JEE Main 2024 30Jan Shift 1 Question Paper (Available)- Download Solution PDF with Answer Key

Let \( g : \mathbb{R} \rightarrow \mathbb{R} \) be a non-constant twice differentiable function such that \( g'\left(\frac{1}{2}\right) = g'\left(\frac{3}{2}\right) \). If a real-valued function \( f \) is defined as \[ f(x) = \frac{1}{2} \left[ g(x) + g(2 - x) \right], \] then

• (1) \( f''(x) = 0 \) for at least two \( x \) in \( (0, 2) \)
• (2) \( f''(x) = 0 \) for exactly one \( x \) in \( (0, 1) \)
• (3) \( f''(x) = 0 \) for no \( x \) in \( (0, 1) \)
• (4) \( f'\left(\frac{3}{2}\right) + f'\left(\frac{1}{2}\right) = 1 \)

The value of \(\lim_{{n \to \infty}} \sum_{{k=1}}^{n} \frac{n^3}{{(n^2 + k^2)(n^2 + 3k^2)}}\) is 

• (1) \( \frac{13\pi}{8(4\sqrt{3} + 3)} \)
• (2) \( \frac{(2\sqrt{3} + 3) \pi}{24} \)
• (3) \( \frac{13(2\sqrt{3} - 3) \pi}{8} \)
• (4) \( \frac{\pi}{8(2\sqrt{3} + 3)} \)

If the circles \( (x+1)^2 + (y+2)^2 = r^2 \) and \( x^2 + y^2 - 4x - 4y + 4 = 0 \) intersect at exactly two distinct points, then

• (1) \( 3 < r < 7 \)
• (2) \( 0 < r < 7 \)
• (3) \( 5 < r < 9 \)
• (4) \( \frac{1}{2} < r < 7 \)

Let \( \vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \) and \( \vec{b} = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \) be two vectors such that \( |\vec{a}| = 1 \), \( \vec{a} \times \vec{b} = 2 \), and \( |\vec{b}| = 4 \). If \( \vec{c} = 2(\vec{a} \times \vec{b}) - 3\vec{b} \), then the angle between \( \vec{b} \) and \( \vec{c} \) is equal to:

• (1) \( \cos^{-1}\left(-\frac{\sqrt{3}}{2}\right) \)
• (2) \( \cos^{-1}\left(-\frac{1}{\sqrt{3}}\right) \)
• (3) \( \cos^{-1}\left(\frac{2}{\sqrt{3}}\right) \)
• (4) \( \cos^{-1}\left(\frac{2}{3}\right) \)

The maximum area of a triangle whose one vertex is at \( (0, 0) \) and the other two vertices lie on the curve \( y = -2x^2 + 54 \) at points \( (x, y) \) and \( (-x, y) \) where \( y > 0 \) is:

• (1) 88
• (2) 108
• (3) 92
• (4) 122

If \[ f(x) = \begin{vmatrix} 2 \cos^4 x & 2 \sin^4 x & 3 + \sin^2 2x \\ 3 + 2 \cos^4 x & 2 \sin^4 x & \sin^2 2x \\ 2 \cos^4 x & 3 + 2 \sin^4 x & \sin^2 2x \end{vmatrix} \] then \( \frac{1}{5} f'(0) \) is equal to

• (1) 0
• (2) 1
• (3) 2
• (4) 6

Let \(\alpha, \beta, \gamma\) be the foot of perpendicular from the point \((1, 2, 3)\) on the line \(\frac{x + 3}{5} = \frac{y - 1}{2} = \frac{z + 4}{3}\). Then \(19(\alpha + \beta + \gamma)\) is equal to:

• (1) 102
• (2) 101
• (3) 99
• (4) 100

A line passing through the point \( A(9, 0) \) makes an angle of \( 30^\circ \) with the positive direction of the x-axis. If this line is rotated about \( A \) through an angle of \( 15^\circ \) in the clockwise direction, then its equation in the new position is:

• (1) \(\frac{x}{\sqrt{3} + 2} + y = 9\)
• (2) \(\frac{y}{\sqrt{3} - 2} + x = 9\)
• (3) \(\frac{x}{\sqrt{3} + 2} + y = 9\)
• (4) \(\frac{x}{\sqrt{3} - 2} + y = 9\)

Consider the system of linear equations \( x + y + z = 4\mu \), \( x + 2y + 2\lambda z = 10\mu \), \( x + 3y + 4\lambda^2 z = \mu^2 + 15 \), where \( \lambda, \mu \in \mathbb{R} \). Which one of the following statements is NOT correct?

