JEE Main 28 Jan 2026, Shift 2

75 questions. Answers are hidden — tap “Show answer” under each one.

Physics

Q1 · MCQ · NucleiOpen
A nucleus has mass number $\alpha$ and radius $R_\alpha$ . Another nucleus has mass number $\beta$ and radius $R_\beta$ . If $\beta = 8\alpha$ then $R_\alpha/R_\beta$ is:
1. (1)2
2. (2)8
3. (3)1
4. (4)0.5
Q2 · MCQ · Electromagnetic WavesOpen
A plane electromagnetic wave is moving in free space with velocity $c = 3\times10^{8}$ m/s and its electric field is given as $\vec E = 54\sin(kz - \omega t)\,\hat j\ \mathrm{V/m}$ , where $\hat j$ is the unit vector along y-axis. The magnetic field vector $\vec B$ of the wave is:
1. (1) $-1.8\times10^{-7}\sin(kz - \omega t)\,\hat i\ \mathrm{T}$
2. (2) $1.4\times10^{-7}\sin(kz - \omega t)\,\hat k\ \mathrm{T}$
3. (3) $1.4\times10^{-7}\sin(kz - \omega t)\,\hat i\ \mathrm{T}$
4. (4) $+1.8\times10^{-7}\sin(kz - \omega t)\,\hat i\ \mathrm{T}$
Q3 · MCQ · Ray OpticsOpen
A biconvex lens is formed by using two thin planoconvex lenses, as shown in the figure. The refractive index and radius of curved surfaces are also mentioned in figure. When an object is placed on the left side of lens at a distance of 30 cm from the biconvex lens, the magnification of the image will be:
1. (1) $-2$
2. (2) $+2$
3. (3) $+2.5$
4. (4) $-2.5$
Q4 · MCQ · Kinetic Theory of GasesOpen
The mean free path of a molecule of diameter $5\times10^{-10}$ m at the temperature $41^\circ$ C and pressure $1.38\times10^{5}$ Pa, is given as m. (Given $k_B = 1.38\times10^{-23}$ J/K).
1. (1) $2\sqrt2\times10^{-10}$
2. (2) $10\sqrt2\times10^{-8}$
3. (3) $2\sqrt2\times10^{-8}$
4. (4) $2\times10^{-8}$
Q5 · MCQ · Semiconductor ElectronicsOpen
Two p-n junction diodes $D_1$ and $D_2$ are connected as shown in figure. A and B are input signals and C is the output. The given circuit will function as a .
1. (1)OR Gate
2. (2)NOR Gate
3. (3)NAND Gate
4. (4)AND Gate
Q6 · MCQ · Current ElectricityOpen
A wheatstone bridge is initially at room temperature and all arms of the bridge have same value of resistances ( $R_1 = R_2 = R_3 = R_4$ ). When $R_3$ resistance is heated to some temperature, its resistance value has gone up by 10%. The potential difference $(V_a - V_b)$ (after $R_3$ is heated) is V.
1. (1)1.05
2. (2)0
3. (3)0.95
4. (4)2
Q7 · MCQ · Units and MeasurementsOpen
In an experiment, a set of readings are obtained 1.24 mm, 1.25 mm, 1.23 mm, 1.21 mm. The expected least count of the instrument used in recording these readings is mm.
1. (1)0.01
2. (2)0.001
3. (3)0.1
4. (4)0.05
We corrected a printing error in the original paper
Source printed “a set of reading are obtained $-1.24$ mm” (stray minus sign on the first reading).

Q8 · MCQ · Motion in a Straight LineOpen

A particle starts moving from time

t = 0

and its coordinate is given as

x(t) = 4t^3 - 3t

A.	The particle returns to its original position (origin) 0.866 units later
B.	The particle is 1 unit away from origin at its turning point.
C.	Acceleration of the particle is non-negative.
D.	The particle is 0.5 units away from origin at its turning point.
E.	Particle never turns back as acceleration is non-negative.

Choose the correct answer from the options given below:

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

Q9 · MCQ · WavesOpen
The speed of a longitudinal wave in a metallic bar is 400 m/s. If the density and Young's modulus of the bar material are increased by 0.5% and 1% respectively then the speed of the wave is changed approximately to m/s.
1. (1)399
2. (2)398
3. (3)402
4. (4)401

Q10 · MCQ · Electric Potential and CapacitanceOpen

Identify the correct statements:

A.	Effective capacitance of a series combination of capacitors is always smaller than the smallest capacitance of the capacitor in the combination.
B.	When a dielectric medium is placed between the charged plates of a capacitor, displacement of charges cannot occur due to insulation property of dielectric.
C.	Increasing of area of capacitor plate or decreasing of thickness of dielectric is an alternate method to increase the capacitance.
D.	For a point charge, concentric spherical shells centered at the location of the charge are equipotential surfaces.

Choose the correct answer from the options given below.

