JEE Main 2024 4 April Shift 2 Question Paper (Available)- Download Solution PDF with Answer Key

Consider a hyperbola \( H \) having its centre at the origin and foci on the \( x \)-axis. Let \( C_1 \) be the circle touching the hyperbola \( H \) and having its centre at the origin. Let \( C_2 \) be the circle touching the hyperbola \( H \) at its vertex and having its centre at one of its foci. If the areas (in square units) of \( C_1 \) and \( C_2 \) are \( 36\pi \) and \( 4\pi \), respectively, then the length (in units) of the latus rectum of \( H \) is:

• (1) \( \frac{28}{3} \)
• (2) \( \frac{14}{3} \)
• (3) \( \frac{10}{3} \)
• (4) \( \frac{11}{3} \)

If the coefficients of \( x^4 \), \( x^5 \), and \( x^6 \) in the expansion of \( (1 + x)^n \) are in arithmetic progression, then the maximum value of \( n \) is:

• (1) 14
• (2) 21
• (3) 28
• (4) 7

The value of \(\frac{1 \times 2^2 + 2 \times 3^2 + \dots + 100 \times (101)^2}{1^2 \times 2 + 2^2 \times 3 + \dots + 100^2 \times 101}\) is:
 

• (1) \( \frac{306}{305} \)
• (2) \( \frac{305}{301} \)
• (3) \( \frac{32}{31} \)
• (4) \( \frac{31}{30} \)

If the function \( f(x) = \begin{cases} \frac{72^x - 9^x - 8^x + 1}{\sqrt{2} - \sqrt{1 + \cos x}}, & x \neq 0 \\ a \log_e 2 \log_e 3, & x = 0 \end{cases} \) is continuous at \( x = 0 \), then the value of \( a^2 \) is equal to

• (1) 968
• (2) 1152
• (3) 746
• (4) 1250

Let \( P \) be the point of intersection of the lines \[ \frac{x - 2}{1} = \frac{y - 4}{5} = \frac{z - 2}{1} \quad \text{and} \quad \frac{x - 3}{2} = \frac{y - 2}{3} = \frac{z - 3}{2}. \] Then, the shortest distance of \( P \) from the line \( 4x = 2y = z \) is:

• (1) \( \frac{5\sqrt{14}}{7} \)
• (2) \( \frac{\sqrt{14}}{7} \)
• (3) \( \frac{3\sqrt{14}}{7} \)
• (4) \( \frac{6\sqrt{14}}{7} \)

Let \( f(x) = 3\sqrt{x - 2} + \sqrt{4 - x} \) be a real-valued function. If \( \alpha \) and \( \beta \) are respectively the minimum and the maximum values of \( f \), then \( \alpha^2 + 2\beta^2 \) is equal to:

• (1) 44
• (2) 42
• (3) 24
• (4) 38

Let \[ f(x) = \int_{0}^{x} \left( t + \sin\left(1 - e^t\right) \right) \, dt, \, x \in \mathbb{R}. \] Then \[ \lim_{x \to 0} \frac{f(x)}{x^3} \] is equal to:

• (1) \( \frac{1}{6} \)
• (2) \( \frac{-1}{6} \)
• (3) \( \frac{-2}{3} \)
• (4) \( \frac{2}{3} \)

The area (in square units) of the region described by: \[ \{(x, y) : y^2 \leq 2x, \, \text{and} \, y \geq 4x - 1\} \] is:

• (1) \( \frac{11}{32} \)
• (2) \( \frac{8}{9} \)
• (3) \( \frac{11}{12} \)
• (4) \( \frac{9}{32} \)

Given the inverse trigonometric function assumes principal values only. Let \( x, y \) be any two real numbers in \( [-1, 1] \) such that \[ \cos^{-1}x - \sin^{-1}y = \alpha, \, -\frac{\pi}{2} \leq \alpha \leq \pi. \] Then, the minimum value of \( x^2 + y^2 + 2xy \sin \alpha \) is:

