JEE Main 2023 Question Paper Jan 31 Shift 1- Download PDF

JEE Main 2023 Questions with Solutions

Section A

A bar magnet with a magnetic moment 5.0 Am² is placed in parallel position relative to a magnetic field of 0.4 T. The amount of required work done in turning the magnet from parallel to antiparallel position relative to the field direction is:

• (1) 4 J
• (2) 1 J
• (3) 2 J
• (4) Zero

If a source of electromagnetic radiation having power 15 kW produces \( 10^{16} \) photons per second, the radiation belongs to a part of spectrum is:

• (1) Micro waves
• (2) Ultraviolet rays
• (3) Gamma rays
• (4) Radio waves

The amplitude of \( 15 \sin(1000 \pi t) \) is modulated by \( 10 \sin(4 \pi t) \) signal. The amplitude modulated signal contains frequency(ies) of:

(A) 500 Hz

(B) 2 Hz

(C) 250 Hz

(D) 498 Hz

(E) 502 Hz

Choose the correct answer from the options given below:

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

As shown in figure, a 70 kg garden roller is pushed with a force of \( F = 200 \, N \) at an angle of 30° with horizontal. The normal reaction on the roller is: (Given \( g = 10 \, m/s^2 \))


 

• (1) \( 800 \sqrt{2} \, N \)
• (2) 600 N
• (3) 800 N
• (4) \( 200 \sqrt{3} \, N \)

The initial speed of a projectile fired from ground is \( u \). At the highest point during its motion, the speed of the projectile is \( \frac{\sqrt{5}}{2} u \). The time of flight of the projectile is:

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

Spherical insulating ball and a spherical metallic ball of the same size and mass are dropped from the same height. Choose the correct statement out of the following (Assume negligible air friction):

• (1) Time taken by them to reach the earth's surface will be independent of the properties of their materials
• (2) Insulating ball will reach the earth's surface earlier than the metal ball
• (3) Both will reach the earth's surface simultaneously
• (4) Metal ball will reach the earth's surface earlier than the insulating ball

A free neutron decays into a proton but a free proton does not decay into neutron. This is because:

• (1) Neutron is an uncharged particle
• (2) Proton is a charged particle
• (3) Neutron is a composite particle made of a proton and an electron
• (4) Neutron has larger rest mass than proton

The effect of increase in temperature on the number of electrons in conduction band (\( n_e \)) and resistance of a semiconductor will be:

• (1) Both \( n_e \) and resistance decrease
• (2) Both \( n_e \) and resistance increase
• (3) \( n_e \) increases, resistance decreases
• (4) \( n_e \) decreases, resistance increases

The maximum potential energy of a block executing simple harmonic motion is 25 J. \( A \) is amplitude of oscillation. At \( \frac{A}{2} \), the kinetic energy of the block is:

• (1) 37.5 J
• (2) 9.75 J
• (3) 18.75 J
• (4) 12.5 J

The pressure of a gas changes linearly with volume from A to B as shown in the figure. If no heat is supplied to or extracted from the gas, then change in the internal energy of the gas will be:

• (1) 6 J
• (2) Zero
• (3) -4.5 J
• (4) 4.5 J

Which of the following correctly represents the variation of electric potential (\( V \)) of a charged spherical conductor of radius (\( R \)) with radial distance (\( r \)) from the centre?

• (1)
• (2)
• (3)
• (4)

Correct Answer: (3)
View Solution

Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R
Assertion A: The beam of electrons shows wave nature and exhibits interference and diffraction.
Reason R: Davisson Germer experimentally verified the wave nature of electrons.
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) A is not correct but R is correct
• (3) Both A and R are correct but R is not the correct explanation of A
• (4) Both A and R are correct and R is the correct explanation of A

The drift velocity of electrons for a conductor connected in an electrical circuit is \( V_d \). The conductor is now replaced by another conductor with the same material and same length but double the area of cross-section. The applied voltage remains the same. The new drift velocity of electrons will be:

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

At a certain depth \( d \) below the surface of the Earth, the value of acceleration due to gravity becomes four times that of its value at a height 3R above the Earth's surface. Where \( R \) is the radius of Earth (Take \( R = 6400 \, km \)). The depth \( d \) is equal to:

• (1) 5260 km
• (2) 640 km
• (3) 2560 km
• (4) 4800 km

If 1000 droplets of water of surface tension 0.07 N/m, having same radius 1 mm each, combine to form a single drop. In the process, the released surface energy is:
(Take \( \pi = \frac{22}{7} \))

• (1) \( 7.92 \times 10^{-6} \, J \)
• (2) \( 7.92 \times 10^{-4} \, J \)
• (3) \( 9.68 \times 10^{-4} \, J \)
• (4) \( 8.8 \times 10^{-5} \, J \)

A rod with circular cross-section area 2 cm\(^2\) and length 40 cm is wound uniformly with 400 turns of an insulated wire. If a current of 0.4 A flows in the wire windings, the total magnetic flux produced inside the windings is \( 4 \times 10^{-6} \, Wb \). The relative permeability of the rod is:
(Given: Permeability of vacuum \( \mu_0 = 4 \pi \times 10^{-7} \, N/A^2 \))

• (1) 12.5
• (2) \( \frac{32}{5} \)
• (3) 125
• (4) \( \frac{5}{16} \)

The correct relation between \( \gamma = \frac{C_p}{C_v} \) and temperature \( T \) is:

• (1) \( \gamma \propto \frac{1}{\sqrt{T}} \)
• (2) \( \gamma \propto T^0 \)
• (3) \( \gamma \propto \frac{1}{T} \)
• (4) \( \gamma \propto T \)

Two polaroide A and B are placed in such a way that the pass-axis of polaroids are perpendicular to each other. Now, another polaroid C is placed between A and B bisecting the angle between them. If intensity of unpolarised light is \( I_0 \), then intensity of transmitted light after passing through polaroid B will be:

• (1) \( \frac{I_0}{4} \)
• (2) \( \frac{I_0}{2} \)
• (3) \( \frac{I_0}{8} \)
• (4) Zero

If \( R \), \( X_L \), and \( X_C \) represent resistance, inductive reactance, and capacitive reactance, then which of the following is dimensionless:

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

100 balls each of mass \( m \) moving with speed \( v \) simultaneously strike a wall normally and are reflected back with the same speed, in time \( t \). The total force exerted by the balls on the wall is:

• (1) \( \frac{100mv}{t} \)
• (2) \( \frac{200mv}{t} \)
• (3) \( \frac{200mvt}{t} \)
• (4) \( \frac{mv}{100t} \)

A thin rod having a length of 1 m and area of cross-section \( 3 \times 10^{-6} \, m^2 \) is suspended vertically from one end. The rod is cooled from 210°C to 160°C. After cooling, a mass \( M \) is attached at the lower end of the rod such that the length of the rod again becomes 1 m. Young's modulus and the coefficient of linear expansion of the rod are \( 2 \times 10^{11} \, N/m^2 \) and \( 2 \times 10^{-5} \, K^{-1} \), respectively. The value of \( M \) is ______ kg. (Take \( g = 10 \, m/s^2 \))

The speed of a swimmer is 4 km/h in still water. If the swimmer makes his strokes normal to the flow of the river of width 1 km, he reaches a point 750 m down the stream on the opposite bank. The speed of the river water is ........ km/h.

In the figure given below, a block of mass \( M = 490 \, g \) is placed on a frictionless table is connected with two springs having the same spring constant (\( K = 2 \, N/m \)). If the block is horizontally displaced through \( X \) m, then the number of complete oscillations it will make in 14t seconds will be:
\begin{figure[h]
\centering

\end{figure

In a medium, the speed of light wave decreases to 0.2 times its speed in free space. The ratio of relative permittivity to the refractive index of the medium is \( x : 1 \). The value of \( x \) is .......
(Given speed of light in free space \( c = 3 \times 10^8 \, m/s \) and for the given medium \( \mu_r = 1 \))

A solid sphere of mass 1 kg rolls without slipping on a plane surface. Its kinetic energy is \( 7 \times 10^3 \) J. The speed of the centre of mass of the sphere is ....... cm/s.

