B-I, C-II, D-IV
• (2) A-IV, B-I, C-II, D-III
• (3) A-II, B-III, C-IV, D-I
• (4) A-I, B-II, C-III, D-IV

If 279 g of aniline is reacted with one equivalent of benzenediazonium chloride, the mximum amount of aniline yellow formed will be _____ g. (nearest integer)
(consider complete conversion)

Consider the following reaction
\[A + B \rightarrow C\]
The time taken for A to become 1/4th of its initial concentration is twice the time taken to become 1/2 of the same. Also, when the change of concentration of B is plotted against time, the resulting graph gives a straight line with a negative slope and a positive intercept on the concentration axis.
The overall order of the reaction is ____.

Major product B of the following reaction has ____ \(\pi\)-bond.

A hypothetical electromagnetic wave is show below.

A solution containing \(10 \, \text{g}\) of an electrolyte \(AB_2\) in \(100 \, \text{g}\) of water boils at \(100.52^\circ \text{C}\). The degree of ionization of the electrolyte (\(\alpha\)) is _____ \(\times 10^{-1}\). (nearest integer)
\([ \text{Given: Molar mass of } AB_2 = 200 \, \text{g mol}^{-1}, \, K_b \, (\text{molal boiling point elevation const. of water}) = 0.52 \, \text{K kg mol}^{-1}, \text{ boiling point of water} = 100^\circ \text{C}; \, AB_2 \text{ ionises as } AB_2 \rightarrow A^{2+} + 2B^- ]\)

Number of molecules from the following which are exceptions to octet rule is ____.
\(\text{CO}_2, \, \text{NO}_2, \, \text{H}_2\text{SO}_4, \, \text{BF}_3, \, \text{CH}_4, \, \text{SiF}_4, \, \text{ClO}_2, \, \text{PCl}_5, \, \text{BeF}_2, \, \text{C}_2\text{H}_6, \, \text{CHCl}_3, \, \text{CBr}_4\)

The number of optical isomers in following compound is : _____.

The 'spin only' magnetic moment value of \(MO_4^{2-}\) is ____ \(\text{BM}\). (Where \(M\) is a metal having least metallic radii among \(\text{Sc, Ti, V, Cr, Mn, Zn}\)). \([ \text{Given atomic number: Sc = 21, Ti = 22, V = 23, Cr = 24, Mn = 25, Zn = 30} ]\)

Number of amine compounds from the following giving solids which are soluble in NaOH upon reaction with Hinsberg's reagent is _____.