• (1) -1
• (2) 0
• (3) \( \frac{-1}{2} \)
• (4) \( \frac{1}{2} \)

\[\text{If } \lambda>0, \text{ let } \theta \text{ be the angle between the vectors }\vec{a} = \hat{i} + \lambda \hat{j} - 3 \hat{k} \text{ and } \vec{b} = 3 \hat{i} - \hat{j} + 2 \hat{k}.\text{ If the vectors } \vec{a} + \vec{b} \text{ and } \vec{a} - \vec{b} \text{ are mutually perpendicular, then the value of } (14 \cos \theta)^2 \text{ is equal to.}\]

• (1) 25
• (2) 20
• (3) 45
• (4) 40

Let \( y = y(x) \) be the solution of the differential equation: \[ (x^2 + 4)^2 \, dy + \left( 2x^3 y + 8xy - 2 \right) dx = 0. \] If \( y(0) = 0 \), then \( y(2) \) is equal to:

• (1) \( \frac{\pi}{8} \)
• (2) \( \frac{\pi}{16} \)
• (3) \( 2\pi \)
• (4) \( \frac{\pi}{32} \)

If the mean of the following probability distribution of a random variable \( X \): \[ \begin{array}{|c|c|c|c|c|c|} \hline X & 0 & 2 & 4 & 6 & 8 \\ \hline P(X) & a & 2a & a+b & 2b & 3b \\ \hline \end{array} \] is \( \frac{46}{9} \), then the variance of the distribution is:

• (1) \( \frac{581}{81} \)
• (2) \( \frac{566}{81} \)
• (3) \( \frac{173}{27} \)
• (4) \( \frac{151}{27} \)

Let \( PQ \) be a chord of the parabola \( y^2 = 12x \) and the midpoint of \( PQ \) be at \( (4, 1) \). Then, which of the following points lies on the line passing through the points \( P \) and \( Q \)?

• (1) (3, -3)
• (2) \( \left( \frac{3}{2}, -16 \right) \)
• (3) (2, -9)
• (4) \( \left( \frac{1}{2}, -20 \right) \)

Let C be a circle with radius \( \sqrt{10} \) units and centre at the origin. Let the line \( x + y = 2 \) intersects the circle C at the points P and Q. Let MN be a chord of C of length 2 unit and slope \(-1\). Then, a distance (in units) between the chord PQ and the chord MN is

• (1) \( 2 - \sqrt{3} \)
• (2) \( 3 - \sqrt{2} \)
• (3) \( \sqrt{2} - 1 \)
• (4) \( \sqrt{2} + 1 \)

The area (in square units) of the region \[ S = \{ z \in \mathbb{C} : |z - 1| \leq 2; (z + \bar{z}) + i (z - \bar{z}) \leq 2, \, \operatorname{Im}(z) \geq 0 \} \] is:

• (1) \( \frac{7\pi}{3} \)
• (2) \( \frac{3\pi}{2} \)
• (3) \( \frac{17\pi}{8} \)
• (4) \( \frac{7\pi}{4} \)

If the value of the integral \[ \int_{-1}^{1} \frac{\cos \alpha x}{1 + 3^x} \, dx = \frac{2}{\pi}, \] then a value of \( \alpha \) is:

• (1) \( \frac{\pi}{6} \)
• (2) \( \frac{\pi}{2} \)
• (3) \( \frac{\pi}{3} \)
• (4) \( \frac{\pi}{4} \)

Let three real numbers a,b,c be in arithmetic progression and a + 1, b, c + 3 be in geometric progression. If a>10 and the arithmetic mean of a,b and c is 8, then the cube of the geometric mean of a,b and c is

• (1) 120
• (2) 312
• (3) 316
• (4) 128

Let a relation \( R \) on \( \mathbb{N} \times \mathbb{N} \) be defined as: \[(x_1, y_1) \, R \, (x_2, y_2) \text{ if and only if } x_1 \leq x_2 \text{ or } y_1 \leq y_2.\] 
Consider the two statements:
[(I)] \( R \) is reflexive but not symmetric. 
[(II)] \( R \) is transitive.
Then which one of the following is true:

• (1) Only (II) is correct.
• (2) Only (I) is correct.
• (3) Both (I) and (II) are correct.
• (4) Neither (I) nor (II) is correct.