An inductor of 0.5 mH, a capacitor of 20 μF, and resistance of 20 Ω are connected in series with a 220 V AC source. If the current is in phase with the emf, the amplitude of current of the circuit is \( \sqrt{K} \) A. The value of \( x \) is:

Expression for an electric field is given by \( \vec{E} = 4000 \hat{i} \, V/m \). The electric flux through the cube of side 20 cm when placed in the electric field (as shown in the figure) is ....... Vcm.
\begin{figure[h]
\centering

\end{figure

A lift of mass \( M = 500 \, kg \) is descending with a speed of \( 2 \, m/s \). Its supporting cable begins to slip, thus allowing it to fall with a constant acceleration of \( 2 \, m/s^2 \). The kinetic energy of the lift at the end of fall through a distance of 6 m will be ....... kJ.

For hydrogen atom, \( \lambda_1 \) and \( \lambda_2 \) are the wavelengths corresponding to the transitions 1 and 2, respectively, as shown in the figure. The ratio of \( \lambda_1 \) and \( \lambda_2 \) is \( \frac{x}{32} \). The value of \( x \) is:
\begin{figure[h]
\centering

\end{figure

Two identical cells, when connected either in parallel or in series, give the same current in an external resistance of 5Ω. The internal resistance of each cell will be ........ \Omega.

The methods NOT involved in concentration of ore are

(A) Liquation

(B) Leaching

(C) Electrolysis

(D) Hydraulic washing

(E) Froth flotation


Choose the correct answer from the options given below:

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

Consider the following reaction \[ Propanal + Methanal \xrightarrow{(i) dil. NaOH} \xrightarrow{(ii) A} \xrightarrow{(iii) NaCN} \xrightarrow{(iv) H_2O} Product B \, (C_5H_8O_3) \]

The correct statement for product B is:

• (1) Optically active and adds one mole of bromine
• (2) Racemic mixture and is neutral
• (3) Racemic mixture and gives a gas with saturated NaHCO₃ solution
• (4) Optically active alcohol and is neutral

The correct order of basicity of oxides of vanadium is:

• (1) \( V_2O_3 > V_2O_4 > V_2O_5 \)
• (2) \( V_2O_3 > V_2O_5 > V_2O_4 \)
• (3) \( V_2O_5 > V_2O_3 > V_2O_4 \)
• (4) \( V_2O_4 > V_2O_3 > V_2O_5 \)

When \( Cu^{2+} \) ion is treated with KI, a white precipitate, X appears in solution. The solution is titrated with sodium thiosulphate, the compound Y is formed. X and Y respectively are:

• (1) \( X = CuI_2, \, Y = Na_2S_4O_5 \)
• (2) \( X = CuI_2, \, Y = Na_2S_4O_6 \)
• (3) \( X = CuI, \, Y = Na_2S_4O_3 \)
• (4) \( X = CuI, \, Y = Na_2S_4O_6 \)

The reaction sequence is:

• (1)
• (2)
• (3)
• (4)

Correct Answer: (4)
View Solution

Cobalt chloride when dissolved in water forms pink colored complex X which has octahedral geometry. This solution on treating with cone HCl forms deep blue complex Y which has a Z geometry. X, Y and Z, respectively, are:

• (1) \( X = [Co(H_2O)_6]^{2+}, \, Y = [CoCl_4]^{2-}, \, Z = Tetrahedral \)
• (2) \( X = [Co(H_2O)_6]^{3+}, \, Y = [CoCl_3]^{3-}, \, Z = Octahedral \)
• (3) \( X = [Co(H_2O)_6]^{3+}, \, Y = [CoCl_4]^{2-}, \, Z = Octahedral \)
• (4) \( X = [Co(H_2O)_6]^{2+}, \, Y = [CoCl_4]^{2-}, \, Z = Tetrahedral \)