• (1) The system has a unique solution if \( \lambda \neq \frac{1}{2} \) and \( \mu \neq 1, 15 \).
• (2) The system has an infinite number of solutions if \( \lambda = \frac{1}{2} \) and \( \mu = 15 \).
• (3) The system is inconsistent if \( \lambda = \frac{1}{2} \) and \( \mu \neq 1 \).
• (4) The system is consistent if \( \lambda \neq \frac{1}{2} \).

Let \( f : \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \rightarrow \mathbb{R} \) be a differentiable function such that \( f(0) = \frac{1}{2} \). If the \( \lim_{x \to 0} \frac{\int_{0}^{x} f(t) \, dt}{e^{x^2} - 1} = \alpha \), then \( 8\alpha^2 \) is equal to:

• (1) 16
• (2) 2
• (3) 1
• (4) 4

If the length of the minor axis of an ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is.

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

• (1) 416
• (2) 104
• (3) 52
• (4) 208

Two integers \( x \) and \( y \) are chosen with replacement from the set \( \{0, 1, 2, 3, \ldots, 10\} \). Then the probability that \( |x - y| > 5 \) is:

• (1) \( \frac{30}{121} \)
• (2) \( \frac{62}{121} \)
• (3) \( \frac{60}{121} \)
• (4) \( \frac{31}{121} \)

Let \( A (2, 3, 5) \) and \( C(-3, 4, -2) \) be opposite vertices of a parallelogram \( ABCD \). If the diagonal \( \overrightarrow{BD} = \hat{i} + 2 \hat{j} + 3 \hat{k} \), then the area of the parallelogram is equal to

• (1) \( \frac{1}{2} \sqrt{410} \)
• (2) \( \frac{1}{2} \sqrt{474} \)
• (3) \( \frac{1}{2} \sqrt{586} \)
• (4) \( \frac{1}{2} \sqrt{306} \)

Let \(y = y(x)\) be the solution of the differential equation \(\sec(x) \frac{dy}{dx} + [2(1 - x) \tan(x) + x(2 - x)] = 0\) such that \(y(0) = 2\). Then \(y(2)\) is equal to:

• (1) 2
• (2) 2\(1 - sin2\)
• (3) 2\(\sin(2) + 1\)
• (4) 1

If \(z = x + iy, \quad xy \neq 0\) satisfies the equation \(z^2 + iz = 0\), then \(|z^2|\); equal to
 

If \( 2 \sin^3 x + \sin 2x \cos x + 4 \sin x - 4 = 0 \) has exactly 3 solutions in the interval \( \left[ 0, \frac{n \pi}{2} \right] \), \( n \in \mathbb{N} \), then the roots of the equation \( x^2 + nx + (n - 3) = 0 \) belong to:

• (1) \( (0, \infty) \)
• (2) \( (-\infty, 0) \)
• (3) \( \left( -\frac{\sqrt{17}}{2}, \frac{\sqrt{17}}{2} \right) \)
• (4) \( \mathbb{Z} \)

If the domain of the function \[ f(x) = \cos^{-1} \left( \frac{2 - |x|}{4} \right) + \left( \log_e (3 - x) \right)^{-1} \] is \([-\alpha, \beta) - \{ \gamma \}\), then \( \alpha + \beta + \gamma \) is equal to:

• (1) 12
• (2) 9
• (3) 11
• (4) 8

Let \( S_n \) denote the sum of the first \( n \) terms of an arithmetic progression. If \( S_{20} = 790 \) and \( S_{10} = 145 \), then \( S_{15} - S_5 \) is:

• (1) 390
• (2) 395
• (3) 405
• (4) 410

The area (in square units) of the region bounded by the parabola \(y^2 = 4(x - 2)\) and the line \(y = 2x - 8\) is:

• (1) 8
• (2) 9
• (3) 6
• (4) 7

A group of 40 students appeared in an examination of 3 subjects – Mathematics, Physics & Chemistry. It was found that all students passed in at least one of the subjects, 20 students passed in Mathematics, 25 students passed in Physics, 16 students passed in Chemistry, at most 11 students passed in both Mathematics and Physics, at most 15 students passed in both Physics and Chemistry, at most 15 students passed in both Mathematics and Chemistry. The maximum number of students passed in all the three subjects is ________.