Let \( A = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix} \) and \[B = I + \text{adj}(A) + (\text{adj}(A))^2 + \dots + (\text{adj}(A))^{10}.\]Then, the sum of all the elements of the matrix \( B \) is:

• (1) -110
• (2) 22
• (3) -88
• (4) -124

Let \[ \vec{a} = \hat{i} + \hat{j} + \hat{k}, \quad \vec{b} = 2\hat{i} + 4\hat{j} - 5\hat{k}, \quad \text{and} \quad \vec{c} = x\hat{i} + 2\hat{j} + 3\hat{k}, \, x \in \mathbb{R}. \] If \( \vec{d} \) is the unit vector in the direction of \( \vec{b} + \vec{c} \) such that \( \vec{a} \cdot \vec{d} = 1 \), then \( (\vec{a} \times \vec{b}) \cdot \vec{c} \) is equal to:

• (1) 9
• (2) 6
• (3) 3
• (4) 11

Let \[ S = \{ \sin^2 2\theta : (\sin^4 \theta + \cos^4 \theta)x^2 + (\sin 2\theta)x + (\sin^6 \theta + \cos^6 \theta) = 0 \, \text{has real roots} \}. \] If \( \alpha \) and \( \beta \) are the smallest and largest elements of the set \( S \), respectively, then \[ 3 \big((\alpha - 2)^2 + (\beta - 1)^2 \big) \] equals:

If \[ \int \cosec^5 x \, dx = \alpha \cot x \cosec x \left( \cosec^2 x + \frac{3}{2} \right) + \beta \log_e \left| \tan \frac{x}{2} \right| + C, \] where \( \alpha, \beta \in \mathbb{R} \) and \( C \) is the constant of integration, then the value of \( 8(\alpha + \beta) \) equals:

Let \( A \) be a \( 2 \times 2 \) symmetric matrix such that \[ A \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \\ 7 \end{bmatrix} \] and the determinant of \( A \) be 1. If \( A^{-1} = \alpha A + \beta I \), where \( I \) is the identity matrix of order \( 2 \times 2 \), then \( \alpha + \beta \) equals \( \dots \).

Consider a triangle \( \triangle ABC \) having the vertices \( A(1, 2) \), \( B(\alpha, \beta) \), and \( C(\gamma, \delta) \) and angles \( \angle ABC = \frac{\pi}{6} \) and \( \angle BAC = \frac{2\pi}{3} \). If the points \( B \) and \( C \) lie on the line \( y = x + 4 \), then \( \alpha^2 + \gamma^2 \) is equal to \( \dots \).

In a tournament, a team plays 10 matches with probabilities of winning and losing each match as \( \frac{1}{3} \) and \( \frac{2}{3} \), respectively. Let \( x \) be the number of matches that the team wins, and \( y \) be the number of matches that the team loses. If the probability \( P(|x - y| \leq 2) \) is \( p \), then \( 3^9 p \) equals .

Consider the function \( f : \mathbb{R} \to \mathbb{R} \) defined by \[ f(x) = \frac{2x}{\sqrt{1 + 9x^2}}. \] If the composition of \( f \), \[ (f \circ f \circ f \circ \dots \circ f)(x) \quad \text{(10 times)} = \frac{2^{10}x}{\sqrt{1 + 9\alpha x^2}}, \] then the value of \( \sqrt{3\alpha + 1} \) is equal to \( \dots \).