Identify X, Y, and Z in the following reaction. (Equation not balanced) \[ ClO + NO_2 \rightarrow X \xrightarrow{H_2O} Y + Z \]

• (1) \( X = ClONO_2, \, Y = HOCl, \, Z = NO_2 \)
• (2) \( X = ClONO_2, \, Y = HCl, \, Z = HNO_3 \)
• (3) \( X = ClONO_2, \, Y = HOCl, \, Z = HNO_3 \)
• (4) \( X = ClO_3, \, Y = Cl_2, \, Z = NO_2 \)

The correct order of melting point of dichlorobenzenes is:

• (1)
• (2)
• (3)
• (4)

Correct Answer: (4)
View Solution

A protein 'X' with molecular weight of 70,000 u, on hydrolysis gives amino acids. One of these amino acids is

• (1)
• (2)
• (3)
• (4)

Correct Answer: (2)
View Solution

Which transition in the hydrogen spectrum would have the same wavelength as the Balmer type transition from \( n = 4 \) to \( n = 2 \) of He\(^{+}\) spectrum?

• (1) \( n = 2 \, to \, 1 \)
• (2) \( n = 1 \, to \, 3 \)
• (3) \( n = 1 \, to \, 2 \)
• (4) \( n = 3 \, to \, 4 \)

Match items of column I and II

Column I (Mixture of compounds)                            Column II (Separation Technique)
A. H₂O/CH₃Cl                                                                            i. Crystallization
B.                                                            ii. Differential solvent extraction
C. Kerosene/Naphthalene                                                        iii. Column chromatography

D. C₆H₅O/NaCl                                                                            iv. Fractional Distillation

Correct match is:

• (1) A-(iii), B-(iv), C-(ii), D-(i)
• (2) A-(ii), B-(iii), C-(ii), D-(iv)
• (3) A-(ii), B-(iii), C-(iv), D-(i)
• (4) A-(ii), B-(iv), C-(i), D-(iii)

The correct increasing order of the ionic radii is:

• (1) \( Cl^- < Ca^{2+} < K^+ < S^{2-} \)
• (2) \( K^+ < S^{2-} < Ca^{2+} < Cl^- \)
• (3) \( S^{2-} < Cl^- < Ca^{2+} < K^+ \)
• (4) \( Ca^{2+} < K^+ < Cl^- < S^{2-} \)

H\(_2\)O\(_2\) acts as a reducing agent in:

• (1) \( 2NaOCl + H_2O_2 \rightarrow 2NaCl + H_2O + O_2 \)
• (2) \( 2Fe^{2+} + 2H_2O_2 \rightarrow 2Fe^{3+} + 2H_2O \)
• (3) \( Mn^{2+} + 2H_2O_2 \rightarrow MnO_2 + 2H_2O \)
• (4) \( Na_2S_4O_6 + 4H_2O_2 \rightarrow Na_2SO_4 + 4H_2O \)

Which of the following artificial sweeteners has the highest sweetness value in comparison to cane sugar?

• (1) Aspartame
• (2) Sucralose
• (3) Alitame
• (4) Saccharin

Match List I with List II

List I                                                                      List II
A. \(XeF_4\)                                                         I. See-saw
B. \(SF_4\)                                                           II. Square planar
C. \(NH_4^+\)                                                      III. Bent T-shaped
D. \(BrF_3\)                                                          IV. Tetrahedral

Choose the correct answer from the options given below:

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

Choose the correct set of reagents for the following conversion \[ trans (Ph-CH=CH-CH_3) \rightarrow cis (Ph-CH=CH-CH_3) \]

• (1) \( Br_2, \, alc. KOH, \, NaNH_2, \, Na(Liq NH_3) \)
• (2) \( Br_2, \, aq KOH, \, NaNH_2, \, H_2 \, Lindlar Catalyst \)
• (3) \( Br_2, \, aq KOH, \, NaNH_2, \, Na(Liq NH_3) \)
• (4) \( Br_2, \, alc. KOH, \, NaNH_2, \, Lindlar Catalyst \)

Adding surfactants in non-polar solvent, the micelles structure will look like (Surfactant structure):

• (1) b
• (2) c
• (3) a
• (4) d

An organic compound 'A' with empirical formula \(C_6H_6\)O gives sooty flame on burning. Its reaction with bromine solution in low polarity solvent results in high yield of B. B is:

• (1)
• (2)
• (3)
• (4)

Correct Answer: (1)
View Solution

Which one of the following statements is correct for electrolysis of brine solution?

