JEE Main 21 Jan 2026, Shift 2

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

Physics

Q1 · MCQ · Electric Potential and CapacitanceOpen
Consider two identical metallic spheres of radius $R$ each having charge $Q$ and mass $m$ . Their centers have an initial separation of $4R$ . Both the spheres are given an initial speed of $u$ towards each other. The minimum value of u, so that they can just touch each other is : (Take $k = \dfrac{1}{4\pi\epsilon_0}$ and assume $kQ^2 > Gm^2$ where G is the Gravitational constant)
1. (1) $\sqrt{\dfrac{kQ^2}{4mR}\left(1 - \dfrac{Gm^2}{kQ^2}\right)}$
2. (2) $\sqrt{\dfrac{kQ^2}{4mR}\left(1 + \dfrac{Gm^2}{kQ^2}\right)}$
3. (3) $\sqrt{\dfrac{kQ^2}{2mR}\left(1 - \dfrac{Gm^2}{kQ^2}\right)}$
4. (4) $\sqrt{\dfrac{kQ^2}{2mR}\left(1 - \dfrac{Gm^2}{2kQ^2}\right)}$
Q2 · MCQ · Current ElectricityOpen
The charge stored by the capacitor C in the given circuit in the steady state is $\mu$ C.
1. (1)12.5
2. (2)10
3. (3)7.5
4. (4)5
Q3 · MCQ · Current ElectricityOpen
The total length of potentiometer wire AB is 50 cm in the arrangement as shown in figure. If P is the point where the galvanometer shows zero reading then the length AP is cm.
1. (1)15
2. (2)30
3. (3)25
4. (4)20
Q4 · MCQ · Alternating CurrentOpen
A capacitor C is first charged fully with potential difference of $V_0$ and disconnected from the battery. The charged capacitor is connected across an inductor having inductance L. In t s 25% of the initial energy in the capacitor is transferred to the inductor. The value of t is s.
1. (1) $\dfrac{\pi\sqrt{LC}}{3}$
2. (2) $\dfrac{\pi\sqrt{LC}}{6}$
3. (3) $\dfrac{\pi\sqrt{LC}}{2}$
4. (4) $\pi\sqrt{\dfrac{LC}{2}}$
Q5 · MCQ · Kinetic Theory of GasesOpen
The r.m.s speed of oxygen molecules at $47^\circ$ C is equal to that of the hydrogen molecules kept at $^\circ$ C. (Mass of oxygen molecule/mass of hydrogen molecule $= 32/2$ )
1. (1) $-235$
2. (2) $-100$
3. (3) $-253$
4. (4) $-20$
Q6 · MCQ · Rotational MotionOpen
Two cars A and B each of mass $10^{3}$ kg are moving on parallel tracks separated by a distance of 10 m, in same direction with speeds 72 km/h and 36 km/h. The magnitude of angular momentum of car A with respect to car B is J.s.
1. (1) $3.6\times10^{5}$
2. (2) $10^{5}$
3. (3) $3\times10^{5}$
4. (4) $2\times10^{5}$
Q7 · MCQ · Rotational MotionOpen
The pulley shown in figure is made using a thin rim and two rods of length equal to diameter of the rim. The rim and each rod have a mass of M. Two blocks of mass of M and m are attached to two ends of a light string passing over the pulley, which is hinged to rotate freely in vertical plane about its centre. The magnitude of the acceleration experienced by the blocks is (assume no slipping of string on pulley.)
1. (1) $\dfrac{(M-m)g}{\left[\left(\dfrac{13}{6}\right)M + m\right]}$
2. (2) $\dfrac{(M-m)g}{M + m}$
3. (3) $\dfrac{(M-m)g}{\left[\left(\dfrac{8}{3}\right)M + m\right]}$
4. (4) $\dfrac{(M-m)g}{2M + m}$
We corrected a printing error in the original paper
Source printed “The magnitudes of the acceleration … is”.
Q8 · MCQ · OscillationsOpen
The kinetic energy of a simple harmonic oscillator is oscillating with angular frequency of 176 rad/s. The frequency of this simple harmonic oscillator is Hz. $\left[\text{take } \pi = \dfrac{22}{7}\right]$
1. (1)14
2. (2)88
3. (3)28
4. (4)176
Q9 · MCQ · Wave OpticsOpen
Given below are two statements:
Statement I : In a Young's double slit experiment, the angular separation of fringes will increase as the screen is moved away from the plane of the slits
Statement II : In a Young's double slit experiment, the angular separation of fringes will increase when monochromatic source is replaced by another monochromatic source of higher wavelength
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)Both Statement I and Statement II are false
3. (3)Statement I is false but Statement II is true
4. (4)Statement I is true but Statement II is false
Q10 · MCQ · Current ElectricityOpen
A battery with EMF E and internal resistance r is connected across a resistance R. The power consumption in R will be maximum when :
1. (1) $R = 2r$
2. (2) $R = \dfrac{r}{2}$
3. (3) $R = \sqrt{2}\,r$
4. (4) $R = r$
Q11 · MCQ · Units and MeasurementsOpen
Keeping the significant figures in view, the sum of the physical quantities 52.01m, 153.2 m and 0.123 m is :
1. (1)205 m
2. (2)205.333 m
3. (3)205.33 m
4. (4)205.3 m
Q12 · MCQ · Units and MeasurementsOpen
A spherical body of radius r and density $\sigma$ falls freely through a viscous liquid having density $\rho$ and viscosity $\eta$ and attains a terminal velocity $v_0$ . Estimated maximum error in the quantity $\eta$ is : (Ignore errors associated with $\sigma$ , $\rho$ and g, gravitational acceleration)
1. (1) $2\dfrac{\Delta r}{r} - \dfrac{\Delta v_0}{v_0}$
2. (2) $\dfrac{2\Delta r}{r} + \dfrac{\Delta v_0}{v_0}$
3. (3) $2\left[\dfrac{\Delta r}{r} + \dfrac{\Delta v_0}{v_0}\right]$
4. (4) $2\left[\dfrac{\Delta r}{r} - \dfrac{\Delta v_0}{v_0}\right]$
Q13 · MCQ · Mechanical Properties of FluidsOpen
Surface tension of two liquids (having same densities), $T_1$ and $T_2$ , are measured using capillary rise method utilizing two tubes with inner radii of $r_1$ and $r_2$ where $r_1 > r_2$ . The measured liquid heights in these tubes are $h_1$ and $h_2$ respectively. [Ignore the weight of the liquid above the lowest point of meniscus]. The heights $h_1$ and $h_2$ and surface tensions $T_1$ and $T_2$ satisfy the relation :
