JEE Main 6 Apr 2026, Shift 1

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

Physics

Q1 · MCQ · Units and MeasurementsOpen
The density $\rho$ of a uniform cylinder is determined by measuring its mass m, length $l$ and diameter d. The measured values of m, $l$ and d are $97.42 \pm 0.02$ g, $8.35 \pm 0.05$ mm and $20.20 \pm 0.02$ mm, respectively. Calculated percentage fractional error in $\rho$ is .
1. (1)0.63%
2. (2)0.82%
3. (3)0.72%
4. (4)0.25%
Q2 · MCQ · Units and MeasurementsOpen
The potential energy of a particle changes with distance x from a fixed origin as $V = \dfrac{A\sqrt{x}}{x + B}$ , where A and B are constants with appropriate dimensions. The dimensions of AB are .
1. (1) $[\mathrm{M^1L^{5/2}T^{-2}}]$
2. (2) $[\mathrm{M^{3/2}L^{5/2}T^{-2}}]$
3. (3) $[\mathrm{M^1L^2T^{-2}}]$
4. (4) $[\mathrm{M^1L^{7/2}T^{-2}}]$
We corrected a printing error in the original paper
Source printed “A and B are constant with appropriate dimensions. The dimension of AB are”.
Q3 · MCQ · Work, Energy and PowerOpen
The rain drop of mass 1 g, starts with zero velocity from a height of 1 km. It hits the ground with a speed of 5 m/s. The work done by the unknown resistive force is J. (take $g = 10\ \mathrm{m/s^2}$ )
1. (1) $-8.75$
2. (2) $-8.35$
3. (3) $-9.55$
4. (4) $-9.98$
Q4 · MCQ · Newton's Laws of MotionOpen
Two blocks (P and Q) with respectively masses 2 kg and 1.5 kg are joined by a massless thread. These blocks are mounted on a frictionless pulley which is fixed on the edge of a cube (S), as shown in the figure below. Block P is positioned on the top surface which has no friction and block Q is in contact with side-surface, having coefficient of friction $\mu$ . The cube (S) moves towards the right with acceleration of $\dfrac{g}{2}$ , where g is gravitational acceleration. During this movement the blocks P and Q remain stationary. The value of $\mu$ is . (take $g = 10\ \mathrm{m/s^2}$ )
1. (1)0.33
2. (2)0.67
3. (3)1
4. (4)0.5
We corrected a printing error in the original paper
The figure printed the mass of Q as “15 kg” (with 15 kg no option fits); figure label changed to the stem's 1.5 kg (original kept as work/orig-P04-stem.png); source also printed “pully”; “coefficient friction”; “the block P and Q”.
Q5 · MCQ · Mechanical Properties of SolidsOpen
A lift of mass 1600 kg is supported by thick iron wire. If the maximum stress which the wire can withstand is $4\times10^{8}\ \mathrm{N/m^2}$ and its radius is 4 mm, then maximum acceleration the lift can take is $\mathrm{m/s^2}$ . (take $g = 10\ \mathrm{m/s^2}$ and $\pi = 3.14$ )
1. (1)2.56
2. (2)3.89
3. (3)4.32
4. (4)5.16
Q6 · MCQ · Rotational MotionOpen
A solid sphere of radius 4 cm and mass 5 kg is rotating (rotation axis is passing through the centre of the sphere) with an angular velocity of 1200 rpm. It is brought to rest in 10 s by applying a constant torque. The torque applied and the number of rotations it made before it comes to rest are and respectively.
1. (1) $0.128\,\pi$ Nm, 100
2. (2) $0.0128\,\pi$ Nm, 50
3. (3) $0.128\,\pi$ Nm, 50
4. (4) $0.0128\,\pi$ Nm, 100
Q7 · MCQ · Circular MotionOpen
A smooth inclined plane ends in a vertical circular loop, as shown in the figure. A small body is released from height h as shown. If the body exerts a force of three times its weight on the plane at the highest point of circle then the height $h = \alpha R$ . The value of $\alpha$ is
1. (1)2
2. (2)4
3. (3)3
4. (4)6
Q8 · MCQ · Centre of Mass, Momentum and CollisionOpen
The position of center of mass of three masses 2 kg, 3 kg and 15 kg placed with respect to mid point (p) of normal bisector, as shown in the figure is .
1. (1) $\left(\dfrac{\sqrt3}{4}, 1.25\right)$
2. (2) $\left(\dfrac{\sqrt3}{4}, 1.0\right)$
3. (3) $(0, 0)$
4. (4) $(1.25, 0)$
Q9 · MCQ · Mechanical Properties of SolidsOpen
The two wires A and B of equal cross-section but of different materials are joined together. The ratio of Young's modulus of wire A and wire B is 20/11. When the joined wire is kept under certain tension the elongations in the wires A and B are equal. If the length of wire A is 2.2 m, then the length of wire B is m.
1. (1)1.1
2. (2)2.22
3. (3)1.21
4. (4)4.44
Q10 · MCQ · Kinetic Theory of GasesOpen
Two closed vessels of same volume are joined through a narrow tube and both vessels are filled with air of pressure 90 kPa and temperature 400 K. Keeping the temperature of one vessel constant at 400 K the second vessel temperature is raised to 500 K. The final pressure in the vessels is kPa.
1. (1)100
2. (2)120
3. (3)90
4. (4)105
Q11 · MCQ · Wave OpticsOpen
In interference experiment the path difference between two interfering waves at a point A on the screen is $\lambda/3$ , where $\lambda$ is the wavelength of these waves, and at another point B the path difference is $\lambda/6$ . The ratio of intensities at points A and B is .
1. (1)3
2. (2)4
3. (3)1/3
4. (4)1/4
Q12 · MCQ · OscillationsOpen
A particle is executing simple harmonic motion. Its amplitude is A and time period is 5 sec. the time required by it to move from $x = A$ to $x = \dfrac{A}{\sqrt2}$ is sec.
1. (1)1/4
2. (2)5/4
3. (3)5/8
4. (4)3/8
Q13 · MCQ · Electric Charges and FieldsOpen
A thin half ring of radius 35 cm is uniformly charged with a total charge of Q coulomb. If the magnitude of the electric field at centre of the half ring is 100 V/m, then the value of Q is nC. ( $\epsilon_0 = 8.85\times10^{-12}\ \mathrm{C^2/Nm^2}$ and $\pi = 3.14$ )
1. (1)2.14
2. (2)2.44
3. (3)3.25
4. (4)0.7
Q14 · MCQ · Semiconductor ElectronicsOpen
The maximum rated power of the LED is 2 mW and it is used in the circuit with input voltage of 5 V as shown in the figure below. The current through resistance $R_S$ is 0.5 mA.
The minimum value of the resistance of $R_S$ , to ensure that the LED is not damaged is k $\Omega$ .
1. (1)6
2. (2)2
3. (3)4
4. (4)5
Q15 · MCQ · Electromagnetic WavesOpen
A point light source emits E.M. waves in free space. A detector, placed at a distance of L m, measures the intensity as $I_0$ . The detector is now shifted to another location on the same spherical surface ensuring the angle between original location and new location as $45^\circ$ . The measured intensity at new location will be .
1. (1) $\dfrac{I_0}{4}$
2. (2) $I_0$
3. (3) $\dfrac{I_0}{\sqrt2}$
4. (4) $\dfrac{I_0}{2}$
Q16 · MCQ · Ray OpticsOpen
A spherical interface lens of radius R separates two media of refractive indices 1 and 1.4 respectively as shown in the figure below. A point source is placed at a distance of 4R in front of spherical interface. The magnitude of the magnification of point source image is .
1. (1)1.66
2. (2)2.33
3. (3)2.66
4. (4)1.33
Q17 · MCQ · Moving Charges and MagnetismOpen
A small cube of side 1 mm is placed at the centre of a circular loop of radius 10 cm carrying a current of 2 A. The magnetic energy stored inside the cube is $\alpha\times10^{-14}$ J. The value of $\alpha$ is . ( $\mu_0 = 4\pi\times10^{-7}$ Tm/A, $\pi = 3.14$ )
1. (1)6.28
2. (2) $6.28\times10^{-6}$
3. (3)628
4. (4) $6.28\times10^{-4}$
Q18 · MCQ · Electromagnetic InductionOpen
An inductor of inductance 10 mH having resistance of $100\ \Omega$ is connected to battery of E.M.F. 1.0 V through a switch as shown in the figure below. After switch is closed, the ratio of instantaneous induced emfs across the inductor when the current passing through it is 2 mA and 4 mA is .
1. (1)4/3
2. (2)3/4
3. (3)5/3
4. (4)3/5
We corrected a printing error in the original paper
Source printed “the ratio of instantaneous voltages across the inductor” (the terminal voltage of the resistive inductor equals the 1.0 V emf at all times, ratio 1); corrected to the intended “induced emfs” ( $L\,di/dt$ : 0.8 V and 0.6 V).
Q19 · MCQ · AtomsOpen
The ratio of momentum of the photons of the $1^{\text{st}}$ and $2^{\text{nd}}$ line of Balmer series of Hydrogen atoms is $\alpha/\beta$ . The possible values of $\alpha$ and $\beta$ are :-
1. (1)27 and 20
2. (2)3 and 16
3. (3)5 and 36
4. (4)20 and 27
Q20 · MCQ · Alternating CurrentOpen
A LCR series circuit driven with $E_{\mathrm{rms}} = 90$ V at frequency $f_d = 30$ Hz has resistance $R = 80\ \Omega$ , an inductance with inductive reactance $X_L = 20.0\ \Omega$ and capacitance with capacitive reactance $X_C = 80.0\ \Omega$ . The power factor of the circuit is .
1. (1)0.8
2. (2)0.64
3. (3)0.9
4. (4)0.5
Q21 · Numerical · Current ElectricityOpen
Refer to the circuit diagram given below. The heat generated across the $6\ \Omega$ resistance in 100 second is $\dfrac{\alpha}{100}$ J. The value of $\alpha$ is . (Nearest integer)
Q22 · Numerical · Wave OpticsOpen
An unpolarized light of intensity $I_0$ passes through polarizer and then through a certain optically active solution and finally it goes to analyser. If the angle between analyser and polariser is $0^\circ$ and intensity of light emerged from analyser is $\dfrac38 I_0$ , the angle of rotation of the light by the solution with respect to analyser is degrees.
Q23 · Numerical · NucleiOpen
The energy released when $\dfrac{7}{17.13}$ kg of ${}^{7}_{3}\mathrm{Li}$ is converted into ${}^{4}_{2}\mathrm{He}$ by proton bombardment is $\alpha\times10^{32}$ eV. The value of $\alpha$ is . (Nearest integer)
(Mass of ${}^{7}_{3}\mathrm{Li} = 7.0183$ u, mass of ${}^{4}_{2}\mathrm{He} = 4.004$ u, mass of proton $= 1.008$ u and 1 u $= 931\ \mathrm{MeV/c^2}$ and Avogadro number $= 6.0\times10^{23}$ )
Q24 · Numerical · Electric Potential and CapacitanceOpen
A three coulomb charge moves from the point $(0, -2, -5)$ to the point $(5, 1, 2)$ in an electric field expressed as $\vec E = 2x\hat i + 3y^2\hat j + 4\hat k$ N/C. The work done in moving the charge is J.
Q25 · Numerical · ThermodynamicsOpen
A certain gas is isothermally compressed to $\left(\dfrac13\right)^{\text{rd}}$ of its initial volume ( $V_0 = 3$ litre) by applying required pressure. If the bulk modulus of the gas is $3\times10^{5}\ \mathrm{N/m^2}$ , the magnitude of work done on the gas is J.

Chemistry

Q1 · MCQ · Some Basic ConceptsOpen
An oxide of iron contains 69.9% iron, its empirical formula, is :
(Given : Molar mass of Fe and O are 56 and 16 g mol $^{-1}$ respectively.)
1. (1) $\ce{FeO}$
2. (2) $\ce{Fe2O3}$
3. (3) $\ce{Fe3O4}$
4. (4) $\ce{FeO3}$
Q2 · MCQ · Structure of AtomOpen
If shortest wavelength of hydrogen atom in Lyman series is x, then longest wavelength in Balmer series of $\ce{He+}$ is :
1. (1) $\dfrac{9x}{5}$
2. (2) $\dfrac{36x}{5}$
3. (3) $\dfrac{x}{4}$
4. (4) $\dfrac{5x}{9}$

Match the List-I with List-II :

List-I		List-II
Orbital		Radial nodes and nodal plane
A.	2s	I.	1 Radial node + two nodal planes
B.	3s	II.	1 Radial node + one nodal plane
C.	3p	III.	2 Radial nodes + No nodal plane
D.	4d	IV.	1 Radial node + No nodal plane.

Choose the correct answer from the options given below :

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

Q4 · MCQ · Chemical BondingOpen
The pairs among
A = [ $\ce{SO3^2-}$ , $\ce{CO3^2-}$ ], B = [ $\ce{O2^2-}$ , $\ce{F2}$ ], C = [ $\ce{CN-}$ , $\ce{CO}$ ], D = [ $\ce{NH3}$ , $\ce{H3O+}$ ] and E = [ $\ce{MnO4^2-}$ , $\ce{CrO4^2-}$ ] that do not have similar Lewis dot structure are :
1. (1)A, B and E
2. (2)A and E
3. (3)B, C and D
4. (4)C and D
Q5 · MCQ · ThermodynamicsOpen
Arrange the following isothermal processes in order of the magnitude of the work ( $p$ – $V$ ) involved between states 1 and 2.
- (A)Expansion in single stage $w_\mathrm{A}$
- (B)Expansion in multi stages $w_\mathrm{B}$
- (C)Compression in single stage $w_\mathrm{C}$
- (D)Compression in multi stages $w_\mathrm{D}$
1. (1) $\lvert w_\mathrm{B}\rvert > \lvert w_\mathrm{A}\rvert > \lvert w_\mathrm{C}\rvert > \lvert w_\mathrm{D}\rvert$
2. (2) $\lvert w_\mathrm{C}\rvert > \lvert w_\mathrm{D}\rvert > \lvert w_\mathrm{A}\rvert > \lvert w_\mathrm{B}\rvert$
3. (3) $\lvert w_\mathrm{C}\rvert > \lvert w_\mathrm{D}\rvert > \lvert w_\mathrm{B}\rvert > \lvert w_\mathrm{A}\rvert$
4. (4) $\lvert w_\mathrm{B}\rvert > \lvert w_\mathrm{A}\rvert > \lvert w_\mathrm{D}\rvert > \lvert w_\mathrm{C}\rvert$
Q6 · MCQ · SolutionsOpen
When 0.25 moles of a non-volatile, non-ionizable solute was dissolved in 1 mole of a solvent the vapor pressure of solution was x% of vapor pressure of pure solvent. What is x% ?
1. (1)50%
2. (2)60%
3. (3)70%
4. (4)80%
Q7 · MCQ · EquilibriumOpen
One mole each of He and A(g) are taken in a 10 L closed flask and heated to 400 K to establish the following equilibrium.
$\ce{A(g) <=> B(g)}$
$K_\mathrm{C}$ for this reaction at 400 K is 4.0. The partial pressures (in atm) of He and B(g) are respectively (at equilibrium).
(Assume He, A(g) and B(g) behave as ideal gases)
(Given : $R = 0.082$ L atm K $^{-1}$ mol $^{-1}$ )
1. (1)3.28, 2.624
2. (2)2.624, 3.28
3. (3)3.28, 0.656
4. (4)0.656, 6.56
Q8 · MCQ · ElectrochemistryOpen
Consider the following data :
Electrolyte $\boldsymbol{\Lambda^\circ_\mathrm{m}}$ (S cm $^{\mathbf{2}}$ mol $^{\mathbf{-1}}$ )
$\ce{BaCl2}$ $x_1$
$\ce{H2SO4}$ $x_2$
HCl $x_3$
$\ce{BaSO4}$ is sparingly soluble in water. If the conductivity of the saturated $\ce{BaSO4}$ solution is $x$ S cm $^{-1}$ then the solubility product of $\ce{BaSO4}$ can be given as.
(Here $\Lambda_\mathrm{m} = \Lambda^\circ_\mathrm{m}$ )
1. (1) $\dfrac{10^{6}x^{2}}{\alpha^{2}(x_1+x_2-2x_3)^{2}}$
2. (2) $\dfrac{x^{2}}{(x_1+x_2-2x_3)^{2}}$
3. (3) $\dfrac{\alpha^{2}(x_1+x_2-2x_3)^{2}}{10^{6}x^{2}}$
4. (4) $\dfrac{x^{2}}{(x_1+x_2+2x_3)^{2}}$
We corrected a printing error in the original paper
Source printed “(Here $\wedge_\mathrm{m} = \wedge^\circ_\mathrm{m}$ )” (logical-and sign for the Greek $\Lambda$ of the table).
Q9 · MCQ · p-Block (13–14)Open
Given below are two statements :
Statement-I : Aluminium is more electropositive than thallium as the standard electrode potential value of $E^\circ_{\mathrm{Al^{3+}/Al}}$ is negative and $E^\circ_{\mathrm{Tl^{3+}/Tl}}$ is positive.
Statement-II : The sum of first three ionization enthalpies of boron is very high when compared to that of aluminium. Due to this reason boron forms covalent compounds only and aluminium forms $\ce{Al^3+}$ ion.
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.
Q10 · MCQ · d- and f-BlockOpen
The correct statements among the following are :
- (A)Basic vanadium oxide is used in the manufacture of $\ce{H2SO4}$ .
- (B)The spin-only magnetic moment value of the transition metal halide employed in Ziegler-Natta polymerization is 2.84 BM.
- (C)The p-block metal compound employed in Ziegler-Natta polymerization has the metal in +3 oxidation state.
- (D)The number of electrons present in the outer most ‘d' orbital of metal halide employed in Wacker process is 8.
Choose the correct answer from the options given below :
1. (1)A and B only
2. (2)A, C and D only
3. (3)C and D only
4. (4)B, C and D only

Q11 · MCQ · Coordination CompoundsOpen

Match the List-I with List-II

List-I		List-II
Electronic configuration of		Crystal Field
tetrahedral metal ion		Stabilization Energy ( $\boldsymbol{\Delta_\mathrm{t}}$ )
A.	d $^2$	I.	$-0.6$
B.	d $^4$	II.	$-0.8$
C.	d $^6$	III.	$-1.2$
D.	d $^8$	IV.	$-0.4$

Choose the correct answer from the options given below :

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

Q12 · MCQ · Coordination CompoundsOpen
Which of the following are true about the energy of the given d-orbitals of a tetrahedral complex ?
- (A) $\mathrm{d}_{xy} = \mathrm{d}_{xz} > \mathrm{d}_{x^2-y^2}$
- (B) $\mathrm{d}_{xy} = \mathrm{d}_{yz} > \mathrm{d}_{z^2}$
- (C) $\mathrm{d}_{x^2-y^2} > \mathrm{d}_{z^2} > \mathrm{d}_{xz}$
- (D) $\mathrm{d}_{x^2-y^2} = \mathrm{d}_{z^2} < \mathrm{d}_{xz}$
Choose the correct answer from the options given below :
1. (1)A, B and D only
2. (2)A and B only
3. (3)B and D only
4. (4)B, C and D only
Q13 · MCQ · Purification & CharacterisationOpen
$R_\mathrm{f}$ value for 2-methylpropene in a solvent system (Ethyl acetate + ether) is 0.42. 2-methylpropene is treated with dilute $\ce{H2SO4}$ to give major organic product (X). $R_\mathrm{f}$ value for (X) in the same solvent system under identical condition will be :
1. (1)0.42
2. (2)0.82
3. (3)0.62
4. (4)0.12
Q14 · MCQ · Organic Basics (GOC)Open
Given below are two statements :
Statement-I : 2, 6-diethylcyclohexanone and 6-methyl-2-n-propylcyclohexanone are metamers.
Statement-II : 2, 2, 6, 6-tetramethylcyclohexanone exhibits keto-enol tautomerism.
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.
Q15 · MCQ · HydrocarbonsOpen
Given below are two statements :
Statement-I : Methane can be prepared by decarboxylation of sodium ethanoate, Kolbe's electrolysis of sodium acetate and reaction of $\ce{CH3MgBr}$ with water.
Statement-II : Methane cannot be prepared from unsaturated hydrocarbons and by Wurtz reaction.
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.
Q16 · MCQ · Haloalkanes & HaloarenesOpen
Given below are two statements :
Statement-I : 3-phenylpropene reacts with HBr and gives secondary alkyl bromide having a chiral carbon atom as the major product.
Statement-II : Aryl chlorides and aryl cyanides can be prepared by Sandmeyer reaction as well as Gattermann reaction.
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.
Q17 · MCQ · Alcohols, Phenols & EthersOpen
Consider the following sequence of reactions
The major product P is :
1. (1)
2. (2)
3. (3)
4. (4)
Q18 · MCQ · AminesOpen
Arrange the following compounds according to increasing order of boiling points.
n- $\ce{C4H9OH}$ (A), n- $\ce{C4H9NH2}$ (B), n- $\ce{C4H10}$ (C) and $\ce{C2H5NHC2H5}$ (D).
1. (1)C $<$ B $<$ A $<$ D
2. (2)D $<$ C $<$ B $<$ A
3. (3)C $<$ D $<$ B $<$ A
4. (4)D $<$ B $<$ A $<$ C
Q19 · MCQ · BiomoleculesOpen
Match the List-I with List-II :
List-I List-II
Deficiency Disease Vitamin
A. Scurvy I. Pyridoxine
B. Convulsions II. Vitamin A
C. Cheilosis III. Ascorbic Acid
D. Xerophthalmia IV. Riboflavin
Choose the correct answer from the options given below :
1. (1)A-I, B-III, C-II, D-IV
2. (2)A-I, B-III, C-IV, D-II
3. (3)A-III, B-I, C-IV, D-II
4. (4)A-III, B-I, C-II, D-IV

Q20 · MCQ · BiomoleculesOpen

Match the List-I with List-II

List-I		List-II
Amino acid		Positive reaction/Test for
		functional group present in
		side chain of amino acid
A.	Glutamine	I.	Hinsberg's test
B.	Lysine	II.	Neutral $\ce{FeCl3}$ test
C.	Tyrosine	III.	Ceric ammonium nitrate test
D.	Serine	IV.	Hoffman bromamide degradation

Choose the correct answer from the options given below :

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

Q21 · Numerical · Classification & PeriodicityOpen
First and second ionization enthalpies of lithium are 520 kJ mol $^{-1}$ and 7297 kJ mol $^{-1}$ respectively. Energy required to convert 3.5 mg lithium (g) into $\ce{Li^2+}$ (g) [ $\ce{Li(g) -> Li^2+(g)}$ ] is kJ. (nearest integer)
[Molar mass of Li = 7 g mol $^{-1}$ ]
We corrected a printing error in the original paper
Source printed the unit of the answer as “kJ mol $^{-1}$ ”; the energy for 3.5 mg of lithium is in kJ.
Q22 · Numerical · AminesOpen
Consider the following sequence of reactions.
The percentage of nitrogen in the yellow product (X) formed is %.
(Nearest Integer)
(Given Molar mass in g mol $^{-1}$ H : 1, C : 12, N : 14)
Q23 · Numerical · Alcohols, Phenols & EthersOpen
4.7 g of phenol is heated with Zn to give product X. If this reaction goes to 60% completion then the number of moles of compound X formed will be $\times\,10^{-2}$ (Nearest Integer)
(Given molar mass in g mol $^{-1}$ : H : 1, C : 12, O : 16)
Q24 · Numerical · Chemical KineticsOpen
Sucrose hydrolyses in acidic medium into glucose and fructose by first order rate law with $t_{1/2} = 3$ hour. The percentage of sucrose remaining after 6 hours is . (Nearest integer)
(Given : $\log 2 = 0.3010$ and $\log 3 = 0.4771$ )
Q25 · Numerical · ThermodynamicsOpen
Consider the reaction $\ce{X <=> Y}$ at 300 K. If $\Delta H^{\theta}$ and K are 28.40 kJ mol $^{-1}$ and $1.8\times10^{-7}$ at the same temperature, then the magnitude of $\Delta S^{\theta}$ for the reaction in J K $^{-1}$ mol $^{-1}$ is .(Nearest Integer). (Given : $R = 8.3$ J K $^{-1}$ mol $^{-1}$ , $\ln 10 = 2.3$ , $\log 3 = 0.48$ , $\log 2 = 0.30$

Maths

Q1 · MCQ · Inverse TrigonometryOpen
Let $[\cdot]$ denote the greatest integer function. If the domain of the function $f(x) = \sin^{-1}\left(\dfrac{x + [x]}{3}\right)$ is $[\alpha, \beta)$ , then $\alpha^2 + \beta^2$ is equal to:
1. (1)2
2. (2)5
3. (3)10
4. (4)13
Q2 · MCQ · Quadratic EquationsOpen
Let one root of the quadratic equation in $x$ : $(k^2 - 15k + 27)x^2 + 9(k - 1)x + 18 = 0$ be twice the other. Then the length of the latus rectum of the parabola $y^2 = 6kx$ is equal to:
1. (1)4
2. (2)6
3. (3)8
4. (4)12
Q3 · MCQ · Conic SectionsOpen
Let $e_1$ and $e_2$ be two distinct roots of the equation $x^2 - ax + 2 = 0$ . Let the sets $\{a \in \mathbb{R} : e_1$ and $e_2$ are the eccentricities of hyperbolas $\} = (\alpha, \beta)$ , and $\{a \in \mathbb{R} : e_1$ and $e_2$ are the eccentricities of an ellipse and a hyperbola, respectively $\} = (\gamma, \infty)$ . The $\alpha^2 + \beta^2 + \gamma^2$ is equal to:
1. (1)18
2. (2)22
3. (3)26
4. (4)34
Q4 · MCQ · Complex NumbersOpen
Let the set of all values of $k \in \mathbb{R}$ such that the equation $z(\bar{z} + 2 + i) + k(2 + 3i) = 0$ , $z \in \mathbb{C}$ , has at least one solution, be the interval $[\alpha, \beta]$ . Then $9(\alpha + \beta)$ is equal to:
1. (1) $-10$
2. (2) $-8$
3. (3) $10\sqrt{13}$
4. (4) $8\sqrt{13}$
Q5 · MCQ · Sequences & SeriesOpen
The value of $1^3 - 2^3 + 3^3 - \ldots + 15^3$ is:
1. (1)1706
2. (2)1856
3. (3)1982
4. (4)2403
Q6 · MCQ · Sequences & SeriesOpen
The sum of the first ten terms of an A.P. is 160 and the sum of the first two terms of a G.P. is 8. If the first term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to common difference of the A.P., then the sum of all possible values of the first term of the G.P. is:
1. (1) $\dfrac{34}{9}$
2. (2) $\dfrac{34}{13}$
3. (3) $\dfrac{32}{9}$
4. (4) $\dfrac{32}{13}$
Q7 · MCQ · Permutations & CombinationsOpen
The number of 4-letter words, with or without meaning, each consisting of two vowels and two consonants that can be formed from the letters of the word INCONSEQUENTIAL, without repeating any letter, is:
1. (1)2670
2. (2)2840
3. (3)2920
4. (4)3600
Q8 · MCQ · Binomial TheoremOpen
If the coefficients of the middle terms in the binomial expansions of $(1 + \alpha x)^{26}$ and $(1 - \alpha x)^{28}$ , $\alpha \neq 0$ , are equal, then the value of $\alpha$ is:
1. (1)1
2. (2) $\dfrac{14}{13}$
3. (3) $\dfrac{27}{7}$
4. (4) $\dfrac{7}{27}$
Q9 · MCQ · StatisticsOpen
A data consists of 20 observations $x_1, x_2, \ldots, x_{20}$ . If $\displaystyle\sum_{i=1}^{20}(x_i + 5)^2 = 2500$ and $\displaystyle\sum_{i=1}^{20}(x_i - 5)^2 = 100$ , then the ratio of mean to standard deviation of this data is:
1. (1) $2:1$
2. (2) $3:1$
3. (3) $3:2$
4. (4) $4:1$
Q10 · MCQ · ProbabilityOpen
A bag contains $(N + 1)$ coins – $N$ fair coins, and one coin with ‘Head' on both sides. A coin is selected at random and tossed. If the probability of getting ‘Head' is $\dfrac{9}{16}$ , then $N$ is equal to:
1. (1)5
2. (2)7
3. (3)8
4. (4)9
Q11 · MCQ · Conic SectionsOpen
If the eccentricity $e$ of the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ , passing through $\left(6, 4\sqrt3\right)$ , satisfies $15(e^2 + 1) = 34e$ , then the length of the latus rectum of the hyperbola $\dfrac{x^2}{b^2} - \dfrac{y^2}{2(a^2 + 1)} = 1$ is:
1. (1)10
2. (2)20
3. (3)25
4. (4)30
Q12 · MCQ · Conic SectionsOpen
Let chord $PQ$ of length $3\sqrt{13}$ of the parabola $y^2 = 12x$ be such that the ordinates of points $P$ and $Q$ are in the ratio $1:2$ . If the chord $PQ$ subtends an angle $\alpha$ at the focus of the parabola, then $\sin\alpha$ is equal to:
1. (1) $\dfrac35$
2. (2) $\dfrac45$
3. (3) $\dfrac{5}{13}$
4. (4) $\dfrac{12}{13}$
Q13 · MCQ · Inverse TrigonometryOpen
Let $0 < \alpha < 1$ , $\beta = \dfrac{1}{3\alpha}$ and $\tan^{-1}(1 - \alpha) + \tan^{-1}(1 - \beta) = \dfrac{\pi}{4}$ . Then $6(\alpha + \beta)$ is equal to:
1. (1)6
2. (2)7
3. (3)8
4. (4)9
Q14 · MCQ · TrigonometryOpen
Let $S = \left\{\theta \in (-2\pi, 2\pi) : \cos\theta + 1 = \sqrt3\sin\theta\right\}$ . Then $\displaystyle\sum_{\theta\in S}\theta$ is equal to:
1. (1) $-\dfrac{2\pi}{3}$
2. (2) $-\dfrac{4\pi}{3}$
3. (3) $\dfrac{2\pi}{3}$
4. (4) $\dfrac{4\pi}{3}$
Q15 · MCQ · 3D GeometryOpen
Let the image of the point $P(1, 6, a)$ in the line $L : \dfrac{x}{1} = \dfrac{y - 1}{2} = \dfrac{z - a + 1}{b}$ , $b > 0$ , be $\left(\dfrac{a}{3}, 0, a + c\right)$ . If $S(\alpha, \beta, \gamma)$ , $\alpha > 0$ , is the point on $L$ such that the distance of $S$ from the foot of perpendicular from the point $P$ on $L$ is $2\sqrt{14}$ , then $\alpha + \beta + \gamma$ is equal to:
1. (1)19
2. (2)20
3. (3)21
4. (4)22
Q16 · MCQ · 3D GeometryOpen
Let a line $L$ be perpendicular to both the lines $L_1 : \dfrac{x + 1}{3} = \dfrac{y + 3}{5} = \dfrac{z + 5}{7} \quad\text{and}\quad L_2 : \dfrac{x - 2}{1} = \dfrac{y - 4}{4} = \dfrac{z - 6}{7}.$ If $\theta$ is the acute angle between the lines $L$ and $L_3 : \dfrac{x - \frac87}{2} = \dfrac{y - \frac47}{1} = \dfrac{z}{2},$ then $\tan\theta$ is equal to:
1. (1) $\dfrac32\sqrt2$
2. (2) $\dfrac52\sqrt2$
3. (3) $\dfrac53\sqrt2$
4. (4) $\dfrac43\sqrt2$
Q17 · MCQ · Limits, Continuity & DifferentiabilityOpen
The value of $\displaystyle\lim_{x\to0}\left(\frac{x^2\sin^2x}{x^2 - \sin^2x}\right)$ is:
1. (1)2
2. (2)3
3. (3)4
4. (4)6
Q18 · MCQ · Definite IntegrationOpen
The value of the integral $\displaystyle\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\left(\frac{32\cos^4x}{1 + e^{\sin x}}\right)dx$ is:
1. (1) $4\pi + 2$
2. (2) $3\pi + 8$
3. (3) $3\pi + 4$
4. (4) $4\pi + 3$
Q19 · MCQ · Area under CurvesOpen
The area of the region $\{(x, y) : 0 \le y \le 6 - x,\ y^2 \ge 4x - 3,\ x \ge 0\}$ is:
1. (1)8
2. (2)9
3. (3)12
4. (4)15
Q20 · MCQ · Area under CurvesOpen
Let $e$ be the base of natural logarithm and let $f : \{1, 2, 3, 4\} \to \{1, e, e^2, e^3\} \quad\text{and}\quad g : \{1, e, e^2, e^3\} \to \left\{1, \dfrac12, \dfrac13, \dfrac14\right\}$ be two bijective functions such that $f$ is strictly decreasing and $g$ is strictly increasing. If $\phi(x) = \left[f^{-1}\left\{g^{-1}\left(\dfrac12\right)\right\}\right]^x,$ then the area of the region $R = \{(x, y) : x^2 \le y \le \phi(x),\ 0 \le x \le 1\}$ is:
1. (1) $\dfrac{3 - \log_e(2)}{3\log_e(2)}$
2. (2) $\dfrac{1}{3\log_e(2)}$
3. (3) $3 + \log_e(2)$
4. (4) $\dfrac{3 + \log_e(2)}{2 + \log_e(3)}$
Q21 · Numerical · Matrices & DeterminantsOpen
Let $A = \begin{bmatrix} -1 & 1 & -1\\ 1 & 0 & 1\\ 0 & 0 & 1\end{bmatrix}$ satisfy $A^2 + \alpha(\operatorname{adj}(\operatorname{adj}(A))) + \beta(\operatorname{adj}(A)(\operatorname{adj}(\operatorname{adj}(A)))) = \begin{bmatrix} 2 & -2 & 2\\ -2 & 0 & -1\\ 0 & 0 & -1\end{bmatrix}$ for Some $\alpha, \beta \in \mathbb{R}$ . Then $(\alpha - \beta)^2$ is equal to .
Q22 · Numerical · CirclesOpen
Let the centre of the circle $x^2 + y^2 + 2gx + 2fy + 25 = 0$ be in the first quadrant and lie on the line $2x - y = 4$ . Let the area of an equilateral triangle inscribed in the circle be $27\sqrt3$ . Then the square of the length of the chord of the circle on the line $x = 1$ is .
Q23 · Numerical · VectorsOpen
If $\vec a = \hat i + \hat j + \hat k$ , $\vec b = \hat j - \hat k$ and $\vec c$ be three vectors such that $\vec a\times\vec c = \vec b$ and $\vec a\cdot\vec c = 3$ , then $\vec c\cdot(\vec a - 2\vec b)$ is equal to .
Q24 · Numerical · TrigonometryOpen
For the functions $f(\theta) = \alpha\tan^2\theta + \beta\cot^2\theta$ , and $g(\theta) = \alpha\sin^2\theta + \beta\cos^2\theta$ , $\alpha > \beta > 0$ , let $\displaystyle\min_{0<\theta<\frac{\pi}{2}} f(\theta) = \max_{0<\theta<\pi} g(\theta)$ . If the first term of a G.P. is $\left(\dfrac{\alpha}{2\beta}\right)$ , its common ratio is $\left(\dfrac{2\beta}{\alpha}\right)$ and the sum of its first 10 terms is $\dfrac{m}{n}$ , $\gcd(m, n) = 1$ , then $m + n$ is equal to .
Q25 · Numerical · Differential EquationsOpen
Let $y = y(x)$ be the solution of the differential equation $\left(x^2 - x\sqrt{x^2 - 1}\right)dy + \left(y\left(x - \sqrt{x^2 - 1}\right) - x\right)dx = 0,$ $x \ge 1$ . If $y(1) = 1$ , then the greatest integer less than $y\left(\sqrt5\right)$ is .

Electrolyte	$\boldsymbol{\Lambda^\circ_\mathrm{m}}$ (S cm $^{\mathbf{2}}$ mol $^{\mathbf{-1}}$ )
$\ce{BaCl2}$	$x_1$
$\ce{H2SO4}$	$x_2$
HCl	$x_3$

List-I		List-II
Deficiency Disease		Vitamin
A.	Scurvy	I.	Pyridoxine
B.	Convulsions	II.	Vitamin A
C.	Cheilosis	III.	Ascorbic Acid
D.	Xerophthalmia	IV.	Riboflavin