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

Q11 · MCQ · Dual Nature of Radiation and MatterOpen
Number of photons of equal energy emitted per second by a 6 mW laser source operating at 663 nm is . (Given: $h = 6.63\times10^{-34}$ J.s and $c = 3\times10^{8}$ m/s)
1. (1) $5\times10^{16}$
2. (2) $5\times10^{15}$
3. (3) $10\times10^{15}$
4. (4) $2\times10^{16}$
Q12 · MCQ · Rotational MotionOpen
When the position vector $\vec r = x\hat i + y\hat j + z\hat k$ changes sign as $-\vec r$ , which one of the following vectors will not flip under sign change?
1. (1)Linear momentum
2. (2)Velocity
3. (3)Acceleration
4. (4)Angular momentum
We corrected a printing error in the original paper
Source printed “which one of the following vector”.
Q13 · MCQ · Electric Potential and CapacitanceOpen
Which one of the following is not a measurable quantity?
1. (1)Voltage difference
2. (2)Resistance
3. (3)Voltage
4. (4)Displacement current

Q14 · MCQ · Moving Charges and MagnetismOpen

A long cylindrical conductor with large cross section carries an electric current distributed uniformly over its cross-section. Magnetic field due to this current is:

A.	maximum at either ends of the conductor and minimum at the midpoint
B.	maximum at the axis of the conductor
C.	minimum at the surface of the conductor
D.	minimum at the axis of the conductor
E.	same at all points in the cross-section of the conductor

Choose the correct answer from the options given below:

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

Q15 · MCQ · Newton's Laws of MotionOpen
A small block of mass $m$ slides down from the top of a frictionless inclined surface, while the inclined plane is moving towards left with constant acceleration $a_0$ . The angle between the inclined plane and ground is $\theta$ and its base length is $L$ . Assuming that initially the small block is at the top of the inclined plane, the time it takes to reach the lowest point of the inclined plane is .
1. (1) $\sqrt{\dfrac{2L}{g\sin2\theta - a_0(1+\cos2\theta)}}$
2. (2) $\sqrt{\dfrac{4L}{g\sin2\theta - a_0(1+\cos2\theta)}}$
3. (3) $\sqrt{\dfrac{4L}{g\cos^2\theta - a_0\sin\theta\cos\theta}}$
4. (4) $\sqrt{\dfrac{2L}{g\sin\theta - a_0\cos\theta}}$

Q16 · MCQ · Electric Charges and FieldsOpen

Identify the correct statements:

A.	Electrostatic field lines form closed loops.
B.	The electric field lines point radially outward when charge is greater than zero.
C.	The Gauss-Law is valid only for inverse-square force.
D.	The work done in moving a charged particle in a static electric field around a closed path is zero.
E.	The motion of a particle under Coulomb's force must take place in a plane.

Choose the correct answer from the options given below:

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

We corrected a printing error in the original paper

Source printed “The workdone”.

Q17 · MCQ · OscillationsOpen
As shown in the figure, a spring is kept in a stretched position with some extension by holding the masses 1 kg and 0.2 kg with a separation more than spring natural length and are released. Assuming the horizontal surface to be frictionless, the angular frequency (in SI unit) of the system is:
1. (1)30
2. (2)27
3. (3)20
4. (4)5
Q18 · MCQ · Ray OpticsOpen
For a transparent prism, if the angle of minimum deviation is equal to its refracting angle, the refractive index $n$ of the prism satisfies.
1. (1) $\sqrt2 < n < 2\sqrt2$
2. (2) $1 < n < 2$
3. (3) $n \ge 2$
4. (4) $\sqrt2 < n < 2$
Q19 · MCQ · Units and MeasurementsOpen
The time period of a simple harmonic oscillator is $T = 2\pi\sqrt{\dfrac{m}{k}}$ . The measured value of mass ( $m$ ) of the object is 10 g with an accuracy of 10 mg, and time for 50 oscillations of the spring is found to be 60 s using a watch of 2 s resolution. Percentage error in determination of spring constant( $k$ ) is %.
1. (1)3.43
2. (2)3.35
3. (3)7.60
4. (4)6.76
We corrected a printing error in the original paper
Source printed “ $T = 2\pi\sqrt{k/m}$ ” ( $k$ and $m$ interchanged).

Q20 · MCQ · Units and MeasurementsOpen

Match List-I with List-II.

	List-I		List-II
A.	Coefficient of viscosity	I.	$[\mathrm{ML^{-1}T^{-2}}]$
B.	Surface tension	II.	$[\mathrm{ML^{2}T^{-2}}]$
C.	Pressure	III.	$[\mathrm{ML^{0}T^{-2}}]$
D.	Surface energy	IV.	$[\mathrm{ML^{-1}T^{-1}}]$

Choose the correct answer from the options given below:

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

Q21 · Numerical · WavesOpen
Two tuning forks A and B are sounded together giving rise to 8 beats in 2 s. When fork A is loaded with wax, the beat frequency is reduced to 4 beats in 2 s. If the original frequency of tuning fork B is 380 Hz, then the original frequency of tuning fork A is Hz.
Q22 · Numerical · ThermodynamicsOpen
A thermodynamic system is taken through the cyclic process ABC as shown in the figure. The total work done by the system during the cycle ABC is J.
Q23 · Numerical · Electromagnetic InductionOpen
An inductor stores 16 J of magnetic field energy and dissipates 32 W of thermal energy due to its resistance when an a.c. current of 2 A (rms) and frequency 50 Hz flows through it. The ratio of inductive reactance to its resistance is . ( $\pi = 3.14$ )
Q24 · Numerical · Wave OpticsOpen
A beam of light consisting of wavelengths 650 nm and 550 nm illuminates the Young's double slits with separation of 2 mm such that the interference fringes are formed on a screen, placed at a distance of 1.2 m from the slits. The least distance of a point from the central maximum, where the bright fringes due to both the wavelengths coincide, is $\times10^{-5}$ m.
Q25 · Numerical · Rotational MotionOpen
A fly wheel having mass 3 kg and radius 5 m is free to rotate about a horizontal axis. A string having negligible mass is wound around the wheel and the loose end of the string is connected to a 3 kg mass. The mass is kept at rest initially and released. Kinetic energy of the wheel when the mass descends by 3 m is J. ( $g = 10\ \mathrm{m/s^2}$ ; take the fly wheel as a uniform disc)
We corrected a printing error in the original paper
Assumption “take the fly wheel as a uniform disc” added; the source leaves the shape unstated (a ring would give 45 J).

Chemistry

Q1 · MCQ · HydrocarbonsOpen
Identify the correct statements :
The presence of – $\ce{NO2}$ group in benzene ring
- (A)activates the ring towards electrophilic substitutions.
- (B)deactivates the ring towards electrophilic substitutions.
- (C)activates the ring towards nucleophilic substitutions.
- (D)deactivates the ring towards nucleophilic substitutions.
1. (1)B and D Only
2. (2)C and A Only
3. (3)A and D Only
4. (4)B and C Only
Q2 · MCQ · p-Block (15–18)Open
Given below are two statements :
Statement I : The increasing order of boiling point of hydrogen halides is HCl $<$ HBr $<$ HI $<$ HF.
Statement II : The increasing order of melting point of hydrogen halides is HCl $<$ HBr $<$ HF $<$ HI.
In the light of the above statements, choose the correct answer from the options given below :
1. (1)Both Statement I and Statement II are true
2. (2)Statement I is true but Statement II is false
3. (3)Both Statement I and Statement II are false
4. (4)Statement I is false but Statement II is true
Q3 · MCQ · Classification & PeriodicityOpen
Consider the elements N, P, O, S, Cl and F. The number of valence electrons present in the elements with most and least metallic character from the above list is respectively.
1. (1)7 and 5
2. (2)5 and 6
3. (3)5 and 7
4. (4)6 and 7
Q4 · MCQ · EquilibriumOpen
Observe the following equilibrium in a 1 L flask.
$\ce{A(g) <=> B(g)}$
At T(K), the equilibrium concentrations of A and B are 0.5 M and 0.375 M respectively. 0.1 moles of A is added into the flask and heated to T(K) to establish the equilibrium again. The new equilibrium concentrations (in M) of A and B are respectively.
1. (1)0.367, 0.275
2. (2)0.53, 0.4
3. (3)0.742, 0.557
4. (4)0.557, 0.418
Q5 · MCQ · ThermodynamicsOpen
The plot of $\log_{10}K$ vs $\dfrac{1}{T}$ gives a straight line. The intercept and slope respectively are (where K is equilibrium constant).
1. (1) $\dfrac{2.303R}{\Delta H^\circ}$ , $\dfrac{2.303R}{\Delta S^\circ}$
2. (2) $\dfrac{\Delta S^\circ}{2.303R}$ , $-\dfrac{\Delta H^\circ}{2.303R}$
3. (3) $-\dfrac{\Delta S^\circ R}{2.303}$ , $\dfrac{\Delta H^\circ R}{2.303}$
4. (4) $-\dfrac{\Delta H^\circ}{2.303R}$ , $\dfrac{\Delta S^\circ}{2.303R}$
Q6 · MCQ · HydrocarbonsOpen
The reactions which produce alcohol as the product are :
- (A) $\ce{CH4 + O2}\xrightarrow[\Delta]{\ce{Mo2O3}}$
- (B) $\ce{2CH3CH3 + 3O2}\xrightarrow[\Delta]{\ce{(CH3COO)2Mn}}$
- (C) $\ce{(CH3)3CH}\xrightarrow{\ce{KMnO4}}$
- (D) $\ce{2CH4 + O2}\xrightarrow{\mathrm{Cu}/523\,\mathrm{K}/100\,\mathrm{atm.}}$
- (E) $\ce{CH3-CH=CH-CH3}\xrightarrow{\ce{KMnO4}/\ce{H+}}$
Choose the correct answer from the options given below :
1. (1)A and D Only
2. (2)A, C and E Only
3. (3)C and D Only
4. (4)B, D and E Only
We corrected a printing error in the original paper
Source printed “the product area”.
Q7 · MCQ · d- and f-BlockOpen
Consider the following statements about manganate and permanganate ions. Identify the correct statements :
- (A)The geometry of both manganate and permanganate ions is tetrahedral.
- (B)The oxidation states of Mn in manganate and permanganate are $+7$ and $+6$ , respectively.
- (C)Oxidation of Mn(II) salt by peroxodisulphate gives manganate ion as the final product.
- (D)Manganate ion is paramagnetic and permanganate ion is diamagnetic.
- (E)Acidified permanganate ion reduces oxalate, nitrite and iodide ions.
Choose the correct answer from the options given below:
1. (1)A, C and D Only
2. (2)A, B and C Only
3. (3)A, D and E Only
4. (4)A and D Only
We corrected a printing error in the original paper
Statement D: source printed “permanganate ions is diamagnetic”.
Q8 · MCQ · Haloalkanes & HaloarenesOpen
Which of the following reaction is NOT correctly represented ?
1. (1)
2. (2)
3. (3)
4. (4)
Q9 · MCQ · Structure of AtomOpen
The wavelength of photon ‘A' is 400 nm. The frequency of photon ‘B' is $10^{16}\,\mathrm{s^{-1}}$ . The wave number of photon ‘C' is $10^{4}\,\mathrm{cm^{-1}}$ . The correct order of energy of these photons is :
1. (1)C $>$ B $>$ A
2. (2)B $>$ A $>$ C
3. (3)A $>$ B $>$ C
4. (4)A $>$ C $>$ B
Q10 · MCQ · Organic Basics (GOC)Open
The cyclic cations having the same number of hyperconjugation are :
- (A)
- (B)
- (C)
- (D)
Choose the correct answer from the options given below :
1. (1)A and C Only
2. (2)B and C Only
3. (3)A and B Only
4. (4)A, C and D only
Q11 · MCQ · BiomoleculesOpen
Structures of four disaccharides are given below. Among the given disaccharides, the non-reducing sugar is :
1. (1)
2. (2)
3. (3)
4. (4)
We corrected a printing error in the original paper
Option (1): source printed “OH” on the fructose ring carbon next to the ring oxygen (C-5), which makes that carbon a hemiacetal and leaves no non-reducing option; corrected in the figure to the intended “CH $_2$ OH” (sucrose, source key (1)); glyphs copied from the same drawing.

