JEE Main 4 Apr 2026, Shift 2

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

Physics

Match the LIST-I with LIST-II

List-I		List-II
A.	Planck's constant	I.	$\mathrm{ML^2T^{-2}}$
B.	Stopping potential	II.	$\mathrm{T^{-1}}$
C.	Work function	III.	$\mathrm{ML^2T^{-1}}$
D.	Threshold frequency	IV.	$\mathrm{ML^2T^{-3}A^{-1}}$

Choose the correct answer from the options given below:

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

Q2 · MCQ · Relative MotionOpen
Two cars A and B are moving in the same direction along a straight line with speeds 100 km/h and 80 km/h, respectively such that car A is moving ahead of car B. A person in car B throws a stone with a speed $v$ so that it hits the car A with a speed of 5 m/s. The value of $v$ is km/h.
1. (1)18
2. (2)28
3. (3)38
4. (4)48
Q3 · MCQ · Newton's Laws of MotionOpen
At $t = 0$ , a body of mass 100 g starts moving under the influence of a force $(5\hat i + 10\hat j)$ N. After 2 s its position is $(2x\hat i + 5y\hat j)$ m. The ratio $x : y$ is .
1. (1) $1 : 2$
2. (2) $2 : 5$
3. (3) $5 : 2$
4. (4) $5 : 4$
Q4 · MCQ · Motion in a PlaneOpen
If $x$ and $y$ coordinates of a projectile as a function of time ( $t$ ) are given as $24t$ and $43.6t - 4.9t^2$ , respectively, then the angle (in degrees) made by the projectile with horizontal when $t = 2$ s is .
1. (1)60
2. (2)45
3. (3)30
4. (4)75
Q5 · MCQ · GravitationOpen
The height in terms of radius of the earth ( $R$ ), at which the acceleration due to gravity becomes $\dfrac{g}{9}$ , where $g$ is acceleration due to gravity on earth's surface, is .
1. (1) $\sqrt3 R$
2. (2) $2\sqrt2 R$
3. (3) $2R$
4. (4) $\dfrac49 R$
Q6 · MCQ · Mechanical Properties of SolidsOpen
A metal string A is suspended from a rigid support and its free end is attached to a block of mass $M$ . Second block having mass $2M$ is suspended at the bottom of the first block using a string B. The area of cross sections of strings A and B are same. The ratio of lengths of strings of A to B is 2 and the ratio of their Young's moduli ( $Y_A/Y_B$ ) is 0.5. The ratio of elongations in A to B is .
1. (1)1
2. (2)4
3. (3)8
4. (4)6
Q7 · MCQ · Mechanical Properties of FluidsOpen
A water spray gun is attached to a hose of cross sectional area $30\ \mathrm{cm^2}$ . The gun comprises of 10 perforations each of cross sectional area of $15\ \mathrm{mm^2}$ . If the water flows in the hose with the speed of 50 cm/s, calculate the speed at which the water flows out from each perforation. (Neglect any edge effects)
1. (1)100 m/s
2. (2)10 m/s
3. (3)1000 m/s
4. (4) $15\times10^{2}$ m/s
Q8 · MCQ · Kinetic Theory of GasesOpen
Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A: If the average kinetic energy of $\mathrm{H_2}$ and $\mathrm{O_2}$ molecules, kept in two different sized containers are same, then their temperatures will be same.
Reason R: The r.m.s. speed of $\mathrm{H_2}$ and $\mathrm{O_2}$ molecules are same at same temperature.
Choose the correct answer from the options given below
1. (1)Both A and R are true and R is the correct explanation of A
2. (2)Both A and R are true but R is NOT the correct explanation of A
3. (3)A is true but R is false
4. (4)A is false but R is true
Q9 · MCQ · Thermal Properties of MatterOpen
The temperature of a metal strip having coefficient of linear expansion $\alpha$ is increased from $T_1$ to $T_2$ resulting in increase of its length by $\Delta L_1$ . The temperature is further increased from $T_2$ to $T_3$ such that the increase in its length is $\Delta L_2$ .
Given $T_3 + T_1 = 2T_2$ and $T_2 - T_1 = \Delta T$ , the value of $\Delta L_2$ is .
1. (1) $\Delta L_1[1 + 2\alpha^2(\Delta T)^2]$
2. (2) $\Delta L_1[1 + \alpha^2(\Delta T)^2]$
3. (3) $\Delta L_1[1 + 2\alpha\,\Delta T]$
4. (4) $\Delta L_1[1 + \alpha\,\Delta T]$
Q10 · MCQ · OscillationsOpen
A uniform disc of radius $R$ and mass $M$ is free to oscillate about the axis A as shown in the figure. For small oscillations the time period is . ( $g$ is acceleration due to gravity)
1. (1) $2\pi\sqrt{\dfrac{5R}{4g}}$
2. (2) $2\pi\sqrt{\dfrac{2R}{3g}}$
3. (3) $2\pi\sqrt{\dfrac{3R}{2g}}$
4. (4) $2\pi\sqrt{\dfrac{3R}{g}}$
Q11 · MCQ · Electric Charges and FieldsOpen
A rigid dipole undergoes a simple harmonic motion about its centre in the presence of an electric field $\vec E_1 = E_0\hat x$ . If another electric field $\vec E_2 = 2E_0(\hat y + \hat z)$ is introduced to the system, what will be the percentage change in the frequency of the oscillation (approximate)?
1. (1)73%
2. (2)63%
3. (3)83%
4. (4)53%
Q12 · MCQ · Electric Potential and CapacitanceOpen
From the circuit given below, the capacitance between terminals A and B shown in the circuit is $\mu$ F. (take $C_1 = C_2 = C_3 = 1\ \mu$ F and $C_4 = 2\ \mu$ F.)
1. (1)2
2. (2)7/2
3. (3)7/3
4. (4)5/2
We corrected a printing error in the original paper
Answer corrected; the source key is (2) — the top wire joins A to the $C_2$ / $C_3$ junction, so $C_1 \parallel C_2 = 2\ \mu$ F in series with $C_4 = 2\ \mu$ F gives $1\ \mu$ F, plus $C_3 = 1\ \mu$ F across A–B: $C_{AB} = 2\ \mu$ F (the source's own solution also ends at $2\ \mu$ F).
Q13 · MCQ · Electric Potential and CapacitanceOpen
Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A: In electrostatics, a conductor does not store any net charge inside.
Reason R: Inside the capacitor (with no dielectric medium), the free charge carriers, if placed between the plates of capacitor, experience force and drift.
Choose the correct answer from the options given below.
1. (1)Both A and R are true and R is the correct explanation of A.
2. (2)Both A and R are true but R is NOT the correct explanation of A.
3. (3)A is true but R is false
4. (4)A is false but R is true
Q14 · MCQ · Magnetism and MatterOpen
A solenoid has a core made of material with relative permeability 400. The magnetic field produced in the interior of solenoid is 1.0 T. The magnetic intensity in SI units is $\alpha\times10^{5}$ . The value of $\alpha$ is . (Free space permeability $\mu_0 = 4\pi\times10^{-7}$ SI units).
1. (1) $\dfrac{25}{\pi}$
2. (2) $\dfrac{1}{16\pi}$
3. (3) $\dfrac{1}{\pi}$
4. (4) $\dfrac{1}{4\pi}$
We corrected a printing error in the original paper
Source printed “The magnetic intensity is SI units”.
Q15 · MCQ · Electromagnetic WavesOpen
A magnetic field vector in an electromagnetic wave is represented by, $\vec B = B_0\sin\left(2\pi\nu t - \frac{2\pi x}{\lambda}\right)\hat j.$ Its associated electric field vector is .
1. (1) $\vec E = -\nu\lambda B_0\sin\left(2\pi\nu t - \dfrac{2\pi x}{\lambda}\right)\hat k$
2. (2) $\vec E = -\nu\lambda B_0\sin\left(2\pi\nu t - \dfrac{2\pi x}{\lambda}\right)\hat i$
3. (3) $\vec E = \nu\lambda B_0\sin\left(2\pi\nu t - \dfrac{2\pi x}{\lambda}\right)\hat k$
4. (4) $\vec E = \nu\lambda B_0\sin\left(2\pi\nu t - \dfrac{2\pi x}{\lambda}\right)\hat i$
We corrected a printing error in the original paper
Source printed the frequency as a Latin “v” ( $2\pi\mathrm{v}t$ , $\mathrm{v}\lambda B_0$ ) instead of $\nu$ .
Q16 · MCQ · Ray OpticsOpen
A convex lens is made from glass material having refractive index of 1.4 with same radius of curvature on both sides. The ratio of its focal length and radius of curvature is .
1. (1)0.5
2. (2)2.5
3. (3)0.8
4. (4)1.25
Q17 · MCQ · Wave OpticsOpen
An unpolarized light of certain intensity passes through a combination of two polarizers whose transmission axes are at $30^\circ$ and $90^\circ$ , respectively, with respect to the horizontal axis. A third polarizer with its transmission axis at $60^\circ$ with the horizontal axis is placed between the two existing polarizers. The ratio of the output intensities with and without the third polarizer is .
1. (1)3/4
2. (2)4/3
3. (3)9/4
4. (4)4/9
Q18 · MCQ · AtomsOpen
In Rutherford's alpha-particle scattering experiment, only a few alpha particles rebound back because
(A) The size of gold nucleus is very small as compared to the size of gold atom.
(B) Alpha particle and gold nucleus have equal charge.
(C) The impact parameter is minimum for a few alpha particles.
(D) A few alpha particles have very high kinetic energy.
(E) Only a few alpha particles undergo head-on collision with the nuclei.
Choose the correct answer from the options given below:
1. (1)A, B Only
2. (2)B, E Only
3. (3)C, D Only
4. (4)A, C, E Only
Q19 · MCQ · Dual Nature of Radiation and MatterOpen
The de Broglie wavelength associated with an electron accelerated through a potential difference $V$ is $\lambda_e$ and the de Broglie wavelength associated with a proton accelerated through the same potential difference is $\lambda_p$ . If their corresponding masses are $m_e$ and $m_p$ , respectively, then the ratio of their de Broglie wavelengths $\left(\dfrac{\lambda_e}{\lambda_p}\right)$ is .
1. (1) $\sqrt{\dfrac{m_p}{m_e}}$
2. (2) $\sqrt{\dfrac{m_e}{m_p}}$
3. (3) $\dfrac{m_p}{m_e}$
4. (4) $\left(\dfrac{m_p}{m_e}\right)^2$
Q20 · MCQ · Semiconductor ElectronicsOpen
Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A: A diode under reverse-biased condition provides very small current which is nearly independent of voltage until a critical limit at which the current increases drastically.
Reason R: Below the critical voltage limit, only majority charge carriers flow which increases drastically above critical voltage.
Choose the correct answer from the options given below.
1. (1)Both A and R are true and R is the correct explanation of A.
2. (2)Both A and R are true but R is NOT the correct explanation of A.
3. (3)A is true but R is false
4. (4)A is false but R is true
Q21 · Numerical · Semiconductor ElectronicsOpen
A diode has Zener voltage of 10 V and maximum power dissipation of 0.5 W, then the minimum resistance to be used in series with this diode for safety when it is connected to a 25 V power supply is $\Omega$ .
Q22 · Numerical · Motion in a PlaneOpen
A gun mounted on the ground fires bullets in all directions with same speed. The farthest distance the bullets could reach is 6.4 m. The speed of the bullets from the gun is m/s. (take $g = 10\ \mathrm{m/s^2}$ )
Q23 · Numerical · Magnetism and MatterOpen
Two identical small bar magnets each of dipole moment $3\sqrt5\ \mathrm{J/T}$ are placed at a center to center separation of 10 cm, with their axes perpendicular to each other as shown in figure. The value of magnetic field at the point P midway between the magnets is $\alpha\times10^{-3}$ T. The value of $\alpha$ is . ( $\mu_0 = 4\pi\times10^{-7}$ Tm/A)
Q24 · Numerical · Moving Charges and MagnetismOpen
A circular coil of radius 2 cm and 125 turns carries a current of 1 A. The coil is placed in a uniform magnetic field of magnitude 0.4 T. The axis of the coil makes an angle of $30^\circ$ with the direction of the magnetic field. The torque acting on the coil is $\alpha\times10^{-4}$ N.m. The value of $\alpha$ is . ( $\pi = 3.14$ )
Q25 · Numerical · Wave OpticsOpen
In a double slit experiment, when one of the slits is covered by a transparent mica sheet of refractive index 1.56, the central fringe shifts to the position of $7^{\text{th}}$ bright fringe, obtained with both slits uncovered. If the light source wavelength is 450 nm, the thickness of mica sheet is $\alpha\times10^{-9}$ m. The value of $\alpha$ is .

Chemistry

Q1 · MCQ · Some Basic ConceptsOpen
The correct order of total number of atoms present in
- (A)2 moles of cyclohexane
- (B)684 g of sucrose
- (C)90.8 L of dihydrogen at STP
1. (1)C $>$ A $>$ B
2. (2)C $>$ B $>$ A
3. (3)B $>$ C $>$ A
4. (4)B $>$ A $>$ C
We corrected a printing error in the original paper
Answer corrected; the source key is (3) — (A) 2 mol C $_6$ H $_{12}$ = 36 mol atoms, (B) 684 g sucrose = 2 mol = 90 mol atoms, (C) 90.8 L H $_2$ = 4 mol = 8 mol atoms, so B $>$ A $>$ C; also source printed “total number of atoms of present in” and option (4) “B $>$ A C”.
Q2 · MCQ · Structure of AtomOpen
The species having identical radii according to the Bohr's theory are :
- (A)H (first orbit)
- (B) $\ce{He+}$ (first orbit)
- (C) $\ce{He+}$ (second orbit)
- (D) $\ce{Li^2+}$ (first orbit)
- (E) $\ce{Be^3+}$ (Second orbit)
Choose the correct answer from the options given below :
1. (1)A and C Only
2. (2)A and E Only
3. (3)B and E Only
4. (4)C and D Only
Q3 · MCQ · Chemical BondingOpen
Which of the following pictorial diagram most correctly represents the $\pi^*$ ( $\pi$ – antibonding) molecular orbital between two atoms if the internuclear axis is taken to be in the z-direction ( $\xrightarrow{\text{z--axis}}$ ) ?
1. (1)
2. (2)
3. (3)
4. (4)
Q4 · MCQ · SolutionsOpen
At 27 $^\circ$ C, 0.1 M, 1 L $\ce{K4[Fe(CN)6]}$ aqueous solution and 0.1 M, 1 L $\ce{FeCl3}$ aqueous solution are placed in a container separated by a semi permeable membrane AB. Assume complete dissociation of both the solutes. Which of the following statement is correct ?
1. (1)Blue color is formed on both sides.
2. (2)Ionic solutes in aqueous solution can pass through semi-permeable membrane.
3. (3)Solution on side ‘y' is hypotonic.
4. (4)To cause the reverse flow of solvent during osmosis, external pressure (any value) should be applied to side ‘x'.
Q5 · MCQ · EquilibriumOpen
20 mL of a solution of acetic acid required 28.4 mL of 0.1 M NaOH for its neutralization. A solution (X) was prepared by mixing 20 mL of the above acetic acid and 14.2 mL of 0.1 M NaOH solution. What is the pH of the solution (X) ? ( $\mathrm{p}K_\mathrm{a}$ value of acetic acid is 4.75).
1. (1)7.0
2. (2)4.75
3. (3)3.5
4. (4)4.82
We corrected a printing error in the original paper
Source printed option (2) as “4.45” (no option was then correct); corrected to the intended “4.75” — 14.2 mL is exactly half the 28.4 mL needed, so pH = p $K_\mathrm{a}$ = 4.75 (source key (2)).

Q6 · MCQ · Organic Basics (GOC)Open

Match the LIST-I with LIST-II

List-I		List-II
Reaction		Mechanism
A.	Williamson Synthesis	I.	Electrophilic addition
B.	Friedel Crafts Reaction	II.	Free radical substitution
C.	Bromination of vinyl benzene	III.	Nucleophilic substitution
D.	Chlorination of toluene in light	IV.	Electrophilic substitution

Choose the correct answer from the options given below:

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

We corrected a printing error in the original paper

Source printed “Friedel Craft Reaction”.

Q7 · MCQ · Classification & PeriodicityOpen
The 1 $^\mathrm{st}$ ionization enthalpy for Mg is $+737$ kJ/mol. The most probable estimated value of the 2 $^\mathrm{nd}$ ionization enthalpy of Mg is .
1. (1) $-906$ kJ/mol
2. (2) $-856$ kJ/mol
3. (3) $+1450$ kJ/mol
4. (4) $+590$ kJ/mol
Q8 · MCQ · p-Block (13–14)Open
The electronegativity of a group 13 element ‘E' is same as that of Ge (on Pauling scale and upto one decimal point). The CORRECT statements about $\ce{E^3+}$ are
- (A)It can act as a reducing agent.
- (B)It can act as an oxidizing agent.
- (C) $\ce{E^3+}$ is more stable than $\ce{E+}$ .
- (D)The standard electrode potential value for $\ce{E^3+}$ /E is positive.
Choose the correct answer from the options given below
1. (1)A and C Only
2. (2)B and C Only
3. (3)B and D Only
4. (4)A and D Only
Q9 · MCQ · d- and f-BlockOpen
Pairs of elements with the same number of electrons in their respective 4f orbital are
[Atomic number, Eu-63, Gd-64, Dy-66, Ho-67, Tm-69, Yb-70, Lu-71, Hf-72]
- (A)(Eu and Gd)
- (B)(Dy and Ho)
- (C)(Yb and Hf)
- (D)(Lu and Tm)
Choose the correct answer from the options given below:
1. (1)B and C Only
2. (2)A and B Only
3. (3)A and D Only
4. (4)A and C Only
Q10 · MCQ · Coordination CompoundsOpen
Consider the metal complexes $\ce{[Ni(en)3]^2+}$ (A), $\ce{[NiCl4]^2-}$ (B) and $\ce{[Ni(NH3)6]^2+}$ (C). Choose the CORRECT option by considering the number of unpaired electron present in (A), (B) and (C) respectively and the order of frequency of absorption.
1. (1)2, 2, 2 and (A) $>$ (C) $>$ (B)
2. (2)0, 2, 0 and (A) $>$ (C) $>$ (B)
3. (3)2, 2, 0 and (B) $>$ (C) $>$ (A)
4. (4)2, 2, 2 and (C) $>$ (A) $>$ (B)
Q11 · MCQ · Organic Basics (GOC)Open
Consider the following molecules/species:
The correct order of carbon – oxygen double bond length is:
1. (1)x $>$ y $>$ z
2. (2)y $>$ z $>$ x
3. (3)z $>$ x $>$ y
4. (4)x $>$ z $>$ y
We corrected a printing error in the original paper
In structure (y) the source printed the lower group as “CH”; corrected in the figure to “CH $_3$ ” (propanone), the subscript copied from the upper CH $_3$ of the same drawing.
Q12 · MCQ · d- and f-BlockOpen
Consider $\lvert x\rvert$ is the difference in oxidation states of Mn in highest manganese fluoride and highest manganese oxide. The ions with $\lvert x\rvert$ number of unpaired electrons from the following are:
- (A) $\ce{Sc^3+}$
- (B) $\ce{Zn^2+}$
- (C) $\ce{V^2+}$
- (D) $\ce{Fe^2+}$
- (E) $\ce{Co^2+}$
Choose the correct answer from the options given below:
1. (1)A and B Only
2. (2)C, D and E Only
3. (3)C and E Only
4. (4)B and E Only
Q13 · MCQ · Chemical KineticsOpen
Consider the given graph showing variation of reactant concentration with time.
Three different reactions were started with identical initial concentration of reactants. Which of the following statement is correct ?
1. (1)The order of all the three reactions is same.
2. (2)The rate constant of reaction 3 is larger than the rate constant of reaction 2 if the order of reaction is same for both.
3. (3)The SI unit of rate constant of reaction 1 is s $^{-1}$ .
4. (4)Thermal decomposition of HI on gold surface is an example of reaction 2.
Q14 · MCQ · HydrocarbonsOpen
Compound (X) is subjected to the sequence of reactions as shown above. Molar mass of the major product (Y) formed is g mol $^{-1}$ .
(Given molar mass in g mol $^{-1}$ C : 12, H : 1, O : 16)
1. (1)90
2. (2)118
3. (3)160
4. (4)125
Q15 · MCQ · Organic Basics (GOC)Open
The following structures are
and
1. (1)enantiomer
2. (2)Identical molecules
3. (3)Diastereomers
4. (4)meso compound
Q16 · MCQ · Organic Basics (GOC)Open
The descending order of acidity among the following compounds is
(A) (B)
(C) (D)
(E)
Choose the correct answer from the option given below options
1. (1)B $>$ D $>$ E $>$ A $>$ C
2. (2)D $>$ B $>$ E $>$ A $>$ C
3. (3)C $>$ A $>$ B $>$ D $>$ E
4. (4)D $>$ E $>$ B $>$ A $>$ C
Q17 · MCQ · AminesOpen
The strongest conjugate acid will result from
1. (1)
2. (2)
3. (3)
4. (4)
Q18 · MCQ · BiomoleculesOpen
A D-aldotetrose on oxidation with concentrated $\ce{HNO3}$ resulted in optically inactive dicarboxylic acid. The structure of the D-aldotetrose is :
1. (1)
2. (2)
3. (3)
4. (4)
Q19 · MCQ · Practical / Qualitative AnalysisOpen
Among $\ce{Fe^3+}$ , $\ce{Pb^2+}$ , $\ce{Cu^2+}$ and $\ce{Mn^2+}$ , identify the one that gets precipitated out while passing $\ce{H2S}$ in presence of $\ce{NH4OH}$ as group reagent. The highest possible oxidation state of the corresponding metal is
1. (1) $+3$
2. (2) $+4$
3. (3) $+2$
4. (4) $+7$
Q20 · MCQ · Practical / Qualitative AnalysisOpen
Match the LIST-I with LIST-II
List-I List-II
Compound Test
A. I Hinsberg's reagent test
B. II Phthalein dye test
C. III. Lucas test
D. IV. Tollen's test
Choose the correct answer from the options given below :
1. (1)A-III, B-I, C-IV, D-II
2. (2)A-III, B-IV, C-I, D-II
3. (3)A-I, B-III, C-II, D-IV
4. (4)A-I, B-II, C-III, D-IV
Q21 · Numerical · ThermodynamicsOpen
If 3.365 g of ethanol ( $l$ ) is burnt completely in a bomb calorimeter at 298.15 K, the heat produced is 99.472 kJ. The $\lvert\Delta H_\mathrm{f}^\circ\rvert$ of ethanol at 298.15 K is $\times\,10^{2}$ kJ mol $^{-1}$ . (Nearest integer)
Given :
Standard enthalpy of combustion of graphite $= -393.5$ kJ mol $^{-1}$
Standard enthalpy of formation of water ( $l$ ) $= -285.8$ kJ mol $^{-1}$
Molar mass in g mol $^{-1}$ of C, H, O are 12, 1 and 16 respectively.
We corrected a printing error in the original paper
Answer corrected; the source key is 282 — $\Delta U_\mathrm{c} = -1359.8$ kJ mol $^{-1}$ , $\Delta H_\mathrm{c} = -1362.3$ kJ mol $^{-1}$ , $\Delta_\mathrm{f}H = -282.1$ kJ mol $^{-1}$ = $-2.82\times10^{2}$ , so the blank before “ $\times\,10^{2}$ ” is 3; also source printed “If 3.365 g of ethanol ( $l$ ) burnt completely”.
Q22 · Numerical · ThermodynamicsOpen
For the following reaction at 50 $^\circ$ C and 2 atm pressure,
$\ce{2N2O5(g) <=> 2N2O4(g) + O2(g)}$
$\ce{N2O5}$ is 50% dissociated
The magnitude of standard free energy change at this temperature is x.
x = J mol $^{-1}$ [Nearest integer]
Given : $R = 8.314$ J mol $^{-1}$ K $^{-1}$ , $\log 2 = 0.30$ , $\log 3 = 0.48$ , $\ln 10 = 2.303$ , $^\circ$ C + 273 = K
We corrected a printing error in the original paper
Source printed “R = 8.314 mol $^{-1}$ K $^{-1}$ ” (unit J missing).
Q23 · Numerical · ElectrochemistryOpen
An electrochemical cell consists of the following two redox couples,
$\mathrm{M}^{x+}$ (aq)/M(s) [ $E^\ominus_\mathrm{red} = +0.15$ V] and
$\ce{Fe^3+}$ (aq)/Fe(s) [ $E^\ominus_\mathrm{red} = -0.036$ V]
The cell EMF ( $E_\mathrm{cell}$ ) is recorded to be 0.2057 V.
If the reaction quotient of the electrochemical reaction is found to be $10^{-2}$ , then the value of x is (Nearest integer)
[Given : M is a p-block metal and $\dfrac{2.303RT}{F} = 0.059$ V]
We corrected a printing error in the original paper
Source printed “An electrochemical cell. consist of”.
Q24 · Numerical · Chemical KineticsOpen
For a first order reaction $\ce{A -> B}$
t/ min [A]/M
0 0.6500
x 0.0650
20 0.00065
x = min. (Nearest integer)
Q25 · Numerical · Purification & CharacterisationOpen
In sulphur estimation, $2.0\times10^{-3}$ mol of an organic compound (X) (molar mass 76 g mol $^{-1}$ ) gave 0.4813 g of barium sulphate (molar mass 233 g mol $^{-1}$ ). The percentage of sulphur in the compound (X) is $\times\,10^{-1}$ %
(Nearest integer)

Maths

Q1 · MCQ · Sets, Relations & FunctionsOpen
For the function $f : [1, \infty) \to [1, \infty)$ defined by $f(x) = (x-1)^4 + 1$ , among the two statements:
(I) The set $S = \{x \in [1, \infty) : f(x) = f^{-1}(x)\}$ contains exactly two elements, and
(II) The set $S = \{x \in [1, \infty) : f(x) = f^{-1}(x+1)\}$ is an empty set,
1. (1)only (I) is TRUE
2. (2)only (II) is TRUE
3. (3)both (I) and (II) are TRUE
4. (4)neither (I) nor (II) is TRUE
Q2 · MCQ · Complex NumbersOpen
Let $S = \{z \in \mathbb{C} : z^2 + 4z + 16 = 0\}$ . Then $\displaystyle\sum_{z\in S}\left\lvert z + \sqrt3\,i\right\rvert^2$ is equal to:
1. (1)42
2. (2)23
3. (3)27
4. (4)38
Q3 · MCQ · Matrices & DeterminantsOpen
If the system of equations: $\begin{aligned} x + y + z &= 5\\ x + 2y + 3z &= 9\\ x + 3y + \lambda z &= \mu\end{aligned}$ has infinitely many solutions, then the value of $\lambda + \mu$ is:
1. (1)16
2. (2)18
3. (3)19
4. (4)21
Q4 · MCQ · Quadratic EquationsOpen
If $\alpha = 1$ and $\beta = 1 + i\sqrt2$ , where $i = \sqrt{-1}$ are two roots of the equation $x^3 + ax^2 + bx + c = 0$ , $a, b, c \in \mathbb{R}$ , then $\displaystyle\int_{-1}^{1}\left(x^3 + ax^2 + bx + c\right)dx$ is equal to:
1. (1) $-2$
2. (2) $-4$
3. (3) $-8$
4. (4) $-10$
Q5 · MCQ · Quadratic EquationsOpen
If the quadratic equation $(\lambda + 2)x^2 - 3\lambda x + 4\lambda = 0$ , $\lambda \neq -2$ , has two positive roots, then the number of possible integral values of $\lambda$ is:
1. (1)1
2. (2)2
3. (3)3
4. (4)4
Q6 · MCQ · Matrices & DeterminantsOpen
Let $A = \begin{bmatrix} 1 & 2 & 7\\ 4 & -2 & 8\\ 3 & 8 & -7\end{bmatrix}$ and $\det(A - \alpha I) = 0$ , where $\alpha$ is a real number. If the largest possible value of $\alpha$ is $p$ , then the circle $(x-p)^2 + (y-2p)^2 = 320$ , intersects the co-ordinate axes at
1. (1)1 point
2. (2)2 points
3. (3)3 points
4. (4)4 points
Q7 · MCQ · Sequences & SeriesOpen
Let $\alpha = \dfrac14 + \dfrac18 + \dfrac1{16} + \ldots\infty$ and $\beta = \dfrac13 + \dfrac19 + \dfrac1{27} + \ldots\infty$ . Then the value of $(0.2)^{\log_{\sqrt5}(\alpha)} + (0.04)^{\log_5(\beta)}$ is equal to:
1. (1)4
2. (2)5
3. (3)8
4. (4)25
Q8 · MCQ · StatisticsOpen
For 10 observations $x_1, x_2, \ldots, x_{10}$ , if $\displaystyle\sum_{i=1}^{10}(x_i + 2)^2 = 180$ and $\displaystyle\sum_{i=1}^{10}(x_i - 1)^2 = 90$ , then their standard deviation is:
1. (1)2
2. (2) $\sqrt3$
3. (3) $2\sqrt2$
4. (4)3
Q9 · MCQ · Binomial TheoremOpen
In the expansion of $\left(9x - \dfrac{1}{3\sqrt{x}}\right)^{18}$ , $x > 0$ , if the term independent of $x$ is $(221)k$ . then $k$ is equal to:
1. (1)84
2. (2)78
3. (3)168
4. (4)198
Q10 · MCQ · Conic SectionsOpen
Let $P(3\cos\alpha, 2\sin\alpha)$ , $\alpha \neq 0$ , be a point on the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1$ , $Q$ be a point on the circle $x^2 + y^2 - 14x - 14y + 82 = 0$ and $R$ be a point on the line $x + y = 5$ such that the centroid of the triangle $PQR$ is $\left(2 + \cos\alpha, 3 + \dfrac23\sin\alpha\right)$ . Then the sum of the ordinates of all possible points $R$ is:
1. (1)6
2. (2)2
3. (3)4
4. (4)8
Q11 · MCQ · Conic SectionsOpen
Let $H : \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ be a hyperbola such that the distance between its foci is 6 and the distance between its directrices is $\dfrac83$ . If the line $x = \alpha$ intersects the hyperbola $H$ at the points $A$ and $B$ such that the area of the triangle $AOB$ is $4\sqrt{15}$ , where $O$ is the origin, then $\alpha^2$ equals
1. (1)12
2. (2)16
3. (3)24
4. (4)25
Q12 · MCQ · Applications of DerivativesOpen
$\displaystyle\max_{0 \le x \le \pi}\left(16\sin\left(\frac{x}{2}\right)\cos^3\left(\frac{x}{2}\right)\right)$ is equal to:
1. (1) $\dfrac{3\sqrt3}{2}$
2. (2) $3\sqrt3$
3. (3) $4\sqrt3$
4. (4) $6\sqrt3$
Q13 · MCQ · 3D GeometryOpen
The shortest distance between the lines $\vec r = \left(\frac13\hat i + 2\hat j + \frac83\hat k\right) + \lambda\left(2\hat i - 5\hat j + 6\hat k\right)$ and $\vec r = \left(-\dfrac23\hat i - \dfrac13\hat k\right) + \mu\left(\hat j - \hat k\right)$ , $\lambda, \mu \in \mathbb{R}$ , is:
1. (1) $\sqrt5$
2. (2)3
3. (3) $2\sqrt3$
4. (4) $\sqrt{15}$
Q14 · MCQ · 3D GeometryOpen
If $\left(2\alpha + 1, \alpha^2 - 3\alpha, \dfrac{\alpha - 1}{2}\right)$ is the image of $(\alpha, 2\alpha, 1)$ in the line $\dfrac{x-2}{3} = \dfrac{y-1}{2} = \dfrac{z}{1}$ , then the possible value(s) of $\alpha$ is(are):
1. (1)Only 3
2. (2)Only 3 and $-1$
3. (3)Only 3, $\dfrac14$ and $-1$
4. (4)Only 3 and $\dfrac14$
Q15 · MCQ · VectorsOpen
Let $\hat u$ and $\hat v$ be unit vectors inclined at an acute angle such that $\lvert\hat u \times \hat v\rvert = \dfrac{\sqrt3}{2}$ . If $\vec A = \lambda\hat u + \hat v + (\hat u \times \hat v)$ . Then $\lambda$ is equal to:
1. (1) $\dfrac43\left(\vec A\cdot\hat u\right) - \dfrac23\left(\vec A\cdot\hat v\right)$
2. (2) $\dfrac23\left(\vec A\cdot\hat u\right) - \dfrac13\left(\vec A\cdot\hat v\right)$
3. (3) $\dfrac43\left(\vec A\cdot\hat u\right) + \dfrac23\left(\vec A\cdot\hat v\right)$
4. (4) $\left(\vec A\cdot\hat u\right) - \dfrac12\left(\vec A\cdot\hat v\right)$
Q16 · MCQ · Sets, Relations & FunctionsOpen
Let for some $\alpha \in \mathbb{R}$ , $f : \mathbb{R} \to \mathbb{R}$ be a function satisfying $f(x + y) = f(x) + 2y^2 + y + \alpha xy$ for all $x, y \in \mathbb{R}$ . If $f(0) = -1$ and $f(1) = 2$ , then the value of $\displaystyle\sum_{n=1}^{5}\left(\alpha + f(n)\right)$ is:
1. (1)110
2. (2)140
3. (3)150
4. (4)170
Q17 · MCQ · Permutations & CombinationsOpen
Let $A = \{(a, b, c) : a, b, c$ are non-negative integers and $a + b + 2c = 22\}$ . The $n(A)$ is equal to:
1. (1)121
2. (2)124
3. (3)144
4. (4)169
Q18 · MCQ · Area under CurvesOpen
The area of the region bounded by the curves $x + 3y^2 = 0$ and $x + 4y^2 = 1$ is equal to:
1. (1) $\dfrac13$
2. (2) $\dfrac23$
3. (3) $\dfrac43$
4. (4) $\dfrac53$
Q19 · MCQ · Differential EquationsOpen
Let $y = y(x)$ be the solution of the differential equation: $\frac{dy}{dx} + \left(\frac{6x^2 + \left(3x^2 + 2x^3 + 4\right)e^{-2x}}{\left(x^3 + 2\right)\left(2 + e^{-2x}\right)}\right)y = 2 + e^{-2x},$ $x \in (-1, 2)$ , satisfying $y(0) = \dfrac32$ . If $y(1) = \alpha\left(2 + e^{-2}\right)$ , then $\alpha$ is equal to:
1. (1) $\dfrac{13}{8}$
2. (2) $\dfrac{6}{13}$
3. (3) $\dfrac{12}{13}$
4. (4) $\dfrac{13}{12}$
We corrected a printing error in the original paper
Source printed “than $\alpha$ is equal to”.
Q20 · MCQ · Definite IntegrationOpen
The integral $\displaystyle\int_0^1\cot^{-1}\left(1 + x + x^2\right)dx$ is equal to:
1. (1) $2\tan^{-1}2 + \dfrac12\log_e\left(\dfrac54\right) + \dfrac{\pi}{2}$
2. (2) $2\tan^{-1}2 + \dfrac12\log_e\left(\dfrac54\right) - \dfrac{\pi}{2}$
3. (3) $2\tan^{-1}2 - \dfrac12\log_e\left(\dfrac54\right) + \dfrac{\pi}{2}$
4. (4) $2\tan^{-1}2 - \dfrac12\log_e\left(\dfrac54\right) - \dfrac{\pi}{2}$
Q21 · Numerical · ProbabilityOpen
From a month of 31 days, 3 different dates are selected at random. If the probability that these dates are in an increasing A.P. is equal to $\dfrac{a}{b}$ , where $a, b \in \mathbb{N}$ and $\gcd(a, b) = 1$ , then $a + b$ is equal to .
Q22 · Numerical · Limits, Continuity & DifferentiabilityOpen
Let $f(x) = \begin{cases} e^{x-1}, & x < 0\\ x^2 - 5x + 6, & x \ge 0\end{cases}$ and $g(x) = f(\lvert x\rvert) + \lvert f(x)\rvert$ . If the number of points where $g$ is not continuous and is not differentiable are $\alpha$ and $\beta$ respectively, then $\alpha + \beta$ is equal to .
Q23 · Numerical · Straight LinesOpen
Let $A$ , $B$ be points on the two half-lines $x - \sqrt3\,\lvert y\rvert = \alpha$ , $\alpha > 0$ at a distance of $\alpha$ from their point of intersection $P$ . The line segment $AB$ meets the angle bisector of the given half-lines at the point $Q$ . If $PQ = \dfrac92$ and $R$ is the radius of the circumcircle of $\triangle PAB$ , then $\dfrac{\alpha^2}{R}$ is equal to .
Q24 · Numerical · Conic SectionsOpen
Let $A$ , $B$ and $C$ be the vertices of a variable right angled triangle inscribed in the parabola $y^2 = 16x$ . Let the vertex $B$ containing the right angle be $(4, 8)$ and the locus of the centroid of $\triangle ABC$ be a conic $C_0$ . Then three times the length of latus rectum of $C_0$ is .
Q25 · Numerical · Differential EquationsOpen
Let $f$ be a twice differentiable function such that $f(x) = \int_0^x\tan(t - x)\,dt - \int_0^x f(t)\tan t\,dt,\quad x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ Then $f''\left(\dfrac{\pi}{6}\right) + 12f'\left(-\dfrac{\pi}{6}\right) + f\left(\dfrac{\pi}{6}\right)$ is equal to .

List-I	List-II
Compound	Test
A.	I	Hinsberg's reagent test
B.	II	Phthalein dye test
C.	III.	Lucas test
D.	IV.	Tollen's test

(A)	(B)
(C)	(D)
(E)

t/ min	[A]/M
0	0.6500
x	0.0650
20	0.00065