JEE Main 21 Jan 2026, Shift 1

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

Physics

Q1 · MCQ · Work, Energy and PowerOpen
Potential energy ( $V$ ) versus distance ( $x$ ) is given by the graph, Rank various regions as per the magnitudes of the force (F) acting on a particle from high to low.
1. (1) $F_{BC} > F_{CD} > F_{DE} > F_{AB}$
2. (2) $F_{CD} > F_{AB} > F_{BC} > F_{DE}$
3. (3) $F_{CD} > F_{DE} > F_{AB} > F_{BC}$
4. (4) $F_{BC} > F_{AB} > F_{DE} > F_{CD}$
Q2 · MCQ · Thermal Properties of MatterOpen
A gas based geyser heats water flowing at the rate of 5.0 litres per minute from $27^\circ$ C to $87^\circ$ C. The rate of consumption of the gas is g/s. (Take heat of combustion of gas $= 5.0\times10^{4}$ J/g) specific heat capacity of water $= 4200$ J/kg. $^\circ$ C
1. (1)2.1
2. (2)4.2
3. (3)0.42
4. (4)0.21
Q3 · MCQ · Electromagnetic InductionOpen
A conducting circular loop of area $1.0\ \mathrm{m^2}$ is placed perpendicular to a magnetic field which varies as $B = \sin(100\,t)$ Tesla. If the resistance of the loop is $100\ \Omega$ , then the average thermal energy dissipated in the loop in one period is J.
1. (1) $\dfrac{\pi}{2}$
2. (2) $2\pi$
3. (3) $\pi$
4. (4) $\pi^2$
Q4 · MCQ · Mechanical Properties of FluidsOpen
Water flows through a horizontal tube as shown in the figure. The difference in height between the water columns in vertical tubes is 5 cm and the areas of cross-sections at $A$ and $B$ are $6\ \mathrm{cm^2}$ and $3\ \mathrm{cm^2}$ respectively. The rate of flow will be $\mathrm{cm^3/s}$ . (take $g = 10\ \mathrm{m/s^2}$ )
1. (1) $\dfrac{200}{\sqrt3}$
2. (2) $200\sqrt6$
3. (3) $200\sqrt3$
4. (4) $100\sqrt3$
We corrected a printing error in the original paper
Source printed “the area of cross-sections at $A$ and $B$ are”.
Q5 · MCQ · Units and MeasurementsOpen
In an experiment the values of two spring constants were measured as $k_1 = (10 \pm 0.2)$ N/m and $k_2 = (20 \pm 0.3)$ N/m. If these springs are connected in parallel, then the percentage error in equivalent spring constant is:
1. (1)2.67%
2. (2)2.33%
3. (3)1.33%
4. (4)1.67%
Q6 · MCQ · Newton's Laws of MotionOpen
A 4 kg mass moves under the influence of a force $\vec F = (4t^3\hat i - 3t\hat j)$ N where $t$ is the time in second. If mass starts from rest at the origin at $t = 0$ , the velocity and position after $t = 2$ s will be:
1. (1) $\vec v = 3\hat i + \dfrac32\hat j,\quad \vec r = \dfrac65\hat i + \hat j$
2. (2) $\vec v = 4\hat i - \dfrac32\hat j,\quad \vec r = \dfrac85\hat i - \hat j$
3. (3) $\vec v = 4\hat i + \dfrac52\hat j,\quad \vec r = \dfrac85\hat i + 2\hat j$
4. (4) $\vec v = 4\hat i - \dfrac32\hat j,\quad \vec r = \dfrac65\hat i - \hat j$
We corrected a printing error in the original paper
Assumption “from rest” added; the source leaves it unstated (“starts from origin at $t = 0$ ”).
Q7 · MCQ · Units and MeasurementsOpen
Consider a modified Bernoulli equation. $\left(P + \frac{A}{Bt^2}\right) + \rho g(h + Bt) + \frac12\rho V^2 = \text{constant}$ If $t$ has the dimension of time then the dimensions of $A$ and $B$ are , respectively.
1. (1) $[\mathrm{ML^0T^{-1}}]$ and $[\mathrm{M^0LT}]$
2. (2) $[\mathrm{ML^0T^{-1}}]$ and $[\mathrm{M^0LT^{-1}}]$
3. (3) $[\mathrm{ML^0T^{-2}}]$ and $[\mathrm{M^0LT^{-2}}]$
4. (4) $[\mathrm{ML^0T^{-2}}]$ and $[\mathrm{M^0LT^{-1}}]$
Q8 · MCQ · Moving Charges and MagnetismOpen
A current carrying solenoid is placed vertically and a particle of mass $m$ with charge $Q$ is released from rest. The particle moves along the axis of solenoid. If $g$ is acceleration due to gravity then the acceleration ( $a$ ) of the charged particle will satisfy:
1. (1) $a = g$
2. (2) $a > g$
3. (3) $a = 0$
4. (4) $0 < a < g$
We corrected a printing error in the original paper
Source printed “A current carrying is placed vertically” (“solenoid” missing).
Q9 · MCQ · Electric Potential and CapacitanceOpen
A parallel plate capacitor has capacitance $C$ , when there is vacuum within the parallel plates. A sheet having thickness $\left(\frac13\right)^{\text{rd}}$ of the separation between the plates and relative permittivity $K$ is introduced between the plates. The new capacitance of the system is:
1. (1) $\dfrac{3KC}{2K+1}$
2. (2) $\dfrac{CK}{2+K}$
3. (3) $\dfrac{3CK^2}{(2K+1)^2}$
4. (4) $\dfrac{4KC}{3K-1}$
Q10 · MCQ · Electromagnetic WavesOpen
The electric field of a plane electromagnetic wave is given by: $E_y = 69\sin[0.6\times10^{3}x - 1.8\times10^{11}t]\ \mathrm{V/m}.$ The expression for magnetic field associated with this electromagnetic wave is T.
1. (1) $B_z = 2.3\times10^{-7}\sin[0.6\times10^{3}x - 1.8\times10^{11}t]$
2. (2) $B_z = 2.3\times10^{-7}\sin[0.6\times10^{3}x + 1.8\times10^{11}t]$
3. (3) $B_y = 69\sin[0.6\times10^{3}x + 1.8\times10^{11}t]$
4. (4) $B_y = 2.3\times10^{-7}\sin[0.6\times10^{3}x - 1.8\times10^{11}t]$
We corrected a printing error in the original paper
Source printed “The electric field a plane electromagnetic wave”.
Q11 · MCQ · Wave OpticsOpen
In a double slit experiment the distance between the slits is 0.1 cm and the screen is placed at 50 cm from the slits plane. When one slit is covered with a transparent sheet having thickness $t$ and refractive index $n\,(=1.5)$ , the central fringe shifts by 0.2 cm. The value of $t$ is cm.
1. (1) $8\times10^{-4}$
2. (2) $6.0\times10^{-3}$
3. (3) $5.6\times10^{-4}$
4. (4) $5.0\times10^{-3}$
Q12 · MCQ · Dual Nature of Radiation and MatterOpen
A light wave described by $E = 60[\sin(3\times10^{15}t) + \sin(12\times10^{15}t)]$ (in SI units) falls on a metal surface of work function 2.8 eV. The maximum kinetic energy of ejected photoelectron is (approximately) eV. ( $h = 6.6\times10^{-34}$ J-s. and $e = 1.6\times10^{-19}$ C)
1. (1)5.1
2. (2)3.8
3. (3)6.0
4. (4)7.8
We corrected a printing error in the original paper
Source printed “ $E = 60\sin(3\times10^{15})t + \sin(12\times10^{15})t]$ ”.
Q13 · MCQ · AtomsOpen
If an alpha particle with energy 7.7 MeV is bombarded on a thin gold foil, the closest distance from nucleus it can reach is m. (Atomic number of gold $= 79$ and $\dfrac{1}{4\pi\epsilon_0} = 9\times10^{9}$ in SI units)
1. (1) $2.95\times10^{-14}$
2. (2) $2.95\times10^{-16}$
3. (3) $3.85\times10^{-16}$
4. (4) $3.85\times10^{-14}$
Q14 · MCQ · Rotational MotionOpen
A uniform rod of mass $m$ and length $l$ is suspended by means of two identical inextensible light strings as shown in figure. Tension in one string immediately after the other string is cut, is . ( $g$ acceleration due to gravity)
1. (1) $mg/2$
2. (2) $mg/4$
3. (3) $mg/3$
4. (4) $mg$
We corrected a printing error in the original paper
Source printed “length $l$ suspended by means of”.
Q15 · MCQ · Thermal Properties of MatterOpen
Aluminium and steel rods having same lengths and cross-sections are joined to make total length of 120 cm at $30^\circ$ C. The coefficients of linear expansion of aluminium and steel are $24\times10^{-6}/^\circ$ C and $1.2\times10^{-5}/^\circ$ C, respectively. The length of this composite rod when its temperature is raised to $100^\circ$ C, is cm.
1. (1)120.20
2. (2)120.15
3. (3)120.03
4. (4)120.06
We corrected a printing error in the original paper
Source printed “An aluminium and steel rods” and “The coefficient of linear expansion … are”.
Q16 · MCQ · Electromagnetic InductionOpen
A 1 m long metal rod AB completes the circuit as shown in figure. The area of circuit is perpendicular to the magnetic field of 0.10 T. If the resistance of the total circuit is $2\ \Omega$ then the force needed to move the rod towards right with constant speed ( $v$ ) of 1.5 m/s is N.
1. (1) $7.5\times10^{-2}$
2. (2) $5.7\times10^{-3}$
3. (3) $5.7\times10^{-2}$
4. (4) $7.5\times10^{-3}$
Q17 · MCQ · Semiconductor ElectronicsOpen
The given circuit works as:
1. (1)AND gate
2. (2)NOR gate
3. (3)NAND gate
4. (4)OR gate
Q18 · MCQ · WavesOpen
Two strings (A, B) having linear densities $\mu_A = 2\times10^{-4}$ kg/m and $\mu_B = 4\times10^{-4}$ kg/m and lengths $L_A = 2.5$ m and $L_B = 1.5$ m respectively are joined. Free ends of A and B are tied to two rigid supports C and D, respectively creating a tension of 500 N in the wire. Two identical pulses, sent from C and D ends, take time $t_1$ and $t_2$ , respectively, to reach the joint. The ratio $t_1/t_2$ is:
1. (1)1.08
2. (2)1.90
3. (3)1.67
4. (4)1.18
Q19 · MCQ · GravitationOpen
Initially a satellite of 100 kg is in a circular orbit of radius $1.5R_E$ . This satellite can be moved to a circular orbit of radius $3R_E$ by supplying $\alpha\times10^{6}$ J of energy. The value of $\alpha$ is . (Take Radius of Earth $R_E = 6\times10^{6}$ m and $g = 10\ \mathrm{m/s^2}$ )
1. (1)150
2. (2)500
3. (3)100
4. (4)1000
Q20 · MCQ · Electric Potential and CapacitanceOpen
A point charge of $10^{-8}$ C is placed at origin. The work done in moving a point charge $2\ \mu$ C from point A(4, 4, 2) m to B(2, 2, 1) m is J. ( $\dfrac{1}{4\pi\epsilon_0} = 9\times10^{9}$ in SI units)
1. (1) $45\times10^{-6}$
2. (2)0
3. (3) $30\times10^{-6}$
4. (4) $15\times10^{-6}$
Q21 · Numerical · Ray OpticsOpen
A collimated beam of light of diameter 2 mm is propagating along x-axis. The beam is required to be expanded in a collimated beam of diameter 14 mm using a system of two convex lenses. If first lens has focal length 40 mm, then the focal length of second lens is mm.
Q22 · Numerical · Current ElectricityOpen
The heat generated in 1 minute between points A and B in the given circuit, when a battery of 9 V with internal resistance of $1\ \Omega$ is connected across these points is J.
Q23 · Numerical · Rotational MotionOpen
Two identical thin rods of mass $M$ kg and length $L$ m are connected as shown in figure. Moment of inertia of the combined rod system about an axis passing through point P and perpendicular to the plane of the rods is $\dfrac{x}{12}ML^2\ \mathrm{kg\,m^2}$ . The value of $x$ is .
We corrected a printing error in the original paper
Source printed “ $\dfrac{x}{2}ML^2$ ” (which gives $x = 17/6$ ); corrected to the intended “ $\dfrac{x}{12}ML^2$ ” ( $I = 17ML^2/12$ , source key 17).
Q24 · Numerical · ThermodynamicsOpen
10 mole of oxygen is heated at constant volume from $30^\circ$ C to $40^\circ$ C. The change in the internal energy of the gas is cal. (The molecular specific heat of oxygen at constant pressure, $C_p = 7$ cal./mol $^\circ$ C and $R = 2$ cal./mol $^\circ$ C.)
Q25 · Numerical · Ray OpticsOpen
In a microscope the objective is having focal length $f_0 = 2$ cm and eye-piece is having focal length $f_e = 4$ cm. The tube length is 32 cm. The magnification produced by this microscope for normal adjustment is .

Chemistry

Q1 · MCQ · Practical / Qualitative AnalysisOpen
Consider the following reactions.
$\ce{PbCl2 + K2CrO4 -> A + 2KCl}$
(Hot solution)
$\ce{A + NaOH <=> B + Na2CrO4}$
$\ce{PbSO4 + 4CH3COONH4 -> (NH4)2SO4 + X}$
In the above reactions, A, B and X are respectively.
1. (1) $\ce{Na2[Pb(OH)2]}$ , $\ce{PbCrO4}$ and $\ce{(NH4)2[Pb(CH3COO)4]}$
2. (2) $\ce{PbCrO4}$ , $\ce{Na2[Pb(OH)4]}$ and $\ce{[Pb(NH3)4]SO4}$
3. (3) $\ce{Na2[Pb(OH)2]}$ , $\ce{PbCrO4}$ and $\ce{[Pb(NH3)4]SO4}$
4. (4) $\ce{PbCrO4}$ , $\ce{Na2[Pb(OH)4]}$ and $\ce{(NH4)2[Pb(CH3COO)4]}$
Q2 · MCQ · Classification & PeriodicityOpen
Which of the following represents the correct trend for the mentioned property ?
- (A)F $>$ P $>$ S $>$ B – First Ionization Energy
- (B)Cl $>$ F $>$ S $>$ P – Electron Affinity
- (C)K $>$ Al $>$ Mg $>$ B – Metallic character
- (D) $\ce{K2O}$ $>$ $\ce{Na2O}$ $>$ $\ce{MgO}$ $>$ $\ce{Al2O3}$ – Basic character
Choose the correct answer from the option given below.
1. (1)A, B and D only
2. (2)A, B, C and D
3. (3)A and B only
4. (4)B and C only
We corrected a printing error in the original paper
Source printed “Frist Ionization Energy”.
Q3 · MCQ · HydrocarbonsOpen
Identify A in the following reaction.
1. (1)
2. (2)
3. (3)
4. (4)
Q4 · MCQ · HydrocarbonsOpen
A hydrocarbon ‘P' ( $\ce{C4H8}$ ) on reaction with HCl gives an optically active compound ‘Q' ( $\ce{C4H9Cl}$ ) which on reaction with one mole of ammonia gives compound ‘R' ( $\ce{C4H11N}$ ). ‘R' on diazotization followed by hydrolysis gives ‘S'. Identify P, Q, R and S.
1. (1)
2. (2)
3. (3)
4. (4)
We corrected a printing error in the original paper
Option (3): source printed R as “CH $_3$ –CH $_2$ =CH(NH $_2$ )–CH $_3$ ” (a pentavalent carbon); the “=” in the figure is corrected to the intended single bond “CH $_3$ –CH $_2$ –CH(NH $_2$ )–CH $_3$ ” (butan-2-amine).
Q5 · MCQ · p-Block (13–14)Open
Given below are two statements :
Statement I : The number of pairs among [ $\ce{SiO2}$ , $\ce{CO2}$ ], [ $\ce{SnO}$ , $\ce{SnO2}$ ], [ $\ce{PbO}$ , $\ce{PbO2}$ ] and [ $\ce{GeO}$ , $\ce{GeO2}$ ], which contain oxides that are both amphoteric is 2.
Statement II : $\ce{BF3}$ is an electron deficient molecule that can act as a lewis acid, forms adduct with $\ce{NH3}$ and has a trigonal planar geometry.
In the light of the above statement, choose the correct answer from the option 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.
We corrected a printing error in the original paper
Source printed “an electron deficient molecule can act as a lewis acid” and, in option (4), “Statement I is false Statement II is true”.
Q6 · MCQ · Some Basic ConceptsOpen
80 mL of a hydrocarbon on mixing with 264 mL of oxygen in a closed U-tube undergoes complete combustion. The residual gases after cooling to 273 K occupy 224 mL. When the system is treated with KOH solution, the volume decreases to 64 mL. The formula of the hydrocarbon is :
1. (1) $\ce{C2H4}$
2. (2) $\ce{C4H10}$
3. (3) $\ce{C2H2}$
4. (4) $\ce{C2H6}$
Q7 · MCQ · Some Basic ConceptsOpen
14.0 g of calcium metal is allowed to react with excess HCl at 1.0 atm pressure and 273 K.
Which of the following statements is incorrect ?
[Given : Molar mass in g mol $^{-1}$ of Ca–40, Cl–35.5, H–1]
1. (1)0.35 mol of $\ce{H2}$ gas is evolved.
2. (2)7.84 L of $\ce{H2}$ gas is evolved.
3. (3)33.3 g of $\ce{CaCl2}$ is produced.
4. (4)The limiting reagent is calcium metal.
We corrected a printing error in the original paper
Source printed “33.3 of $\ce{CaCl2}$ is produced” (unit missing).
Q8 · MCQ · Purification & CharacterisationOpen
In Carius method, 0.75 g of an organic compound gave 1.2 g of barium sulphate, find percentage of sulphur (molar mass 32 g mol $^{-1}$ ). Molar mass of barium sulphate is 233 g mol $^{-1}$ .
1. (1)4.55%
2. (2)10.30%
3. (3)21.97%
4. (4)16.48%
Q9 · MCQ · SolutionsOpen
Elements P and Q form two types of non-volatile, non-ionizable compounds PQ and $\ce{PQ2}$ . When 1 g of PQ is dissolved in 50 g of solvent ‘A'. $\Delta T_\mathrm{b}$ was 1.176 K while when 1 g of $\ce{PQ2}$ is dissolved in 50 g of solvent ‘A', $\Delta T_\mathrm{b}$ was 0.689 K. ( $K_\mathrm{b}$ of ‘A' $= 5$ K kg mol $^{-1}$ ). The molar masses of elements P and Q (in g mol $^{-1}$ ) respectively, are :
1. (1)70, 110
2. (2)65, 145
3. (3)60, 25
4. (4)25, 60
Q10 · MCQ · AminesOpen
An organic compound (P) on treatment with aqueous ammonia under hot condition forms compound (Q) which on heating with $\ce{Br2}$ and KOH forms compound (R) having molecular formula $\ce{C6H7N}$ . Names of P, Q and R respectively are.
1. (1)Benzoic acid, benzamide, aniline
2. (2)Toluic acid, methylbenzamide, 2-methylaniline
3. (3)Benzoic acid, 4-methylbenzamide, 4-methylaniline.
4. (4)Phenylethanoic acid, phenylethanamide, benzamine
We corrected a printing error in the original paper
Source printed “molecular formula $\ce{C9H7N}$ ” (fits no option); corrected to the intended “ $\ce{C6H7N}$ ” (aniline, source key (1)).
Q11 · MCQ · Aldehydes, Ketones & AcidsOpen
An organic compound “P” of molecular formula $\ce{C6H12O3}$ gives positive Iodoform test but negative Tollen's test. When “P” is treated with dilute acid, it produces “Q”. “Q” gives positive Tollen's test and also iodoform test. The structure of “P” is :
1. (1)
2. (2)
3. (3)
4. (4)
Q12 · MCQ · Organic Basics (GOC)Open
From the following, the least stable structure is :
1. (1)
2. (2)
3. (3)
4. (4)
Q13 · MCQ · d- and f-BlockOpen
$\ce{MnO4^2-}$ , in acidic medium, disproportionates to :
1. (1) $\ce{Mn2O7}$ and $\ce{MnO2}$
2. (2) $\ce{MnO4-}$ and $\ce{MnO}$
3. (3) $\ce{MnO4-}$ and $\ce{MnO2}$
4. (4) $\ce{Mn2O7}$ and $\ce{MnO}$
Q14 · MCQ · Chemical BondingOpen
Given below are two statements:
Statement I: The number of species among $\ce{SF4}$ , $\ce{NH4+}$ , $\ce{[NiCl4]^2-}$ , $\ce{XeF4}$ , $\ce{[PtCl4]^2-}$ , $\ce{SeF4}$ and $\ce{[Ni(CN)4]^2-}$ , that have tetrahedral geometry is 3.
Statement II: In the set [ $\ce{NO2}$ , $\ce{BeH2}$ , $\ce{BF3}$ , $\ce{AlCl3}$ ], all the molecules have incomplete octet around central atom.
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 false
3. (3)Statement I is false but Statement II is true
4. (4)Both Statement I and Statement II are true
We corrected a printing error in the original paper
Source printed “[NiCl $_4$ ]” (charge missing), “[PtC1 $_4$ ] $^{2-}$ ” and “AIC1 $_3$ ” (digit 1 and capital I for l).
Q15 · MCQ · Coordination CompoundsOpen
Given below are two statements:
Statement I: Among $\ce{[Cu(NH3)4]^2+}$ , $\ce{[Ni(en)3]^2+}$ , $\ce{[Ni(NH3)6]^2+}$ and $\ce{[Mn(H2O)6]^2+}$ , $\ce{[Mn(H2O)6]^2+}$ , has the maximum number of unpaired electrons.
Statement II: The number of pairs among { $\ce{[NiCl4]^2-}$ , $\ce{[Ni(CO)4]}$ }, { $\ce{[NiCl4]^2-}$ , $\ce{[Ni(CN)4]^2-}$ } and { $\ce{[Ni(CO)4]}$ , $\ce{[Ni(CN)4]^2-}$ } that contain only diamagnetic species is two.
In the light of the above statements, choose the correct answer from the options given below:
1. (1)Statement I is false but Statement II is true
2. (2)Both Statement I and Statement II are true
3. (3)Both Statement I and Statement II are false
4. (4)Statement I is true but Statement II is false
We corrected a printing error in the original paper
Source printed “[Cu(NH $_3$ ) $^{2+}$ ” (subscript 4 and closing bracket missing), “[Ni(en) $_3$ )] $^{2+}$ ” and the second pair “[NiCl $_4$ ] $^{2-}$ , [Ni(CN) $_4$ ] $^{2-}$ ” without its closing brace.
Q16 · MCQ · Organic Basics (GOC)Open
Identify correct statement from the following :
- (A)Propanal and propanone are functional isomers.
- (B)Ethoxyethane and methoxypropane are metamers.
- (C)But-2-ene shows optical isomerism.
- (D)But-1-ene and but-2-ene are functional isomers.
- (E)Pentane and 2, 2-dimethyl propane are chain isomers.
Choose the correct answer from the options given below :
1. (1)B, C and D only
2. (2)A, B and C only
3. (3)A, B and E only
4. (4)C, D and E only
Q17 · MCQ · BiomoleculesOpen
Identify the correct statements.
- (A)Arginine and Tryptophan are essential amino acids.
- (B)Histidine does not contain heterocyclic ring in its structure.
- (C)Proline is a six membered cyclic ring amino acid.
- (D)Glycine does not have chiral centre.
- (E)Cysteine has characteristic feature of side chain as $\ce{MeS-CH2-CH2-}$ .
Choose the correct answer from the options given below:
Option
1. (1)C and E Only
2. (2)B and E Only
3. (3)C and D Only
4. (4)A and D Only
Q18 · MCQ · ThermodynamicsOpen
Which of the following graphs between pressure ‘P' versus volume ‘V' represent the maximum work done ?
1. (1)
2. (2)
3. (3)
4. (4)
Q19 · MCQ · ThermodynamicsOpen
For the reaction, $\ce{N2O4 <=> 2NO2}$ , graph is plotted as shown below. Identify correct statements.
- (A)Standard free energy change for the reaction is $-5.40$ kJ mol $^{-1}$ .
- (B)As $\Delta G^\ominus$ in graph is positive, $\ce{N2O4}$ will not dissociate into $\ce{NO2}$ at all.
- (C)Reverse reaction will go to completion.
- (D)When 1 mole of $\ce{N2O4}$ changes into equilibrium mixture, value of $\Delta G^\ominus = -0.84$ kJ mol $^{-1}$
- (E)When 2 mole of $\ce{NO2}$ , changes into equilibrium mixture, $\Delta G^\ominus$ for equilibrium mixture is $-6.24$ kJ mol $^{-1}$ .
Choose the correct answer from the options given below :
1. (1)D and E only
2. (2)C and E only
3. (3)A and D only
4. (4)B and C only
We corrected a printing error in the original paper
Source printed “As AG $^\ominus$ in graph is positive” and printed statement E twice.
Q20 · MCQ · Structure of AtomOpen
Given below are two statements:
Statement I: When an electric discharge is passed through gaseous hydrogen, the hydrogen molecules dissociate and the energetically excited hydrogen atoms produce electromagnetic radiation of discrete frequencies.
Statement II: The frequency of second line of Balmer series obtained from $\ce{He+}$ is equal to that of first line of Lyman series obtained from hydrogen atom.
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
Q21 · Numerical · Chemical KineticsOpen
Pre-exponential factors of two different reactions of same order are identical. Let activation energy of first reaction exceed the activation energy of second reaction by 20 kJ mol $^{-1}$ . If $k_1$ and $k_2$ are the rate constants of first and second reaction respectively at 300 K, then $\ln\dfrac{k_2}{k_1}$ will be . (nearest integer) [ $R = 8.3$ J K $^{-1}$ mol $^{-1}$ ]
We corrected a printing error in the original paper
Source printed “Let activation energy of first reaction exceeds” and “In $\dfrac{k_2}{k_1}$ ” (capital I for the natural logarithm ln).
Q22 · Numerical · ElectrochemistryOpen
The pH and conductance of a weak acid (HX) were found to be 5 and $4\times10^{-5}$ S, respectively. The conductance was measured under standard condition using a cell where the electrode plates having a surface area of 1 cm $^2$ were at a distance of 15 cm apart. The value of the limiting molar conductivity is S m $^2$ mol $^{-1}$ . (nearest integer)
(Given: degree of dissociation of the weak acid ( $\alpha$ ) $\ll 1$ )
We corrected a printing error in the original paper
Source printed “The pH and conductance of a weak acid (HX) was found” and “the weak acid (a) $\ll$ 1”.

Use the following data :

Substance	$\dfrac{\Delta_\mathrm{f}H^\ominus(500\,\mathrm{K})}{\mathrm{kJ\,mol^{-1}}}$	$\dfrac{S^\ominus(500\,\mathrm{K})}{\mathrm{J\,K^{-1}\,mol^{-1}}}$
AB(g)	32	222
$\ce{A2}$ (g)	6	146
$\ce{B2}$ (g)	X	280

One mole each of $\ce{A2}$ (g) and $\ce{B2}$ (g) are taken in a 1L closed flask and allowed to establish the equilibrium at 500K.

$\ce{A2(g) + B2(g) <=> 2AB(g)}$

The value of X (in kJ mol $^{-1}$ ) is . (Nearest integer)

(Given: $\log K = 2.2$ , $R = 8.3$ J K $^{-1}$ mol $^{-1}$ )

We corrected a printing error in the original paper

Source printed “The value of x” (the table names the quantity X).

Q24 · Numerical · AminesOpen
Consider the following reaction sequence
The percentage of nitrogen in product ‘T' formed is %. (Nearest integer)
(Given molar mass in g mol $^{-1}$ H:1, C:12, N:14, O:16)
We corrected a printing error in the original paper
Source printed the reagent over the third arrow as “(CH $_3$ CO) $_2$ ”; corrected in the figure to acetic anhydride “(CH $_3$ CO) $_2$ O”.
Q25 · Numerical · d- and f-BlockOpen
Consider the following reactions:
$\ce{NaCl + K2Cr2O7 + H2SO4 -> A + KHSO4 + NaHSO4 + H2O}$
$\ce{A + NaOH -> B + NaCl + H2O}$
$\ce{B + H2SO4 + H2O2 -> C + Na2SO4 + H2O}$
In the product ‘C', ‘X' is the number of $\ce{O2^2-}$ units, ‘Y' is the total number of oxygen atoms present and ‘Z' is the oxidation state of Cr. The value of X + Y + Z is .
We corrected a printing error in the original paper
Source printed “the total number oxygen atoms”.

Maths

Q1 · MCQ · Inverse TrigonometryOpen
If the domain of the function $f(x) = \cos^{-1}\left(\dfrac{2x-5}{11-3x}\right) + \sin^{-1}(2x^2 - 3x + 1)$ is the interval $[\alpha, \beta]$ , then $\alpha + 2\beta$ is equal to:
1. (1)1
2. (2)3
3. (3)5
4. (4)2
Q2 · MCQ · Area under CurvesOpen
The area of the region, inside the ellipse $x^2 + 4y^2 = 4$ and outside the region bounded by the curves $y = \lvert x\rvert - 1$ and $y = 1 - \lvert x\rvert$ , is:
1. (1) $2(\pi - 1)$
2. (2) $2\pi - \dfrac12$
3. (3) $3(\pi - 1)$
4. (4) $2\pi - 1$
Q3 · MCQ · Sets, Relations & FunctionsOpen
The number of relations, defined on the set $\{a, b, c, d\}$ , which are both reflexive and symmetric, is equal to:
1. (1)256
2. (2)16
3. (3)1024
4. (4)64
Q4 · MCQ · Straight LinesOpen
Let a point $A$ lie between the parallel lines $L_1$ and $L_2$ such that its distances from $L_1$ and $L_2$ are 6 and 3 units, respectively. Then the area (in sq. units) of the equilateral triangle $ABC$ , where the points $B$ and $C$ lie on the lines $L_1$ and $L_2$ respectively, is:
1. (1) $15\sqrt6$
2. (2)27
3. (3) $21\sqrt3$
4. (4) $12\sqrt2$
Q5 · MCQ · VectorsOpen
Let $\vec a = -\hat i + 2\hat j + 2\hat k$ , $\vec b = 8\hat i + 7\hat j - 3\hat k$ and $\vec c$ be a vector such that $\vec a\times\vec c = \vec b$ . If $\vec c\cdot(\hat i + \hat j + \hat k) = 4$ , then $\lvert\vec a + \vec c\rvert^2$ is equal to:
1. (1)33
2. (2)30
3. (3)35
4. (4)27
Q6 · MCQ · Sequences & SeriesOpen
Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing positive terms such that $a_2\cdot a_3\cdot a_4 = 64$ and $a_1 + a_3 + a_5 = \dfrac{813}{7}$ . Then $a_3 + a_5 + a_7$ is equal to:
1. (1)3256
2. (2)3252
3. (3)3244
4. (4)3248
Q7 · MCQ · VectorsOpen
Let $\vec c$ and $\vec d$ be vectors such that $\lvert\vec c + \vec d\rvert = \sqrt{29}$ and $\vec c\times(2\hat i + 3\hat j + 4\hat k) = (2\hat i + 3\hat j + 4\hat k)\times\vec d$ . If $\lambda_1, \lambda_2\ (\lambda_1 > \lambda_2)$ are the possible values of $(\vec c + \vec d)\cdot(-7\hat i + 2\hat j + 3\hat k)$ , then the equation $K^2x^2 + (K^2 - 5K + \lambda_1)xy + \left(3K + \dfrac{\lambda_2}{2}\right)y^2 - 8x + 12y + \lambda_2 = 0$ represents a circle, for $K$ equal to:
1. (1)4
2. (2)1
3. (3) $-1$
4. (4)2
We corrected a printing error in the original paper
Source printed “represents a circle, for $k$ equal to” (lower-case $k$ ; the equation uses $K$ ).
Q8 · MCQ · Differential EquationsOpen
Let $y = y(x)$ be the solution curve of the differential equation $(1 + x^2)\,dy + (y - \tan^{-1}x)\,dx = 0$ , $y(0) = 1$ . Then the value of $y(1)$ is:
1. (1) $\dfrac{2}{e^{\frac{\pi}{4}}} + \dfrac{\pi}{4} - 1$
2. (2) $\dfrac{2}{e^{\frac{\pi}{4}}} - \dfrac{\pi}{4} - 1$
3. (3) $\dfrac{4}{e^{\frac{\pi}{4}}} + \dfrac{\pi}{2} - 1$
4. (4) $\dfrac{4}{e^{\frac{\pi}{4}}} - \dfrac{\pi}{2} - 1$
Q9 · MCQ · Permutations & CombinationsOpen
The number of strictly increasing functions $f$ from the set $\{1, 2, 3, 4, 5, 6\}$ to the set $\{1, 2, 3, \ldots, 9\}$ such that $f(i) \neq i$ for $1 \le i \le 6$ , is equal to:
1. (1)21
2. (2)27
3. (3)22
4. (4)28
We corrected a printing error in the original paper
Source printed the second set as “ $(1, 2, 3, \ldots, 9)$ ” (parentheses instead of set braces).
Q10 · MCQ · Limits, Continuity & DifferentiabilityOpen
Let $f : \mathbb{R} \to (0, \infty)$ be a twice differentiable function such that $f(3) = 18$ , $f'(3) = 0$ and $f''(3) = 4$ . Then $\lim_{x\to1}\left(\log_e\left(\dfrac{f(2+x)}{f(3)}\right)^{\frac{18}{(x-1)^2}}\right)$ is equal to:
1. (1)1
2. (2)9
3. (3)2
4. (4)18
Q11 · MCQ · Conic SectionsOpen
Let the foci of hyperbola coincide with the foci of the ellipse $\dfrac{x^2}{36} + \dfrac{y^2}{16} = 1$ . If the eccentricity of the hyperbola is 5, then the length of its latus rectum is:
1. (1)12
2. (2)16
3. (3) $\dfrac{96}{\sqrt5}$
4. (4) $24\sqrt5$
Q12 · MCQ · Definite IntegrationOpen
The value of $\displaystyle\int_{-\pi/6}^{\pi/6}\left(\frac{\pi + 4x^{11}}{1 - \sin(\lvert x\rvert + \pi/6)}\right)dx$ is equal to
1. (1) $2\pi$
2. (2) $4\pi$
3. (3) $8\pi$
4. (4) $6\pi$
Q13 · MCQ · StatisticsOpen
Let the mean and variance of 7 observations 2, 4, 10, $x$ , 12, 14, $y$ , $x > y$ , be 8 and 16 respectively. Two numbers are chosen from $\{1, 2, 3, x-4, y, 5\}$ one after another without replacement, then the probability, that the smaller number among the two chosen numbers is less than 4, is:
1. (1) $\dfrac35$
2. (2) $\dfrac45$
3. (3) $\dfrac25$
4. (4) $\dfrac13$
Q14 · MCQ · 3D GeometryOpen
Let $(\alpha, \beta, \gamma)$ be the co-ordinates of the foot of the perpendicular drawn from the point $(5, 4, 2)$ on the line $\vec r = (-\hat i + 3\hat j + \hat k) + \lambda(2\hat i + 3\hat j - \hat k)$ . Then the length of the projection of the vector $\alpha\hat i + \beta\hat j + \gamma\hat k$ on the vector $6\hat i + 2\hat j + 3\hat k$ is:
1. (1) $\dfrac{15}{7}$
2. (2)4
3. (3) $\dfrac{18}{7}$
4. (4)3
Q15 · MCQ · CirclesOpen
Let $PQ$ and $MN$ be two straight lines touching the circle $x^2 + y^2 - 4x - 6y - 3 = 0$ at the points $A$ and $B$ respectively. Let $O$ be the centre of the circle and $\angle AOB = \pi/3$ . Then the locus of the point of intersection of the lines $PQ$ and $MN$ is:
1. (1) $3(x^2 + y^2) - 18x - 12y + 25 = 0$
2. (2) $x^2 + y^2 - 12x - 18y - 25 = 0$
3. (3) $x^2 + y^2 - 18x - 12y - 25 = 0$
4. (4) $3(x^2 + y^2) - 12x - 18y - 25 = 0$
Q16 · MCQ · Binomial TheoremOpen
If the coefficient of $x$ in the expansion of $(ax^2 + bx + c)(1 - 2x)^{26}$ is $-56$ and the coefficients of $x^2$ and $x^3$ are both zero, then $a + b + c$ is equal to
1. (1)1300
2. (2)1500
3. (3)1403
4. (4)1483
Q17 · MCQ · Complex NumbersOpen
If $x^2 + x + 1 = 0$ , then the value of $\left(x + \frac1x\right)^4 + \left(x^2 + \frac{1}{x^2}\right)^4 + \left(x^3 + \frac{1}{x^3}\right)^4 + \ldots + \left(x^{25} + \frac{1}{x^{25}}\right)^4$ is:
1. (1)128
2. (2)162
3. (3)175
4. (4)145
Q18 · MCQ · TrigonometryOpen
The value of $\operatorname{cosec}10^\circ - \sqrt3\,\sec10^\circ$ is equal to:
1. (1)4
2. (2)2
3. (3)8
4. (4)6
Q19 · MCQ · Quadratic EquationsOpen
The sum of all the roots of the equation $(x-1)^2 - 5\lvert x - 1\rvert + 6 = 0$ , is:
1. (1)4
2. (2)3
3. (3)1
4. (4)5
Q20 · MCQ · Conic SectionsOpen
Let $O$ be the vertex of the parabola $x^2 = 4y$ and $Q$ be any point on it. Let the locus of the point $P$ , which divides the line segment $OQ$ internally in the ratio $2:3$ be the conic $C$ . Then the equation of the chord of $C$ , which is bisected at the point $(1, 2)$ , is:
1. (1) $5x - y - 3 = 0$
2. (2) $4x - 5y + 6 = 0$
3. (3) $x - 2y + 3 = 0$
4. (4) $5x - 4y + 3 = 0$
We corrected a printing error in the original paper
Source labels the options “(1), (2), (4), (4)”; the third option ( $x - 2y + 3 = 0$ ) is printed as “(4)”, the key (4) is the last option.
Q21 · Numerical · Differential EquationsOpen
Let $f : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that the quadratic equation $f(x)m^2 - 2f'(x)m + f''(x) = 0$ in $m$ , has two equal roots for every $x \in \mathbb{R}$ . If $f(0) = 1$ , $f'(0) = 2$ and $(\alpha, \beta)$ is the largest interval in which the function $f(\log_e x - x)$ is increasing, then $\alpha + \beta$ is equal to .
Q22 · Numerical · Sequences & SeriesOpen
Let $a_1 = 1$ and for $n \ge 1$ , $a_{n+1} = \dfrac12 a_n + \dfrac{n^2 - 2n - 1}{n^2(n+1)^2}$ . Then $\left\lvert\displaystyle\sum_{n=1}^{\infty}\left(a_n - \frac{2}{n^2}\right)\right\rvert$ is equal to .
Q23 · Numerical · Permutations & CombinationsOpen
Let $S = \{(m, n) : m, n \in \{1, 2, 3, \ldots, 50\}\}$ . If the number of elements $(m, n)$ in $S$ such that $6^m + 9^n$ is a multiple of 5 is $p$ and the number of elements $(m, n)$ in $S$ such that $m + n$ is a square of a prime number is $q$ , then $p + q$ is equal to .
Q24 · Numerical · Matrices & DeterminantsOpen
For some $\alpha, \beta \in \mathbb{R}$ , let $A = \begin{bmatrix} \alpha & 2\\ 1 & 2\end{bmatrix} \quad\text{and}\quad B = \begin{bmatrix} 1 & 1\\ 1 & \beta\end{bmatrix}$ be such that $A^2 - 4A + 2I = B^2 - 3B + I = O$ . Then $(\det(\operatorname{adj}(A^3 - B^3)))^2$ is equal to .
Q25 · Numerical · Definite IntegrationOpen
$6\displaystyle\int_0^{\pi}\left\lvert(\sin 3x + \sin 2x + \sin x)\right\rvert dx$ is equal to .