• (1) \( Cl_2 \) is formed at cathode
• (2) \( O_2 \) is formed at cathode
• (3) \( H_2 \) is formed at anode
• (4) \( OH^- \) is formed at cathode

SECTION B

Question 51:

The logarithm of equilibrium constant for the reaction \[ Pd^{2+} (aq) + 4Cl^- (aq) \rightleftharpoons PdCl_4^{2-} (aq) \]
is:

Given: \[ 2.303RT = 0.06\, V, \quad F = 96,485\, C/mol, \quad E^\circ = 0.83\, V, \quad E^\circ = 0.65\, V \]

\[ \textbf{A} \rightarrow \textbf{B} \]

The rate constants of the above reaction at 200 K and 300 K are 0.03 min\(^{-1}\) and 0.05 min\(^{-1}\) respectively.

The activation energy for the reaction is \hspace{2cm} J (Nearest integer)

Given: \[ \text{ln} \, 10 = 2.3, \quad R = 8.3 \, \text{J K}^{-1} \text{mol}^{-1}, \quad \log 5 = 0.70, \quad \log 3 = 0.48, \quad \log 2 = 0.30 \]

The enthalpy change for the conversion of \[ \frac{1}{2} Cl_2(g) \rightarrow Cl^-(aq) \, is \, (-) \, ...... kJ mol^{-1} \, (Nearest integer) \]

Given: \[ \Delta_{f}H^\circ(Cl_2(g)) = 240 \, kJ mol^{-1}, \quad \Delta_{g}H^\circ(Cl_2(g)) = -350 \, kJ mol^{-1}, \quad \Delta_{hyd}H^\circ(Cl^-) = -380 \, kJ mol^{-1} \]

On complete combustion, 0.492 g of an organic compound gave 0.792 g of \(CO_2\). The % of carbon in the organic compound is \underline{\hspace{2cm}} (Nearest integer).

Given: \[ \text{Weight of carbon in 0.792 g CO}_2 \]

At 27°C, a solution containing 2.5 g of solute in 250.0 mL of solution exerts an osmotic pressure of 400 Pa. The molar mass of the solute is \hspace{2cm} g mol\(^{-1}\) (Nearest integer).

Given: \[ R = 0.083 \, \text{L bar K}^{-1} \text{mol}^{-1}, \quad \pi = \text{osmotic pressure}, \quad T = 300 \, \text{K}, \quad \text{osmotic pressure} = 400 \, \text{Pa}, \quad 1 \, \text{bar} = 10^5 \, \text{Pa} \]

Zinc reacts with hydrochloric acid to give hydrogen and zinc chloride. The volume of hydrogen gas produced at STP from the reaction of 11.5 g of zinc with excess HCl is \hspace{2cm} L (Nearest integer).

Given: \[ \text{Molar mass of Zn} = 65.4 \, \text{g mol}^{-1}, \quad \text{Molar volume of H}_2 \, \text{at STP} = 22.7 \, \text{L/mol} \]

How many of the transformations given below would result in aromatic amines?

For reaction: \[ \text{SO}_2(g) + \frac{1}{2} \text{O}_2(g) \rightleftharpoons \text{SO}_3(g) \] \( K_p = 2 \times 10^{12} \) at 27°C and 1 atm pressure. The \( K_c \) for the same reaction is \hspace{2cm} (Nearest integer).

Given: \[ R = 0.082 \, \text{L atm K}^{-1} \text{mol}^{-1} \]

The oxidation state of phosphorus in hypophosphoric acid is ________________.