1. (1) $h_1 < h_2$ and $T_1 = T_2$
2. (2) $h_1 = h_2$ and $T_1 = T_2$
3. (3) $h_1 > h_2$ and $T_1 = T_2$
4. (4) $h_1 > h_2$ and $T_1 < T_2$
We corrected a printing error in the original paper
Source printed “about the lowest point of miniscus”.
Q14 · MCQ · Relative MotionOpen
A river of width 200 m is flowing from west to east with a speed of 18 km/h. A boat, moving with speed of 36 km/h in still water, is made to travel one-round trip (bank to bank of the river). Minimum time taken by the boat for this journey and also the displacement along the river bank are and respectively.
1. (1)20 s and 100 m
2. (2)40 s and 0 m
3. (3)40 s and 200 m
4. (4)40 s and 100 m
Q15 · MCQ · Current ElectricityOpen
Two known resistances of R $\Omega$ and 2R $\Omega$ and one unknown resistance X $\Omega$ are connected in a circuit as shown in the figure. If the equivalent resistance between points A and B in the circuit is X $\Omega$ , then the value of X is $\Omega$ .
1. (1) $\left(\sqrt3 - 1\right)R$
2. (2) $R$
3. (3) $2\left(\sqrt3 - 1\right)R$
4. (4) $\left(\sqrt3 + 1\right)R$
We corrected a printing error in the original paper
Source printed “Two known resistance”; “and and”.
Q16 · MCQ · AtomsOpen
The energy of an electron in an orbit of the Bohr's atom is $0.04E_0$ where $E_0$ is the ground state energy. If L is the angular momentum of the electron in this orbit and h is the Planck's constant, then $\dfrac{2\pi L}{h}$ is :
1. (1)2
2. (2)4
3. (3)5
4. (4)6
We corrected a printing error in the original paper
Source printed “ $-0.04E_0$ eV” (positive, i.e. unbound, since $E_0$ is negative; and a doubled unit).
Q17 · MCQ · Moving Charges and MagnetismOpen
An infinitely long straight wire carrying current I is bent in a planar shape as shown in the diagram. The radius of the circular part is r. The magnetic field at the centre O of the circular loop is :
1. (1) $\dfrac{\mu_0}{2\pi}\dfrac{I}{r}(\pi + 1)\,\hat i$
2. (2) $-\dfrac{\mu_0}{2\pi}\dfrac{I}{r}(\pi - 1)\,\hat i$
3. (3) $\dfrac{\mu_0}{2\pi}\dfrac{I}{r}(\pi - 1)\,\hat i$
4. (4) $-\dfrac{\mu_0}{2\pi}\dfrac{I}{r}(\pi + 1)\,\hat i$
We corrected a printing error in the original paper
Source printed “planer”.
Q18 · MCQ · Circular MotionOpen
A large drum having radius R is spinning around its axis with angular velocity $\omega$ , as shown in figure. The minimum value of $\omega$ so that a body of mass M remains stuck to the inner wall of the drum, taking the coefficient of friction between the drum surface and mass M is $\mu$ , is :
1. (1) $\sqrt{\dfrac{\mu g}{R}}$
2. (2) $\sqrt{\dfrac{2g}{\mu R}}$
3. (3) $\sqrt{\dfrac{g}{2\mu R}}$
4. (4) $\sqrt{\dfrac{g}{\mu R}}$
Q19 · MCQ · Work, Energy and PowerOpen
A body of mass 2 kg is moving along x-direction such that its displacement as function of time is given by $x(t) = \alpha t^2 + \beta t + \gamma$ , where $\alpha = 1\ \mathrm{m/s^2}$ , $\beta = 1$ m/s and $\gamma = 1$ m. The work done on the body during the time interval $t = 2$ s to $t = 3$ s, is J.
1. (1)49
2. (2)42
3. (3)24
4. (4)12
We corrected a printing error in the original paper
Source printed “ $x(t) = \alpha t^2 + \beta t + \gamma\,\mathrm{m}$ ” (unit m doubled on $\gamma = 1$ m).
Q20 · MCQ · Ray OpticsOpen
As shown in the diagram, when the incident ray is parallel to base of the prism, the emergent ray grazes along the second surface.
If refractive index of the material of prism is $\sqrt2$ , the angle $\theta$ of prism is.
1. (1) $60^\circ$
2. (2) $75^\circ$
3. (3) $90^\circ$
4. (4) $45^\circ$
Q21 · Numerical · Electromagnetic WavesOpen
An electromagnetic wave of frequency 100 MHz propagates through a medium of conductivity, $\sigma = 10$ mho/m. The ratio of maximum conducting current density to maximum displacement current density is . $\left[\text{Take } \dfrac{1}{4\pi\epsilon_0} = 9\times10^{9}\ \mathrm{Nm^2/C^2}\right]$
Q22 · Numerical · Mechanical Properties of FluidsOpen
The terminal velocity of a metallic ball of radius 6 mm in a viscous fluid is 20 cm/s. The terminal velocity of another ball of same material and having radius 3 mm in the same fluid will be cm/s.
Q23 · Numerical · Dual Nature of Radiation and MatterOpen
A particle having electric charge $3\times10^{-19}$ C and mass $6\times10^{-27}$ kg is accelerated by applying an electric potential of 1.21 V. Wavelength of the matter wave associated with the particle is $\alpha\times10^{-12}$ m. The value of $\alpha$ is . (Take Planck's constant $= 6.6\times10^{-34}$ J.s)
Q24 · Numerical · Wave OpticsOpen
In a Young's double slit experiment set up, the two slits are kept 0.4 mm apart and screen is placed at 1 m from slits. If a thin transparent sheet of thickness 20 $\mu$ m is introduced in front of one of the slits then central bright fringe shifts by 20 mm on the screen. The refractive index of transparent sheet is given by $\dfrac{\alpha}{10}$ , where $\alpha$ is .
We corrected a printing error in the original paper
Source printed “0.4. mm”; “centre bring fringe”.
Q25 · Numerical · ThermodynamicsOpen
A diatomic gas ( $\gamma = 1.4$ ) does 100 J of work when it is expanded isobarically. Then the heat given to the gas is J.
We corrected a printing error in the original paper
Source printed “the heat given to the gas ___ J” (“is” missing).