Q12 · MCQ · Chemical BondingOpen

Match List-I with List-II according to shape.

List-I		List-II
A.	$\ce{XeO3}$	I.	$\ce{BrF5}$
B.	$\ce{XeF2}$	II.	$\ce{NH3}$
C.	$\ce{XeO2F2}$	III.	$\ce{[I3]-}$
D.	$\ce{XeOF4}$	IV.	$\ce{SF4}$

Choose the correct answer from the options given below :

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

Q13 · MCQ · AminesOpen
A student performed analysis of aliphatic organic compound ‘X' which on analysis gave C $= 61.01\%$ , H $= 15.25\%$ , N $= 23.74\%$ .
This compound, on treatment with $\ce{HNO2}$ / $\ce{H2O}$ produced another compound ‘Y' which did not contain any nitrogen atom. However, the compound ‘Y' upon controlled oxidation produced another compound ‘Z' that responded to iodoform test.
The structure of ‘X' is:
1. (1) $\ce{CH3CH2CH2NH2}$
2. (2)
3. (3)
4. (4)
Q14 · MCQ · SolutionsOpen
Consider the following aqueous solutions.
- (I)2.2 g Glucose in 125 mL of solution.
- (II)1.9 g Calcium chloride in 250 mL of solution.
- (III)9.0 g Urea in 500 mL of solution.
- (IV)20.5 g Aluminium sulphate in 750 mL of solution.
The correct increasing order of boiling point of these solutions will be:
[Given: Molar mass in g mol $^{-1}$ : H $=1$ , C $=12$ , N $=14$ , O $=16$ , Cl $=35.5$ , Ca $=40$ , Al $=27$ and S $=32$ ]
1. (1)I $<$ II $<$ III $<$ IV
2. (2)III $<$ I $<$ II $<$ IV
3. (3)II $<$ III $<$ I $<$ IV
4. (4)II $<$ III $<$ IV $<$ I
Q15 · MCQ · Aldehydes, Ketones & AcidsOpen
The correct order of acidic strength of the major products formed in the given reactions, is :
- (A) $\mathrm{PhNH_2}\xrightarrow[\substack{\text{(2) }\ce{CuCN}\\ \text{(3) }\ce{H3O+}/\Delta}]{\text{(1) }\ce{NaNO2 + HCl}\,(<5^\circ\mathrm{C})}[\mathrm{A}]$
- (B) $\ce{CH3CH2CHO}\xrightarrow[\Delta]{\ce{[Ag(NH3)2]+}\text{, }\ce{OH-}}[\mathrm{B}]$
- (C) $\ce{CH4 + O2}\xrightarrow[\text{(ii) }\ce{Na2Cr2O7}/\ce{H+}]{\text{(i) }\ce{Mo2O3}}[\mathrm{C}]$
- (D) $\ce{PhCH2MgBr + CO2}\xrightarrow[\ce{H3O+}]{\text{Dry ether}}[\mathrm{D}]$
Choose the correct answer from the options given below :
1. (1)C $>$ B $>$ A $>$ D
2. (2)A $>$ D $>$ C $>$ B
3. (3)A $>$ D $>$ B $>$ C
4. (4)C $>$ A $>$ D $>$ B
Q16 · MCQ · AminesOpen
Total number of alkali insoluble solid sulphonamides obtained by reaction of given amines with Hinsberg's reagent is ……… .
Aniline, N-Methylaniline, Methanamine, N, N-Dimethylmethanamine, N-Methyl methanamine, Phenylmethanamine, N-propylaniline, N-phenylaniline, N, N-Dimethylaniline, Allyl amine, Isopropyl amine
1. (1)4
2. (2)2
3. (3)8
4. (4)5
Q17 · MCQ · p-Block (13–14)Open
Consider the following reactions.
$\ce{Na2B4O7}\xrightarrow{\Delta}\ce{2X + Y}$
$\ce{CuSO4 + Y}\xrightarrow{\text{Non-Luminous flame}}\ce{Z + SO3}$
$\ce{2Z + 2X + Carbon}\xrightarrow{\text{Luminous flame}}\ce{2Q + Na2B4O7 + CO}$
The oxidation states of Cu in Z and Q, respectively are :
1. (1) $+2$ and $+2$
2. (2) $+2$ and $+1$
3. (3) $+1$ and $+2$
4. (4) $+1$ and $+1$
Q18 · MCQ · Some Basic ConceptsOpen
For the given reaction;
$\ce{CaCO3 + 2HCl -> CaCl2 + H2O + CO2}$
If 90 g $\ce{CaCO3}$ is added to 300 mL of HCl which contains 38.55% HCl by mass and has density 1.13 g mL $^{-1}$ , then which of the following option is correct?
Given molar mass of H, Cl, Ca and O are 1, 35.5, 40 and 16 g mol $^{-1}$ respectively.
1. (1)64.97 g of HCl remains unreacted
2. (2)32.85 g of $\ce{CaCO3}$ remains unreacted
3. (3)97.30 g of HCl reacted
4. (4)60.32 g of HCl remains unreacted
We corrected a printing error in the original paper
Source printed “CaCO $_3$ + 2HC1” (digit 1 for l) and, in option (4), “reamains”.
Q19 · MCQ · Coordination CompoundsOpen