The total pressure of a mixture of non-reacting gases X (0.6 g) and Y (0.45 g) in a vessel is 740 mm of Hg. The partial pressure of the gas X is \hspace{2cm} mm of Hg (Nearest Integer).

Given: \[ \textbf{Molar mass of X} = 20 \, \textbf{g mol}^{-1}, \quad \textbf{Molar mass of Y} = 45 \, \textbf{g mol}^{-1} \]

For all \( z \in \mathbb{C} \) on the curve \( C \) such that \( | z | = 1 \), let the locus of the point \( z + \frac{1}{z} \) be the curve \( C_1 \). Then:

• (1) the curves \( C_1 \) and \( C_2 \) intersect at 4 points
• (2) the curves \( C_1 \) lies inside \( C_2 \)
• (3) the curves \( C_1 \) and \( C_2 \) intersect at 2 points
• (4) the curves \( C_2 \) lies inside \( C_1 \)

A wire of length 20 m is to be cut into two pieces. A piece of length \( \ell_1 \) is bent to make a square of area \( A_1 \), and the other piece of length \( \ell_2 \) is made into a circle of area \( A_2 \). If \( 2A_1 + 3A_2 \) is minimum, then \( \frac{\ell_1}{\ell_2} \) is equal to:

• (1) \( 6 : 1 \)
• (2) \( 3 : 1 \)
• (3) \( 1 : 6 \)
• (4) \( 4 : 1 \)

For the system of linear equations:

\( x + y + z = 6 \)

\( \alpha x + \beta y + 7z = 3 \)

\( x + 2y + 3z = 14 \)

Which of the following is NOT true?

• (1) If \( \alpha = \beta \), then the system has no solution
• (2) If \( \alpha = \beta \) and \( \alpha \neq 7 \), then the system has a unique solution.
• (3) There is a unique point \( (\alpha, \beta) \) on the line \( x + 2y + 18 = 0 \) for which the system has infinitely many solutions.
• (4) For every point \( (\alpha, \beta) \neq (7, 7) \) on the line \( x = 2y + 7 \), the system has infinitely many solutions.

Let the shortest distance between the lines \[ L: \frac{x - 5}{2} = \frac{y - \lambda}{0} = \frac{z + 1}{1}, \quad \lambda \geq 0 \quad and \quad L_1: x + 1 = y - 1 = 4 - z = 2\sqrt{6} \]
If \( (\alpha, \beta, \gamma) \) lies on \( L \), then which of the following is NOT possible?

• (1) \( \alpha + 2\gamma = 24 \)
• (2) \( 2\alpha + \gamma = 7 \)
• (3) \( 2\alpha - \gamma = 9 \)
• (4) \( \alpha - 2\gamma = 19 \)

Let \( y = f(x) \) represent a parabola with focus \( \left( -\frac{1}{2}, 0 \right) \) and directrix \( y = -\frac{1}{2} \).

Then \[ S = \left\{ x \in \mathbb{R} : \tan^{-1} \left( \sqrt{f(x)} + \sin \left( \sqrt{f(x) + 1} \right) \right) = \frac{\pi}{2} \right\} \]
contains:

• (1) Exactly two elements
• (2) Exactly one element
• (3) An infinite set
• (4) An empty set

Let \[ A = \begin{pmatrix} 1 & 0 & 0
0 & 4 & -1
0 & 12 & -3 \end{pmatrix} \]
Then the sum of the diagonal elements of the matrix \( (A + I)^{11} \) is equal to:

• (1) 6144
• (2) 4094
• (3) 4097
• (4) 2050

Let \( R \) be a relation on \( \mathbb{N} \times \mathbb{N} \) defined by \[ (a, b) \, R \, (c, d) \quad if and only if \quad ad(b - c) = bc(a - d). \]
Then \( R \) is:

• (1) Symmetric but neither reflexive nor transitive
• (2) Transitive but neither reflexive nor symmetric
• (3) Reflexive and symmetric but not transitive
• (4) Symmetric and transitive but not reflexive