Chemistry

Q1 · MCQ · Structure of AtomOpen
Consider the following spectral lines for atomic hydrogen:
- (A)First line of Paschen series
- (B)Second line of Balmer series
- (C)Third line of Paschen series
- (D)Fourth line of Brackett series
The correct arrangement of the above lines in ascending order of energy is:
1. (1)D $<$ C $<$ A $<$ B
2. (2)A $<$ B $<$ C $<$ D
3. (3)C $<$ D $<$ B $<$ A
4. (4)D $<$ A $<$ C $<$ B
We corrected a printing error in the original paper
Source printed “Fourth line of Bracket sereis”.

Match List-I with List-II.

	List-I (Pair of Compounds)		List-II (Type of Isomers)
A.	2-Methylpropene and but-1-ene	I.	Stereoisomers
B.	Cis-but-2-ene and trans-but-2-ene	II.	Position isomers
C.	2-Butanol and diethyl ether	III.	Chain isomers
D.	But-1-ene and but-2-ene	IV.	Functional group isomers

Choose the correct answer from the options given below:

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

We corrected a printing error in the original paper

Answer corrected; the source key is (2) — 2-butanol and diethyl ether are functional group isomers (C-IV) and but-1-ene and but-2-ene are position isomers (D-II), which is option (1).