The correct increasing order of spin-only magnetic moment values of the complex ions $\ce{[MnBr4]^2-}$ (A), $\ce{[Cu(H2O)6]^2+}$ (B), $\ce{[Ni(CN)4]^2-}$ (C) and $\ce{[Ni(H2O)6]^2+}$ (D) is:
1. (1)A $=$ B $<$ C $<$ D
2. (2)A $=$ B $<$ D $<$ C
3. (3)C $=$ D $<$ B $<$ A
4. (4)C $<$ B $<$ D $<$ A
We corrected a printing error in the original paper
Source printed “[Ni(CN $_4$ )] $^{2-}$ ”.
Q20 · MCQ · Purification & CharacterisationOpen
A student has been given 0.314 g of an organic compound and asked to estimate Sulphur. During the experiment, the student has obtained 0.4813 g of barium sulphate. The percentage of sulphur present in the compound is .
(Given Molar mass in g mol $^{-1}$ S:32, $\ce{BaSO4}$ : 233)
1. (1)42.10%
2. (2)63.15%
3. (3)21.05%
4. (4)48.24%
Q21 · Numerical · Structure of AtomOpen
Two positively charged particles $m_1$ and $m_2$ have been accelerated across the same potential difference of 200 keV as shown below.
[Given mass of $m_1 = 1$ amu and $m_2 = 4$ amu]
The deBroglie wavelength of $m_1$ will be x times of $m_2$ . The value of x is . (nearest integer)
Q22 · Numerical · Chemical KineticsOpen
$\ce{A -> B}$ (first reaction)
$\ce{C -> D}$ (second reaction)
Consider the above two first-order reactions. The rate constant for first reaction at 500 K is double of the same at 300 K. At 500 K, 50% of the reaction becomes complete in 2 hour. The activation energy of the second reaction is half of that of first reaction. If the rate constant at 500 K of the second reaction becomes double of the rate constant of first reaction at the same temperature; then rate constant for the second reaction at 300 K is $\times 10^{-1}$ hour $^{-1}$ (nearest integer).
Q23 · Numerical · ElectrochemistryOpen
For strong electrolyte $\Lambda_\mathrm{m}$ increases slowly with dilution and can be represented by the equation $\Lambda_\mathrm{m} = \Lambda^\circ_\mathrm{m} - Ac^{1/2}$ Molar conductivity values of the solutions of strong electrolyte AB at 18 $^\circ$ C are given below :
$c$ [mol L $^{-1}$ ] 0.04 0.09 0.16 0.25
$\Lambda_\mathrm{m}$ [S cm $^2$ mol $^{-1}$ ] 96.1 95.7 95.3 94.9
The value of constant A based on the above data [in S cm $^2$ mol $^{-1}$ /(mol/L) $^{1/2}$ ] unit is .
Q24 · Numerical · ElectrochemistryOpen
A volume of x mL of 5 M $\ce{NaHCO3}$ solution was mixed with 10 mL of 2 M $\ce{H2CO3}$ solution to make an electrolytic buffer. If the same buffer was used in the following electrochemical cell to record a cell potential of 235.3 mV, then the value of x $=$ mL (nearest integer).
$\ce{Sn(s) | Sn(OH)6^2-}$ (0.5 M) $|$ $\ce{HSnO2-}$ (0.05 M) $|$ $\ce{OH-}$ $|$ $\ce{Bi2O3(s) | Bi(s)}$
Consider upto one place of decimal for intermediate calculations $\left[\begin{array}{ll} \text{Given :}\qquad & E^\circ_{\ce{Sn(OH)6^2-}|\ce{HSnO2-}} = -0.9\ \mathrm{V} \\[2pt] & E^\circ_{\ce{Bi2O3}|\ce{Bi}} = -0.44\ \mathrm{V} \\[2pt] & \mathrm{p}K_{\mathrm{a}(\ce{H2CO3})} = 6.11 \\[2pt] & \dfrac{2.303RT}{F} = 0.059\ \mathrm{V} \\[6pt] & \text{Antilog}\,(1.29) = 19.5 \end{array}\right]$
We corrected a printing error in the original paper
Source printed the first datum as “ $E^\circ$ with subscript HSnO $_2^-$ $|$ Sn(OH) $_6^{2-}$ $= -0.9$ V” (couple written reduced form first, readable as an oxidation potential); corrected to the reduction couple Sn(OH) $_6^{2-}$ $|$ HSnO $_2^-$ , the value the key uses.
Q25 · Numerical · Coordination CompoundsOpen
The number of isoelectronic species among $\ce{Sc^3+}$ , $\ce{Cr^2+}$ , $\ce{Mn^3+}$ , $\ce{Co^3+}$ and $\ce{Fe^3+}$ is ‘n'. If ‘n' moles of AgCl is formed during the reaction of complex with formula $\ce{CoCl3(en)2NH3}$ with excess of $\ce{AgNO3}$ solution, then the number of electrons present in the $t_{2g}$ orbital of the complex is .

Maths

Q1 · MCQ · Binomial TheoremOpen
Given below two statements :
Statement I : $25^{13} + 20^{13} + 8^{13} + 3^{13}$ is divisible by 7.
Statement II : The integral part of $\left(7 + 4\sqrt3\right)^{25}$ is an odd number.
In the light of the above statements, choose the correct answer from the options given below :
1. (1)Both Statement I and Statement II are false.
2. (2)Both Statement I and Statement II are true.
3. (3)Statement I is false but Statement II is true.
4. (4)Statement I is true but Statement II is false.
Q2 · MCQ · Binomial TheoremOpen
The sum of the coefficients of $x^{499}$ and $x^{500}$ in $(1 + x)^{1000} + x(1 + x)^{999} + x^2(1 + x)^{998} + \cdots + x^{1000}$ is
1. (1) ${}^{1001}C_{501}$
2. (2) ${}^{1002}C_{500}$
3. (3) ${}^{1002}C_{501}$
4. (4) ${}^{1000}C_{501}$
Q3 · MCQ · Conic SectionsOpen
Let $A$ be the focus of the parabola $y^2 = 8x$ . Let the line $y = mx + c$ intersect the parabola at two distinct points $B$ and $C$ . If the centroid of the triangle $ABC$ is $\left(\dfrac73, \dfrac43\right)$ , then $(BC)^2$ is equal to :
1. (1)41
2. (2)80
3. (3)89
4. (4)32

Q4 · MCQ · ProbabilityOpen

The probability distribution of a random variable

X

is given below :

$X$	$4k$	$\dfrac{30}{7}k$	$\dfrac{32}{7}k$	$\dfrac{34}{7}k$	$\dfrac{36}{7}k$	$\dfrac{38}{7}k$	$\dfrac{40}{7}k$	$6k$
$P(X)$	$\dfrac{2}{15}$	$\dfrac{1}{15}$	$\dfrac{2}{15}$	$\dfrac{1}{5}$	$\dfrac{1}{15}$	$\dfrac{2}{15}$	$\dfrac{1}{5}$	$\dfrac{1}{15}$

E(X) = \dfrac{263}{15}

, then

P(X < 20)

is equal to :