Let \[ y = f(x) = \sin^3 \left( \frac{\pi}{3} \cos \left( \frac{\pi}{3\sqrt{2}} \left( -4x^3 + 5x^2 + 1 \right)^{\frac{3}{2}} \right) \right) \]
Then, at \( x = 1 \),

• (1) \( 2y' + 3y = 0 \)
• (2) \( 2y' + 3y' = 0 \)
• (3) \( y' - 3y' = 0 \)
• (4) \( y' + 3y' = 0 \)

If the sum and product of four positive consecutive terms of a G.P. are 126 and 1296, respectively, then the sum of common ratios of all such GPs is:

• (1) 7
• (2) \( \frac{9}{2} \)
• (3) 3
• (4) 14

The number of real roots of the equation \[ \sqrt{x^2 - 4x + 3} + \sqrt{x^2 - 9} = \sqrt{4x^2 - 14x + 6} \]
is:

• (1) 0
• (2) 1
• (3) 3
• (4) 2

Let a differentiable function \(f\) satisfy \[ f(x) + \int_3^x f(t) \, dt = \sqrt{x+1}, \quad x \geq 3. \]
Then \(f(8)\) is equal to:

• (1) 34
• (2) 19
• (3) 17
• (4) 1

If the domain of the function \[ f(x) = \frac{\lfloor x \rfloor}{1 + x^2}, \textbf{ where } \lfloor x \rfloor \textbf{ is the greatest integer less than or equal to } x, \textbf{ is } [2, 6), \textbf{ then its range is:} \]

• (1) \( \left[ \frac{5}{26}, \frac{9}{27} \right] \)
• (2) \( \left[ \frac{5}{26}, \frac{9}{29} \right] \)
• (3) \( \left[ \frac{9}{37}, \frac{29}{109} \right] \)
• (4) \( \left[ \frac{5}{37}, \frac{53}{37} \right] \)

Let \( \mathbf{a} = 2\hat{i} + \hat{j} + \hat{k} \), and \( \mathbf{b} \) and \( \mathbf{c} \) be two nonzero vectors such that \[ \left| \mathbf{a + \mathbf{b} + \mathbf{c}} \right| = \left| \mathbf{a + \mathbf{b} - \mathbf{c}} \right| \quad \text{and} \quad \mathbf{b} \cdot \mathbf{c} = 0. \]

Consider the following two statements:

(B) \( \mathbf{a} \) and \( \mathbf{c} \) are always parallel.

• (1) only (B) is correct
• (2) neither (A) nor (B) is correct
• (3) only (A) is correct
• (4) both (A) and (B) are correct

Let \( \alpha \in (0, 1) \) and \( \beta = \log(1 - \alpha) \). Let \[ P_n(x) = x + \frac{x^2}{2} + \frac{x^3}{3} + \cdots + \frac{x^n}{n}, \quad x \in (0, 1). \]

Then the integral

\[ \int_0^\alpha \frac{1}{1 - t} \, dt \] is equal to:

• (1) \( \beta - P_0(\alpha) \)
• (2) \( -(\beta + P_0(\alpha)) \)
• (3) \( P_0(\alpha) - \beta \)
• (4) \( \beta + P_0(\alpha) \)

If \( \sin^{-1} \left( \frac{\alpha}{17} \right) + \cos^{-1} \left( \frac{4}{5} \right) - \tan^{-1} \left( \frac{77}{36} \right) = 0, \quad 0 < \alpha < 13, \)
then \( \sin^{-1} (\sin \alpha) + \cos^{-1} (\cos \alpha) \) is equal to:

• (1) \( \pi \)
• (2) 16
• (3) 0
• (4) \( 16 - 5\pi \)

Let a circle \( C_1 \) be obtained on rolling the circle \[ x^2 + y^2 - 4x - 6y + 11 = 0 \] upwards 4 units on the tangent \( T \) to it at the point (3, 2). Let \( C_2 \) be the image of \( C_1 \) in \( T \).

