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JEE Main 2026 April 08, Shift 2 Question Paper with Solutions

All 72 questions from the JEE Main 2026 (April 08, Shift 2) shift — Physics (24), Chemistry (24) and Mathematics (24) — with the correct answer and a step-by-step solution for every question.

Physics24 questions

Q26Single correctUnits and Measurements
A new unit (α\alpha) of length is chosen such that it is equal to the speed of light in vacuum. What is the distance between Venus and Earth in terms of α\alpha units if light takes 6 min. 40s to cover this distance?
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Q27Single correctUnits and Measurements
Consider the equation H=xpϵqErtsH = \dfrac{x^{p}\epsilon^{q}E^{r}}{t^{s}} Where H = magnetic field; E = electric field, ϵ\epsilon = permittivity, x = distance, t = time. The values of p, q, r and s respectively are:
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Q28Single correctLaws of Motion
A car moving with a speed of 54 km/h takes a turn of radius 20 m. A simple pendulum is suspended from the ceiling of the car. Determine the angle made by the string of the pendulum with the vertical during the turning. (Take g = 10 m/s2s^{2})
(A)
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Q29Single correctKinematics
A gas balloon is going up with a constant velocity of 10 m/s. When this balloon reached a height of 75 m, a stone is dropped from it and balloon keeps moving up with the same velocity. The height of the balloon when the stone hits the ground is ...... m. (Take g = 10 m/s2s^{2})
(A)
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Q30Single correctOptics
A thin biconvex lens is prepared from the glass (μ=1.5\mu = 1.5) both curved surfaces of which have equal radii of 20 cm each. Left side surface of the lens is silvered from outside to make it reflecting. To have the position of image and object at the same place, the object should be placed, from the lens at a distance of ......... cm.
(A)
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Q31Single correctKinematics
Two identical bodies, projected with the same speed at two different angles cover the same horizontal range R. If the time of flight of these bodies are 5 s and 10 s, respectively, then the value of R is ________ m. (Take g = 10 m/s2s^{2})
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Q32Single correctRotational Motion
A solid cylinder having radius R and length L is slipping on a rough horizontal plane. At time t = 0 the cylinder has a translational velocity v0=49v_0 = 49 m/s, perpendicular to its axis and a rotational velocity v0/4Rv_0/4R about the centre. The time taken by the cylinder to start rolling is ____ seconds. (coefficient of kinetic friction μK=0.25\mu_K = 0.25 and g = 9.8 m/s2s^{2})
(A)
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Q33Single correctProperties of Solids and Liquids
A liquid of density 600 kg/m3m^{3} flowing steadily in a tube of varying cross-section. The cross-section at a point A is 1.0 cm2m^{2} and that at B is 20 mm2m^{2}. Both the points A and B are in same horizontal plane, the speed of the liquid at A is 10 cm/s. The difference in pressures at A and B points is ______ Pa.
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Q34Single correctProperties of Solids and Liquids
A spherical liquid drop of radius R acquires the terminal velocity v1v_1 when falls through a gas of viscosity η\eta. Now the drop is broken into 64 identical droplets and each droplet acquires terminal velocity v2v_2 falling through the same gas. The ratio of terminal velocities v1/v2v_1/v_2 is ____ .
(A)
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Q36Single correctThermodynamics
Initial pressure and volume of a monoatomic ideal gas are P and V. The change in internal energy of this gas in adiabatic expansion to volume Vfinal=27VV_{final} = 27V is ____ J.
(A)
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Q37Single correctOscillations and Waves
The frequency of oscillation of a mass m suspended by a spring is υ1\upsilon_1. If the length of the spring is cut to half, the same mass oscillates with frequency υ2\upsilon_2. The value of υ2/υ1\upsilon_2/\upsilon_1 is ______ .
(A)
(B)
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Q38Single correctDual Nature of Matter and Radiation
A monochromatic source of light operating at 1515 kW emits 2.5×10222.5 \times 10^{22} photons/s. The region of an electromagnetic spectrum to which the emitted electromagnetic radiation belongs to ____. (Take h=6.6×1034h = 6.6 \times 10^{-34} J.s and c=3×108c = 3 \times 10^{8} m/s).
(A)
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Q39Single correctMagnetic Effects of Current and Magnetism
A current carrying circular loop of radius 22 cm with unit normal n^=k^+i^2\hat{n} = \dfrac{\hat{k}+\hat{i}}{\sqrt{2}} is placed in a magnetic field, B=Bo(3i^+2k^)\vec{B} = B_o(3\hat{i}+2\hat{k}). If Bo=4×103B_o = 4\times 10^{-3} T and current I=1002I = 100\sqrt{2} A, the torque experienced by the loop is ____ Wb.A. (π=3.14\pi = 3.14).
(A)
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Q40Single correctElectromagnetic Induction and Alternating Currents
A 3030 cm long solenoid has 1010 turns per cm and area of 55 cm2m^2. The current through the solenoid coil varies from 22 A to 44 A in 3.143.14 s. The e.m.f. induced in the coil is α×105\alpha\times 10^{-5} V. The value of α\alpha is ____ .
(A)
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Q41Single correctElectrostatics
Two point charges q1=3μCq_1 = 3\,\mu C and q2=4μCq_2 = -4\,\mu C are placed at points (2i^+3j^+3k^)(2\hat{i}+3\hat{j}+3\hat{k}) and (i^+j^+k^)(\hat{i}+\hat{j}+\hat{k}) respectively. Force on charge q2q_2 is ____ N. (Take 14πϵ0=9×109\dfrac{1}{4\pi\epsilon_0} = 9\times 10^{9} SI Units).
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Q42Single correctOptics
Light ray incident along a vector AO (AO=2i^3j^)\vec{AO}\ (\vec{AO} = 2\hat{i}-3\hat{j}) emerges out along vector OB (OB=Ci^4j^)\vec{OB}\ (\vec{OB} = C\hat{i}-4\hat{j}) as shown in the figure below. The value of C is ____ . (Medium 1 above the interface has μ1=1\mu_1 = 1, medium 2 below has μ2=1.5\mu_2 = 1.5.)
A horizontal interface separates medium 1 (mu1 = 1, above) from medium 2 (mu2 = 1.5, below). An incident ray from point A strikes point O on the interface making angle alpha with the vertical normal, and the refracted ray continues from O to point B in the lower medium making angle beta with the normal; beta < alpha indicating bending toward the normal.
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Q43Single correctDual Nature of Matter and Radiation
K1K_1 and K2K_2 be the maximum kinetic energies of photoelectrons emitted from a surface of a given material for the light of wavelength λ1\lambda_1 and λ2\lambda_2, respectively. If λ1=2λ2\lambda_1 = 2\lambda_2 then the work function of material is given by :
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Q44Single correctAtoms and Nuclei
Two radioactive substances A and B of mass numbers 200200 and 212212 respectively, shows spontaneous α\alpha-decay with same Q value of 11 MeV. The ratio of energies of α\alpha-rays produced by A and B is ____ .
(A)
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Q45Single correctElectronic Devices
The output YY for the given inputs AA and BB to the circuit is :
A logic circuit: inputs A and B feed two AND gates whose outputs combine into a final OR gate giving output Y. Alongside are the input waveforms: A is high (1) during the interval t = 1 to 2 s and low otherwise; B is high (1) during t = 0 to 1 s and low otherwise. Time axis marked at 0, 1, 2, 3 s.
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Q46NumericalElectrostatics
A parallel plate capacitor is having separation between plates 0.885 mm0.885\ \text{mm}. It has a capacitance of 1 μF1\ \mu\text{F} when the space between the plates is filled with an insulating material of resistivity 1×1013 Ωm1\times10^{13}\ \Omega\text{m} and resistance 17.7×1014 Ω17.7\times10^{14}\ \Omega. Relative permittivity of the insulating material is α×107\alpha\times10^{7}. The value of α\alpha is ________. (Take permittivity of free space =8.85×1012 F/m=8.85\times10^{-12}\ \text{F/m})
Q47NumericalOptics
Some distant star is to be observed by some telescope of diameter of objective lens a, at an angular resolution of 3.0×1073.0\times10^{-7} radian. If the wavelength of light from the star reaching the telescope is 500 nm500\ \text{nm}, the minimum diameter of the objective lens of the telescope is ______ cm. (nearest integer)
Q48NumericalMagnetic Effects of Current and Magnetism
A 5 mg5\ \text{mg} particle carrying a charge of 5π×106 C5\pi\times10^{-6}\ \text{C} is moving with velocity of (3i^+2k^)×102 m/s(3\hat{i}+2\hat{k})\times10^{-2}\ \text{m/s} in a region having magnetic field B=0.1k^ Wb/m2\vec{B}=0.1\hat{k}\ \text{Wb/m}^2. It moves a distance of α\alpha meter along k^\hat{k} when it completes 5 revolutions. The value of α\alpha is ______.
Q49NumericalCurrent Electricity
The stored charge in the capacitor in steady state of the following circuit is ______ μC\mu\text{C}.
Circuit with a 12 V source connected to a ladder network of resistors. The top branch has 5 ohm, 4 ohm and 10 ohm resistors in series; the bottom branch has 2 ohm and 2 ohm resistors; vertical branches contain 12 ohm, 10 ohm and 4 ohm resistors, and a 100 microfarad capacitor is connected on the right side across the last vertical branch.
Q50NumericalKinematics
Two masses of 3.4 kg3.4\ \text{kg} and 2.5 kg2.5\ \text{kg} are accelerated from an initial speed of 5 m/s5\ \text{m/s} and 12 m/s12\ \text{m/s}, respectively. The distances traversed by the masses in the 5th5^{\text{th}} second are 104 m104\ \text{m} and 129 m129\ \text{m}, respectively. The ratio of their momenta after 10 s10\ \text{s} is x8\dfrac{x}{8}. The value of x is ______.

Chemistry24 questions

Q51Single correctSome Basic Concepts in Chemistry
Match List - I with List - II.
List - I Mass of substanceList - II Number of atoms
A. 1.8 mg waterI. 2×104×NA2\times10^{-4}\times \mathrm{N_A}
B. 9.8 mg sulphuric acidII. 1.5×104×NA1.5\times10^{-4}\times \mathrm{N_A}
C. 1.8 mg carbonIII. 3×104×NA3\times10^{-4}\times \mathrm{N_A}
D. 5.85 mg salt (NaCl)IV. 7×104×NA7\times10^{-4}\times \mathrm{N_A}
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Q52Single correctSolutions
Given below are two statements:
Given: Molar mass of C, H, O, Cl are 12, 1, 16 and 35.5 gmol1\mathrm{g\,mol^{-1}}, respectively.
Statement I: In 30%(w/w) solution of methanol in CCl4\mathrm{CCl_4} (at T K), the mole fraction of CCl4\mathrm{CCl_4} is equal to 0.33.
Statement II: Mixture of methanol and CCl4\mathrm{CCl_4} shows positive deviation from Raoult's law.
In the light of the above statements, choose the correct answer from the options given below:
(A)
(B)
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(D)
Q53Single correctChemical Bonding and Molecular Structure
Bromine trifluoride autoionizes to form BrF2\mathrm{BrF_2^{\oplus}} and BrF4\mathrm{BrF_4^{\ominus}}. The shapes of the cation and anion are respectively ______, and ______.
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Q54Single correctSolutions
Which of the following statements are not correct?
A. For water, magnitude of Kb\mathrm{K_b} is more than the magnitude of Kf\mathrm{K_f}.
B. The elevation in boiling point of water when a non-volatile solute is added to it is larger in magnitude than its depression in freezing point.
C. Osmotic pressure measurement is preferred over any other colligative property to determine molar mass of proteins and polymers.
D. The dimerised form of benzoic acid in benzene is
Dimerised form of benzoic acid in benzene shown as two benzoic acid molecules linked by hydrogen bonds: C6H5-C(=O)-OH ... O=C-(OH)-C6H5
(A)
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Q55Single correctEquilibrium
Consider the following reactions in which all the reactants and products are present in gaseous state:
2xyx2+y22xy \rightleftharpoons x_2 + y_2 ; K1=2.5×105K_1 = 2.5\times10^{5}
xy+12z2xyzxy + \dfrac{1}{2}z_2 \rightleftharpoons \text{xyz} ; K2=5×103K_2 = 5\times10^{-3}
The value of K3K_3 for the equilibrium 12x2+12y2+12z2xyz\dfrac{1}{2}x_2 + \dfrac{1}{2}y_2 + \dfrac{1}{2}z_2 \rightleftharpoons \text{xyz} is:
(A)
(B)
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Q56Single correctRedox Reactions and Electrochemistry
Given at 298 K:
EFe2+/Fe=XE^{\ominus}_{\mathrm{Fe^{2+}/Fe}} = X Volt
EFe3+/Fe=YE^{\ominus}_{\mathrm{Fe^{3+}/Fe}} = Y Volt
The EFe3+/Fe2+E^{\ominus}_{\mathrm{Fe^{3+}/Fe^{2+}}} in Volt at 298 K is given by:
(A)
(B)
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(D)
Q57Single correctChemical Kinetics
Given below are two statements:
R=8.314JK1mol1R = 8.314\,\mathrm{J\,K^{-1}\,mol^{-1}} and 1cal=4.2J1\,\mathrm{cal} = 4.2\,\mathrm{J}
Statement I: When Ea=12.6kcal/molE_a = 12.6\,\mathrm{kcal/mol}, the room temperature rate constant is doubled by a 10C10^\circ\text{C} increase in temperature (298 K to 308 K).
Statement II: For a first order reaction ABA \to B, the plot of t1/2t_{1/2}/s versus [A]0[A]_0/mol L1L^{-1} is a straight line passing through the origin with positive slope. Here [A]0[A]_0 is the initial concentration of A and t1/2t_{1/2} is the half life of reaction.
In the light of the above statements, choose the correct answer from the options given below:
Graph with y-axis t(1/2)/s and x-axis [A]0/mol L^-1, showing a straight line of positive slope passing through the origin (as drawn in Statement II).
(A)
(B)
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Q58Single correctClassification of Elements and Periodicity in Properties
Match List - I with List - II.
List - I Electronic configuration of neutral atom (where n = 2)List - II 1st Ionization Energy (kJmol1\mathrm{kJ\,mol^{-1}})
A. ns2ns^2I. 2080
B. ns2np1ns^2np^1II. 899
C. ns2np3ns^2np^3III. 800
D. ns2np6ns^2np^6IV. 1402
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Q59Single correctp-Block Elements
Find the correct statements related to group 15 hydrides.
A. Reducing nature increases from NH3\mathrm{NH_3} to BiH3\mathrm{BiH_3}.
B. Tendency to donate lone pair of electrons decreases from NH3\mathrm{NH_3} to BiH3\mathrm{BiH_3}.
C. The stability of hydrides decreases from NH3\mathrm{NH_3} to BiH3\mathrm{BiH_3}.
D. HEH bond angle decreases from NH3\mathrm{NH_3} to SbH3\mathrm{SbH_3} (E = Elements of group 15).
Choose the correct answer from the options given below:
(A)
(B)
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Q60Single correctd- and f-Block Elements
Given below are two statements:
Statement I: The number of pairs among [Ti4+,V2+][\mathrm{Ti^{4+}, V^{2+}}], [V2+,Mn2+][\mathrm{V^{2+}, Mn^{2+}}], [Mn2+,Fe3+][\mathrm{Mn^{2+}, Fe^{3+}}] and [V2+,Cr2+][\mathrm{V^{2+}, Cr^{2+}}] in which both ions are coloured is 3.
Statement II: The number of pairs among [La3+,Yb2+][\mathrm{La^{3+}, Yb^{2+}}], [Lu3+,Ce4+][\mathrm{Lu^{3+}, Ce^{4+}}] and [Ac3+,Lr3+][\mathrm{Ac^{3+}, Lr^{3+}}] ions in which both are diamagnetic is 3.
In the light of the above statements, choose the correct from the options given below:
(A)
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Q61Single correctd- and f-Block Elements
Given below are two statements for catalytic properties of transition metals.
Statement I: First row transition metals which act as catalyst utilise their 3d electrons only for formation of bonds between reactant molecules and atoms on the surface of catalyst.
Statement II: There is increase in the concentration of reactants on the surface of catalyst which strengthens the bonds in reacting molecules.
In the light of the above statements, choose the correct answer from the options given below :
(A)
(B)
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(D)
Q62Single correctPurification and Characterisation of Organic Compounds
Given below are two statements :
Statement I: Vapours of the liquid with higher boiling point condense before vapours of the liquid with lower boiling points in fractional distillation.
Statement II: The vapours rising up in the fractionating column become richer in high boiling component of the mixture.
In the light of the above statements, choose the correct answer from the options given below :
(A)
(B)
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(D)
Q63Single correctSome Basic Principles of Organic Chemistry
The major product of which of the following reaction is not obtained by rearrangement reaction?
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Q65Single correctHydrocarbons
n-Butane on monochlorination under photochemical condition gives an optically active compound "P". "P" on further chlorination gives dichloro compounds.
The number of dichloro compounds obtained (ignore stereoisomers) is :
(A)
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Q66Single correctOrganic Compounds Containing Halogens
Given below are two statements :
Statement I: Due to increase in van der Waals forces, the order of boiling points is CH3CH2CH2I>CH3CH2I>CH3I\mathrm{CH_3CH_2CH_2I > CH_3CH_2I > CH_3I}.
(A)
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Q67Single correctOrganic Compounds Containing Nitrogen
Consider the following reaction.
A piperidine ring with a carbonyl (lactam/amide C=O on ring carbon adjacent to N), the nitrogen bearing a CH(CH3)-CHO side chain (2-methyl aldehyde). Reagents: (i) NaBH4/MeOH, (ii) NaOH(aq.), heat, (iii) H3O+ giving product P.
(A)
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Q68Single correctOrganic Compounds Containing Nitrogen
Which statements are True?
A. In Hoffmann bromamide degradation, 4 moles of NaOH and 2 moles of Br2\mathrm{Br_2} are consumed per mole of an amide
B. Hoffmann bromamide reaction is not given by alkyl amides
C. Primary amines can be synthesized by Hoffmann bromamide degradation.
D. Secondary amide on reaction with Br2\mathrm{Br_2} and NaOH will give secondary amine.
E. The by-products of Hoffmann degradation are Na2CO3\mathrm{Na_2CO_3} and H2O\mathrm{H_2O}.
Choose the correct answer from the options given below :
(A)
(B)
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Q69Single correctBiomolecules
The incorrect statement from the following with respect to carbohydrates is :
(A)
(B)
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Q70Single correctBiomolecules
Which of the following amino acid will give violet coloured complex with neutral ferric chloride solution?
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Q71NumericalCoordination Compounds
Number of paramagnetic complexes among the following is ______.
[MnBr4]2, [NiCl4]2, [Ni(CN)4]2, [Ni(CO)4], [CoF6]3, [Fe(CN)6]4, [Mn(CN)6]3, [Ti(CN)6]3, [Cu(H2O)6]2+, [Co(C2O4)3]3[\mathrm{MnBr_4}]^{2-},\ [\mathrm{NiCl_4}]^{2-},\ [\mathrm{Ni(CN)_4}]^{2-},\ [\mathrm{Ni(CO)_4}],\ [\mathrm{CoF_6}]^{3-},\ [\mathrm{Fe(CN)_6}]^{4-},\ [\mathrm{Mn(CN)_6}]^{3-},\ [\mathrm{Ti(CN)_6}]^{3-},\ [\mathrm{Cu(H_2O)_6}]^{2+},\ [\mathrm{Co(C_2O_4)_3}]^{3-}
Q72NumericalOrganic Compounds Containing Oxygen
'x' is the product which is obtained from benzene by reacting it with carbon monoxide and hydrogen chloride in the presence of cuprous chloride. 'y' is the major product obtained from the benzene by reacting it with ethanoyl chloride in the presence of anhydrous AlCl3\mathrm{AlCl_3}. Product (major) obtained by heating x and y in the presence of alkali is z. Total number of π(pi)\pi\,(pi) electrons in z is ______
Q73NumericalAtomic Structure
Consider two radiations of wavelengths
1. λ1=2000 A˚\lambda_1 = 2000\ \text{\AA}
2. λ2=6000 A˚\lambda_2 = 6000\ \text{\AA}
The ratio of the energies of these two radiations (E1E2)\left(\dfrac{E_1}{E_2}\right) is ______ (Nearest integer).
Q74NumericalChemical Thermodynamics
Consider the reaction:
2H2S(g)+3O2(g)2H2O(l)+2SO2(g)2\mathrm{H_2S}(g) + 3\mathrm{O_2}(g) \rightarrow 2\mathrm{H_2O}(l) + 2\mathrm{SO_2}(g)
The magnitude of enthalpy change for the reaction in kJ mol1\mathrm{kJ~mol^{-1}} is ______. (Nearest integer)
Given: ΔfH(H2S)=20.1 kJ mol1\Delta_f H^{\ominus}(\mathrm{H_2S}) = -20.1~\mathrm{kJ~mol^{-1}}, ΔfH(H2O)=286.0 kJ mol1\Delta_f H^{\ominus}(\mathrm{H_2O}) = -286.0~\mathrm{kJ~mol^{-1}}, ΔfH(SO2)=297.0 kJ mol1\Delta_f H^{\ominus}(\mathrm{SO_2}) = -297.0~\mathrm{kJ~mol^{-1}}
Q75NumericalEquilibrium
Solid carbon, CaO and CaCO3\mathrm{CaCO_3} are mixed and allowed to attain equilibrium at T K.
CaCO3(s)CaO(s)+CO2(g)Kp1=0.08 atm\mathrm{CaCO_3(s)} \rightleftharpoons \mathrm{CaO(s)} + \mathrm{CO_2(g)} \qquad Kp_1 = 0.08\ \mathrm{atm}
C(s)+CO2(g)2CO(g)Kp2=2 atm\mathrm{C(s)} + \mathrm{CO_2(g)} \rightleftharpoons 2\,\mathrm{CO(g)} \qquad Kp_2 = 2\ \mathrm{atm}
The partial pressure of CO is\text{The partial pressure of CO is} ____ ×101\times 10^{-1} atm

Mathematics24 questions

Q1Single correctSets, Relations and Functions
Consider the relation R on the set {2,1,0,1,2}\{-2,-1,0,1,2\} defined by (a,b)R(a,b) \in R if and only if 1+ab>01+ab > 0. Then, among the statements :
I. The number of elements in R is 17
II. R is an equivalence relation
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Q2Single correctComplex Numbers and Quadratic Equations
The number of values of zCz \in \mathbb{C}, satisfying the equations z(4+8i)=10|z-(4+8i)| = \sqrt{10} and z(3+5i)+z(5+11i)=45|z-(3+5i)| + |z-(5+11i)| = 4\sqrt{5}, is :
(A)
(B)
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(D)
Q3Single correctMatrices and Determinants
If the system of linear equations:
x+y+z=6x+y+z=6
x+2y+5z=10x+2y+5z=10
2x+3y+λz=μ2x+3y+\lambda z=\mu
has infinitely many solutions, then the value of λ+μ\lambda+\mu equals:
(A)
(B)
(C)
(D)
Q4Single correctMatrices and Determinants
Let A=[α12230045]A = \begin{bmatrix} \alpha & 1 & 2 \\ 2 & 3 & 0 \\ 0 & 4 & 5 \end{bmatrix} and B=[10005α004α2α]+adj(A)B = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -5\alpha & 0 \\ 0 & 4\alpha & -2\alpha \end{bmatrix} + \mathrm{adj}(A). If det(B)=66\det(B) = 66, then det(adj(A))\det(\mathrm{adj}(A)) equals :
(A)
(B)
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(D)
Q5Single correctTrigonometry
Let α=3+4+8+9+13+14+\alpha = 3+4+8+9+13+14+\ldots upto 40 terms. If (tanβ)α1020(\tan\beta)^{\frac{\alpha}{1020}} is a root of the equation x2+x2=0x^2 + x - 2 = 0, β(0,π2)\beta \in \left(0, \frac{\pi}{2}\right), then sin2β+3cos2β\sin^2\beta + 3\cos^2\beta is equal to :
(A)
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(D)
Q6Single correctStatistics and Probability
A candidate has to go to the examination centre to appear in an examination. The candidate uses only one means of transportation for the entire distance out of bus, scooter and car. The probabilities of the candidate going by bus, scooter and car, respectively, are 25,15\frac{2}{5}, \frac{1}{5} and 25\frac{2}{5}. The probabilities that the candidate reaches late at the examination centre are 15,13\frac{1}{5}, \frac{1}{3} and 14\frac{1}{4} if the candidate uses bus, scooter and car, respectively. Given that the candidate reached late at the examination centre, the probability that the candidate travelled by bus is :
(A)
(B)
(C)
(D)
Q7Single correctStatistics and Probability
A set of four observations has mean 1 and variance 13. Another set of six observations has mean 2 and variance 1. Then, the variance of all these 10 observations is equal to :
(A)
(B)
(C)
(D)
Q8Single correctBinomial Theorem and its Simple Applications
If 26(233(12C2)+255(12C4)+277(12C6)++21313(12C12))=313α26\left(\frac{2^3}{3}\left({}^{12}C_2\right) + \frac{2^5}{5}\left({}^{12}C_4\right) + \frac{2^7}{7}\left({}^{12}C_6\right) + \cdots + \frac{2^{13}}{13}\left({}^{12}C_{12}\right)\right) = 3^{13} - \alpha, then α\alpha is equal to :
(A)
(B)
(C)
(D)
Q9Single correctPermutations and Combinations
A person has three different bags and four different books. The number of ways, in which he can put these books in the bags so that no bag is empty, is :
(A)
(B)
(C)
(D)
Q10Single correctCo-ordinate Geometry
If a straight line drawn through the point of intersection of the lines 4x+3y1=04x+3y-1=0 and 3x+4y1=03x+4y-1=0, meets the co-ordinate axes at the points P and Q, then the locus of the mid point of PQ is :
(A)
(B)
(C)
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Q11Single correctCo-ordinate Geometry
Let O be the vertex of the parabola y2=4xy^2 = 4x and its chords OP and OQ are perpendicular to each other. If the locus of the mid-point of the line segment PQ is a conic C, then the length of its latus rectum is :
(A)
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Q12Single correctTrigonometry
Let α=3sin1(611)\alpha = 3\sin^{-1}\left(\dfrac{6}{11}\right) and β=3cos1(49)\beta = 3\cos^{-1}\left(\dfrac{4}{9}\right), where inverse trigonometric functions take only the principal values. Given below are two statements :
Statement I: cos(α+β)>0\cos(\alpha+\beta) > 0.
Statement II: cos(α)<0\cos(\alpha) < 0.
In the light of the above statements, choose the correct answer from the options given below :
(A)
(B)
(C)
(D)
Q13Single correctLimit, Continuity and Differentiability
For the function f(x)=esinxx, xRf(x)=e^{\sin|x|}-|x|,\ x\in\mathbf{R}, consider the following statements:
Statement I : f is differentiable for all xRx\in\mathbf{R}.
Statement II: f is increasing in (π,π2)\left(-\pi,-\dfrac{\pi}{2}\right).
In the light of the above statements, choose the correct answer from the options given below :
(A)
(B)
(C)
(D)
Q14Single correctVector Algebra
Let a=4i^j^+3k^\vec{a}=4\hat{i}-\hat{j}+3\hat{k}, b=10i^+2j^k^\vec{b}=10\hat{i}+2\hat{j}-\hat{k} and a vector c\vec{c} be such that 2(a×b)+3(b×c)=02(\vec{a}\times\vec{b})+3(\vec{b}\times\vec{c})=\vec{0}. If ac=15\vec{a}\cdot\vec{c}=15, then c(i^+j^3k^)\vec{c}\cdot(\hat{i}+\hat{j}-3\hat{k}) is equal to :
(A)
(B)
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Q15Single correctThree Dimensional Geometry
Let the foot of perpendicular from the point (λ,2,3)(\lambda,2,3) on the line x41=y92=z51\dfrac{x-4}{1}=\dfrac{y-9}{2}=\dfrac{z-5}{1} be the point (1,μ,2)(1,\mu,2). Then the distance between the lines x12=y23=z+46\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z+4}{6} and xλ2=yμ3=z+56\dfrac{x-\lambda}{2}=\dfrac{y-\mu}{3}=\dfrac{z+5}{6} is equal to:
(A)
(B)
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(D)
Q17Single correctDifferential Equations
Let y=y(x)y=y(x) be the solution of the differential equation x1x2dy+(y1x2xcos1x)dx=0, x(0,1), limx1y(x)=1x\sqrt{1-x^2}\,dy+\left(y\sqrt{1-x^2}-x\cos^{-1}x\right)dx=0,\ x\in(0,1),\ \displaystyle\lim_{x\to1}y(x)=1. Then y(12)y\left(\dfrac{1}{2}\right) equals:
(A)
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Q18Single correctIntegral Calculus
Let f:(1,)Rf:(1,\infty)\to\mathbf{R} be a function defined as f(x)=x1x+1f(x)=\dfrac{x-1}{x+1}. Let fi+1(x)=f(fi(x)), i=1,2,,25f^{i+1}(x)=f\left(f^{i}(x)\right),\ i=1,2,\ldots,25, where f1(x)=f(x)f^{1}(x)=f(x). If g(x)+f26(x)=0, x(1,)g(x)+f^{26}(x)=0,\ x\in(1,\infty), then the area of the region bounded by the curves y=g(x), 2y=2x3, y=0y=g(x),\ 2y=2x-3,\ y=0 and x=4x=4 is:
(A)
(B)
(C)
(D)
Q19Single correctLimit, Continuity and Differentiability
Let f(x)={13, xπ/2b(1sinx)(π2x)2, x>π/2f(x)=\begin{cases}\dfrac{1}{3} & ,\ x\le \pi/2\\[2mm]\dfrac{b(1-\sin x)}{(\pi-2x)^2} & ,\ x>\pi/2\end{cases}. If f is continuous at x=π/2x=\pi/2, then the value of 03b6x2+2x3dx\displaystyle\int_0^{3b-6}\left|x^2+2x-3\right|dx is :
(A)
(B)
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(D)
Q20Single correctCo-ordinate Geometry
Let x2f(a2+7a+3)+y2f(3a+15)=1\dfrac{x^2}{f\left(a^2+7a+3\right)}+\dfrac{y^2}{f(3a+15)}=1 represent an ellipse with major axis along y-axis, where f is a strictly decreasing positive function on R\mathbf{R}. If the set of all possible values of a is R[α,β]\mathbf{R}-[\alpha,\beta], then α2+β2\alpha^2+\beta^2 is equal to :
(A)
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Q21NumericalComplex Numbers and Quadratic Equations
The sum of squares of all the real solutions of the equation log(x+1)(2x2+5x+3)=4log(2x+3)(x2+2x+1)\log_{(x+1)}\left(2x^2+5x+3\right)=4-\log_{(2x+3)}\left(x^2+2x+1\right) is equal to
Q22NumericalIntegral Calculus
If π/6π/4(cot(xπ3)cot(x+π3)+1)dx=αloge(31)\displaystyle\int_{\pi/6}^{\pi/4}\left(\cot\left(x-\frac{\pi}{3}\right)\cot\left(x+\frac{\pi}{3}\right)+1\right)dx=\alpha\log_e(\sqrt3-1), then 9α29\alpha^2 is equal to
Q23NumericalThree Dimensional Geometry
Let a line L1L_1 pass through the origin and be perpendicular to the lines L2:r=(3+t)i^+(2t1)j^+(2t+4)k^L_2:\vec{r}=(3+t)\hat{i}+(2t-1)\hat{j}+(2t+4)\hat{k} and L3:r=(3+2s)i^+(3+2s)j^+(2+s)k^, t,sRL_3:\vec{r}=(3+2s)\hat{i}+(3+2s)\hat{j}+(2+s)\hat{k},\ t,s\in\mathbf{R}. If (a,b,c), aZ(a,b,c),\ a\in\mathbf{Z}, is the point on L3L_3 at a distance of 17\sqrt{17} from the point of intersection of L1L_1 and L2L_2, then (a+b+c)2(a+b+c)^2 is equal to .................
Q24NumericalCo-ordinate Geometry
Consider the circle C: x2+y26x8y11=0C:\ x^2+y^2-6x-8y-11=0. Let a variable chord AB of the circle C subtend a right angle at the origin. If the locus of the foot of the perpendicular drawn from the origin on the chord AB is the circle x2+y2αxβyγ=0x^2+y^2-\alpha x-\beta y-\gamma=0, then α+β+2γ\alpha+\beta+2\gamma is equal to......
Q25NumericalSequence and Series
Let f be a polynomial function such that log2(f(x))=(log2(2+23+29+))log3(1+f(x)f(1/x)), x>0\log_2(f(x))=\left(\log_2\left(2+\frac{2}{3}+\frac{2}{9}+\ldots\ldots\infty\right)\right)\cdot\log_3\left(1+\frac{f(x)}{f(1/x)}\right),\ x>0 and f(6)=37f(6)=37. Then n=110f(n)\displaystyle\sum_{n=1}^{10} f(n) is equal to ______.

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