(1) $\dfrac35$
(2) $\dfrac{8}{15}$
(3) $\dfrac{11}{15}$
(4) $\dfrac{14}{15}$

Q5 · MCQ · Limits, Continuity & DifferentiabilityOpen
Let $f(x) = \lim_{n\to\infty}\left(\dfrac{\cos\pi x - x^{2n}\sin(x-1)}{1 + x^{2n}(x-1)}\right),\ x \in \mathbb{R}.$ Consider the following two statements :
(I) $f(x)$ is discontinuous at $x = 1$ .
(II) $f(x)$ is continuous at $x = -1$ .
Then,
1. (1)Neither (I) nor (II) is True
2. (2)Both (I) and (II) are True
3. (3)Only (II) is True
4. (4)Only (I) is True
We corrected a printing error in the original paper
Source printed “discontinous” and “continous”; source printed the limit as $\theta\to0$ with $x^{(2/\theta)}$ in place of $x^{2n}$ (undefined for $x<0$ , and the one-sided limits in $\theta$ differ for $0<\lvert x\rvert<1$ ); corrected to the intended $n\to\infty$ with $x^{2n}$ , the reading the key uses.
Q6 · MCQ · Inverse TrigonometryOpen
Considering the principal values of inverse trigonometric functions, the value of the expression $\tan\left(2\sin^{-1}\left(\dfrac{2}{\sqrt{13}}\right) - 2\cos^{-1}\left(\dfrac{3}{\sqrt{10}}\right)\right)$ is equal to :
1. (1) $-\dfrac{33}{56}$
2. (2) $\dfrac{33}{56}$
3. (3) $\dfrac{16}{63}$
4. (4) $-\dfrac{16}{63}$
Q7 · MCQ · Quadratic EquationsOpen
Let the arithmetic mean of $\dfrac1a$ and $\dfrac1b$ be $\dfrac{5}{16}$ , $a > 2$ . If $\alpha$ is such that $a, 4, \alpha, b$ are in A.P., then the equation $\alpha x^2 - ax + 2(\alpha - 2b) = 0$ has :
1. (1)One root in $(1, 4)$ and another in $(-2, 0)$
2. (2)One root in $(0, 2)$ and another in $(-4, -2)$
3. (3)Complex roots of magnitude less than 2
4. (4)Both roots in the interval $(-2, 0)$
Q8 · MCQ · Sets, Relations & FunctionsOpen
Given below are two statements :
Statement I : The function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \dfrac{x}{1 + \lvert x\rvert}$ is one-one.
Statement II : The function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \dfrac{x^2 + 4x - 30}{x^2 - 8x + 18}$ is many-one.
In the light of the above statements, choose the correct answer from the options given below :
1. (1)Both Statement I and Statement II are false.
2. (2)Both Statement I and Statement II are true.
3. (3)Statement I is false but Statement II is true.
4. (4)Statement I is true but Statement II is false.
Q9 · MCQ · Conic SectionsOpen
An ellipse has its center at $(1, -2)$ , one focus at $(3, -2)$ and one vertex at $(5, -2)$ . Then the length of its latus rectum is :
1. (1) $\dfrac{16}{\sqrt3}$
2. (2)6
3. (3) $4\sqrt3$
4. (4) $6\sqrt3$
Q10 · MCQ · Conic SectionsOpen
Let the ellipse $E : \dfrac{x^2}{144} + \dfrac{y^2}{169} = 1$ and the hyperbola $H : \dfrac{x^2}{16} - \dfrac{y^2}{\lambda^2} = -1$ have the same foci. If $e$ and $L$ respectively denote the eccentricity and the length of the latus rectum of $H$ , then the value of $24(e + L)$ is :
1. (1)296
2. (2)126
3. (3)148
4. (4)67
Q11 · MCQ · Area under CurvesOpen
Let $P_1 : y = 4x^2$ and $P_2 : y = x^2 + 27$ be two parabolas. If the area of the bounded region enclosed between $P_1$ and $P_2$ is six times the area of the bounded region enclosed between the line $y = \alpha x$ , $\alpha > 0$ and $P_1$ , then $\alpha$ is equal to :
1. (1)8
2. (2)15
3. (3)12
4. (4)6
Q12 · MCQ · CirclesOpen
Let the circle $x^2 + y^2 = 4$ intersect $x$ -axis at the points $A(a, 0)$ , $a > 0$ and $B(b, 0)$ . Let $P(2\cos\alpha, 2\sin\alpha)$ , $0 < \alpha < \dfrac{\pi}{2}$ and $Q(2\cos\beta, 2\sin\beta)$ be two points such that $(\alpha - \beta) = \dfrac{\pi}{2}$ . Then the point of intersection of $AQ$ and $BP$ lies on :
1. (1) $x^2 + y^2 - 4y - 4 = 0$
2. (2) $x^2 + y^2 - 4x - 4 = 0$
3. (3) $x^2 + y^2 - 4x - 4y = 0$
4. (4) $x^2 + y^2 - 4x - 4y - 4 = 0$
Q13 · MCQ · Definite IntegrationOpen
Let $[\cdot]$ denote the greatest integer function. Then $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\dfrac{12(3 + [x])}{3 + [\sin x] + [\cos x]}\right)dx$ is equal to:
1. (1) $15\pi + 4$
2. (2) $11\pi + 2$
3. (3) $13\pi + 1$
4. (4) $12\pi + 5$