Let \( y = y(x) \) be the solution of the differential equation \[ (x + y + 2)^2 \, dx = dy, \quad y(0) = -2. \] Let the maximum and minimum values of the function \( y = y(x) \) in \( \left[ 0, \frac{\pi}{3} \right] \) be \( \alpha \) and \( \beta \), respectively. If \[ (3\alpha + \pi)^2 + \beta^2 = \gamma + \delta\sqrt{3}, \quad \gamma, \delta \in \mathbb{Z}, \] then \( \gamma + \delta \) equals \( \dots \).

Let \( f : \mathbb{R} \to \mathbb{R} \) be a thrice differentiable function such that \[ f(0) = 0, \, f(1) = 1, \, f(2) = -1, \, f(3) = 2, \, \text{and} \, f(4) = -2. \] Then, the minimum number of zeros of \( (3f' f' + f'') (x) \) is:

Consider a line \( L \) passing through the points \( P(1, 2, 1) \) and \( Q(2, 1, -1) \). If the mirror image of the point \( A(2, 2, 2) \) in the line \( L \) is \( (\alpha, \beta, \gamma) \), then \( \alpha + \beta + 6\gamma \) is equal to .

There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If 4 persons are selected from each group, then the number of ways of selecting 4 men and 4 women is

Applying the principle of homogeneity of dimensions, determine which one is correct. Where \( T \) is the time period, \( G \) is the gravitational constant, \( M \) is the mass, and \( r \) is the radius of the orbit.

• (1) \( T^2 = \frac{4\pi^2 r}{GM^2} \)
• (2) \( T^2 = 4\pi^2 r^3 \)
• (3) \( T^2 = \frac{4\pi^2 r^3}{GM} \)
• (4) \( T^2 = \frac{4\pi^2 r^2}{GM} \)

A 2 kg brick begins to slide over a surface which is inclined at an angle of \( 45^\circ \) with respect to the horizontal axis. The coefficient of static friction between their surfaces is:

• (1) 1
• (2) \( \frac{1}{\sqrt{3}} \)
• (3) 0.5
• (4) 1.7

Match List I with List II

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

Which of the diode circuit shows correct biasing used for the measurement of dynamic resistance of p-n junction diode :

• (1)

• (2)

• (3)

• (4)

A sample of gas at temperature \( T \) is adiabatically expanded to double its volume. Adiabatic constant for the gas is \( \gamma = \frac{3}{2} \). The work done by the gas in the process is (\( \mu = 1 \) mole):

• (1) \( RT\left[\sqrt{2} - 2\right] \)
• (2) \( RT\left[1 - 2\sqrt{2}\right] \)
• (3) \( RT\left[2\sqrt{2} - 1\right] \)
• (4) \( RT\left[2 - \sqrt{2}\right] \)

The magnetic moment of a bar magnet is \( 0.5 \, \text{Am}^2 \). It is suspended in a uniform magnetic field of \( 8 \times 10^{-2} \, \text{T} \). The work done in rotating it from its most stable to most unstable position is:

• (1) \( 16 \times 10^{-2} \, \text{J} \)
• (2) \( 8 \times 10^{-2} \, \text{J} \)
• (3) \( 4 \times 10^{-2} \, \text{J} \)
• (4) \( \text{Zero} \)

In simple harmonic motion, the total mechanical energy of the given system is \( E \). If the mass of the oscillating particle \( P \) is doubled, then the new energy of the system for the same amplitude is:

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

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): Number of photons increases with increase in frequency of light.
Reason (R): Maximum kinetic energy of emitted electrons increases with the frequency of incident radiation.
In the light of the above statements, choose the most appropriate answer from the options given below:

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

A 90 kg body placed at \( 2R \) distance from the surface of the earth experiences gravitational pull of:
(\( R \) = Radius of Earth, \( g = 10 \, \text{ms}^{-2} \)).

• (1) 300 N
• (2) 225 N
• (3) 120 N
• (4) 100 N

According to Bohr's theory, the moment of momentum of an electron revolving in the 4^th orbit of a hydrogen atom is:

• (1) \( \frac{8h}{\pi} \)
• (2) \( \frac{h}{\pi} \)
• (3) \( \frac{2h}{\pi} \)
• (4) \( \frac{h}{2\pi} \)

A cyclist starts from the point P of a circular ground of radius 2 km and travels along its circumference to the point S. The displacement of a cyclist is :

• (1) 6 km
• (2) √8 km
• (3) 4 km
• (4) 8 km

A body of m kg slides from rest along the curve of vertical circle from point A to B in friction less path. The velocity of the body at B is :

• (1) 19.8 m/s
• (2) 21.9 m/s
• (3) 16.7 m/s
• (4) 10.6 m/s

The width of one of the two slits in a Young's double slit experiment is 4 times that of the other slit. The ratio of the maximum of the minimum intensity in the interference pattern is :

• (1) 9 :1
• (2) 16 : 1
• (3) 1 : 1
• (4) 4 : 1

The translational degrees of freedom (\(f_t\)) and rotational degrees of freedom (\(f_r\)) of \( \text{CH}_4 \) molecule are:

• (1) \( f_t = 2 \, \text{and} \, f_r = 2 \)
• (2) \( f_t = 3 \, \text{and} \, f_r = 3 \)
• (3) \( f_t = 3 \, \text{and} \, f_r = 2 \)
• (4) \( f_t = 2 \, \text{and} \, f_r = 3 \)

A charge q is placed at the center of one of the surface of a cube. The flux linked with the cube is :-

• (1) \( \frac{q}{4\epsilon_0} \)
• (2) \( \frac{q}{2\epsilon_0} \)
• (3) \( \frac{q}{8\epsilon_0} \)
• (4) Zero

Correct formula for height of a satellite from earths surface is :

• (1) \( \left( \frac{T^2 R^2 g}{4 \pi} \right)^{1/2} - R \)
• (2) \( \left( \frac{T^2 R^2 g}{4 \pi^2} \right)^{1/3} - R \)
• (3) \( \left( \frac{T^2 R^2}{4 \pi^2 g} \right)^{1/3} - R \)
• (4) \( \left( \frac{T^2 R^2}{4 \pi^2 g} \right)^{-1/3} + R \)

Given below are two statements :
Statement I : The contact angle between a solid and a liquid is a property of the material of the solid and liquid as well.
Statement II : The rise of a liquid in a capillary tube does not depend on the inner radius of the tube.
In the light of the above statements, choose the correct answer from the options given below :

• (1) Both Statement I and Statement II are false
• (2) Statement I is false but Statement II is true.
• (3) Statement I is true but Statement II is false.
• (4) Both Statement I and Statement II are true.

An electric bulb rated 50 W – 200 V is connected across a 100 V supply. The power dissipation of the bulb is :

• (1) 12.5 W
• (2) 25 W
• (3) 50 W
• (4) 100 W

Arrange the following in the ascending order of wavelength:
(A) Gamma rays (\( \lambda_1 \))
(B) X-ray (\( \lambda_2 \))
(C) Infrared waves (\( \lambda_3 \))
(D) Microwaves (\( \lambda_4 \))
Choose the most appropriate answer from the options given below:

• (1) \( \lambda_4<\lambda_3<\lambda_1<\lambda_2 \)
• (2) \( \lambda_4<\lambda_3<\lambda_2<\lambda_1 \)
• (3) \( \lambda_1<\lambda_2<\lambda_3<\lambda_4 \)
• (4) \( \lambda_2<\lambda_1<\lambda_4<\lambda_3 \)

Identify the logic gate given in the circuit :

• (1) NAND - gate
• (2) OR - gate
• (3) AND gate
• (4) NOR gate

In a system, two particles of masses \( m_1 = 3 \, \text{kg} \) and \( m_2 = 2 \, \text{kg} \) are placed at a certain distance from each other. The particle of mass \( m_1 \) is moved towards the center of mass of the system through a distance \( 2 \, \text{cm} \). In order to keep the center of mass of the system at the original position, the particle of mass \( m_2 \) should move towards the center of mass by the distance ______ \( \, \text{cm} \).

Two parallel long current carrying wire separated by a distance \(2r\) are shown in the figure. The ratio of magnetic field at \(A\) to the magnetic field produced at \(C\) is \(\frac{x}{7}\). The value of \(x\) is ___.

A bus moving along a straight highway with speed of 72 km/h is brought to halt within 4s after applying the brakes. The distance travelled by the bus during this time (Assume the retardation is uniform) is _______m.

A rod of length 60 cm rotates with a uniform angular velocity \(20 \, \text{rad} \, \text{s}^{-1}\) about its perpendicular bisector, in a uniform magnetic field \(0.5 \, \text{T}\). The direction of the magnetic field is parallel to the axis of rotation. The potential difference between the two ends of the rod is _____ V.

The disintegration energy Q for the nuclear fission of ²³⁵U → ¹⁴⁰Ce + ⁹⁴Zr + n is _____ MeV.

Given atomic masses:
²³⁵U = 235.0439 u 
¹⁴⁰Ce = 139.9054 u 
⁹⁴Zr = 93.9063 u 
n = 1.0086 u 

Value of c² = 931 MeV/u.

Mercury is filled in a tube of radius \(2 \, \text{cm}\) up to a height of \(30 \, \text{cm}\). The force exerted by mercury on the bottom of the tube is ___N.
(Given, atmospheric pressure = \(10^5 \, \text{N/m}^2\), density of mercury = \(1.36 \times 10^4 \, \text{kg/m}^3\), \(g = 10 \, \text{m/s}^2\), \(\pi = \frac{22}{7}\))

A light ray is incident on a glass slab of thickness \(4\sqrt{3}\) cm and refractive index \(\sqrt{2}\). The angle of incidence is equal to the critical angle for the glass slab with air. The lateral displacement of the ray after passing through the glass slab is _____ cm.
(Given \(\sin 15^\circ = 0.25\))

Two wires A and B are made up of the same material and have the same mass. Wire A has radius \(2.0 \, \text{mm}\) and wire B has radius \(4.0 \, \text{mm}\). The resistance of wire B is \(2 \, \Omega\). The resistance of wire A is _____ \( \Omega \).

The displacement of a particle executing SHM is given by \( x = 10 \sin\left(\omega t + \frac{\pi}{3}\right) \, \text{m} \). The time period of motion is \( 3.14 \, \text{s} \). The velocity of the particle at \( t = 0 \) is ______ m/s.

A parallel plate capacitor of capacitance \( 12.5 \, \text{pF} \) is charged by a battery connected between its plates to a potential difference of \( 12.0 \, \text{V} \). The battery is now disconnected and a dielectric slab (\( \varepsilon_r = 6 \)) is inserted between the plates. The change in its potential energy after inserting the dielectric slab is ______ \( \times 10^{-12} \, \text{J} \).

Fuel cell, using hydrogen and oxygen as fuels,
A. has been used in spaceship
B. has as efficiency of 40% to produce electricity
C. uses aluminium as catalysts
D. is eco-friendly
E. is actually a type of Galvanic cell only

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

Match List I with List II

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

If an iron (III) complex with the formula \[\left[ \text{Fe}(\text{NH}_3)_x (\text{CN})_y \right]^-\]has no electron in its \( e_g \) orbital, then the value of \( x + y \) is:

• (1) 5
• (2) 6
• (3) 3
• (4) 4

The number of unpaired d-electrons in \([ \text{Co}(\text{H}_2\text{O})_6 ]^{3+}\) is ______

