JEE Main 24 Jan 2026, Shift 2

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

Physics

Q1 · MCQ · Current ElectricityOpen
The reading of the ammeter (A) in steady state in the following circuit (assuming negligible internal resistance of the ammeter) is A.
1. (1)2
2. (2)1
3. (3)1/2
4. (4)0
Q2 · MCQ · Units and MeasurementsOpen
In a vernier callipers, 50 vernier scale divisions are equal to 48 main scale divisions. If one main scale division $= 0.05$ mm, then the least count of the vernier callipers is mm.
1. (1)0.002
2. (2)0.05
3. (3)0.02
4. (4)0.005
Q3 · MCQ · Ray OpticsOpen
Five persons $P_1$ , $P_2$ , $P_3$ , $P_4$ and $P_5$ recorded object distance ( $u$ ) and image distance ( $v$ ) using same convex lens having power $+5$ D as (25, 96), (30, 62), (35, 37), (45, 35) and (50, 32) respectively. Identify correct statement
1. (1)Readings recorded by all persons are correct
2. (2)Reading recorded by $P_3$ persons are incorrect
3. (3)Reading recorded by $P_3$ and $P_2$ persons are incorrect
4. (4)Reading recorded by $P_4$ and $P_5$ persons are incorrect
Q4 · MCQ · Current ElectricityOpen
A regular hexagon is formed by six wires each of resistance $r\ \Omega$ and the corners are joined to centre by wires of same resistance. If the current enters at one corner and leaves at the opposite corner, the equivalent resistance of the hexagon between the two opposite corners will be
1. (1) $\dfrac45 r$
2. (2) $\dfrac58 r$
3. (3) $\dfrac34 r$
4. (4) $\dfrac35 r$
Q5 · MCQ · Circular MotionOpen
In case of vertical circular motion of a particle by a thread of length $r$ if the tension in the thread is zero at an angle $30^\circ$ shown in figure, the velocity at the bottom point (A) of the circular path is
( $g$ = gravitational acceleration)
1. (1) $\sqrt{5gr}$
2. (2) $\sqrt{\dfrac72 gr}$
3. (3) $\sqrt{4gr}$
4. (4) $\sqrt{\dfrac52 gr}$
Q6 · MCQ · Rotational MotionOpen
A thin uniform rod (X) of mass $M$ and length $L$ is pivoted at a height $\left(\dfrac{L}{3}\right)$ as shown in the figure. The rod is allowed to fall from a vertical position and lie horizontally on the table. The angular velocity of this rod when it hits the table top, is . ( $g$ = gravitational acceleration)
1. (1) $\sqrt{\dfrac32\dfrac{g}{L}}$
2. (2) $\dfrac{3}{\sqrt2}\sqrt{\dfrac{g}{L}}$
3. (3) $\dfrac{1}{\sqrt2}\sqrt{\dfrac{g}{L}}$
4. (4) $\sqrt{\dfrac{3g}{L}}$
Q7 · MCQ · Moving Charges and MagnetismOpen
Two identical circular loops P and Q each of radius $r$ are lying in parallel planes such that they have common axis. The currents through P and Q are $I$ and $4I$ respectively in clockwise direction as seen from O. The net magnetic field at O is:
1. (1) $\dfrac{3\mu_0 I}{4\sqrt2\, r}$ toward P
2. (2) $\dfrac{\mu_0 I}{4\sqrt2\, r}$ toward P
3. (3) $\dfrac{\mu_0 I}{4\sqrt2\, r}$ towards Q
4. (4) $\dfrac{3\mu_0 I}{4\sqrt2\, r}$ towards Q
We corrected a printing error in the original paper
Source printed “The current through P and Q are”.
Q8 · MCQ · Dual Nature of Radiation and MatterOpen
When a light of a given wavelength falls on a metallic surface the stopping potential for photoelectrons is 3.2 V. If a second light having wavelength twice of first light is used, the stopping potential drops to 0.7 V. The wavelength of first light is m.
( $h = 6.63\times10^{-34}$ J.s, $e = 1.6\times10^{-19}$ C, $c = 3\times10^{8}$ m/s)
1. (1) $2.9\times10^{-8}$
2. (2) $2.2\times10^{-8}$
3. (3) $3.1\times10^{-7}$
4. (4) $2.5\times10^{-7}$
Q9 · MCQ · WavesOpen
The fifth harmonic of a closed organ pipe is found to be in unison with the first harmonic of an open pipe. The ratio of lengths of closed pipe to that of the open pipe is $5/x$ . The value of $x$ is .