Q3 · MCQ · AminesOpen
Consider the above sequence of reactions. The number of bromine atom(s) in the final product (P) will be:
1. (1)1
2. (2)6
3. (3)5
4. (4)3
Q4 · MCQ · Some Basic ConceptsOpen
Aqueous HCl reacts with $\ce{MnO2(s)}$ to form $\ce{MnCl2(aq)}$ , $\ce{Cl2(g)}$ and $\ce{H2O}$ ( $l$ ). What is the weight (in g) of $\ce{Cl2}$ liberated when 8.7 g of $\ce{MnO2(s)}$ is reacted with excess aqueous HCl solution ? (Given Molar mass in g mol $^{-1}$ Mn $= 55$ , Cl $= 35.5$ , O $= 16$ , H $= 1$ )
1. (1)7.1
2. (2)71
3. (3)21.3
4. (4)14.2
Q5 · MCQ · Purification & CharacterisationOpen
By usual analysis, 1.00g of compound (X) gave 1.79g of magnesium pyrophosphate. The percentage of phosphorus in compound (X) is: (nearest integer)
(Given, molar mass in g mol $^{-1}$ : O $= 16$ , Mg $= 24$ , P $= 31$ )
1. (1)50
2. (2)30
3. (3)20
4. (4)40
Q6 · MCQ · ThermodynamicsOpen
Consider the following data:
$\Delta_\mathrm{f}H^\ominus$ (methane, g) $= -X$ kJ mol $^{-1}$
Enthalpy of sublimation of graphite $= Y$ kJ mol $^{-1}$
Dissociation enthalpy of $\ce{H2}$ $= Z$ kJ mol $^{-1}$
The bond enthalpy of C – H bond is given by:
1. (1) $\dfrac{X+Y+2Z}{4}$
2. (2) $\dfrac{X+Y+4Z}{2}$
3. (3) $X+Y+Z$
4. (4) $\dfrac{-X+Y+Z}{4}$

Q7 · MCQ · Aldehydes, Ketones & AcidsOpen

Match List-I with List-II.

	List-I (Reagents)		List-II (Reaction Name (Involving aldehydes))
A.	$\ce{H2}$ , Pd- $\ce{BaSO4}$	I.	Etard Reaction
B.	$\ce{SnCl2}$ , HCl	II.	Rosenmund Reduction
C.	$\ce{CrO2Cl2}$ , $\ce{CS2}$	III.	Gatterman – Koch Reaction
D.	CO, HCl, Anhyd. $\ce{AlCl3}$	IV.	Stephen Reaction

Choose the correct answer from the options given below:

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

Q8 · MCQ · Chemical KineticsOpen
Decomposition of A is a first order reaction at T(K) and is given by $\ce{A(g) -> B(g) + C(g)}$ . In a closed 1 L vessel, 1 bar A(g) is allowed to decompose at T(K). After 100 minutes, the total pressure was 1.5 bar. What is the rate constant (in min $^{-1}$ ) of the reaction ? ( $\log 2 = 0.3$ )
1. (1) $6.9\times10^{-1}$
2. (2) $6.9\times10^{-3}$
3. (3) $6.9\times10^{-2}$
4. (4) $6.9\times10^{-4}$
Q9 · MCQ · Haloalkanes & HaloarenesOpen
The correct order of reactivity of the following benzyl halides towards reaction with KCN is:
1. (1)a $>$ b $>$ c $>$ d
2. (2)b $>$ a $>$ d $>$ c
3. (3)b $>$ a $>$ c $>$ d
4. (4)a $>$ b $>$ d $>$ c
We corrected a printing error in the original paper
Option (4): source printed “a $>$ b $>$ d $>$ d” (c missing, d twice); corrected to the intended “a $>$ b $>$ d $>$ c” (“a $>$ b $>$ c $>$ d” is already option (1)).
Q10 · MCQ · Classification & PeriodicityOpen
Given below are two statements:
Statement-I: The correct order in terms of atomic/ionic radii is Al $>$ Mg $>$ $\ce{Mg^{2+}}$ $>$ $\ce{Al^{3+}}$ .
Statement-II: The correct order in terms of the magnitude of electron gain enthalpy is Cl $>$ Br $>$ S $>$ O.
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)Statement I is false but Statement II is true
3. (3)Statement I is true but Statement II is false
4. (4)Both Statement I and Statement II are true
Q11 · MCQ · BiomoleculesOpen
The correct statements are:
- (A)Activation energy for enzyme catalysed hydrolysis of sucrose is lower than that of acid catalysed hydrolysis.
- (B)During denaturation, secondary and tertiary structures of a protein are destroyed but primary structure remains intact.
- (C)Nucleotides are joined together by glycosidic linkage between C $_1$ and C $_4$ carbons of the pentose sugar
- (D)Quaternary structure of proteins represents overall folding of the polypeptide chain.
Choose the correct answer from the options given below:
1. (1)A, C and D Only
2. (2)A, B and D Only
3. (3)A and B Only
4. (4)B and C Only
Q12 · MCQ · Haloalkanes & HaloarenesOpen
The correct order of the rate of the reaction for the following reaction with respect to nucleophiles is:
$\ce{CH3Br + Nu^{$\ominus$} -> CH3Nu + Br^{$\ominus$}}$
1. (1) $\ce{PhO^-}$ $>$ $\ce{^{-}OH}$ $>$ $\ce{CH3COO^-}$ $>$ $\ce{ClO4^-}$
2. (2) $\ce{ClO4^-}$ $>$ $\ce{CH3COO^-}$ $>$ $\ce{^{-}OH}$ $>$ $\ce{PhO^-}$
3. (3) $\ce{CH3COO^-}$ $>$ $\ce{PhO^-}$ $>$ $\ce{^{-}OH}$ $>$ $\ce{ClO4^-}$
4. (4) $\ce{^{-}OH}$ $>$ $\ce{PhO^-}$ $>$ $\ce{CH3COO^-}$ $>$ $\ce{ClO4^-}$
Q13 · MCQ · Coordination CompoundsOpen
Given below are two statements:
Statement I: Crystal Field Stabilization Energy (CFSE) of $\ce{[Cr(H2O)6]^{2+}}$ is greater than that of $\ce{[Mn(H2O)6]^{2+}}$ .
Statement II: Potassium ferricyanide has a greater spin-only magnetic moment than sodium Ferrocyanide.
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)Both Statement I and Statement II are false
3. (3)Statement I is true but Statement II is false
4. (4)Statement I is false but Statement II is true
Q14 · MCQ · Chemical BondingOpen
The correct increasing order of C – H(A), C – O(B), C $=$ O(C) and C $\equiv$ N(D) bonds in terms of covalent bond length is:
1. (1)A $<$ B $<$ C $<$ D
2. (2)A $<$ D $<$ C $<$ B
3. (3)D $<$ C $<$ B $<$ A
4. (4)D $<$ C $<$ A $<$ B
Q15 · MCQ · Haloalkanes & HaloarenesOpen
Given below are four compounds:
- (a)n-propyl chloride
- (b)iso-propyl chloride
- (c)sec-butyl chloride
- (d)neo-pentyl chloride
Percentage of carbon in the one which exhibits optical isomerism is:
1. (1)52
2. (2)56
3. (3)46
4. (4)40
We corrected a printing error in the original paper
Source printed “n-propyl choride”.
Q16 · MCQ · d- and f-BlockOpen
Given below are some of the statements about Mn and $\ce{Mn2O7}$ . Identify the correct statements
- (A)Mn forms the oxide $\ce{Mn2O7}$ in which Mn is in its highest oxidation state.
- (B)Oxygen stabilizes the Mn in higher oxidation states by forming multiple bonds with Mn
- (C) $\ce{Mn2O7}$ is an ionic oxide.
- (D)The structure of $\ce{Mn2O7}$ consists of one bridged oxygen.
