JEE Main 2 April Shift 2 Question Paper (Available)- Download Solutions and Answer Key

If the image of the point \( P(1, 0, 3) \) in the line joining the points \( A(4, 7, 1) \) and \( B(3, 5, 3) \) is \( Q(\alpha, \beta, \gamma) \), then \( \alpha + \beta + \gamma \) is equal to:

(1) 47/3
(2) 46/3
(3) 18
(4) 13

Let \( f : [1, \infty) \to [2, \infty) \) be a differentiable function,
If \( \int_{1}^{x} f(t) \, dt = 5x f(x) - x^5 - 9 \) for all \( x \geq 1 \), then the value of \( f(3) \) is :

• (1) 18
• (2) 32
• (3) 22
• (4) 26

The number of terms of an A.P. is even; the sum of all the odd terms is 24, the sum of all the even terms is 30 and the last term exceeds the first by \( \frac{21}{2} \). Then the number of terms which are integers in the A.P. is:

• (1) 4
• (2) 10
• (3) 6
• (4) 8

Let \( A = \{1, 2, 3, \dots, 10\} \) and \( R \) be a relation on \( A \) such that \( R = \{(a, b) : a = 2b + 1\} \). Let \( (a_1, a_2), (a_3, a_4), (a_5, a_6), \dots, (a_k, a_{k+1}) \) be a sequence of \( k \) elements of \( R \) such that the second entry of an ordered pair is equal to the first entry of the next ordered pair. Then the largest integer \( k \), for which such a sequence exists, is equal to:

• (1) 6
• (2) 7
• (3) 5
• (4) 8

If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is:

• (1) \( \frac{4}{\sqrt{17}} \)
• (2) \( \frac{\sqrt{5}}{16} \)
• (3) \( \frac{3}{\sqrt{19}} \)
• (4) \( \frac{\sqrt{5}}{7} \)

The line \( L_1 \) is parallel to the vector \( \mathbf{a} = -3\hat{i} + 2\hat{j} + 4\hat{k} \) and passes through the point \( (7, 6, 2) \), and the line \( L_2 \) is parallel to the vector \( \mathbf{b} = 2\hat{i} + \hat{j} + 3\hat{k} \) and passes through the point \( (5, 3, 4) \). The shortest distance between the lines \( L_1 \) and \( L_2 \) is:

• (1) \( \frac{23}{\sqrt{38}} \)
• (2) \( \frac{21}{\sqrt{57}} \)
• (3) \( \frac{23}{\sqrt{57}} \)
• (4) \( \frac{21}{\sqrt{38}} \)

Let \( (a, b) \) be the point of intersection of the curve \( x^2 = 2y \) and the straight line \( y = 2x - 6 \) in the second quadrant. Then the integral \[ I = \int_a^b \frac{9x^2}{1 + 5x^3} \, dx \]
is equal to:

• (1) 24
• (2) 27
• (3) 18
• (4) 21

If the system of equation \[ 2x + \lambda y + 3z = 5
3x + 2y - z = 7
4x + 5y + \mu z = 9 \]
has infinitely many solutions, then \( \lambda^2 + \mu^2 \) is equal to:

• (1) 22
• (2) 18
• (3) 26
• (4) 30

If \( \theta \in \left[ \frac{7\pi}{6}, \frac{4\pi}{3} \right] \), then the number of solutions of \[ \sqrt{3} \csc^2\theta - 2(\sqrt{3} - 1) \csc\theta - 4 = 0, \] is equal to:

(1) 6

(2) 8

(3) 10

(4) 7

Given three identical bags each containing 10 balls, whose colours are as follows:

Bag I 3 Red 2 Blue 5 Green
Bag II 4 Red 3 Blue 3 Green
Bag III 5 Red 1 Blue 4 Green

A person chooses a bag at random and takes out a ball. If the ball is Red, the probability that it is from Bag I is \( p \) and if the ball is Green, the probability that it is from Bag III is \( q \), then the value of \( \frac{1}{p} + \frac{1}{q} \) is:

(1) 6

(2) 9

(3) 7

(4) 8

If the mean and the variance of 6, 4, 8, 8, b, 12, 10, 13 are 9 and 9.25 respectively, then \( a + b + ab \) is equal to:

(1) 105

(2) 103

(3) 100

(4) 106

If the domain of the function \( f(x) = \frac{1}{\sqrt{10 + 3x - x^2}} \) is \( (a, b) \), then \( (1 + a)^2 + b \) is equal to:

(1) 26

(2) 29

(3) 25

(4) 30

\[ \int \frac{1}{\sqrt{3+x^2}+\sqrt{1+x^2}} \, dx - 3 \log \left( \sqrt{3} \right) \] is equal to:

(1) \( 2 + \sqrt{2} + \log \left( 1 + \sqrt{2} \right) \)

(2) \( 2 - \sqrt{2} - \log \left( 1 + \sqrt{2} \right) \)

(3) \( 2 + \sqrt{2} - \log \left( 1 + \sqrt{2} \right) \)

(4) \( 2 - \sqrt{2} + \log \left( 1 + \sqrt{2} \right) \)

If \( \lim_{x \to 0} \frac{\cos(2x) + a \cos(4x) - b}{x^4} \) is finite, then \( (a + b) \) is equal to:

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

If   \sum_{r=0}^{10} \left( 10^{r+1} - 1 \right) \binom{10}{r} = \alpha^{11} - 1 then \( \alpha \) is equal to :

• (1) 15
• (2) 11
• (3) 24
• (4) 20

The number of ways, in which the letters A, B, C, D, E can be placed in the 8 boxes of the figure below so that no row remains empty and at most one letter can be placed in a box, is:

• (1) 5880
• (2) 960
• (3) 840
• (4) 5760

Let the point \( P \) of the focal chord \( PQ \) of the parabola \( y^2 = 16x \) be \( (1, -4) \). If the focus of the parabola divides the chord \( PQ \) in the ratio \( m : n \), gcd(\(m, n\)) = 1, then \( m^2 + n^2 \) is equal to:

• (1) 17
• (2) 10
• (3) 37
• (4) 26

Let \( \mathbf{a} = 2\hat{i} - 3\hat{j} + \hat{k}, \, \mathbf{b} = 3\hat{i} + 2\hat{j} + 5\hat{k} \) and a vector \( \mathbf{c} \) be such that \( (\mathbf{a} - \mathbf{c}) \times \mathbf{b} = -18\hat{i} - 3\hat{j} + 12\hat{k} \) and \( \mathbf{a} \cdot \mathbf{c} = 3 \). If \( \mathbf{b} \times \mathbf{c} = \mathbf{a} \), then \( |\mathbf{a} \cdot \mathbf{c}| \) is equal to:

• (1) 18
• (2) 12
• (3) 9
• (4) 15

Let the area of the triangle formed by a straight line \( L: x + by + c = 0 \) with co-ordinate axes be 48 square units. If the perpendicular drawn from the origin to the line \( L \) makes an angle of \( 45^\circ \) with the positive x-axis, then the value of \( b^2 + c^2 \) is:

• (1) 90
• (2) 93
• (3) 97
• (4) 83

Let \( A \) be a \( 3 \times 3 \) real matrix such that \( A^2(A - 2I) - 4(A - I) = O \), where \( I \) and \( O \) are the identity and null matrices, respectively. If \( A^3 = \alpha A^2 + \beta A + \gamma I \), where \( \alpha \), \( \beta \), and \( \gamma \) are real constants, then \( \alpha + \beta + \gamma \) is equal to:

• (1) 12
• (2) 20
• (3) 76
• (4) 4

Let \( y = y(x) \) be the solution of the differential equation \[ \frac{dy}{dx} + 2y \sec^2 x = 2 \sec^2 x + 3 \tan x \sec^2 x \]
such that \( y(0) = \frac{5}{4} \). Then \( 12 \left( y \left( \frac{\pi}{4} \right) - e^2 \right) \) is equal to:

• (1) 21
• (2) 22
• (3) 20
• (4) 25

If the sum of the first 10 terms of the series \[ \frac{4.1}{1 + 4.1^4} + \frac{4.2}{1 + 4.2^4} + \frac{4.3}{1 + 4.3^4} + \cdots \]
is \( \frac{m}{n} \), where \( \gcd(m, n) = 1 \), then \( m + n \) is equal to .....…

(1) 15

(2) 24

(3) 41

(4) 76

If \( y = \cos \left( \frac{\pi}{3} + \cos^{-1} \frac{x}{2} \right) \), then \( (x - y)^2 + 3y^2 \) is equal to ________.

• (1) 6
• (2) 8
• (3) 3 \
• (4) 7

Let \( A(4, -2), B(1, 1) \) and \( C(9, -3) \) be the vertices of a triangle ABC. Then the maximum area of the parallelogram AFDE, formed with vertices D, E and F on the sides BC, CA and AB of the triangle ABC respectively, is __________.

• (1) 4
• (2) 6
• (3) 3
• (4) 9

If the set of all \( a \in \mathbb{R} \setminus \{1\} \), for which the roots of the equation \( (1 - a)x^2 + 2(a - 3)x + 9 = 0 \) are positive is \( (-\infty, -\alpha] \cup [\beta, \gamma] \), then \( 2\alpha + \beta + \gamma \) is equal to ...........

• (1) 7
• (2) 10
• (3) 3
• (4) 9

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A): Net dipole moment of a polar linear isotropic dielectric substance is not zero even in the absence of an external electric field.

Reason (R): In absence of an external electric field, the different permanent dipoles of a polar dielectric substance are oriented in random directions.

In the light of the above statements, choose the most appropriate answer from the options given below:
(1) (A) is correct but (R) is not correct

(2) Both (A) and (R) are correct but (R) is not the correct explanation of (A)

(3) Both (A) and (R) are correct and (R) is the correct explanation of (A)

(4) (A) is not correct but (R) is correct

In a moving coil galvanometer, two moving coils \( M_1 \) and \( M_2 \) have the following particulars: \( R_1 = 5 \, \Omega \), \( N_1 = 15 \), \( A_1 = 3.6 \times 10^{-3} \, m^2 \), \( B_1 = 0.25 \, T \) \( R_2 = 7 \, \Omega \), \( N_2 = 21 \), \( A_2 = 1.8 \times 10^{-3} \, m^2 \), \( B_2 = 0.50 \, T \)

Assuming that torsional constant of the springs are same for both coils, what will be the ratio of voltage sensitivity of \( M_1 \) and \( M_2 \)?

(1) 1 : 1

(2) 1 : 4

(3) 1 : 3

(4) 1 : 2

The moment of inertia of a circular ring of mass \( M \) and diameter \( r \) about a tangential axis lying in the plane of the ring is:

(1) \( \frac{1}{2} M r^2 \)

(2) \( \frac{3}{8} M r^2 \)

(3) \( \frac{3}{2} M r^2 \)

(4) \( 2 M r^2 \)

Two water drops each of radius \( r \) coalesce to form a bigger drop. If \( T \) is the surface tension, the surface energy released in this process is:

(1) \( 4\pi r^2 T \left[ 2^2 - 2^3 \right] \)

(2) \( 4\pi r^2 T \left[ 2^{-1} - 2^3 \right] \)

(3) \( 4\pi r^2 T \left[ 1 + \sqrt{2} \right] \)

(4) \( 4\pi r^2 T \left[ \sqrt{2} - 1 \right] \)

An electron with mass \( m \) with an initial velocity \( (t = 0) \, \vec{v} = \vec{v_0} \, (v_0 > 0) \) enters a magnetic field \( \vec{B} = B \hat{j} \). If the initial de-Broglie wavelength at \( t = 0 \) is \( \lambda_0 \), then its value after time \( t \) would be:

(1) \( \frac{\lambda_0}{\sqrt{1 - \frac{e^2 B^2 t^2}{m^2}}} \)

(2) \( \frac{\lambda_0}{\sqrt{1 + \frac{e^2 B^2 t^2}{m^2}}} \)

(3) \( \lambda_0 \sqrt{1 + \frac{e^2 B^2 t^2}{m^2}} \)

(4) \( \lambda_0 \)

A sinusoidal wave of wavelength 7.5 cm travels a distance of 1.2 cm along the x-direction in 0.3 sec. The crest P is at \( x = 0 \) at \( t = 0 \) sec and maximum displacement of the wave is 2 cm. Which equation correctly represents this wave?

(1) \( y = 2 \cos(0.83x - 3.35t) \, cm \)

(2) \( y = 2 \sin(0.83x - 3.5t) \, cm \)

(3) \( y = 2 \cos(3.35x - 0.83t) \, cm \)

(4) \( y = 2 \cos(0.13x - 0.5t) \, cm \)

Given a charge \( q \), current \( I \) and permeability of vacuum \( \mu_0 \). Which of the following quantity has the dimension of momentum?

(1) \( qI / \mu_0 \)

(2) \( q \mu_0 I \)

(3) \( q^2 \mu_0 I \)

(4) \( q \mu_0 / I \)

A solenoid having area \( A \) and length \( \ell \) is filled with a material having relative permeability 2. The magnetic energy stored in the solenoid is:

(1) \( \frac{B^2A}{\mu_0} \)

(2) \( \frac{B^2A}{2\mu_0} \)

(3) \( \frac{B^2A}{\mu_0} \)

(4) \( \frac{B^2A}{4\mu_0} \)

Two large plane parallel conducting plates are kept 10 cm apart as shown in figure. The potential difference between them is \( V \). The potential difference between the points A and B (shown in the figure) is:



(1) \( \frac{1}{4} V \)

(2) \( \frac{2}{5} V \)

(3) \( \frac{3}{4} V \)

(4) \( 1 V \)

Correct Answer: (2)
View Solution

Identify the characteristics of an adiabatic process in a monatomic gas.

(A) Internal energy is constant. (B) Work done in the process is equal to the change in internal energy. (C) The product of temperature and volume is a constant. (D) The product of pressure and volume is a constant. (E) The work done to change the temperature from \( T_1 \) to \( T_2 \) is proportional to \( (T_2 - T_1) \).

Choose the correct answer from the options given below:

(1) (A), (C), (D) only

(2) (A), (C), (E) only

(3) (B), (E) only

(4) (B), (D) only

Assuming the validity of Bohr's atomic model for hydrogen-like ions, the radius of \( Li^{2+} \) ion in its ground state is given by \( \frac{1}{X} a_0 \), where \( a_0 \) is the first Bohr's radius.

(1) 2

(2) 1

(3) 3

(4) 9

Energy released when two deuterons \( (H_2) \) fuse to form a helium nucleus \( (He_4) \) is:

(1) 8.1 MeV

(2) 5.9 MeV

(3) 23.6 MeV

(4) 26.8 MeV

In the digital circuit shown in the figure, for the given inputs the P and Q values are:





(1) \( P = 1, Q = 1 \)

(2) \( P = 0, Q = 0 \)

(3) \( P = 0, Q = 1 \)

(4) \( P = 1, Q = 0 \)

Two identical objects are placed in front of convex mirror and concave mirror having same radii of curvature of 12 cm, at same distance of 18 cm from the respective mirrors. The ratio of sizes of the images formed by convex mirror and by concave mirror is:

(1) \( \frac{1}{2} \)

(2) 2

(3) 3

(4) \( \frac{1}{3} \)

Correct Answer: (1)
View Solution

A sportsman runs around a circular track of radius \( r \) such that he traverses the path ABAB. The distance travelled and displacement, respectively, are:




(1) \( 2r, 3\pi r \)

(2) \( 3\pi r, \pi r \)

(3) \( \pi r, 3r \)

(4) \( 3\pi r, 2r \)