1. (1)4
2. (2)2
3. (3)1
4. (4)3
Q10 · MCQ · Electric Potential and CapacitanceOpen
Three parallel plate capacitors each with area $A$ and separation $d$ are filled with two dielectrics ( $k_1$ and $k_2$ ) in the following fashion. Which of the following is true? ( $k_1 > k_2$ )
1. (1) $C_B > C_C > C_A$
2. (2) $C_C > C_B > C_A$
3. (3) $C_C > C_A > C_B$
4. (4) $C_A > C_C > C_B$
We corrected a printing error in the original paper
Source printed “filled with two dielectric”.
Q11 · MCQ · NucleiOpen
The binding energies for the following nuclear reactions are expressed in MeV. ${}_{2}\mathrm{He}^{3} + {}_{0}\mathrm{n}^{1} \rightarrow {}_{2}\mathrm{He}^{4} + 20\ \mathrm{MeV}$ ${}_{2}\mathrm{He}^{4} + {}_{0}\mathrm{n}^{1} \rightarrow {}_{2}\mathrm{He}^{5} - 0.9\ \mathrm{MeV}$ If $X_3$ , $X_4$ , $X_5$ denote the stability of ${}_{2}\mathrm{He}^{3}$ , ${}_{2}\mathrm{He}^{4}$ and ${}_{2}\mathrm{He}^{5}$ , respectively, then the correct order is:
1. (1) $X_4 > X_5 > X_3$
2. (2) $X_4 = X_5 = X_3$
3. (3) $X_4 > X_5 < X_3$
4. (4) $X_4 < X_5 < X_3$
We corrected a printing error in the original paper
Source printed “The binding energy for the following nuclear reactions are”.
Q12 · MCQ · Mechanical Properties of FluidsOpen
A cubical block of density $\rho_b = 600\ \mathrm{kg/m^3}$ floats in a liquid of density $\rho_e = 900\ \mathrm{kg/m^3}$ . If the height of block is $H = 8.0$ cm then height of the submerged part is cm.
1. (1)7.3
2. (2)4.3
3. (3)6.3
4. (4)5.3
Q13 · MCQ · Moving Charges and MagnetismOpen
A moving coil galvanometer of resistance $100\ \Omega$ shows a full scale deflection for a current of 1 mA. The value of resistance required to convert this galvanometer into an ammeter, showing full scale deflection for a current of 5 mA, is $\Omega$
1. (1)25
2. (2)10
3. (3)0.5
4. (4)2.5
Q14 · MCQ · Semiconductor ElectronicsOpen
Identify the correct truth table of the given logical circuit.
1. (1)
  A B Y
  0 0 0
  0 1 1
  1 0 1
  1 1 0
2. (2)
  A B Y
  0 0 1
  0 1 0
  1 0 1
  1 1 0
3. (3)
  A B Y
  0 0 1
  0 1 1
  1 0 1
  1 1 0
4. (4)
  A B Y
  0 0 0
  0 1 0
  1 0 1
  1 1 0
Q15 · MCQ · Newton's Laws of MotionOpen
A flexible chain of mass $m$ hangs between two fixed points at the same level. The inclination of the chain with the horizontal at the two points of support is $30^\circ$ . Considering the equilibrium of each half of the chain, the tension of the chain at the lowest point is .
1. (1) $\dfrac{\sqrt3}{2}mg$
2. (2) $\dfrac12 mg$
3. (3) $mg$
4. (4) $\sqrt3\,mg$
Q16 · MCQ · WavesOpen
A point source is kept at the center of a spherically enclosed detector. If the volume of the detector increased by 8 times, the intensity will
1. (1)increase by 8 times
2. (2)increase by 64 times
3. (3)decrease by 8 times
4. (4)decrease by 4 times
Q17 · MCQ · Wave OpticsOpen
In the Young's double slit experiment the intensity produced by each one of the individual slits is $I_0$ . The distance between two slits is 2 mm. The distance of screen from slits is 10 m. The wavelength of light is 6000 Å. The intensity of light on the screen in front of one of the slits is .
1. (1) $2I_0$
2. (2) $I_0$
3. (3) $\dfrac{I_0}{2}$
4. (4) $4I_0$
Q18 · MCQ · ThermodynamicsOpen
10 mole of an ideal gas is undergoing the process shown in the figure. The heat involved in the process from $P_1$ to $P_2$ is $\alpha$ Joule ( $P_1 = 21.7$ Pa and $P_2 = 30$ Pa, $C_v = 21$ J/K.mol, $R = 8.3$ J/mol.K). The value of $\alpha$ is .
1. (1)24
2. (2)15
3. (3)21
4. (4)28
Q19 · MCQ · Motion in a Straight LineOpen
The velocity ( $v$ ) – Distance ( $x$ ) graph is shown in figure. Which graph represents acceleration ( $a$ ) versus distance ( $x$ ) variation of this system?
1. (1)
2. (2)
3. (3)
4. (4)
Q20 · MCQ · Ray OpticsOpen
Distance between an object and three times magnified real image is 40 cm. The focal length of the mirror used is cm.
1. (1) $-15/2$
2. (2) $-10$
3. (3) $-20$
4. (4) $-15$
Q21 · Numerical · ThermodynamicsOpen
When 300 J of heat given to an ideal gas with $C_p = \dfrac72 R$ its temperature raises from $20\ ^\circ$ C to $50\ ^\circ$ C keeping its volume constant. The mass of the gas is (approximately) g. ( $R = 8.314$ J/mol.K).
Flawed question. Asks for the mass of the gas but no molar mass is given (only $n \approx 0.481$ mol follows); the printed key 481 matches no stated quantity, so there is no evident intended version. It is shown for completeness and is left out of practice tests.
Q22 · Numerical · Electric Charges and FieldsOpen
A point charge $q = 1\ \mu$ C is located at a distance 2 cm from one end of a thin insulating wire of length 10 cm having a charge $Q = 24\ \mu$ C, distributed uniformly along its length, as shown in figure. Force between $q$ and wire is N.
(Use $\dfrac{1}{4\pi\epsilon_0} = 9\times10^{9}\ \mathrm{N.m^2/C^2}$ )
Q23 · Numerical · Current ElectricityOpen
In a meter bridge experiment to determine the value of unknown resistance, first the resistances $2\ \Omega$ and $3\ \Omega$ are connected in the left and right gaps of the bridge and the null point is obtained at a distance $l$ cm from the left. Now when an unknown resistance $x\ \Omega$ is connected in parallel to $3\ \Omega$ resistance, the null point is shifted by 10 cm to the right of wire. The value of unknown resistance $x$ is $\Omega$ .
Q24 · Numerical · Rotational MotionOpen
A uniform solid cylinder of length $L$ and radius $R$ has moment of inertia about its axis equal to $I_1$ . A small concentric cylinder of length $L/2$ and radius $R/3$ carved from this cylinder has moment of inertia about its axis equal to $I_2$ . The ratio $I_1/I_2$ is .
We corrected a printing error in the original paper
Source printed “A small co-centric cylinder” and “about its axis equals to $I_2$ ”.
Q25 · Numerical · Mechanical Properties of FluidsOpen
A soap bubble of surface tension 0.04 N/m is blown to a diameter of 7 cm. If $(15000 - x)\ \mu$ J of work is done in blowing it further to make its diameter 14 cm, then the value of $x$ is .
( $\pi = 22/7$ )

Chemistry

Q1 · MCQ · p-Block (13–14)Open
Choose the INCORRECT statement
1. (1)Among the isotopes of carbon, $^{13}$ C is a radioactive isotope.
2. (2)Carbon exhibits negative oxidation states along with $+4$ and $+2$ .
3. (3)Carbon cannot exceed its covalency more than four.
4. (4) $\ce{CO2}$ is the most acidic oxide among the dioxides of group 14 elements.
We corrected a printing error in the original paper
Option (4): source printed “the dioxides of group of 14 elements”.
Q2 · MCQ · Aldehydes, Ketones & AcidsOpen
The unsaturated ether on acidic hydrolysis produces carbonyl compounds as shown below:-
$\ce{CH3-CH=CH-O-CH=CH2}$
$\downarrow$ $\ce{H3O^{(+)}}$
$\ce{CH3-CH2-CH=O}$ and $\ce{O=CH-CH3}$
Based on this, predict the solution / reagent that will help to distinguish “P” and “Q” obtained in the following reaction.
1. (1)Lucas reagent
2. (2)2,4-DNP reagent
3. (3)Saturated $\ce{NaHSO3}$ solution
4. (4)Fehling solution
Q3 · MCQ · BiomoleculesOpen
The number of possible tripeptides formed involving alanine (ala), glycine (gly) and valine (val), where no amino acid has been used more than once is :
1. (1)6
2. (2)3
3. (3)4
4. (4)8
Q4 · MCQ · SolutionsOpen
Two liquids A and B form an ideal solution at temperature T K. At T K, the vapour pressures of pure A and B are 55 and 15 kN m $^{-2}$ respectively. What is the mole fraction of A in solution of A and B in equilibrium with a vapour in which the mole fraction of A is 0.8 ?
1. (1)0.5217
2. (2)0.480
3. (3)0.663
4. (4)0.340
Q5 · MCQ · EquilibriumOpen
Consider the following gaseous equilibrium in a closed container of volume “V” at T(K).
$\ce{P2(g) + Q2(g) <=> 2PQ(g)}$
2 moles each of $\ce{P2(g)}$ , $\ce{Q2(g)}$ and PQ (g) are present at equilibrium. Now one mole each of ‘ $\ce{P2}$ ' and ‘ $\ce{Q2}$ ' are added to the equilibrium keeping the temperature at T(K). The number of moles of $\ce{P2}$ , $\ce{Q2}$ and PQ at the new equilibrium, respectively, are -
1. (1)2.67, 2.67, 2.67
2. (2)1.21, 2.24, 1.56
3. (3)1.66, 1.66, 1.66
4. (4)2.56, 1.62, 2.24
Q6 · MCQ · Chemical BondingOpen
Pair of species among the following having same bond order as well as paramagnetic character will be-
1. (1) $\ce{O2+}$ , $\ce{N2^2-}$
2. (2) $\ce{O2-}$ , $\ce{N2+}$
3. (3) $\ce{O2+}$ , $\ce{N2-}$
4. (4) $\ce{O2-}$ , $\ce{N2-}$
Q7 · MCQ · Coordination CompoundsOpen
The wavelengths of light absorbed for the following complexes are in the order. $\underset{\text{(I)}}{\ce{[Co(NH3)6]^3+}}\ ;\ \underset{\text{(II)}}{\ce{[Co(H2O)6]^3+}}\ ;\ \underset{\text{(III)}}{\ce{[Co(CN)6]^3-}}\ ;\ \underset{\text{(IV)}}{\ce{[Co(NH3)5(H2O)]^3+}}\ ;\ \underset{\text{(V)}}{\ce{[CoF6]^3-}}$
1. (1)III $<$ I $<$ II $<$ IV $<$ V
2. (2)III $<$ I $<$ IV $<$ V $<$ II
3. (3)III $<$ IV $<$ I $<$ II $<$ V
4. (4)III $<$ I $<$ IV $<$ II $<$ V
We corrected a printing error in the original paper
Source printed “The wavelength of light absorbed for the following complexes are”.
Q8 · MCQ · p-Block (15–18)Open
One mole of $\ce{Cl2}$ (g) was passed into 2 L of cold 2M KOH solution. After the reaction, the concentrations of $\ce{Cl-}$ , $\ce{ClO-}$ and $\ce{OH-}$ are respectively (assume volume remains constant)
1. (1)0.75 M, 0.75 M, 1 M
2. (2)0.5 M, 0.5 M, 0.5 M
3. (3)0.5 M, 0.5 M, 1 M
4. (4)1 M, 1 M, 1 M
Q9 · MCQ · Some Basic ConceptsOpen
A student has planned to prepare acetanilide from aniline using acetic anhydride. The student has started from 9.3 g of aniline. However, the student has managed to obtain 11 g of dry acetanilide.
The % yield of this reaction is :-
1. (1)81.5%
2. (2)97.5%
3. (3)59.5%
4. (4)72.5%
Q10 · MCQ · Organic Basics (GOC)Open
Find out the statements which are not true.
- (A)Resonating structures with more number of covalent bonds and lesser charge separation are more stable.
- (B)In electromeric effect, an unsaturated system shows $+$ E effect with nucleophile and $-$ E effect with electrophile.
- (C)Inductive effect is responsible for high melting point, boiling point and dipole moment of polar compounds.
- (D)The greater the number of alkyl groups attached to the doubly bonded carbon atoms, higher is the heat of hydrogenation.
- (E)Stability of carbanion increases with the increase in s-character of the carbon carrying the negative charge.
Choose the correct answer from the options given below.
1. (1)A, D & E only
2. (2)B, D & E only
3. (3)A, C & D only
4. (4)B & D only
We corrected a printing error in the original paper
Statement A: source printed “Resonating structure with more number of covalent bonds and lesser charge separation are more stable”.
Q11 · MCQ · Classification & PeriodicityOpen
The correct order of C, N, O and F in terms of second ionisation potential is
1. (1)F $<$ N $<$ C $<$ O
2. (2)C $<$ O $<$ N $<$ F
3. (3)C $<$ N $<$ F $<$ O
4. (4)C $<$ F $<$ N $<$ O
We corrected a printing error in the original paper
Answer corrected; the source key is (2) — second ionisation enthalpies are C 2353, N 2856, F 3374, O 3388 kJ mol $^{-1}$ ( $\ce{O+}$ loses an electron from the stable half-filled 2p $^3$ ), so C $<$ N $<$ F $<$ O, option (3); the source's own solution also gives O $>$ F $>$ N $>$ C.
Q12 · MCQ · Practical / Qualitative AnalysisOpen
In the Group analysis of cations, $\ce{Ba^2+}$ & $\ce{Ca^2+}$ are precipitated respectively as
1. (1)sulphide & sulphide
2. (2)hydroxide & carbonate
3. (3)carbonate & carbonate
4. (4)chromate & sulphide
Q13 · MCQ · Structure of AtomOpen
The wavelength of spectral line obtained in the spectrum of $\ce{Li^2+}$ ion, when the transition takes place between two levels whose sum is 4 and difference is 2, is
1. (1) $2.28\times10^{-7}$ cm
2. (2) $2.28\times10^{-6}$ cm
3. (3) $1.14\times10^{-7}$ cm
4. (4) $1.14\times10^{-6}$ cm
We corrected a printing error in the original paper
Source printed “LI $^{2+}$ ion”.
Q14 · MCQ · Aldehydes, Ketones & AcidsOpen
Given below are two statements :
Statement I : Cross aldol condensation between two different aldehydes will always produce four different products.
Statement II : When semicarbazide reacts with a mixture of benzaldehyde and acetophenone under optimum pH, it forms a condensation product with acetophenone only.
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)Both Statement I and Statement II are true
4. (4)Statement I is true but Statement II is false
Q15 · MCQ · AminesOpen
Given below are two statements :
Statement I : The dipole moment of $\ce{R-CN}$ is greater than $\ce{R-NC}$ and $\ce{R-NC}$ can undergo hydrolysis under acidic medium to produce
Statement II : $\ce{R-CN}$ hydrolyses under acidic medium to produce a compound which on treatment with $\ce{SOCl2}$ , followed by the addition of $\ce{NH3}$ gives another compound(x). This compound (x) on treatment with NaOCl/NaOH gives a product, that on treatment with $\ce{CHCl3}$ /KOH/ $\Delta$ produces $\ce{R-NC}$
In the light of the above statements, choose the correct answer from the options given below :
1. (1)Both Statement I and Statement II are false
2. (2)Both Statement I and Statement II are true
3. (3)Statement I is true but Statement II is false
4. (4)Statement I is false but Statement II is true
Q16 · MCQ · Alcohols, Phenols & EthersOpen
From the following, how many compounds contain at least one secondary alcohol ?
Choose the correct answer from the options given below :
1. (1)Five
2. (2)Three
3. (3)Four
4. (4)two
Q17 · MCQ · ThermodynamicsOpen
The heats of atomisation of methane and ethane are ‘x' kJ mol $^{-1}$ and ‘y' kJ mol $^{-1}$ respectively. The longest wavelength ( $\lambda$ ) of light capable of breaking the C-C bond can be expressed in SI unit as :
1. (1) $\dfrac{hc}{1000}\left(\dfrac{y-6x}{4}\right)^{-1}$
2. (2) $\dfrac{N_\mathrm{A}hc}{250(4y-6x)}$
3. (3) $\dfrac{N_\mathrm{A}hc}{250(y-6x)}$
4. (4) $N_\mathrm{A}hc\left(y-\dfrac{6x}{4}\right)^{-1}$
We corrected a printing error in the original paper
Source printed “The heat of atomisation of methane and ethane are”.
Q18 · MCQ · SolutionsOpen
At 298 K, the mole percentage of $\ce{N2(g)}$ in air is 80%. Water is in equilibrium with air at a pressure of 10 atm. What is the mole fraction of $\ce{N2(g)}$ in water at 298 K ? ( $K_\mathrm{H}$ for $\ce{N2}$ is $6.5\times10^{7}$ mm Hg)
1. (1) $1.23\times10^{-7}$
2. (2) $1.17\times10^{-4}$
3. (3) $9.35\times10^{5}$
4. (4) $9.35\times10^{-5}$
Q19 · MCQ · d- and f-BlockOpen
“X” is an oxoanion of the lightest element of group 7 (in the periodic table). The metal is in $+6$ oxidation state in “X”. The color of the potassium salt of X is
1. (1)green
2. (2)purple
3. (3)yellow
4. (4)orange
Q20 · MCQ · HydrocarbonsOpen
Given below are two statements :
Statement I : There are several conformers for n-butane. Out of those conformers,
is the least stable and most stable conformer is
Statement II : As the dihedral angle increases, torsional strain decreases from (X) to (Y).
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
Q21 · Numerical · Purification & CharacterisationOpen
0.25 g of an organic compound “A” containing carbon, hydrogen and oxygen was analysed using the combustion method. There was an increase in mass of $\ce{CaCl2}$ tube and potash tube at the end of the experiment. The amount was found to be 0.15 g and 0.1837 g, respectively. The percentage of oxygen in compound A is %. (Nearest integer)
(Given : molar mass in g mol $^{-1}$ H : 1, C : 12, O : 16)
Q22 · Numerical · Chemical KineticsOpen
The half-life of $^{65}$ Zn is 245 days. After x days, 75% of original activity remained. The value of x in days is . (Nearest integer)
(Given : $\log 3 = 0.4771$ and $\log 2 = 0.3010$ )
Q23 · Numerical · Coordination CompoundsOpen
A chromium complex with a formula $\ce{CrCl3.6H2O}$ has a spin only magnetic moment value of 3.87 BM and its solution conductivity corresponds to 1 : 2 electrolyte. 2.75 g of the complex solution was initially passed through a cation exchanger. The solution obtained after the process was reacted with excess of $\ce{AgNO3}$ . The amount of AgCl formed in the above process is g. (Nearest integer)
[Given : Molar mass in g mol $^{-1}$ Cr : 52; Cl : 35.5, Ag : 108, O : 16, H : 1]
Q24 · Numerical · ElectrochemistryOpen
Molar conductivity of a weak acid HQ of concentration 0.18 M was found to be 1/30 of the molar conductivity of another weak acid HZ with concentration of 0.02 M. If $\lambda^{0}_{\mathrm{Q}^-}$ happened to be equal with $\lambda^{0}_{\mathrm{Z}^-}$ , then the difference of the $\mathrm{p}K_\mathrm{a}$ values of the two weak acids ( $\mathrm{p}K_\mathrm{a}(\mathrm{HQ})-\mathrm{p}K_\mathrm{a}(\mathrm{HZ})$ ) is (Nearest integer).
[Given : degree of dissociation ( $\alpha$ ) $\ll 1$ for both weak acids, $\lambda^\circ$ : limiting molar conductivity of ions]
We corrected a printing error in the original paper
Source printed “with concentration of 0.02 of M”.
Q25 · Numerical · Aldehydes, Ketones & AcidsOpen
Grignard reagent RMgBr(P) reacts with water and forms a gas (Q). One gram of Q occupies 1.4 dm $^3$ at STP. (P) on reaction with dry ice in dry ether followed by $\ce{H3O+}$ forms a compound (Z). 0.1 mole of (Z) will weigh g. (Nearest integer)

Maths

Q1 · MCQ · Matrices & DeterminantsOpen
Let $f(x) = \displaystyle\int\dfrac{7x^{10} + 9x^8}{(1 + x^2 + 2x^9)^2}\,dx$ , $x > 0$ , $\displaystyle\lim_{x\to0} f(x) = 0$ and $f(1) = \dfrac14$ . If $A = \begin{bmatrix} 0 & 0 & 1\\[2pt] \dfrac14 & f'(1) & 1\\[2pt] \alpha^2 & 4 & 1\end{bmatrix}$ and $B = \operatorname{adj}(\operatorname{adj} A)$ be such that $\lvert B\rvert = 81$ , then $\alpha^2$ is equal to
1. (1)2
2. (2)3
3. (3)1
4. (4)4
Q2 · MCQ · Conic SectionsOpen
Let the length of the latus rectum of an ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ , $(a > b)$ , be 30. If its eccentricity is the maximum value of the function $f(t) = -\dfrac34 + 2t - t^2$ , then $(a^2 + b^2)$ is equal to
1. (1)516
2. (2)256
3. (3)496
4. (4)276
Q3 · MCQ · Straight LinesOpen
Let the angles made with the positive $x$ -axis by two straight lines drawn from the point $P(2, 3)$ and meeting the line $x + y = 6$ at a distance $\sqrt{\dfrac23}$ from the point $P$ be $\theta_1$ and $\theta_2$ . Then the value of $(\theta_1 + \theta_2)$ is:
1. (1) $\dfrac{\pi}{12}$
2. (2) $\dfrac{\pi}{6}$
3. (3) $\dfrac{\pi}{2}$
4. (4) $\dfrac{\pi}{3}$
Q4 · MCQ · VectorsOpen
Let $\vec a = 2\hat i - \hat j - \hat k$ , $\vec b = \hat i + 3\hat j - \hat k$ and $\vec c = 2\hat i + \hat j + 3\hat k$ . Let $\vec v = \vec a + \lambda\vec b$ , $\lambda \in \mathbb{R}$ , be the vector in the plane of the vectors $\vec a$ and $\vec b$ , such that the length of its projection on the vector $\vec c$ is $\dfrac{1}{\sqrt{14}}$ . Then $\lvert\vec v\rvert$ is equal to
1. (1) $\dfrac{\sqrt{21}}{2}$
2. (2)13
3. (3) $\dfrac{\sqrt{35}}{2}$
4. (4)7
We corrected a printing error in the original paper
Assumption “ $\vec v = \vec a + \lambda\vec b$ , $\lambda \in \mathbb{R}$ ” added; the source states only that $\vec v$ lies in the plane of $\vec a$ and $\vec b$ , which leaves $\lvert\vec v\rvert$ undetermined (the source prints “Ans. (Bonus)” and “NTA (3)”); key (3) holds when the coefficient of $\vec a$ is 1.
Q5 · MCQ · Sequences & SeriesOpen
Let $\alpha_1, \alpha_2, \alpha_3, \alpha_4$ be an A.P. of four terms such that each term of the A.P. and its common difference $l$ are integers. If $\alpha_1 + \alpha_2 + \alpha_3 + \alpha_4 = 48$ and $\alpha_1\alpha_2\alpha_3\alpha_4 + l^4 = 361$ then the largest term of the A.P. is equal to
1. (1)27
2. (2)24
3. (3)21
4. (4)23
We corrected a printing error in the original paper
Source printed “ $\alpha_1, \alpha_2, \alpha_3, \alpha_4 + l^4 = 361$ ” (commas instead of the product $\alpha_1\alpha_2\alpha_3\alpha_4$ ).
Q6 · MCQ · Conic SectionsOpen
Let the image of parabola $x^2 = 4y$ , in the line $x - y = 1$ be $(y + a)^2 = b(x - c)$ , $a, b, c \in \mathbb{N}$ . Then $a + b + c$ is equal to
1. (1)12
2. (2)4
3. (3)6
4. (4)8
We corrected a printing error in the original paper
Source printed “ $(y + \alpha)^2 = b(x - c)$ ” (Greek $\alpha$ for the $a$ of “ $a, b, c \in \mathbb{N}$ ” and “ $a + b + c$ ”).
Q7 · MCQ · Sequences & SeriesOpen
$\left(\frac13 + \frac47\right) + \left(\frac{1}{3^2} + \frac13\times\frac47 + \frac{4^2}{7^2}\right) + \left(\frac{1}{3^3} + \frac{1}{3^2}\times\frac47 + \frac13\times\frac{4^2}{7^2} + \frac{4^3}{7^3}\right) + \ldots$ upto infinite terms is equal to
1. (1) $\dfrac52$
2. (2) $\dfrac74$
3. (3) $\dfrac43$
4. (4) $\dfrac65$
Q8 · MCQ · Matrices & DeterminantsOpen
Let $P = [p_{ij}]$ and $Q = [q_{ij}]$ be two square matrices of order 3 such that $q_{ij} = 2^{(i + j - 1)}p_{ij}$ and $\det(Q) = 2^{10}$ . Then the value of $\det(\operatorname{adj}(\operatorname{adj} P))$ is:
1. (1)32
2. (2)16
3. (3)81
4. (4)124
Q9 · MCQ · Permutations & CombinationsOpen
The letters of the word “UDAYPUR” are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word “UDAYPUR” is:
1. (1)1580
2. (2)1578
3. (3)1579
4. (4)1581
Q10 · MCQ · Permutations & CombinationsOpen
The largest value of $n$ , for which $40^n$ divides $60!$ , is
1. (1)13
2. (2)11
3. (3)12
4. (4)14
Q11 · MCQ · 3D GeometryOpen
The sum of all values of $\alpha$ , for which the shortest distance between the lines $\frac{x + 1}{\alpha} = \frac{y - 2}{-1} = \frac{z - 4}{-\alpha} \quad\text{and}\quad \frac{x}{\alpha} = \frac{y - 1}{2} = \frac{z - 1}{2\alpha}$ is $\sqrt2$ , is
1. (1)8
2. (2) $-6$
3. (3)6
4. (4) $-8$
Q12 · MCQ · Area under CurvesOpen
Let $f(\alpha)$ denote the area of the region in the first quadrant bounded by $x = 0$ , $x = 1$ , $y^2 = x$ and $y = \lvert\alpha x - 5\rvert - \lvert 1 - \alpha x\rvert + \alpha x^2$ . Then $(f(0) + f(1))$ is equal to
1. (1)9
2. (2)14
3. (3)7
4. (4)12
Q13 · MCQ · Inverse TrigonometryOpen
If the domain of the function $f(x) = \sin^{-1}\left(\dfrac{1}{x^2 - 2x - 2}\right)$ , is $(-\infty, \alpha] \cup [\beta, \gamma] \cup [\delta, \infty)$ , then $\alpha + \beta + \gamma + \delta$ is equal to
1. (1)2
2. (2)4
3. (3)3
4. (4)5
Q14 · MCQ · StatisticsOpen
Let $X = \{x \in \mathbb{N} : 1 \le x \le 19\}$ and for some $a, b \in \mathbb{R}$ , $Y = \{ax + b : x \in X\}$ . If the mean and variance of the elements of $Y$ are 30 and 750, respectively, then the sum of all possible values of $b$ is
1. (1)20
2. (2)80
3. (3)100
4. (4)60
Q15 · MCQ · Applications of DerivativesOpen
Consider the following three statements for the function $f : (0, \infty) \to \mathbb{R}$ defined by $f(x) = \lvert\log_e x\rvert - \lvert x - 1\rvert$ :
(I) $f$ is differentiable at all $x > 0$ .
(II) $f$ is increasing in $(0, 1)$ .
(III) $f$ is decreasing in $(1, \infty)$ .
Then.
1. (1)All (I), (II) and (III) are TRUE.
2. (2)Only (I) is TRUE.
3. (3)Only (II) and (III) are TRUE.
4. (4)Only (I) and (III) are TRUE.
Q16 · MCQ · Differential EquationsOpen
Let $y = y(x)$ be a differentiable function in the interval $(0, \infty)$ such that $y(1) = 2$ . and $\lim_{t\to x}\left(\frac{t^2y(x) - x^2y(t)}{x - t}\right) = 3 \quad\text{for each } x > 0.$ Then $2y(2)$ is equal to
1. (1)18
2. (2)23
3. (3)27
4. (4)12
Q17 · MCQ · Sets, Relations & FunctionsOpen
Let $f$ be a function such that $3f(x) + 2f\left(\dfrac{m}{19x}\right) = 5x$ , $x \neq 0$ , where $m = \displaystyle\sum_{i=1}^{9}(i)^2$ . Then $f(5) - f(2)$ is equal to
1. (1) $-9$
2. (2)36
3. (3)18
4. (4)9
Q18 · MCQ · Quadratic EquationsOpen
The smallest positive integral value of $a$ , for which all the roots of $x^4 - ax^2 + 9 = 0$ are real and distinct, is equal to
1. (1)9
2. (2)3
3. (3)4
4. (4)7
Q19 · MCQ · VectorsOpen
Let $\vec a = 2\hat i - 5\hat j + 5\hat k$ and $\vec b = \hat i - \hat j + 3\hat k$ . If $\vec c$ is a vector such that $2\left(\vec a\times\vec c\right) + 3\left(\vec b\times\vec c\right) = \vec 0 \quad\text{and}\quad \left(\vec a - \vec b\right)\cdot\vec c = -97,$ then $\left\lvert\vec c\times\hat k\right\rvert^2$ is equal to
1. (1)193
2. (2)233
3. (3)218
4. (4)205
We corrected a printing error in the original paper
Source printed “ $2\hat i - 5\hat j + 5k$ ”, “ $\hat i - \hat j + 3k$ ” and “ $\lvert\vec c\times k\rvert^2$ ” (unit vector $\hat k$ printed without its hat).
Q20 · MCQ · Limits, Continuity & DifferentiabilityOpen
Let $[t]$ denote the greatest integer less than or equal to $t$ . If the function $f(x) = \begin{cases} b^2\sin\left(\dfrac{\pi}{2}\left[\dfrac{\pi}{2}(\cos x + \sin x)\cos x\right]\right), & x < 0\\[10pt] \dfrac{\sin x - \frac12\sin 2x}{x^3}, & x > 0\\[10pt] a, & x = 0\end{cases}$ is continuous at $x = 0$ , then $a^2 + b^2$ is equal to
1. (1) $\dfrac58$
2. (2) $\dfrac{9}{16}$
3. (3) $\dfrac34$
4. (4) $\dfrac12$
Q21 · Numerical · Definite IntegrationOpen
If $f(x)$ satisfies the relation $f(x) = e^x + \displaystyle\int_0^1\left(y + xe^x\right)f(y)\,dy$ , then $e + f(0)$ is equal to .
Q22 · Numerical · Conic SectionsOpen
Let $(h, k)$ lie on the circle $C : x^2 + y^2 = 4$ and the point $(2h + 1, 3k + 2)$ lie on an ellipse with eccentricity $e$ . Then the value of $\dfrac{5}{e^2}$ is equal to .
Q23 · Numerical · Complex NumbersOpen
Let $z = (1 + i)(1 + 2i)(1 + 3i)\ldots(1 + ni)$ , where $i = \sqrt{-1}$ . If $\lvert z\rvert^2 = 44200$ , then $n$ is equal to .
Q24 · Numerical · ProbabilityOpen
Let $S$ be a set of 5 elements and $P(S)$ denote the power set of $S$ . Let $E$ be an event of choosing an ordered pair $(A, B)$ from the set $P(S)\times P(S)$ such that $A \cap B = \varnothing$ . If the probability of the event $E$ is $\dfrac{3^p}{2^q}$ , where $p, q \in \mathbb{N}$ , then $p + q$ is equal to .
Q25 · Numerical · TrigonometryOpen
The number of elements in the set $\left\{x \in [0, 180^\circ] : \tan(x + 100^\circ) = \tan(x + 50^\circ)\tan x\tan(x - 50^\circ)\right\}$ is .