Q14 · MCQ · Differential EquationsOpen
Let $y = y(x)$ be the solution of the differential equation $x\dfrac{dy}{dx} - y = x^2\cot x$ , $x \in (0, \pi)$ . If $y\left(\dfrac{\pi}{2}\right) = \dfrac{\pi}{2}$ , then $6y\left(\dfrac{\pi}{6}\right) - 8y\left(\dfrac{\pi}{4}\right)$ is equal to :
1. (1) $3\pi$
2. (2) $-3\pi$
3. (3) $-\pi$
4. (4) $\pi$
Q15 · MCQ · TrigonometryOpen
The sum of all the elements in the range of $f(x) = \operatorname{Sgn}(\sin x) + \operatorname{Sgn}(\cos x) + \operatorname{Sgn}(\tan x) + \operatorname{Sgn}(\cot x)$ , $x \neq \dfrac{n\pi}{2}$ , $n \in \mathbb{Z}$ , where $\operatorname{Sgn}(t) = \begin{cases} 1, & \text{if } t > 0\\ -1 & \text{if } t < 0\end{cases}$ , is
1. (1)4
2. (2)2
3. (3) $-2$
4. (4)0
Q16 · MCQ · 3D GeometryOpen
Let $Q(a, b, c)$ be the image of the point $P(3, 2, 1)$ in the line $\dfrac{x-1}{1} = \dfrac{y}{2} = \dfrac{z-1}{1}$ . Then the distance of $Q$ from the line $\dfrac{x-9}{3} = \dfrac{y-9}{2} = \dfrac{z-5}{-2}$ is
1. (1)6
2. (2)8
3. (3)7
4. (4)5
Q17 · MCQ · VectorsOpen
Let $P$ be a point in the plane of the vectors $\overrightarrow{AB} = 3\hat i + \hat j - \hat k$ and $\overrightarrow{AC} = \hat i - \hat j + 3\hat k$ such that $P$ is equidistant from the lines $AB$ and $AC$ . If $\left\lvert\overrightarrow{AP}\right\rvert = \dfrac{\sqrt5}{2}$ , then the area of the triangle $ABP$ is :
1. (1)2
2. (2) $\dfrac32$
3. (3) $\dfrac{\sqrt{30}}{4}$
4. (4) $\dfrac{\sqrt{26}}{4}$
We corrected a printing error in the original paper
Source printed “in the plane of the vector” (two vectors follow).
Q18 · MCQ · Complex NumbersOpen
Let $A = \{z \in \mathbb{C} : \lvert z - 2\rvert \le 4\} \quad\text{and}\quad B = \{z \in \mathbb{C} : \lvert z - 2\rvert + \lvert z + 2\rvert = 5\}.$ Then the $\max\{\lvert z_1 - z_2\rvert : z_1 \in A \text{ and } z_2 \in B\}$ is
1. (1) $\dfrac{15}{2}$
2. (2)8
3. (3) $\dfrac{17}{2}$
4. (4)9
Q19 · MCQ · Indefinite IntegrationOpen
Let $f(x) = \displaystyle\int\frac{dx}{x^{\left(\frac23\right)} + 2x^{\left(\frac12\right)}}$ be such that $f(0) = -26 + 24\log_e(2)$ . If $f(1) = a + b\log_e(3)$ , where $a, b \in \mathbb{Z}$ , then $a + b$ is equal to:
1. (1) $-18$
2. (2) $-5$
3. (3) $-11$
4. (4) $-26$
Q20 · MCQ · Sequences & SeriesOpen
$\dfrac{6}{3^{26}} + \dfrac{10\cdot1}{3^{25}} + \dfrac{10\cdot2}{3^{24}} + \dfrac{10\cdot2^2}{3^{23}} + \cdots + \dfrac{10\cdot2^{24}}{3}$ is equal to
1. (1) $2^{25}$
2. (2) $2^{26}$
3. (3) $3^{25}$
4. (4) $3^{26}$
Q21 · Numerical · Sequences & SeriesOpen
If $\displaystyle\sum_{r=1}^{25}\left(\frac{r}{r^4 + r^2 + 1}\right) = \frac{p}{q}$ , where $p$ and $q$ are positive integers such that $\gcd(p, q) = 1$ , then $p + q$ is equal to .
Q22 · Numerical · Permutations & CombinationsOpen
Three persons enter in a lift at the ground floor. The lift will go upto $10^{\text{th}}$ floor. The number of ways, in which the three persons can exit the lift at three different floors, if the lift does not stop at first, second and third floors, is equal to .
Q23 · Numerical · Applications of DerivativesOpen
Let $f$ be a differentiable function satisfying $f(x) = 1 - 2x + \int_0^x e^{(x-t)}f(t)\,dt,\ x \in \mathbb{R}$ and let $g(x) = \int_0^x (f(t) + 2)^{15}(t - 4)^6(t + 12)^{17}\,dt,\ x \in \mathbb{R}.$ If $p$ and $q$ are respectively the points of local minima and local maxima of $g$ , then the value of $\lvert p + q\rvert$ is equal to .
Q24 · Numerical · 3D GeometryOpen
If the distance of the point $P(43, \alpha, \beta)$ , $\beta < 0$ , from the line $\vec r = 4\hat i - \hat k + \mu(2\hat i + 3\hat k)$ , $\mu \in \mathbb{R}$ along a line with direction ratios $3, -1, 0$ is $13\sqrt{10}$ , then $\alpha^2 + \beta^2$ is equal to .
Q25 · Numerical · Matrices & DeterminantsOpen
Let $A = \begin{bmatrix} 3 & -4\\ 1 & -1\end{bmatrix}$ and $B$ be two matrices such that $A^{100} = 100B + I$ . Then the sum of all the elements of $B^{100}$ is .

$c$ [mol L $^{-1}$ ]	0.04	0.09	0.16	0.25
$\Lambda_\mathrm{m}$ [S cm $^2$ mol $^{-1}$ ]	96.1	95.7	95.3	94.9