Choose the correct answer from the options given below:
1. (1)A, B, C and D
2. (2)A, B and D Only
3. (3)A, C and D Only
4. (4)A, B and C Only
Q17 · MCQ · ElectrochemistryOpen
For a closed circuit Daniell cell, which of the following plots is the accurate one at a given temperature ?
1. (1)
2. (2)
3. (3)
4. (4)
Q18 · MCQ · Organic Basics (GOC)Open
Given below are two statements :
Statement – I : Compound (X), shown below , dissolves in $\ce{NaHCO3}$ solution and has two chiral carbon atoms
Statement – II : Compound (Y), shown below, has two carbons with sp $^3$ hybridization, one carbon with sp $^2$ and one carbon with sp hybridization
In the light of the above statements, choose the correct answer from the options given below :
1. (1)Statement I is true but Statement II is false
2. (2)Statement I is false but Statement II is true
3. (3)Both Statement I and Statement II are true
4. (4)Both Statement I and Statement II are false
Q19 · MCQ · Chemical BondingOpen
Given below are two statements :
Statement I : The correct order in terms of bond dissociation enthalpy is $\ce{Cl2}$ $>$ $\ce{Br2}$ $>$ $\ce{F2}$ $>$ $\ce{I2}$
Statement II : The correct trend in the covalent character of the metal halides is [ $\ce{SnCl4}$ $>$ $\ce{SnCl2}$ ], [ $\ce{PbCl4}$ $>$ $\ce{PbCl2}$ ] and [ $\ce{UF4}$ $>$ $\ce{UF6}$ ]
In the light of the above statements, choose the correct answer from the options given below :
1. (1)Statement I is true but Statement II is false
2. (2)Both Statement I and Statement II are true
3. (3)Statement I is false but Statement II is true
4. (4)Both Statement I and Statement II are false
Q20 · MCQ · Practical / Qualitative AnalysisOpen
On heating a mixture of common salt and $\ce{K2Cr2O7}$ in equal amount along with concentrated $\ce{H2SO4}$ in a test tube, a gas is evolved. Formula of the gas evolved and oxidation state of the central metal atom in the gas respectively are :
1. (1) $\ce{CrO2Cl2}$ and $+5$
2. (2) $\ce{CrO2Cl2}$ and $+6$
3. (3) $\ce{Cr2O2Cl2}$ and $+6$
4. (4) $\ce{Cr2O2Cl2}$ and $+3$
Q21 · Numerical · EquilibriumOpen
The first and second ionization constants of $\ce{H2X}$ are $2.5\times10^{-8}$ and $1.0\times10^{-13}$ respectively. The concentration of $\ce{X^{2-}}$ in 0.1 M $\ce{H2X}$ solution is $\times10^{-15}$ M. (Nearest Integer)
Q22 · Numerical · SolutionsOpen
The osmotic pressure of a living cell is 12 atm at 300 K. The strength of sodium chloride solution that is isotonic with the living cell at this temperature is g L $^{-1}$ . (Nearest integer)
Given : $R = 0.08$ L atm K $^{-1}$ mol $^{-1}$
Assume complete dissociation of NaCl
(Given : Molar mass of Na and Cl are 23 and 35.5 g mol $^{-1}$ respectively.)
We corrected a printing error in the original paper
Source printed “The osmotic pressure of a living cell in 12 atm”.
Q23 · Numerical · SolutionsOpen
A substance ‘X' (1.5 g) dissolved in 150 g of a solvent ‘Y' (molar mass $= 300$ g mol $^{-1}$ ) led to an elevation of the boiling point by 0.5 K. The relative lowering in the vapour pressure of the solvent ‘Y' is $\times10^{-2}$ . (Nearest integer)
[Given : $K_\mathrm{b}$ of the solvent $= 5.0$ K kg mol $^{-1}$ ]
Assume the solution to be dilute and no association or dissociation of X takes place in solution.
Q24 · Numerical · d- and f-BlockOpen
Identify the metal ions among $\ce{Co^{2+}}$ , $\ce{Ni^{2+}}$ , $\ce{Fe^{2+}}$ , $\ce{V^{3+}}$ and $\ce{Ti^{2+}}$ having a spin-only magnetic moment value more than 3.0 BM. The sum of unpaired electrons present in the high spin octahedral complexes formed by those metal ions is .
Q25 · Numerical · ElectrochemistryOpen
MX is a sparingly soluble salt that follows the given solubility equilibrium at 298 K
$\ce{MX(s) <=> M+(aq) + X^-(aq)}$ ; $K_\mathrm{sp} = 10^{-10}$
If the standard reduction potential for
$\ce{M+(aq) ->[+e^-] M(s)}$ is $\left(E^\ominus_{\mathrm{M^+/M}}\right) = 0.79$ V, then the value of the standard reduction potential for the metal/metal insoluble salt electrode $E^\ominus_{\mathrm{X^-/MX(s)/M}}$ is mV. (nearest integer)
[Given: $\dfrac{2.303RT}{F} = 0.059$ V]