A body of mass 1kg is suspended with the help of two strings making angles as shown in the figure. Magnitude of tensions \( T_1 \) and \( T_2 \), respectively, are (in N):

(1) 5, \( 5\sqrt{3} \)

(2) \( 5\sqrt{3} \), 5

(3) \( 5\sqrt{3} \), \( 5\sqrt{3} \)

(4) 5, 5

A bi-convex lens has radius of curvature of both the surfaces same as \( \frac{1}{6} \, cm \). If this lens is required to be replaced by another convex lens having different radii of curvatures on both sides \( (R_1 \neq R_2) \), without any change in lens power then possible combination of \( R_1 \) and \( R_2 \) is:

(1) \( \frac{1}{3} \, cm \) and \( \frac{1}{3} \, cm \)

(2) \( \frac{1}{5} \, cm \) and \( \frac{1}{7} \, cm \)

(3) \( \frac{1}{3} \, cm \) and \( \frac{1}{7} \, cm \)

(4) \( \frac{1}{6} \, cm \) and \( \frac{1}{9} \, cm \)

If \( \mu_0 \) and \( \epsilon_0 \) are the permeability and permittivity of free space, respectively, then the dimension of \( \left( \frac{1}{\mu_0 \epsilon_0} \right) \) is :

(1) \( L T^2 \)

(2) \( L^2 T^2 \)

(3) \( T^2 / L \)

(4) \( T^2 / L^2 \)

Match List-I with List-II:
List-I                                                    List-II

• (A) Heat capacity of body                 (I) \( J \, kg^{-1} \)
• (B) Specific heat capacity of body   (II) \( JK^{-1} \)
• (C) Latent heat                                (III) \( J \, kg^{-1} K^{-1} \)
• (D) Thermal conductivity                 (IV) \( J m^{-1} k^{-1} s^{-1} \)
  Choose the correct answer from the options given below:
• (1) (A)-(III), (B)-(I), (C)-(II), (D)-(IV)
  (2) (A)-(I), (B)-(III), (C)-(II), (D)-(IV)
  (3) (A)-(III), (B)-(IV), (C)-(I), (D)-(II)
  (4) (A)-(I), (B)-(III), (C)-(I), (D)-(IV)

Consider a circular loop that is uniformly charged and has a radius \( \sqrt{2} \). Find the position along the positive \( z \)-axis of the cartesian coordinate system where the electric field is maximum if the ring was assumed to be placed in the \( xy \)-plane at the origin:

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

A wheel of radius \( 0.2 \, m \) rotates freely about its center when a string that is wrapped over its rim is pulled by a force of \( 10 \, N \) as shown in the figure. The established torque produces an angular acceleration of \( 2 \, rad/s^2 \). Moment of inertia of the wheel is............. kg m².

 

• (1) \( 1 \, kg \, m^2 \)
• (2) \( 2 \, kg \, m^2 \)
• (3) \( 3 \, kg \, m^2 \)
• (4) \( 4 \, kg \, m^2 \)

The internal energy of air in \( 4 \, m \times 4 \, m \times 3 \, m \) sized room at 1 atmospheric pressure will be \( \_\_ \times 10^6 \, J \).
(Consider air as a diatomic molecule)

A ray of light suffers minimum deviation when incident on a prism having angle of the prism equal to \(60^\circ\). The refractive index of the prism material is \( \sqrt{2} \). The angle of incidence (in degrees) is ______.

The length of a light string is 1.4 m when the tension on it is 5 N. If the tension increases to 7 N, the length of the string is 1.56 m. The original length of the string is ______ m.

A satellite of mass 1000 kg is launched to revolve around the earth in an orbit at a height of 270 km from the earth's surface. Kinetic energy of the satellite in this orbit is ______ x \(10^{10}\) J.

(Mass of earth = \(6 \times 10^{24}\) kg, Radius of earth = \(6.4 \times 10^6\) m, Gravitational constant = \(6.67 \times 10^{-11}\) Nm\(^2\) kg\(^{-2}\))