Let \( y = y(x) \) be the solution of the differential equation \( (1 - x^2) \, dy = \left[ xy + \left( x^3 + 2 \right) \sqrt{3 \left( 1 - x^2 \right)} \right] dx \), \( -1 < x < 1, y(0) = 0 \). If \( y\left( \frac{1}{2} \right) = \frac{m}{n} \), \( m \) and \( n \) are coprime numbers, then \( m + n \) is equal to \(\_\_\_\_\_\_\_\_\_\).

The value \( 9 \int_{0}^{9} \left\lfloor \frac{10x}{x+1} \right\rfloor \, dx \), where \( \left\lfloor t \right\rfloor \) denotes the greatest integer less than or equal to \( t \), is ________.

Let \( \alpha, \beta \in \mathbb{N} \) be roots of the equation \( x^2 - 70x + \lambda = 0 \), where \( \frac{\lambda}{2}, \frac{\lambda}{3} \notin \mathbb{N} \). If \( \lambda \) assumes the minimum possible value, then \[ \frac{\left( \sqrt{\alpha - 1} + \sqrt{\beta - 1} \right)(\lambda + 35)}{|\alpha - \beta|} \] is equal to____.

Number of integral terms in the expansion of \( \left( \sqrt{7} z + \frac{1}{6 \sqrt{z}} \right)^{824} \) is equal to ______.

Let the latus rectum of the hyperbola \( \frac{x^2}{9} - \frac{y^2}{b^2} = 1 \) subtend an angle of \( \frac{\pi}{3} \) at the center of the hyperbola. If \( b^2 \) is equal to \( \frac{1}{m} \left( 1 + \sqrt{n} \right) \), where \( l \) and \( m \) are coprime numbers, then \( l^2 + m^2 + n^2 \) is equal to \(\_\_\_\_\_\_\_\_\_\).

If \( d_1 \) is the shortest distance between the lines  \(\frac{x+1}{2} = \frac{y-1}{-12} = \frac{z-1}{-7} \quad \text{and} \quad \frac{x-1}{7} = \frac{y+8}{2} = \frac{z-4}{5},\) and \( d_2 \) is the shortest distance between the lines \(\frac{x-1}{2} = \frac{y-2}{1} = \frac{z-6}{-3} \quad \text{and} \quad \frac{x+2}{1} = \frac{y+2}{2} = \frac{z-1}{6},\) then the value of \( \frac{32\sqrt{3}d_1}{d_2} \) is:

Let \( \alpha = 1^2 + 4^2 + 8^2 + 13^2 + 19^2 + 26^2 + \ldots \) up to 10 terms and \( \beta = \sum_{n=1}^{10} n^4 \). If \( 4\alpha - \beta = 55k + 40 \), then \( k \) is equal to \(\_\_\_\_\_\_\_\_\_\).

Let \( A = \{1, 2, 3, \ldots, 7\} \) and let \( P(A) \) denote the power set of \( A \). If the number of functions \( f : A \rightarrow P(A) \) such that \( a \in f(a), \, \forall a \in A \) is \( m^n \), \( m \) and \( n \in \mathbb{N} \) and \( m \) is least, then \( m + n \) is equal to \(\_\_\_\_\_\_\_\_\_\).

If the function \( f(x) = \begin{cases} \frac{1}{|x|}, & |x| \geq 2 \\ ax^2 + 2b, & |x| < 2 \end{cases} \) is differentiable on \( \mathbb{R} \), then \( 48 (a + b) \) is equal to \(\_\_\_\_\_\_\_\_\_\).

Theratio of the magnitude of the kinetic energy to the potential energy of an electron in the 5th excited state of a hydrogen atom is:

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

The work function of a substance is 3.0 eV. The longest wavelength of light that can cause the emission of photoelectrons from this substance is approximately:

• (1) 215 nm
• (2) 400 nm
• (3) 414 nm
• (4) 200 nm

Primary coil of a transformer is connected to 220 V ac. Primary and secondary turns of the transforms are 100 and 10 respectively. Secondary coil of transformer is connected to two series resistance shown in shown in figure. The output voltage (V0) is :

• (1) 15 V
• (2) 7 V
• (3) 44 V
• (4) 22 V

A particle of mass m is projected from ground with speed u at an angle of 30° with the horizontal. Find its angular momentum about the point of projection when it reaches its maximum height.

• (1) \(\frac{mu^3}{16g}\)
• (2) \({\frac{\sqrt3mu^3}{16g}}\)
• (3) \(\frac{mu^3}{3g}\)
• (4) Zero

The diffraction pattern of a light of wavelength 400 nm diffracting from a slit of width 0.2 mm is focused on the focal plane of a convex lens of focal length 100 cm. The width of the 1st secondary maxima will be :

• (1) 2 mm
• (2) 2 cm
• (3) 0.02 mm
• (4) 0.2 mm

A series \( L, R \) circuit connected with an AC source \( E = (25 \sin 1000 \, t) \, \text{V} \) has a power factor of \( \frac{1}{\sqrt{2}} \). If the source of emf is changed to \( E = (20 \sin 2000 \, t) \, \text{V} \), the new power factor of the circuit will be:

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

The electrostatic potential due to an electric dipole at a distance \( r \) varies as:

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

A particle is placed at the point \( A \) of a frictionless track \( ABC \) as shown in the figure. It is gently pushed toward the right. The speed of the particle when it reaches the point \( B \) is: (Take \( g = 10 \, \text{m/s}^2 \)).

• (1) 20 m/s
• (2) \( \sqrt{10} \, \text{m/s} \)
• (3) \( 2 \sqrt{10} \, \text{m/s} \)
• (4) 10 m/s

Two thermodynamical processes are shown in the figure. The molar heat capacity for process \( A \) and \( B \) are \( C_A \) and \( C_B \). The molar heat capacity at constant pressure and constant volume are represented by \( C_P \) and \( C_V \), respectively. Choose the correct statement.

• (1) \( C_B = \infty, \, C_A = 0 \)
• (2) \( C_A = 0 \, \text{and} \, C_B = \infty \)
• (3) \( C_P \gt C_V \gt C_A = C_B \)
• (4) \( C_A \gt C_P \gt C_V \)

The electric field of an electromagnetic wave in free space is represented as \( \vec{E} = E_0 \cos(\omega t - kz) \hat{i} \). The corresponding magnetic induction vector will be:

• (1) \( \vec{B} = E_0 C \cos(\omega t - kz) \hat{j} \)
• (2) \( \vec{B} = \frac{E_0}{C} \cos(\omega t - kz) \hat{j} \)
• (3) \( \vec{B} = E_0 C \cos(\omega t + kz) \hat{j} \)
• (4) \( \vec{B} = \frac{E_0}{C} \cos(\omega t + kz) \hat{j} \)

At which temperature the r.m.s. velocity of a hydrogen molecule equal to that of an oxygen molecule at 47ºC?

• (1) 80 K
• (2) -73 K
• (3) 20 K
• (4) 4 K

Match List-I with List-II. 
\[ \begin{array}{|c|c|c|} \hline \text{List-I} & & \text{List-II} \\ \hline \text{A. Coefficient of viscosity} & & \text{I.} \; [M L^2 T^{-2}] \\ \text{B. Surface Tension} & & \text{II.} \; [M L^2 T^{-1}] \\ \text{C. Angular momentum} & & \text{III.} \; [M L^1 T^{-1}] \\ \text{D. Rotational kinetic energy} & & \text{IV.} \; [M L^0 T^{-2}] \\ \hline \end{array} \]

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

A potential divider circuit is shown in figure. The output voltage V₀ is

• (1) 4V
• (2) 2 mV
• (3) 0.5 V
• (4) 12 mV

A Zener diode of breakdown voltage 10V is used as a voltage regulator as shown in the figure. The current through the Zener diode is

• (1) 50 mA
• (2) 0
• (3) 30 mA
• (4) 20 mA

Two insulated circular loop A and B radius ‘a’ carrying a current of ‘I’ in the anti clockwise direction as shown in figure. The magnitude of the magnetic induction at the centre will be :

• (1) \( \frac{\sqrt{2} \mu_0 I}{a} \)
• (2) \( \frac{\mu_0 I}{2a} \)
• (3) \( \frac{\mu_0 I}{\sqrt{2} a} \)
• (4) \( \frac{2 \mu_0 I}{a} \)

All surfaces shown in figure are assumed to be frictionless and the pulleys and the string are light. The acceleration of the block of mass 2 kg is :

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

An electric toaster has resistance of \( 60 \, \Omega \) at room temperature \( (27^\circ \text{C}) \). The toaster is connected to a 220 V supply. If the current flowing through it reaches 2.75 A, the temperature attained by the toaster is around: (if \( \alpha = 2 \times 10^{-4} / ^\circ \text{C} \))

• (1) 1694\( ^\circ \)C
• (2) 1235\( ^\circ \)C
• (3) 694\( ^\circ \)C
• (4) 1667\( ^\circ \)C

A spherical body of mass 100 g is dropped from a height of 10 m from the ground. After hitting the ground, the body rebounds to a height of 5 m. The impulse of force imparted by the ground to the body is given by: (given \( g = 9.8 \, \text{m/s}^2 \))

• (1) 4.32 kg ms\(^{-1}\)
• (2) 43.2 kg ms\(^{-1}\)
• (3) 23.9 kg ms\(^{-1}\)
• (4) 2.39 kg ms\(^{-1}\)

Young’s modules of material of a wire of length ‘L’ and cross-sectional area A is Y. If the length of the wire is doubled and cross-sectional area is halved then Young’s modules will be :

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

The gravitational potential at a point above the surface of earth is \( -5.12 \times 10^7 \, \text{J} / \text{kg} \) and the acceleration due to gravity at that point is \( 6.4 \, \text{m/s}^2 \). Assume that the mean radius of earth to be \( 6400 \, \text{km} \). The height of this point above the earth’s surface is:

• (1) 1200 km
• (2) 540 km
• (3) 1600 km
• (4) 1000 km

The horizontal component of earth’s magnetic field at a place is \( 3.5 \times 10^{-5} \, \text{T} \). A very long straight conductor carrying current of \( \sqrt{2} \, \text{A} \) in the direction from South East to North West is placed. The force per unit length experienced by the conductor is \( \ldots \times 10^{-6} \, \text{N/m} \).

Two cells are connected in opposition as shown. Cell \( E_1 \) is of 8 V emf and 2 \( \Omega \) internal resistance; the cell \( E_2 \) is of 2 V emf and 4 \( \Omega \) internal resistance. The terminal potential difference of cell \( E_2 \) is:

An electron of a hydrogen atom in an excited state is having energy \( E_n = -0.85 \, \text{eV} \). The maximum number of allowed transitions to lower energy levels is \( \ldots \).

A capacitor of capacitance \( C \) and potential \( V \) has energy \( E \). It is connected to another capacitor of capacitance \( 2C \) and potential \( 2V \). Then the loss of energy is \( \frac{x}{3} E \), where \( x \) is \( \_\_\_\_ \).

The distance between object and its two times magnified real image as produced by a convex lens is 45 cm. The focal length of the lens used is ______ cm.

In a closed organ pipe, the frequency of the fundamental note is 30 Hz. A certain amount of water is now poured in the organ pipe so that the fundamental frequency is increased to 110 Hz. If the organ pipe has a cross-sectional area of 2 cm\(^2\), the amount of water poured in the organ tube is \( \_\_\_\_\_\_ \) g. (Take speed of sound in air as 330 m/s)

Consider a disc of mass 5 kg, radius 2 m, rotating with angular velocity of 10 rad/s about an axis perpendicular to the plane of rotation. An identical disc is kept gently over the rotating disc along the same axis. The energy dissipated so that both the discs continue to rotate together without slipping is \( \_\_\_\_ \) J.