Maths

Q1 · MCQ · Sequences & SeriesOpen
The positive integer $n$ , for which the solutions of the equation $x(x+2) + (x+2)(x+4) + \ldots + (x+2n-2)(x+2n) = \dfrac{8n}{3}$ are two consecutive even integers, is:
1. (1)3
2. (2)6
3. (3)12
4. (4)9
Q2 · MCQ · Applications of DerivativesOpen
Let $f : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that $f''(x) > 0$ for all $x \in \mathbb{R}$ and $f'(a-1) = 0$ , where $a$ is real number. Let $g(x) = f(\tan^2 x - 2\tan x + a)$ , $0 < x < \dfrac{\pi}{2}$ .
Consider the following two statements:
(I) $g$ is increasing in $\left(0, \dfrac{\pi}{4}\right)$
(II) $g$ is decreasing in $\left(\dfrac{\pi}{4}, \dfrac{\pi}{2}\right)$
Then,
1. (1)Neither (I) nor (II) is True
2. (2)Only (II) is True
3. (3)Only (I) is True
4. (4)Both (I) and (II) are True
Q3 · MCQ · Limits, Continuity & DifferentiabilityOpen
Let $f(x) = x^3 + x^2 f'(1) + 2x f''(2) + f'''(3)$ , $x \in \mathbb{R}$ . Then the value of $f'(5)$ is:
1. (1) $\dfrac{62}{5}$
2. (2) $\dfrac{657}{5}$
3. (3) $\dfrac{2}{5}$
4. (4) $\dfrac{117}{5}$
Q4 · MCQ · Conic SectionsOpen
If the line $\alpha x + 4y = \sqrt7$ , where $\alpha \in \mathbb{R}$ , touches the ellipse $3x^2 + 4y^2 = 1$ at the point $P$ in the first quadrant, then one of the focal distances of $P$ is:
1. (1) $\dfrac{1}{\sqrt3} - \dfrac{1}{2\sqrt{11}}$
2. (2) $\dfrac{1}{\sqrt3} + \dfrac{1}{2\sqrt5}$
3. (3) $\dfrac{1}{\sqrt3} - \dfrac{1}{2\sqrt5}$
4. (4) $\dfrac{1}{\sqrt3} + \dfrac{1}{2\sqrt7}$
We corrected a printing error in the original paper
Source printed “In the line” for “If the line”.
Q5 · MCQ · Conic SectionsOpen
Let $y^2 = 12x$ be the parabola with its vertex at $O$ . Let $P$ be a point on the parabola and $A$ be a point on the $x$ -axis such that $\angle OPA = 90^\circ$ . Then the locus of the centroid of such triangles $OPA$ is:
1. (1) $y^2 - 6x + 4 = 0$
2. (2) $y^2 - 9x + 6 = 0$
3. (3) $y^2 - 2x + 8 = 0$
4. (4) $y^2 - 4x + 8 = 0$
Q6 · MCQ · Conic SectionsOpen
Let one end of a focal chord of the parabola $y^2 = 16x$ be $(16, 16)$ . If $P(\alpha, \beta)$ divides this focal chord internally in the ratio $5:2$ , then the minimum value of $\alpha + \beta$ is equal to:
1. (1)22
2. (2)7
3. (3)5
4. (4)16
Q7 · MCQ · 3D GeometryOpen
Let the line $L$ pass through the point $(-3, 5, 2)$ and make equal angles with the positive coordinate axes. If the distance of $L$ from the point $(-2, r, 1)$ is $\sqrt{\dfrac{14}{3}}$ , then the sum of all possible values of $r$ is:
1. (1)12
2. (2)16
3. (3)6
4. (4)10
Q8 · MCQ · 3D GeometryOpen
Let the line $L_1$ be parallel to the vector $-3\hat i + 2\hat j + 4\hat k$ and pass through the point $(2, 6, 7)$ and the line $L_2$ be parallel to the vector $2\hat i + \hat j + 3\hat k$ and pass through the point $(4, 3, 5)$ . If the line $L_3$ is parallel to the vector $-3\hat i + 5\hat j + 16\hat k$ and intersects the lines $L_1$ and $L_2$ at the points $C$ and $D$ , respectively, then $\lvert\overrightarrow{CD}\rvert^2$ is equal to:
1. (1)171
2. (2)290
3. (3)312
4. (4)89
Q9 · MCQ · Quadratic EquationsOpen
Let $\alpha$ and $\beta$ be the roots of equation $x^2 + 2ax + (3a + 10) = 0$ such that $\alpha < 1 < \beta$ . Then the set of all possible values of $a$ is:
1. (1) $\left(-\infty, \dfrac{-11}{5}\right) \cup (5, \infty)$
2. (2) $(-\infty, -2) \cup (5, \infty)$
3. (3) $(-\infty, -3)$
4. (4) $\left(-\infty, \dfrac{-11}{5}\right)$
Q10 · MCQ · ProbabilityOpen
A random variable $X$ takes values 0, 1, 2, 3 with probabilities $\dfrac{2a+1}{30}, \dfrac{8a-1}{30}, \dfrac{4a+1}{30}$ , $b$ respectively, where $a, b \in \mathbb{R}$ . Let $\mu$ and $\sigma$ respectively be the mean and standard deviation of $X$ such that $\sigma^2 + \mu^2 = 2$ . Then $\dfrac{a}{b}$ is equal to:
1. (1)30
2. (2)3
3. (3)60
4. (4)12
Q11 · MCQ · Area under CurvesOpen
If the area of the region $\{(x, y) : 1 - 2x \le y \le 4 - x^2,\ x \ge 0, y \ge 0\}$ is $\dfrac{\alpha}{\beta}$ , $\alpha, \beta, \in \mathbb{N}$ , $\gcd(\alpha, \beta) = 1$ , then the value of $(\alpha + \beta)$ is:
1. (1)73
2. (2)85
3. (3)91
4. (4)67
Q12 · MCQ · Sequences & SeriesOpen
Let $a_1, \dfrac{a_2}{2}, \dfrac{a_3}{2^2}, \ldots, \dfrac{a_{10}}{2^9}$ be a G.P. of common ratio $\dfrac{1}{\sqrt2}$ . If $a_1 + a_2 + \ldots + a_{10} = 62$ , then $a_1$ is equal to:
1. (1) $2\left(\sqrt2 - 1\right)$
2. (2) $2 - \sqrt2$
3. (3) $\sqrt2 - 1$
4. (4) $2(2 - \sqrt2)$
Q13 · MCQ · Sets, Relations & FunctionsOpen
Let $A = \{x : \lvert x^2 - 10\rvert \le 6\}$ and $B = \{x : \lvert x - 2\rvert > 1\}$ . Then
1. (1) $A \cup B = (-\infty, 1] \cup (2, \infty)$
2. (2) $A - B = [2, 3)$
3. (3) $A \cap B = [-4, -2] \cup [3, 4]$
4. (4) $B - A = (-\infty, -4) \cup (-2, 1) \cup (4, \infty)$
Q14 · MCQ · Matrices & DeterminantsOpen
For the matrices $A = \begin{bmatrix} 3 & -4\\ 1 & -1\end{bmatrix} \quad\text{and}\quad B = \begin{bmatrix} -29 & 49\\ -13 & 18\end{bmatrix},$ if $(A^{15} + B)\begin{bmatrix} x\\ y\end{bmatrix} = \begin{bmatrix} 0\\ 0\end{bmatrix}$ , then among the following which one is true?
1. (1) $x = 5$ , $y = 7$
2. (2) $x = 18$ , $y = 11$
3. (3) $x = 11$ , $y = 2$
4. (4) $x = 16$ , $y = 3$
Q15 · MCQ · VectorsOpen
For a triangle $ABC$ , let $\vec p = \overrightarrow{BC}$ , $\vec q = \overrightarrow{CA}$ and $\vec r = \overrightarrow{BA}$ . If $\lvert\vec p\rvert = 2\sqrt3$ , $\lvert\vec q\rvert = 2$ and $\cos\theta = \dfrac{1}{\sqrt3}$ , where $\theta$ is the angle between $\vec p$ and $\vec q$ , then $\left\lvert\vec p \times \left(\vec q - 3\vec r\right)\right\rvert^2 + 3\lvert\vec r\rvert^2$ is equal to:
1. (1)340
2. (2)220
3. (3)410
4. (4)200
We corrected a printing error in the original paper
Source printed “the angle between $\vec P$ and $\vec q$ ” (capital $P$ ; the vector is $\vec p$ ).
Q16 · MCQ · Differential EquationsOpen
Let $y = y(x)$ be the solution of the differential equation $\sec x\dfrac{dy}{dx} - 2y = 2 + 3\sin x$ , $x \in \left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$ , $y(0) = -\dfrac{7}{4}$ . Then $y\left(\dfrac{\pi}{6}\right)$ is equal to:
1. (1) $-\dfrac{5}{2}$
2. (2) $-\dfrac{5}{4}$
3. (3) $-3\sqrt3 - 7$
4. (4) $-3\sqrt2 - 7$
Q17 · MCQ · Sets, Relations & FunctionsOpen
Let $A = \{2, 3, 5, 7, 9\}$ . Let $R$ be the relation on $A$ defined by $x\,R\,y$ if and only if $2x \le 3y$ . Let $\ell$ be the number of elements in $R$ , and $m$ be the minimum number of elements required to be added in $R$ to make it a symmetric relation. Then $\ell + m$ is equal to:
1. (1)23
2. (2)25
3. (3)21
4. (4)27
Q18 · MCQ · Matrices & DeterminantsOpen
If the system of equations
$\begin{align*} 3x + y + 4z &= 3\\ 2x + \alpha y - z &= -3\\ x + 2y + z &= 4 \end{align*}$
has no solution, then the value of $\alpha$ is equal to:
1. (1)19
2. (2)4
3. (3)13
4. (4)23
We corrected a printing error in the original paper
Source printed the third equation as “ $X + 2y + z = 4$ ” (capital $X$ for $x$ ).
Q19 · MCQ · Complex NumbersOpen
Let $z$ be the complex number satisfying $\lvert z - 5\rvert \le 3$ and having maximum positive principal argument. Then $34\left\lvert\dfrac{5z - 12}{5iz + 16}\right\rvert^2$ is equal to:
1. (1)16
2. (2)12
3. (3)26
4. (4)20
Q20 · MCQ · Permutations & CombinationsOpen
The largest $n \in \mathbb{N}$ , for which $7^n$ divides $101!$ , is:
1. (1)16
2. (2)18
3. (3)15
4. (4)19
Q21 · Numerical · Limits, Continuity & DifferentiabilityOpen
Let $[\cdot]$ denote the greatest integer function and $f(x) = \displaystyle\lim_{n\to\infty}\frac{1}{n^3}\sum_{k=1}^{n}\left[\frac{k^2}{3^x}\right]$ . Then $12\displaystyle\sum_{j=1}^{\infty} f(j)$ is equal to .
Q22 · Numerical · Definite IntegrationOpen
If $\displaystyle\int_0^1 4\cot^{-1}(1 - 2x + 4x^2)\,dx = a\tan^{-1}(2) - b\log_e(5)$ , where $a, b \in \mathbb{N}$ , then $(2a + b)$ is equal to .
Q23 · Numerical · Inverse TrigonometryOpen
Let the maximum value of $(\sin^{-1}x)^2 + (\cos^{-1}x)^2$ for $x \in \left[-\dfrac{\sqrt3}{2}, \dfrac{1}{\sqrt2}\right]$ be $\dfrac{m}{n}\pi^2$ , where $\gcd(m, n) = 1$ . Then $m + n$ is equal to .
Q24 · Numerical · Binomial TheoremOpen
If $\left(\frac{1}{{}^{15}C_0} + \frac{1}{{}^{15}C_1}\right)\left(\frac{1}{{}^{15}C_1} + \frac{1}{{}^{15}C_2}\right)\cdots\left(\frac{1}{{}^{15}C_{12}} + \frac{1}{{}^{15}C_{13}}\right) = \frac{\alpha^{13}}{{}^{14}C_0\,{}^{14}C_1\cdots{}^{14}C_{12}},$ then $30\alpha$ is equal to .
Q25 · Numerical · Straight LinesOpen
If $P$ is a point on the circle $x^2 + y^2 = 4$ , $Q$ is a point on the straight line $5x + y + 2 = 0$ and $x - y + 1 = 0$ is the perpendicular bisector of $PQ$ , then 13 times the sum of abscissa of all such point $P$ is .