• (1) 4
• (2) 2
• (3) 0
• (4) 1

Correct order of stability of carbanion is

• (1) c > b > d > a
• (2) a > b > c > d
• (3) d > a > c > b
• (4) d > c > b > a

• (1)

• (2)

• (3)

• (4)

The correct order of the first ionization enthalpy is

• (1) Al > Ga > Tl
• (2) Ga > Al > B
• (3) B > Al > Ga
• (4) Tl > Ga > Al

Given below are two statements:
Statement I: The correct order of first ionization enthalpy values of Li, Na, F, and Cl is \( \text{Na} < \text{Li} < \text{Cl} < \text{F} \).
Statement II: The correct order of negative electron gain enthalpy values of Li, Na, F, and Cl is \( \text{Na} < \text{Li} < \text{F} < \text{Cl} \).
In the light of the above statements, choose the correct answer from the options given below:

• (1) Both Statement I and Statement II are true
• (2) Both Statement I and Statement II are false
• (3) Statement I is false but Statement II is true
• (4) Statement I is true but Statement II is false

For a strong electrolyte, a plot of molar conductivity against \( (\text{concentration})^{1/2} \) is a straight line, with a negative slope, the correct unit for the slope is:

• (1) S cm\(^2\) mol\(^{-3/2}\) L\(^{1/2}\)
• (2) S cm\(^2\) mol\(^{-1}\) L\(^{1/2}\)
• (3) S cm\(^2\) mol\(^{-3/2}\) L
• (4) S cm\(^2\) mol\(^{-3/2}\) L\(^{-1/2}\)

When \( \text{MnO}_2 \) and \( \text{H}_2\text{SO}_4 \) is added to a salt (A), the greenish yellow gas liberated as salt (A) is:

• (1) \( \text{NaBr} \)
• (2) \( \text{CaI}_2 \)
• (3) \( \text{KNO}_3 \)
• (4) \( \text{NH}_4\text{Cl} \)

The correct statement/s about Hydrogen bonding is/are :
A. Hydrogen bonding exists when H is covalently bonded to the highly electro negative atom.
B. Intermolecular H bonding is present in o-nitro phenol
C. Intramolecular H bonding is present in HF.
D. The magnitude of H bonding depends on the physical state of the compound.
E. H-bonding has powerful effect on the structure and properties of compounds.
Choose the correct answer from the options given below :

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

Choose the Incorrect Statement about Dalton's Atomic Theory

• (1) Compounds are formed when atoms of different elements combine in any ratio
• (2) All the atoms of a given element have identical properties including identical mass
• (3) Matter consists of indivisible atoms
• (4) Chemical reactions involve recorganization of atoms

• (1) B = C = 2-Propanol
• (2) B = 2-Propanol C = 1-Propanol
• (3) B = 1-Propanol C = 2-Propanol
• (4) B = C = 1-Propanol

The number of species from the following that have pyramidal geometry around the central atom is ______
\( \text{S}_2\text{O}_3^{2-}, \, \text{SO}_4^{2-}, \, \text{SO}_3^{2-}, \, \text{S}_2\text{O}_7^{2-} \)

• (1) 4
• (2) 3
• (3) 1
• (4) 2

A first row transition metal in its +2 oxidation state has a spin-only magnetic moment value of 3.86 BM. The atomic number of the metal is

• (1) 25
• (2) 26
• (3) 22
• (4) 23

Common name of Benzene-1, 2-diol is

• (1) quinol
• (2) resorcinol
• (3) catechol
• (4) o-cresol

The equilibrium constant for the reaction
\( \text{SO}_3 \, (g) \rightleftharpoons \text{SO}_2 \, (g) + \frac{1}{2} \text{O}_2 \, (g) \)
is \( K_C = 4.9 \times 10^{-2} \). The value of \( K_C \) for the reaction given below is
\( 2 \text{SO}_2 \, (g) + \text{O}_2 \, (g) \rightleftharpoons 2 \text{SO}_3 \, (g) \) is

• (1) 4.9
• (2) 41.6
• (3) 49
• (4) 416

Find out the major product formed from the following reaction.\([ \text{Me: } -\text{CH}_3 ]\)

• (1)

• (2)

• (3)

• (4)

The adsorbent used in adsorption chromatography is/are
A. silica gel
B. alumina
C. quick lime
D. magnesia
Choose the most appropriate answer from the options given below :

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

• (1) \( \text{O}_3, \, \text{Zn}/\text{H}_2\text{O} \, \text{and} \, \text{NaOH}_{(\text{alc.})}/\text{I}_2 \)
• (2) \( \text{H}_2\text{O}, \, \text{H}^+ \, \text{and} \, \text{NaOH}_{(\text{alc.})}/\text{I}_2 \)
• (3) \( \text{H}_2\text{O}, \, \text{H}^+ \, \text{and} \, \text{KMnO}_4 \)
• (4) \( \text{O}_3, \, \text{Zn}/\text{H}_2\text{O} \, \text{and} \, \text{KMnO}_4 \)

The maximum number of orbitals which can be identified with \( n = 4 \) and \( m_l = 0 \) is ______

Number of compounds/species from the following with non-zero dipole moment is ______
\(\text{BeCl}_2, \, \text{BCl}_3, \, \text{NF}_3, \, \text{XeF}_4, \, \text{CCl}_4, \, \text{H}_2\text{O}, \, \text{H}_2\text{S}, \, \text{HBr}, \, \text{CO}_2, \, \text{H}_2, \, \text{HCl}\)

The total number of 'sigma' and 'Pi' bonds in 2-oxohex-4-ynoic acid is____.

From 6.55 g of aniline, the maximum amount of acetanilide that can be prepared will be ___ × \( 10^{-1} \) g.

2.7 Kg of each of water and acetic acid are mixed. The freezing point of the solution will be –x °C. Consider the acetic acid does not dimerise in water, nor dissociates in water. \( x = \) ______ (nearest integer).
[Given: Molar mass of water = \( 18 \, \text{g mol}^{-1} \), acetic acid = \( 60 \, \text{g mol}^{-1} \) \( K_f \, \text{H}_2\text{O} = 1.86 \, \text{K kg mol}^{-1} \) \( K_f \, \text{acetic acid} = 3.90 \, \text{K kg mol}^{-1} \) Freezing point: \( \text{H}_2\text{O} = 273 \, \text{K}, \, \text{acetic acid} = 290 \, \text{K} \)]

Consider the following reaction, the rate expression of which is given below:
\( \text{A} + \text{B} \to \text{C} \)
\(\text{rate} = k [\text{A}]^{1/2} [\text{B}]^{1/2}\)
The reaction is initiated by taking 1M concentration of A and B each. If the rate constant (\(k\)) is \(4.6 \times 10^{-2} \, \text{s}^{-1}\), then the time taken for A to become 0.1 M is ______ sec. (nearest integer)

Three moles of an ideal gas are compressed isothermally from 60 L to 20 L using constant pressure of 5 atm. Heat exchange Q for the compression is – ____ Lit. atm.

Vanillin compound obtained from vanilla beans, has total sum of oxygen atoms and \( \pi \) electrons is ______

Phthalimide is made to undergo following sequence of reactions.

A first row transition metal with the highest enthalpy of atomisation, upon reaction with oxygen at high temperature, forms oxides of formula \( \text{M}_2\text{O}_n \) (where \( n = 3, 4, 5 \)). The 'spin-only' magnetic moment value of the amphoteric oxide from the above oxides is ___ BM (near integer).
(Given atomic number: \( \text{Sc: 21, Ti: 22, V: 23, Cr: 24, Mn: 25, Fe: 26, Co: 27, Ni: 28, Cu: 29, Zn: 30} \))