The displacement and the increase in the velocity of a moving particle in the time interval of \( t \) to \( (t + 1) \) s are 125 m and 50 m/s, respectively. The distance travelled by the particle in \( (t + 2)^{\text{th}} \) s is \( \_\_\_\_ \) m.

Each of three blocks \( P \), \( Q \), and \( R \) shown in the figure has a mass of 3 kg. Each of the wires \( A \) and \( B \) has a cross-sectional area of 0.005 cm\(^2\) and Young's modulus \( 2 \times 10^{11} \, \text{N} \, \text{m}^{-2} \). Neglecting friction, the longitudinal strain on wire \( B \) is \( \ldots \times 10^{-4} \). (Take \( g = 10 \, \text{m/s}^2 \))

A ceiling fan having 3 blades of length 80 cm each is rotating with an angular velocity of 1200 rpm. The magnetic field of earth in that region is 0.5 G and the angle of dip is \( 30^\circ \). The emf induced across the blades is \( N \pi \times 10^{-5} \, \text{V} \). The value of \( N \) is \( \_\_\_\_\_ \).

Aluminium chloride in acidified aqueous solution forms an ion having geometry:

• (1) Square planar
• (2) Tetrahedral
• (3) Octahedral
• (4) Trigonal bipyramidal

Example of vinylic halide is

• (1)

• (2)

• (3)

• (4)

Given below are two statements:
Statement-I: The gas liberated on warming a salt with dil H₂SO₄, turns a piece of paper dipped in lead acetate into black, it is a confirmatory test for sulphide ion.
Statement-II: In statement-I the colour of paper turns black because of formation of lead sulphite. 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 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.

The final product A, formed in the following multistep reaction sequence is:

• (1)

• (2)

• (3)

• (4)

• (1) Etard Reaction
• (2) Stephen's Reaction
• (3) Wolff Kishner Reduction
• (4) Rosenmund reduction

Compound A formed in the following reaction reacts with B gives the product C. Find out A and B.\[ \text{CH}_3 - \text{C} \equiv \text{CH} + \text{Na} \xrightarrow{} \text{A}\xrightarrow{B} \text{CH}_3 - \text{C} \equiv \text{C} - \text{CH}_2 - \text{CH}_2 + \text{NaBr}(C)\]

• (1)

• (2)

• (3)

• (4)

Which of the following molecule/species is most stable?

• (1)

• (2)

• (3)

• (4)

What happens to freezing point of benzene when small quantity of napthalene is added to benzene?

• (1) Increases
• (2) Remains unchanged
• (3) First decreases and then increases
• (4) Decreases

Choose the correct option

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

Match List-I with List-II
\[ \begin{array}{|c|c|} \hline \text{List-I (Species)} & \text{List-II (Electronic distribution)} \\ \hline \text{(A) Cr}^{+2} & \text{(I) } 3d^8 \\ \text{(B) Mn}^{+} & \text{(II) } 3d^3 4s^1 \\ \text{(C) Ni}^{+2} & \text{(III) } 3d^4 \\ \text{(D) V}^{+} & \text{(IV) } 3d^5 4s^1 \\ \hline \end{array} \]

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

The Lassaigne’s extract is boiled with dil HNO\(_3\) before testing for halogens because

• (1) AgCN is soluble in HNO\(_3\)
• (2) Silver halides are soluble in HNO\(_3\)
• (3) Ag\(_2\)S is soluble in HNO\(_3\)
• (4) Na\(_2\)S and NaCN are decomposed by HNO\(_3\)

Given below are two statements:
Statement-I: The orbitals having same energy are called as degenerate orbitals.
Statement-II: In hydrogen atom, 3p and 3d orbitals are not degenerate orbitals.
In the light of the above statements, choose the most appropriate answer from the options given

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

Sugar which does not give reddish brown precipitate with Fehling‟s reagent is:

• (1) Lactose
• (2) Maltose
• (3) Sucrose
• (4) Glucose

In the given reactions identify the reagent A and reagent B

• (1) \( A-\text{CrO}_3 \quad B-\text{CrO}_3\)
• (2) \(A-\text{CrO}_3 \quad B-\text{CrO}_2\text{Cl}_2\)
• (3) \(A-\text{CrO}_2\text{Cl}_2 \quad B-\text{CrO}_2\text{Cl}_2\)
• (4) \(A-\text{CrO}_2\text{Cl}_2 \quad B-\text{CrO}_3\)

Diamagnetic Lanthanoid ions are:

• (1) Nd\(^{3+}\) and Eu\(^{3+}\)
• (2) Nd\(^{3+}\) and Ce\(^{4+}\)
• (3) La\(^{3+}\) and Ce\(^{4+}\)
• (4) Lu\(^{3+}\) and Eu\(^{3+}\)

Choose the correct statements from the following:
A Ethane-1,2-diamine is a chelating ligand.
B Metallic aluminium is produced by electrolysis of aluminium oxide in presence of cryolite.
C Cyanide ion is used as a ligand for leaching of silver.
D Phosphine acts as a ligand in Wilkinson catalyst.
E The stability constants of \( \text{Ca}^{2+} \) and \( \text{Mg}^{2+} \) are similar to EDTA complexes.

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

Following is a confirmatory test for aromatic primary amines. Identify reagent (A) and (B)

• (1)

• (2)

• (3)

• (4)

Given below are two statement one is labeled as Assertion (A) and the other is labeled as Reason (R).
Assertion (A): CH₂ = CH – CH₂ – Cl is an example of allyl halide
Reason (R): Allyl halides are the compounds in which the halogen atom is attached to sp² hybridised carbon atom.
In the light of the two above statements, choose the most appropriate answer from the options given below

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

Given below are the two statements: one is labelled as Assertion (A) and the other is labeled as Reason (R).
Assertion (A): There is a considerable increase in covalent radius from N to P. However from As to Bi only a small increase in covalent radius is observed.
Reason (R): covalent and ionic radii in a particular oxidation state increases down the group.
In the light of the above statement, choose the most appropriate answer from the options given below:

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

Structure of 4-Methylpent-2-enal is

• (1)

• (2)

• (3)

• (4)

The total number of molecular orbitals formed from 2s and 2p atomic orbitals of a diatomic molecule is _________.

The mass of sodium acetate (CH₃COONa) required to prepare 250 mL of 0.35 M aqueous solution is g. (Molar mass of CH₃COONa is 82.02 g/mol)

0.05 cm thick coating of silver is deposited on a plate of 0.05 m² area. The number of silver atoms deposited on plate are _____ × 10²³. (At mass Ag = 108,d = 7.9 g/cm³)

The compound formed by the reaction of ethanal with semicarbazide contains _____number of nitrogen atoms.

The pH at which Mg(OH)\(_2\) \([K_{sp} = 1 \times 10^{-11}]\) begins to precipitate from a solution containing 0.10 M Mg\(^{2+}\) ions is \( \_\_\_\_ \).

\[ 2 \text{MnO}_4^{-} + b \text{I}^{-} + c \text{H}_2\text{O} \rightarrow x \text{I}_2 + y \text{MnO}_2 + z \text{OH}^{-} \]
If the above equation is balanced with integer coefficients, the value of \( z \) is \( \_\_\_\_\_\_ \).

The rate of a first order reaction is \( 0.04 \, \text{mol} \, \text{L}^{-1} \, \text{s}^{-1} \) at 10 minutes and \( 0.03 \, \text{mol} \, \text{L}^{-1} \, \text{s}^{-1} \) at 20 minutes after initiation. Half-life of the reaction is \( \_\_\_\_ \) minutes. (Given \( \log 2 = 0.3010, \, \log 3 = 0.4771 \))

On a thin layer chromatographic plate, an organic compound moved by 3.5 cm, while the solvent moved by 5 cm. The retardation factor of the organic compound is \( \_\_\_\_ \times 10^{-1} \).

If IUPAC name of an element is “Unununnium” then the element belongs to nth group of periodic table. The value of n is______