JEE Main 2024 9 April Shift 2 Question Paper (Available)- Download Solution PDF with Answer Key

Let the foci of a hyperbola \( H \) coincide with the foci of the ellipse \( E : \frac{(x - 1)^2}{100} + \frac{(y - 1)^2}{75} = 1 \) and the eccentricity of the hyperbola \( H \) be the reciprocal of the eccentricity of the ellipse \( E \). If the length of the transverse axis of \( H \) is \( \alpha \) and the length of its conjugate axis is \( \beta \), then \( 3\alpha^2 + 2\beta^2 \) is equal to:

• (1) 242
• (2) 225
• (3) 237
• (4) 205

Let \( z \) be a complex number such that the real part of \[ \frac{z - 2i}{z + 2i} \] is zero. Then, the maximum value of \( |z - (6 + 8i)| \) is equal to:

• (1) 12
• (2) \(\infty\)
• (3) 10
• (4) 8

Let the range of the function \[ f(x) = \frac{1}{2 + \sin 3x + \cos 3x}, \, x \in \mathbb{R} \, \text{be } [a, b]. \] If \( \alpha \) and \( \beta \) are respectively the arithmetic mean (A.M.) and the geometric mean (G.M.) of \( a \) and \( b \), then \( \frac{\alpha}{\beta} \) is equal to:

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

Let \[ \int_{0}^{x} \sqrt{1 - (y'(t))^2} \, dt = \int_{0}^{x} y(t) \, dt, \quad 0 \leq x \leq 3, \, y \geq 0, \, y(0) = 0. \] Then, at \( x = 2 \), \( y'' + y + 1 \) is equal to:

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

Two vertices of a triangle \( \triangle ABC \) are \( A(3, -1) \) and \( B(-2, 3) \), and its orthocentre is \( P(1, 1) \). If the coordinates of the point \( C \) are \( (\alpha, \beta) \) and the centre of the circle circumscribing the triangle \( \triangle PAB \) is \( (h, k) \), then the value of \[ (\alpha + \beta) + 2(h + k) \] equals:

• (1) 51
• (2) 81
• (3) 5
• (4) 15

The integral \[ \int_{1/4}^{3/4} \cos\left( 2 \cot^{-1} \sqrt{\frac{1 - x}{1 + x}} \right) \, dx \] is equal to:

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

Let \( \alpha, \beta; \, \alpha > \beta \), be the roots of the equation \[ x^2 - \sqrt{2}x - \sqrt{3} = 0. \] Let \( P_n = \alpha^n - \beta^n, \, n \in \mathbb{N} \). Then \[ \left( 11\sqrt{3} - 10\sqrt{2} \right) P_{10} + \left( 11\sqrt{2} + 10 \right) P_{11} - 11P_{12} \] is equal to:  

• (1) \( 10\sqrt{2}P_9 \)
• (2) \( 10\sqrt{3}P_9 \)
• (3) \( 11\sqrt{2}P_9 \)
• (4) \( 11\sqrt{3}P_9 \)

If \( \log_e y = 3 \sin^{-1}x \), then \( (1 - x)^2 y'' - xy' \) at \( x = \frac{1}{2} \) is equal to:

• (1) \( 9e^{\pi/6} \)
• (2) \( 3e^{\pi/6} \)
• (3) \( 3e^{\pi/2} \)
• (4) \( 9e^{\pi/2} \)

If an unbiased dice is rolled thrice, then the probability of getting a greater number in the \( i \)-th roll than the number obtained in the \( (i-1) \)-th roll, \( i = 2, 3 \), is equal to:

• (1) \( \frac{3}{54} \)
• (2) \( \frac{2}{54} \)
• (3) \( \frac{5}{54} \)
• (4) \( \frac{1}{54} \)

\[ \lim_{x \to \frac{\pi}{2}} \frac{\int_{x^3}^{\left(\frac{\pi}{2}\right)^3} \left( \sin\left(2t^{1/3}\right) + \cos\left(t^{1/3}\right) \right) \, dt}{\left( x - \frac{\pi}{2} \right)^2} \] is equal to:

• (1) \( \frac{9\pi^2}{8} \)
• (2) \( \frac{11\pi^2}{10} \)
• (3) \( \frac{3\pi^2}{2} \)
• (4) \( \frac{5\pi^2}{9} \)

The value of the integral \[ \int_{-1}^{2} \log_e \left( x + \sqrt{x^2 + 1} \right) \, dx \] is:

• (1) \( \sqrt{5} - \sqrt{2} + \log_e \left( \frac{9 + 4\sqrt{5}}{1 + \sqrt{2}} \right) \)
• (2) \( \sqrt{2} - \sqrt{5} + \log_e \left( \frac{9 + 4\sqrt{5}}{1 + \sqrt{2}} \right) \)
• (3) \( \sqrt{5} - \sqrt{2} + \log_e \left( \frac{7 + 4\sqrt{5}}{1 + \sqrt{2}} \right) \)
• (4) \( \sqrt{2} - \sqrt{5} + \log_e \left( \frac{7 + 4\sqrt{5}}{1 + \sqrt{2}} \right) \)

Between the following two statements: {Statement-I: Let \[ \vec{a} = \hat{i} + 2\hat{j} - 3\hat{k} \quad \text{and} \quad \vec{b} = 2\hat{i} + \hat{j} - \hat{k}. \] Then the vector \( \vec{r} \) satisfying \[ \vec{a} \times \vec{r} = \vec{a} \times \vec{b} \quad \text{and} \quad \vec{a} \cdot \vec{r} = 0 \] is of magnitude \( \sqrt{10} \). {Statement-II: In a triangle \( \triangle ABC \), \[ \cos 2A + \cos 2B + \cos 2C \geq -\frac{3}{2}. \]

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

If the variance of the frequency distribution is 160, then the value of \( c \in \mathbb{N} \) is: \[ \begin{array}{|c|c|c|c|c|c|c|} \hline x & c & 2c & 3c & 4c & 5c & 6c \\ \hline f & 2 & 1 & 1 & 1 & 1 & 1 \\ \hline \end{array} \]

• (1) 5
• (2) 8
• (3) 7
• (4) 6

Let \[ B = \begin{bmatrix} 1 & 3 \\ 1 & 5 \end{bmatrix} \] and \(A\) be a \(2 \times 2\) matrix such that \[ AB^{-1} = A^{-1}. \] If \(BCB^{-1} = A\) and \[ C^4 + \alpha C^2 + \beta I = O, \] then \(2\beta - \alpha\) is equal to: 

• (1) 16
• (2) 2
• (3) 8
• (4) 10

\(\lim_{x \to 0} \frac{e - (1 + 2x)^{\frac{1}{2x}}}{x} \quad \text{is equal to:}\)

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

Consider the line \(L\) passing through the points \((1, 2, 3)\) and \((2, 3, 5)\). The distance of the point \[ \left( \frac{11}{3}, \frac{11}{3}, \frac{19}{3} \right) \] from the line \(L\) along the line \[ \frac{3x - 11}{2} = \frac{3y - 11}{1} = \frac{3z - 19}{2} \] is equal to: 

• (1) 3
• (2) 5
• (3) 4
• (4) 6

The area (in square units) of the region enclosed by the ellipse \[ x^2 + 3y^2 = 18 \] in the first quadrant below the line \( y = x \) is: 

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

Let \( a, ar, ar^2, \dots \) be an infinite G.P. If \[ \sum_{n=0}^\infty ar^n = 57 \quad \text{and} \quad \sum_{n=0}^\infty a^3 r^{3n} = 9747, \] then \( a + 18r \) is equal to: 

• (1) 27
• (2) 46
• (3) 38
• (4) 31

The sum of the coefficients of \( x^{2/3} \) and \( x^{-2/5} \) in the binomial expansion of \[ \left( x^{2/3} + \frac{1}{2} x^{-2/5} \right)^9 \] is: 

• (1) \( \frac{21}{4} \)
• (2) \( \frac{69}{16} \)
• (3) \( \frac{63}{16} \)
• (4) \( \frac{19}{4} \)

Let \[ \vec{a} = 2\hat{i} + \alpha \hat{j} + \hat{k}, \quad \vec{b} = -\hat{i} + \hat{k}, \quad \vec{c} = \beta \hat{j} - \hat{k}, \] where \( \alpha \) and \( \beta \) are integers and \( \alpha \beta = -6 \). Let the values of the ordered pair \( (\alpha, \beta) \) for which the area of the parallelogram of diagonals \( \vec{a} + \vec{b} \) and \( \vec{b} + \vec{c} \) is \( \frac{\sqrt{21}}{2} \), be \( (\alpha_1, \beta_1) \) and \( (\alpha_2, \beta_2) \). Then \( \alpha_1^2 + \beta_1^2 - \alpha_2 \beta_2 \) is equal to:

• (1) 17
• (2) 24
• (3) 21
• (4) 19

Let \[ A = \{(x, y) : 2x + 3y = 23, \, x, y \in \mathbb{N}\} \] and \[ B = \{x : (x, y) \in A\}. \] Then the number of one-one functions from \( A \) to \( B \) is equal to _________ .

Let \( A, B, \) and \( C \) be three points on the parabola \( y^2 = 6x \), and let the line segment \( AB \) meet the line \( L \) through \( C \) parallel to the \( x \)-axis at the point \( D \). Let \( M \) and \( N \) respectively be the feet of the perpendiculars from \( A \) and \( B \) on \( L \). Then \[ \left( \frac{\text{AM} \cdot \text{BN}}{\text{CD}} \right)^2 \] is equal to ______ .

Consider the matrices: \[ A = \begin{bmatrix} 2 & -5 \\ 3 & m \end{bmatrix}, \quad B = \begin{bmatrix} 20 \\ m \end{bmatrix}, \quad \text{and} \quad X = \begin{bmatrix} x \\ y \end{bmatrix}. \] Let the set of all \( m \), for which the system of equations \( AX = B \) has a negative solution (i.e., \( x < 0 \) and \( y <0 \)), be the interval \( (a, b) \). Then \[ 8 \int_a^b |\det(A)| \, dm \] is equal to _____ .

For a differentiable function \( f : \mathbb{R} \to \mathbb{R} \), suppose \[ f'(x) = 3f(x) + \alpha, \] where \( \alpha \in \mathbb{R} \), \( f(0) = 1 \), and \[ \lim_{x \to -\infty} f(x) = 7. \] Then \( 9f(-\log_2 3) \) is equal to __________ .

Consider the circle \( C : x^2 + y^2 = 4 \) and the parabola \( P : y^2 = 8x \). If the set of all values of \( \alpha \), for which three chords of the circle \( C \) on three distinct lines passing through the point \( (\alpha, 0) \) are bisected by the parabola \( P \), is the interval \( (p, q) \), then \( (2q - p)^2 \) is equal to ________ .

The square of the distance of the image of the point \( (6, 1, 5) \) in the line \[ \frac{x - 1}{3} = \frac{y}{2} = \frac{z - 2}{4}, \] from the origin is _____ .

Let the set of all values of \( p \), for which \[ f(x) = (p^2 - 6p + 8)(\sin^2 2x - \cos^2 2x) + 2(2 - p)x + 7 \] does not have any critical point, be the interval \( (a, b) \). Then \( 16ab \) is equal to _____ .

The number of integers, between 100 and 1000 having the sum of their digits equals to 14, is______ .

Let the inverse trigonometric functions take principal values. The number of real solutions of the equation \[ 2 \sin^{-1} x + 3 \cos^{-1} x = \frac{2\pi}{5}, \] is ______ .

If \[ \left( \frac{1}{\alpha + 1} + \frac{1}{\alpha + 2} + \dots + \frac{1}{\alpha + 1012} \right) - \left( \frac{1}{2 \cdot 1} + \frac{1}{4 \cdot 3} + \frac{1}{6 \cdot 5} + \dots + \frac{1}{2024 \cdot 2023} \right) \]

The effective resistance between A and B, if resistance of each resistor is R, will be

• (1) \( \frac{2R}{3} \)
• (2) \( \frac{8R}{3} \)
• (3) \( \frac{5R}{3} \)
• (4) \( \frac{4R}{3} \)

In the truth table of the above circuit the value of X and Y are

• (1) 1,1
• (2) 1,0
• (3) 0,1
• (4) 0,0

Two cars are travelling towards each other at a speed of \( 20 \, \text{m/s} \) each. When the cars are \( 300 \, \text{m} \) apart, both the drivers apply brakes and the cars retard at the rate of \( 2 \, \text{m/s}^2 \). The distance between them when they come to rest is:

• (1) 200 m
• (2) 50 m
• (3) 100 m
• (4) 25 m

A nucleus at rest disintegrates into two smaller nuclei with their masses in the ratio of 2:1. After disintegration they will move

• (1) In opposite directions with speed in the ratio of 1:2 respectively
• (2) In opposite directions with speed in the ratio of 2:1 respectively
• (3) In the same direction with same speed.
• (4) In opposite directions with the same speed.

A satellite of \( 10^3 \, \text{kg} \) mass is revolving in a circular orbit of radius \( 2R \). If \[ \frac{10^4 R}{6} \, \text{J} \] energy is supplied to the satellite, it would revolve in a new circular orbit of radius: \[ \text{(use } g = 10 \, \text{m/s}^2, \, R = \text{radius of earth)}. \]

• (1) 2.5 R
• (2) 3 R
• (3) 4 R
• (4) 6 R

The magnetic field in a plane electromagnetic wave is \[ B_y = (3.5 \times 10^{-7}) \sin \left( 1.5 \times 10^3 x + 0.5 \times 10^{11} t \right) \, \text{T}. \] The corresponding electric field will be:

• (1) \( E_y = 1.17 \sin \left( 1.5 \times 10^3 x + 0.5 \times 10^{11} t \right) \, \text{V/m} \)
• (2) \( E_z = 105 \sin \left( 1.5 \times 10^3 x + 0.5 \times 10^{11} t \right) \, \text{V/m} \)
• (3) \( E_z = 1.17 \sin \left( 1.5 \times 10^3 x + 0.5 \times 10^{11} t \right) \, \text{V/m} \)
• (4) \( E_y = 10.5 \sin \left( 1.5 \times 10^3 x + 0.5 \times 10^{11} t \right) \, \text{V/m} \)

A real gas within a closed chamber at \( 27^\circ \text{C} \) undergoes the cyclic process as shown in the figure. The gas obeys \( PV^3 = RT \) equation for the path \( A \) to \( B \). The net work done in the complete cycle is (assuming \( R = 8 \, \text{J/mol K} \)):

• (1) 225 J
• (2) 205 J
• (3) 20 J
• (4) -20 J

Five charges +q, +5q, –2q, +3q and –4q are situated as shown in the figure. The electric flux due to this configuration through the surface S is

• (1) \( \frac{5q}{\epsilon_0} \)
• (2) \( \frac{4q}{\epsilon_0} \)
• (3) \( \frac{3q}{\epsilon_0} \)
• (4) \( \frac{q}{\epsilon_0} \)

UV light of 4.13 eV is incident on a photosensitive metal surface having work function 3.13 eV. The maximum kinetic energy of ejected photoelectrons will be

• (1) 4.13 eV
• (2) 1 eV
• (3) 3.13 eV
• (4) 7.26 eV

The I-V characteristics of an electronic device shown in the figure. The device is

• (1) a solar cell
• (2) a transistor which can be used as an amplifier
• (3) a zener diode which can be used as voltage regulator
• (4) a diode which can be used as a rectifier

The energy released in the fusion of 2 kg of hydrogen deep in the sun is \(E_H\) and the energy released in the fission of 2 kg of \(^{235}U\) is \(E_U\). The ratio \(\frac{E_H}{E_U}\) is approximately:

• (1) 9.13
• (2) 15.04
• (3) 7.62
• (4) 25.6

A hydrogen atom in ground state is given an energy of 10.2 eV. How many spectral lines will be emitted due to transition of electrons ?

• (1) 6
• (2) 3
• (3) 10
• (4) 1

A proton and a deuteron (\( q = +e, \, m = 2.0u \)) having the same kinetic energies enter a region of uniform magnetic field \( \vec{B} \), moving perpendicular to \( \vec{B} \). The ratio of the radius \( r_d \) of the deuteron path to the radius \( r_p \) of the proton path is:

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

The temperature of a gas is \(-78^\circ\text{C}\) and the average translational kinetic energy of its molecules is \( K \). The temperature at which the average translational kinetic energy of the molecules of the same gas becomes \( 2K \) is:

• (1) \(-39^\circ\text{C}\)
• (2) \(117^\circ\text{C}\)
• (3) \(127^\circ\text{C}\)
• (4) \(-78^\circ\text{C}\)

The excess pressure inside a soap bubble is thrice the excess pressure inside a second soap bubble. The ratio between the volume of the first and the second bubble is :

• (1) 1 : 9
• (2) 1 : 3
• (3) 1 : 81
• (4) 1 : 27

A spherical ball of radius \( 1 \times 10^{-4} \, \text{m} \) and density \( 10^5 \, \text{kg/m}^3 \) falls freely under gravity through a distance \( h \) before entering a tank of water. If after entering in water the velocity of the ball does not change, then the value of \( h \) is approximately: \[ \text{(The coefficient of viscosity of water is } 9.8 \times 10^{-6} \, \text{N s/m}^2 \text{)} \]

• (1) 2296 m
• (2) 2249 m
• (3) 2518 m
• (4) 2396 m

The de-Broglie wavelength associated with a particle of mass \( m \) and energy \( E \) is \[\lambda = \frac{h}{\sqrt{2mE}}.\]The dimensional formula for Planck's constant is:

• (1) \([ML^{-1}T^{-2}]\)
• (2) \([ML^2T^{-1}]\)
• (3) \([MLT^{-2}]\)
• (4) \([M^2L^2T^{-2}]\)

A square loop of side 15 cm being moved towards right at a constant speed of 2 cm/s as shown in figure. The front edge enters the 50 cm wide magnetic field at t = 0. The value of induced emf in the loop at t = 10 s will be :

• (1) 0.3 mV
• (2) 4.5 mV
• (3) zero
• (4) 3 mV

A 1 kg mass is suspended from the ceiling by a rope of length 4m. A horizontal force 'F' is applied at the mid point of the rope so that the rope makes an angle of 45° with respect to the vertical axis as shown in figure. The magnitude of F is

• (1) \( \frac{10}{\sqrt{2}} \, \text{N} \)
• (2) \( 1 \, \text{N} \)
• (3) \( \frac{1}{10 \times \sqrt{2}} \, \text{N} \)
• (4) \( 10 \, \text{N} \)

The following figure represents two biconvex lenses \( L_1 \) and \( L_2 \) having focal lengths 10 cm and 15 cm, respectively. The distance between \( L_1 \) and \( L_2 \) is:

• (1) 10 cm
• (2) 15 cm
• (3) 25 cm
• (4) 35 cm

The resultant of two vectors \( \vec{A} \) and \( \vec{B} \) is perpendicular to \( \vec{A} \) and its magnitude is half that of \( \vec{B} \). The angle between vectors \( \vec{A} \) and \( \vec{B} \) is ________ .

An electric field \( \vec{E} = (2x \hat{i}) \, \text{N C}^{-1} \) exists in space. A cube of side \( 2 \, \text{m} \) is placed in the space as per the figure given below. The electric flux through the cube is __________ \( \text{N m}^2/\text{C} \).

To determine the resistance \( R \) of a wire, a circuit is designed below. The V-I characteristic curve for this circuit is plotted for the voltmeter and the ammeter readings as shown in the figure. The value of \( R \) is \( \dots \dots \dots \Omega \).

A circular disc reaches from top to bottom of an inclined plane of length \( l \). When it slips down the plane, it takes \( t \) s. When it rolls down the plane, then it takes \[ \left( \frac{\alpha}{2} \right)^{1/2} t \, \text{s}, \] where \( \alpha \) is __________ .

A force \( (3x^2 + 2x - 5) \, \text{N} \) displaces a body from \( x = 2 \, \text{m} \) to \( x = 4 \, \text{m} \). The work done by this force is _________ J.

A particle of mass \( 0.50 \, \text{kg} \) executes simple harmonic motion under force \( F = -50 \, (\text{N m}^{-1}) x \). The time period of oscillation is \( \frac{x}{35} \, \text{s} \). The value of \( x \) is \( \dots \dots \dots \dots \). \[ \text{(Given } \pi = \frac{22}{7} \text{)} \]

At room temperature \( 27^\circ\text{C} ,\) the resistance of a heating element is \(50 \, \Omega\). The temperature coefficient of the material is \(2.4 \times 10^{-4}\)  \(^\circ\text{C}^{-1} \). The temperature of the element, when its resistance is \(62 \, \Omega\) , is _________\(\degree C\).

A straight magnetic strip has a magnetic moment of \( 44 \, \text{A m}^2 \). If the strip is bent in a semicircular shape, its magnetic moment will be \( \dots \) A m\(^2\). \[ \text{(Given } \pi = \frac{22}{7} \text{)} \]

Monochromatic light of wavelength \( 500 \, \text{nm} \) is used in Young's double slit experiment. An interference pattern is obtained on a screen. When one of the slits is covered with a very thin glass plate (refractive index \( = 1.5 \)), the central maximum is shifted to a position previously occupied by the 4^th bright fringe. The thickness of the glass plate is ___________ \( \mu \text{m} \).

A capacitor of reactance \( 4 \sqrt{3} \, \Omega \) and a resistor of resistance \( 4 \, \Omega \) are connected in series with an AC source of peak value \( 8 \sqrt{2} \, \text{V} \). The power dissipation in the circuit is ___________ W.

In the above reaction product ’P’ is

• (1)
• (2)
• (3)
• (4)

The correct stability order of the following resonance structures of \( \text{CH}_3 - \text{CH} = \text{CH} - \text{CHO} \) is:

• (1) II > III > I
• (2) III > II > I
• (3) I > II > III
• (4) II > I > III

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

The electronic configuration of Einsteinium is :(Given atomic number of Einsteinium = 99)

• (1) [Rn] \( 5f^{12} 6d^0 7s^2 \)
• (2) [Rn] \( 5f^{11} 6d^0 7s^2 \)
• (3) [Rn] \( 5f^{13} 6d^0 7s^2 \)
• (4) [Rn] \( 5f^{10} 6d^0 7s^2 \)

Which of the following compound can give positive iodoform test when treated with aqueous KOH solution followed by potassium hypoiodite.

• (1)
• (2)
• (3)
• (4)

Given below are two statements :
Statement I : The higher oxidation states are more stable down the group among transition elements unlike p-block elements.
Statement II : Copper can not liberate hydrogen from weak acids.
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) Both Statement I and Statement II are true
• (4) Statement I is true but Statement II is false

Which of the following compounds will give silver mirror with ammoniacal silver nitrate?
(A) Formic acid
(B) Formaldehyde
(C) Benzaldehyde
(D) Acetone
Choose the correct answer from the options given below :

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

Total number of stereo isomers possible for the given structure :

• (1) 4
• (2) 2
• (3) 8
• (4) 3

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

The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency \( 'A' \times 10^{12} \, \text{hertz} \) and that has a radiant intensity in that direction of \( \frac{1}{'B'} \, \text{watt per steradian} \). 'A' and 'B' are respectively:

• (1) 540 and \( \frac{1}{683} \)
• (2) 540 and 683
• (3) 450 and \( \frac{1}{683} \)
• (4) 450 and 683

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

Which out of the following is a correct equation to show change in molar conductivity with respect to concentration for a weak electrolyte, if the symbols carry their usual meaning

• (1) \( \Lambda_m^2 C - K_a \Lambda_m + K_a \Lambda_m^{\circ 2} = 0 \)
• (2) \( \Lambda_m - \Lambda_m^{\circ} + AC \frac{1}{2} = 0 \)
• (3) \( \Lambda_m - \Lambda_m^{\circ} - AC \frac{1}{2} = 0 \)
• (4) \( \Lambda_m^2 C + K_a \Lambda_m^{\circ 2} - K_a \Lambda_m^{\circ} = 0 \)

The correct increasing order for bond angles among \( \text{BF}_3, \, \text{PF}_3, \, \text{and} \, \text{CF}_3 \) is:

• (1) \( \text{PF}_3 \, < \, \text{BF}_3 \, < \, \text{CF}_3 \)
• (2) \( \text{BF}_3 \, < \, \text{PF}_3 \, < \, \text{CF}_3 \)
• (3) \( \text{CF}_3 \, < \, \text{PF}_3 \, < \, \text{BF}_3 \)
• (4) \( \text{BF}_3 \, = \, \text{PF}_3 \, < \, \text{CF}_3 \)

For a sparingly soluble salt \( \text{AB}_2 \), the equilibrium concentrations of \( \text{A}^{2+} \) ions and \( \text{B}^- \) ions are \( 1.2 \times 10^{-4} \, \text{M} \) and \( 0.24 \times 10^{-3} \, \text{M} \), respectively. The solubility product of \( \text{AB}_2 \) is:

• (1) \( 0.069 \times 10^{-12} \)
• (2) \( 6.91 \times 10^{-12} \)
• (3) \( 0.276 \times 10^{-12} \)
• (4) \( 27.65 \times 10^{-12} \)

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

Match List I with List II

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

The incorrect statement about Glucose is :

• (1) Glucose is soluble in water because of having aldehyde functional group
• (2) Glucose remains in multiple isomeric form in its aqueous solution
• (3) Glucose is an aldohexose
• (4) Glucose is one of the monomer unit in sucrose

Major product of the following reaction is

• (1)
• (2)
• (3)
• (4)

The coordination environment of \( \text{Ca}^{2+} \) ion in its complex with \( \text{EDTA}^{4-} \) is:

• (1) tetrahedral
• (2) octahedral
• (3) square planar
• (4) trigonal prismatic

The incorrect statement regarding ethyne is

• (1) The C–C bonds in ethyne is shorter than that in ethene
• (2) Both carbons are sp hybridised
• (3) Ethyne is linear
• (4) The carbon-carbon bonds in ethyne is weaker than that in ethene

Based on Heisenberg's uncertainty principle, the uncertainty in the velocity of the electron to be found within an atomic nucleus of diameter \( 10^{-15} \, \text{m} \) is \( \dots \dots  \times 10^9 \, \text{ms}^{-1} \) (nearest integer). \[ \text{[Given: mass of electron} = 9.1 \times 10^{-31} \, \text{kg, Planck's constant (} h \text{)} = 6.626 \times 10^{-34} \, \text{Js]} \] \[ \text{(Value of } \pi = 3.14) \]

In the given TLC, the distance of spot \( A \) and \( B \) are 5 cm and 7 cm, from the bottom of the TLC plate, respectively. The \( R_f \) value of \( B \) is \( x \times 10^{-1} \) times more than \( A \). The value of \( x \) is _________ .

Number of compounds from the following which cannot undergo Friedel-Crafts reactions is :____
toluene, nitrobenzene, xylene, cumene, aniline, chlorobenzene, m-nitroaniline, m-dinitrobenzene

Total number of electrons present in \( (\pi^*) \) molecular orbitals of \( \text{O}_2 \), \( \text{O}_2^+ \), and \( \text{O}_2^- \) is __________ .

A transition metal 'M' among Sc, Ti, V , Cr, Mn and Fe has the highest second ionisation enthalpy. The spin only magnetic moment value of M+
ion is ______ BM (Near integer) (Given atomic number Sc : 21, Ti : 22, V : 23, Cr : 24, Mn : 25, Fe : 26)

Consider the following first-order gas phase reaction at constant temperature \[ \text{A(g)} \rightarrow 2\text{B(g)} + \text{C(g)} \] If the total pressure of the gases is found to be 200 torr after 23 sec. and 300 torr upon the complete decomposition of \( \text{A} \) after a very long time, then the rate constant of the given reaction is \( \dots \dots \times 10^{-2} \, \text{s}^{-1} \) (nearest integer). \[ \text{[Given: } \log_{10}(2) = 0.301 \text{]} \]

When \( \Delta H_{\text{vap}} = 30 \, \text{kJ/mol} \) and \( \Delta S_{\text{vap}} = 75 \, \text{J mol}^{-1} \text{K}^{-1} \), then the temperature of vapour, at one atmosphere, is \( \dots \dots \dots \, \text{K} \).

Consider the following test for a group-IV cation. \[ \text{M}^{2+} + \text{H}_2\text{S} \rightarrow \text{A (Black precipitate)} + \text{byproduct} \] \[ \text{A + aqua regia} \rightarrow \text{B + NOCl + S + H}_2\text{O} \] \[ \text{B + KNO}_2 + \text{CH}_3\text{COOH} \rightarrow \text{C + byproduct} \] The spin-only magnetic moment value of the metal complex \( \text{C} \) is \( \dots \dots \dots \) BM. (Nearest integer)

The number of oxygen atoms present in chemical formula of fuming sulphuric acid is _______.

The vapour pressure of pure benzene and methyl benzene at \( 27^\circ \text{C} \) is given as 80 Torr and 24 Torr, respectively. The mole fraction of methyl benzene in vapour phase, in equilibrium with an equimolar mixture of those two liquids (ideal solution) at the same temperature, is \( \dots  \times 10^{-2} \) (nearest integer).