Let A and B be the centers of circles \( C_1 \) and \( C_2 \) respectively, and M and N be respectively the feet of perpendiculars drawn from A and B on the x-axis. Then the area of the trapezium AMNB is:

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

(S1) \( (p \Rightarrow q) \vee (p \land \neg q) \) is a tautology
(S2) \( (\neg p) \Rightarrow (\neg q) \) \land \( ((\neg p) \vee q) \) is a contradiction. Then:

• (1) only (S2) is correct
• (2) both (S1) and (S2) are correct
• (3) both (S1) and (S2) are wrong
• (4) only (S1) is correct

The value of \[ \int \frac{(2 + 3 \sin x)}{\sin x (1 + \cos x)} \, dx \]
is equal to:

• (1) \( \frac{7}{2} \sqrt{3} - \log \sqrt{3} \)
• (2) \( 2 + 3\sqrt{3} + \log \sqrt{3} \)
• (3) \( \frac{10}{3} \sqrt{3} - \log \sqrt{3} \)
• (4) \( \sqrt{3} - \log \sqrt{3} \)

A bag contains 6 balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least 5 black balls is:

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

Section – B

Question 81:

Let 5 digit numbers be constructed using the digits 0, 2, 3, 4, 7, 9 with repetition allowed, and are arranged in ascending order with serial numbers.
Then the serial number of the number 42923 is:

Let \( a_1, a_2, \dots, a_n \) be in A.P. If \( a_5 = 2a_1 \text{ and } a_1 = 18, \) then

\[ 12 \left( \frac{1}{\sqrt{a_0} + \sqrt{a_1}} + \frac{1}{\sqrt{a_1} + \sqrt{a_2}} + \cdots + \frac{1}{\sqrt{a_{17}} + \sqrt{a_{18}}} \right) \]

is equal to:

Let \( \theta \) be the angle between the planes

\[ P_1: \vec{r} \cdot ( \hat{i} + \hat{j} + 2 \hat{k}) = 9 \quad \text{and} \quad P_2: \vec{r} \cdot (2 \hat{i} - \hat{j} + \hat{k}) = 15. \]

Let \( L \) be the line that meets \( P_2 \) at the point (4, -2, 5) and makes an angle \( \theta \) with the normal of \( P_2 \). If \( \alpha \) is the angle between \( L \) and \( P_2 \), then

\[ (\tan^2 \theta)(\cot^2 \alpha) \] is equal to:

Question 84:

Let α > 0, be the smallest number such that the expansion of (3/x³ + ²/_x)³⁰ has a term βx^-α, β ∈ ℕ. Then α is equal to:

Let \( \vec{a} \) and \( \vec{b} \) be two vectors such that \[ |\vec{a}| = \sqrt{14}, \quad |\vec{b}| = \sqrt{6}, \quad |\vec{a} \times \vec{b}| = \sqrt{48}. \]
Then \( (\vec{a} \cdot \vec{b})^2 \textbf{ is equal to:} \)

Let the line \( L: \frac{x-1}{2} = \frac{y+1}{-1} = \frac{z-3}{1} \) intersect the plane \[ 2x + y + 3z = 16 \textbf{ at the point } P. \]
Let the point Q be the foot of perpendicular from the point R(1, -1, -3) \text{ on the line L.
If \alpha is the area of triangle PQR, then \alpha^2 is equal to:

The remainder on dividing \( 5^{99} \) by 11 is:

If the variance of the frequency distribution

\[ \begin{array}{|c|c|c|c|c|c|c|c|} \hline x_i & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline f_i & 3 & 6 & 16 & \alpha & 9 & 5 & 6 \\ \hline \end{array} \]

is given, find the value of \( \alpha \).

Let for \( x \in \mathbb{R} \) \[ f(x) = \frac{x + |x|}{2} \quad for \quad x \geq 0 \quad and \quad f(x) = \frac{x}{2} \quad for \quad x < 0 \] \[ g(x) = \begin{cases} x^2 & for \quad x \geq 0,
x & for \quad x < 0 \end{cases} \]
\text{Then the area bounded by the curve \( y = f \circ g(x) \) \text{ and the lines \( y = 0, 2y - x = 15 \) \text{ is equal to:

Number of 4-digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11, is equal to: