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JEE Main 2026 April 06, Shift 1 Question Paper with Solutions

All 75 questions from the JEE Main 2026 (April 06, Shift 1) shift — Physics (25), Chemistry (25) and Mathematics (25) — with the correct answer and a step-by-step solution for every question.

Physics25 questions

Q26Single correctUnits and Measurements
The density ρ\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±0.0297.42 \pm 0.02 g, 8.35±0.058.35 \pm 0.05 mm and 20.20±0.0220.20 \pm 0.02 mm, respectively. Calculated percentage fractional error in ρ\rho is _____ .
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Q27Single correctUnits and Measurements
The potential energy of a particle changes with distance x from a fixed origin as V=Axx+BV = \dfrac{A\sqrt{x}}{x+B}, where A and B are constant with appropriate dimensions. The dimensions of AB are _____ .
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Q28Single correctWork, Energy and Power
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=10g = 10 m/s2s^{2})
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Q29Single correctLaws of Motion
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 friction μ\mu. The cube (S) moves towards the right with acceleration of g2\dfrac{g}{2}, where g is gravitational acceleration. During this movement the block P and Q remain stationary. The value of μ\mu is _____ (take g=10g = 10 m/s2s^{2})
A cube labelled S with block P (2 kg) resting on its frictionless top surface and block Q (15 kg) hanging against the right side surface; P and Q are connected by a massless thread over a small pulley fixed at the top-right edge of the cube.
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Q30Single correctProperties of Solids and Liquids
A lift of mass 1600 kg is supported by thick iron wire. If the maximum stress which the wire can withstand is 4×108N/m24 \times 10^{8}\,\mathrm{N/m^{2}} and its radius is 4 mm, then maximum acceleration the lift can take is_____ m/s2s^{2}. (Take g=10g = 10 m/s2s^{2} and π=3.14\pi = 3.14)
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Q31Single correctRotational Motion
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.
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Q32Single correctWork, Energy and Power
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=αRh = \alpha R. The value of α\alpha is _____ .
An inclined slope leading into a vertical circular loop of radius R; a body is released from height h measured from the horizontal level of the loop centre to the release point at the top of the incline.
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Q33Single correctRotational Motion
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 _____ .
An isosceles triangle: a 15 kg mass at the apex with a 120 degree angle between the two equal sides of length 10 m each, with a 2 kg mass at the bottom-left vertex and a 3 kg mass at the bottom-right vertex. Point p marked on the dashed perpendicular bisector of the base, midway between the apex and the base midpoint.
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Q34Single correctProperties of Solids and Liquids
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 .
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Q35Single correctKinetic Theory of Gases
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 .
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Q36Single correctOptics
In interference experiment the path difference between two interfering waves at a point A on the screen is λ/3\lambda/3, where λ\lambda is the wavelength of these waves, and at another point B the path difference is λ/6\lambda/6. The ratio of intensities at points A and B is _____ .
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Q37Single correctOscillations and Waves
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=Ax = A to x=A2x = \dfrac{A}{\sqrt{2}} is _____ sec.
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Q38Single correctElectrostatics
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 100V/m, then the value of Q is ______ nC . (ϵ0=8.85×1012C2/Nm2\epsilon_0 = 8.85 \times 10^{-12}\,\text{C}^2/\text{Nm}^2 and π=3.14\pi = 3.14)
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Q39Single correctElectronic Devices
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 RSR_S is 0.5 mA . The minimum value of the resistance RSR_S, to ensure that the LED is not damaged is ______ kΩ\Omega.
A 5 V battery connected in series with resistor R_S feeds a parallel combination of a 1 k ohm resistor R (with a diode in series) and an LED rated 2 mW.
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Q40Single correctElectromagnetic Waves
A point light source emits E.M. waves in free space. A detector, placed at a distance of Lm, measures the intensity as I0I_0. The detector is now shifted to another location on the same spherical surface ensuring the angle between original location and new location as 4545^{\circ}. The measured intensity at new location will be ____ .
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Q41Single correctOptics
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 ______ .
A point source P sits 4R to the left of a convex spherical refracting surface of radius R; the medium on the left has n1 = 1 and on the right has n2 = 1.4.
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Q42Single correctMagnetic Effects of Current and Magnetism
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 α×1014\alpha \times 10^{-14} J. The value of α\alpha is ______. (μ0=4π×107\mu_0 = 4\pi \times 10^{-7} Tm / A, π=3.14\pi = 3.14)
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Q43Single correctElectromagnetic Induction and Alternating Currents
An inductor of inductance 10 mH having resistance of 100Ω\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 voltages across the inductor when the current passing through it is 2 mA and 4 mA is ______.
A series RL circuit: an inductor coil and a battery with a switch, connected so that closing the switch lets current grow through the inductor.
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Q44Single correctAtoms and Nuclei
The ratio of momentum of the photons of the 1st^{\text{st}} and 2nd^{\text{nd}} line of Balmer series of Hydrogen atoms is α/β\alpha/\beta. The possible values of α\alpha and β\beta are :
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Q45Single correctElectromagnetic Induction and Alternating Currents
A LCR series circuit driven with ErmsE_{\text{rms}} = 90V at frequency fdf_d = 30Hz has resistance R = 80Ω\Omega, an inductance with inductive reactance XLX_L = 20.0Ω\Omega and capacitance with capacitive reactance XCX_C = 80.0Ω\Omega. The power factor of the circuit is ____ .
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Q46NumericalCurrent Electricity
Refer to the circuit diagram given below. The heat generated across the 6Ω\Omega resistance in 100 second is α100\dfrac{\alpha}{100} J. The value of α\alpha is ______. (Nearest integer)
A 2 V battery and a 3 V battery drive a network containing a 3 ohm resistor, a 6 ohm resistor (in series with the 3 V battery in the top branch) and a 4 ohm resistor (forming a parallel branch).
Q47NumericalOptics
An unpolarized light of intensity I0I_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 00^{\circ} and intensity of light emerged from analyser is 38I0\dfrac{3}{8} I_0, the angle of rotation of the light by the solution with respect to analyser is ______ degrees.
Q48NumericalAtoms and Nuclei
The energy released when 717.13\dfrac{7}{17.13} kg of 37Li^{7}_{3}\text{Li} is converted into 24He^{4}_{2}\text{He} by proton bombardment is α×1032\alpha \times 10^{32} eV. The value of α\alpha is _____. (Nearest integer) (Mass of 37Li=7.0183^{7}_{3}\text{Li} = 7.0183u , mass of 24He=4.004^{4}_{2}\text{He} = 4.004u , mass of proton = 1.008u and 1u = 931 MeV/c2c^{2} and Avogadro number = 6.0×10236.0 \times 10^{23})
Q49NumericalElectrostatics
A three coulomb charge moves from the point (0, -2, -5) to the point (5, 1, 2) in an electric field expressed as E=2xi^+3y2j^+4k^\vec{E} = 2x\hat{i} + 3y^{2}\hat{j} + 4\hat{k} N / C. The work done in moving the charge is _____ J.
Q50NumericalProperties of Solids and Liquids
A certain gas is isothermally compressed to (13)rd\left(\dfrac{1}{3}\right)^{\text{rd}} of its initial volume (V0V_0 = 3 litre) by applying required pressure. If the bulk modulus of the gas is 3 ×\times 1050^{5} N/m2m^{2}, the magnitude of work done on the gas is ___ J.

Chemistry25 questions

Q51Single correctSome Basic Concepts in Chemistry
An oxide of iron contains 69.969.9% iron, its empirical formula, is: (Given : Molar mass of Fe and O are 56 and 16g mol116\,\text{g mol}^{-1} respectively.)
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Q52Single correctAtomic Structure
If shortest wavelength of hydrogen atom in Lyman series is x, then longest wavelength in Balmer series of He+\text{He}^{+} is:
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Q53Single correctAtomic Structure
Match the LIST-I with LIST-II
List-I (Orbital)List-II (Radial nodes and nodal plane)
A. 2sI. 1 Radial node + two nodal planes
B. 3sII. 1 Radial node + one nodal plane
C. 3pIII. 2 Radial nodes + No nodal plane
D. 4dIV. 1 Radial node + No nodal plane
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Q54Single correctChemical Bonding and Molecular Structure
The pairs among A=[SO32,CO32], B=[O22,F2], C=[CN,CO], D=[NH3,H3O+] and E=[MnO42,CrO42]A = [\text{SO}_3^{2-},\,\text{CO}_3^{2-}],\ B = [\text{O}_2^{2-},\,\text{F}_2],\ C = [\text{CN}^{-},\,\text{CO}],\ D = [\text{NH}_3,\,\text{H}_3\text{O}^{+}]\ \text{and}\ E = [\text{MnO}_4^{2-},\,\text{CrO}_4^{2-}] that do not have similar Lewis dot structure are
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Q55Single correctChemical Thermodynamics
Arrange the following isothermal processes in order of the magnitude of the work (pV)(p-V) involved between states 1 and 2.
A. Expansion in single stage wAw_{A}.
B. Expansion in multi stages wBw_{B}.
C. Compression in single stage wCw_{C}.
D. Compression in multi stages wDw_{D}.
Choose the correct option.
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Q56Single correctSolutions
When 0.25 moles of a non-volatile, non-ionizable solute was dissolved in 1 mole of a solvent the vapour pressure of solution was xx% of vapour pressure of pure solvent. What is xx% ?
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Q57Single correctEquilibrium
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. A(g)B(g)A(g) \rightleftharpoons B(g). KcK_{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=0.082L atm K1mol1R = 0.082\,\text{L atm K}^{-1}\,\text{mol}^{-1})
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Q58Single correctRedox Reactions and Electrochemistry
Consider the following data.
Electrolyte | Λm\Lambda_{m}^{\circ} (in S cm2mol1\text{S cm}^{2}\,\text{mol}^{-1}) |
BaCl2\text{BaCl}_2 | x1x_{1} |
H2SO4\text{H}_2\text{SO}_4 | x2x_{2} |
HCl\text{HCl} | x3x_{3} |
BaSO4\text{BaSO}_4 is sparingly soluble in water. If the conductivity of the saturated BaSO4\text{BaSO}_4 solution is xS cm1x\,\text{S cm}^{-1} then the solubility product of BaSO4\text{BaSO}_4 can be given as (Here Λm=Λm\Lambda_{m} = \Lambda_{m}^{\circ})
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Q59Single correctp-Block Elements
Given below are two statements: Statement I: Aluminium is more electropositive than thallium as the standard electrode potential value of EAl3+AlE^{\circ}_{\text{Al}^{3+}|\text{Al}} is negative and ETl3+/TlE^{\circ}_{\text{Tl}^{3+}/\text{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 Al3+\text{Al}^{3+} ion. In the light of the above statements, choose the correct answer from the options given below
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Q60Single correctd- and f-Block Elements
The correct statements among the following are. A. Basic vanadium oxide is used in the manufacture of H2SO4\text{H}_2\text{SO}_4. 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+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:
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Q61Single correctCoordination Compounds
Match the LIST-I with LIST-II
List-I (Electronic configuration of tetrahedral metal ion)List-II (Crystal Field Stabilization Energy (Δt)(\Delta_{t}))
A. d2d^{2}I. 0.6-0.6
B. d4d^{4}II. 0.8-0.8
C. d5d^{5}III. 1.2-1.2
D. d8d^{8}IV. 0.4-0.4
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Q62Single correctCoordination Compounds
Which of the following are true about the energy of the given d-orbitals of a tetrahedral complex? A. dxy=dxz>dx2y2d_{xy} = d_{xz} > d_{x^{2}-y^{2}}. B. dxy=dyz>dz2d_{xy} = d_{yz} > d_{z^{2}}. C. dx2y2>dz2>dxzd_{x^{2}-y^{2}} > d_{z^{2}} > d_{xz}. D. dx2y2=dz2<dxzd_{x^{2}-y^{2}} = d_{z^{2}} < d_{xz}. Choose the correct answer from the given below:
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Q63Single correctPurification and Characterisation of Organic Compounds
RfR_{f} value for 2-methylpropene in a solvent system (Ethyl acetate + ether) is 0.42 . 2-methylpropene is treated with dilute H2SO4\text{H}_2\text{SO}_4 to give major organic product (X). RfR_{f} value for (X) in the same solvent system under identical condition will be:
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Q64Single correctSome Basic Principles of Organic Chemistry
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
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Q65Single correctHydrocarbons
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 CH3MgBr\text{CH}_3\text{MgBr} 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
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Q66Single correctHydrocarbons
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
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Q67Single correctSome Basic Principles of Organic Chemistry
Consider the following sequence of reactions.
The major product P is:
tert-Butyl alcohol undergoes (i) Cu/573 K dehydration to isobutylene, then (ii) H+ and benzoic acid (PhCOOH) addition giving product P.
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Q68Single correctSome Basic Principles of Organic Chemistry
Arrange the following compounds according to increasing order of boiling points.
nC4H9OH(A),nC4H9NH2(B),nC4H10(C) and C2H5NHC2H5(D)\text{n}-\text{C}_4\text{H}_9\text{OH(A)}, \text{n}-\text{C}_4\text{H}_9\text{NH}_2\text{(B)}, \text{n}-\text{C}_4\text{H}_{10}\text{(C)} \text{ and } \text{C}_2\text{H}_5\text{NHC}_2\text{H}_5\text{(D)}
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Q69Single correctSome Basic Principles of Organic Chemistry
Match the LIST-I with LIST-II

Choose the correct answer from the options given below:
List-I (Deficiency Disease)List-II (Vitamin)
A. ScurvyI. Pyridoxine
B. ConvulsionsII. Vitamin A
C. CheilosisIII. Ascorbic Acid
D. XerophthalmiaIV. Riboflavin
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Q70Single correctSome Basic Principles of Organic Chemistry
Match the LIST-I with LIST-II

Choose the correct answer from the options given below:
List-I (Amino acid)List-II (Positive reaction/Test for functional group present in side chain of amino acid)
A. GlutamineI. Hinsberg's test
B. LysineII. Neutral FeCl3\text{FeCl}_3 test
C. TyrosineIII. Ceric ammonium nitrate test
D. SerineIV. Hoffman bromamide degradation
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Q71NumericalClassification of Elements and Periodicity in Properties
First and second ionization enthalpies of lithium are 520kJmol1520\,\text{kJ}\,\text{mol}^{-1} and 7297kJmol17297\,\text{kJ}\,\text{mol}^{-1} respectively. Energy required to convert 3.5 mg lithium (g) into Li2+(g)\text{Li}^{2+}(g) [Li(g)Li2+(g)]\left[\text{Li(g)} \to \text{Li}^{2+}(g)\right] is ____ kJ. (nearest integer) [Molar mass of Li=7gmol1\text{Li} = 7\,\text{g}\,\text{mol}^{-1}]
Q72NumericalSome Basic Principles of Organic Chemistry
Consider the following sequence of reactions.

C6H5N=NNHC6H5(ii) Aniline (in excess)(iii) Warm 4045C(iv) Cool, acetic acid(i) HCl\text{C}_6\text{H}_5 - \text{N} = \text{N} - \text{NH} - \text{C}_6\text{H}_5 \xrightarrow[\substack{(\text{ii}) \text{ Aniline (in excess)} \\ (\text{iii}) \text{ Warm } 40-45\,^\circ\text{C} \\ (\text{iv}) \text{ Cool, acetic acid}}]{(\text{i}) \text{ HCl}} Yellow Product (X).

The percentage of nitrogen in the yellow product (X) formed is ____ %. (Nearest Integer)

(Given Molar mass in gmol1H:1,C:12,N:14\text{g}\,\text{mol}^{-1}\, \text{H}:1, \text{C}:12, \text{N}:14)
Structure of 1,3-diphenyltriazene (diazoaminobenzene): C6H5-N=N-NH-C6H5
Q73NumericalHydrocarbons
4.7 g of phenol is heated with Zn to give product X. If this reaction goes to 6060% completion then the number of moles of compound X formed will be ____ ×102\times 10^{-2}. (Nearest Integer)

(Given molar mass in gmol1:H:1,C:12,O:16\text{g}\,\text{mol}^{-1}: \text{H}:1, \text{C}:12, \text{O}:16)
Q74NumericalChemical Kinetics
Sucrose hydrolyses in acidic medium into glucose and fructose by first order rate law with t1/2=3t_{1/2} = 3 hour. The percentage of sucrose remaining after 6 hours is ____ . (Nearest integer)

(Given : log2=0.3010\log 2 = 0.3010 and log3=0.4771\log 3 = 0.4771)
Q75NumericalChemical Thermodynamics
Consider the reaction XYX \rightleftharpoons Y at 300 K. If ΔH0\Delta H^0 and K are 28.40kJmol128.40\,\text{kJ}\,\text{mol}^{-1} and 1.8×1071.8 \times 10^{-7} at the same temperature, then the magnitude of ΔS0\Delta S^0 for the reaction in JK1mol1\text{J}\,\text{K}^{-1}\,\text{mol}^{-1} is ____ . (Nearest integer)

(Given : R=8.3JK1mol1,ln10=2.3,log3=0.48,log2=0.30R = 8.3\,\text{J}\,\text{K}^{-1}\,\text{mol}^{-1}, \ln 10 = 2.3, \log 3 = 0.48, \log 2 = 0.30)

Mathematics25 questions

Q1Single correctSets, Relations and Functions
Let [][\,\cdot\,] denote the greatest integer function. If the domain of the function f(x)=sin1 ⁣(x+[x]3)f(x)=\sin^{-1}\!\left(\dfrac{x+[x]}{3}\right) is [α,β)[\alpha,\beta), then α2+β2\alpha^{2}+\beta^{2} is equal to :
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Q2Single correctComplex Numbers and Quadratic Equations
Let one root of the quadratic equation in x : (k215k+27)x2+9(k1)x+18=0\left(k^{2}-15k+27\right)x^{2}+9(k-1)x+18=0 be twice the other. Then the length of the latus rectum of the parabola y2=6kxy^{2}=6kx is equal to:
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Q3Single correctCo-ordinate Geometry
Let e1e_{1} and e2e_{2} be two distinct roots of the equation x2ax+2=0x^{2}-ax+2=0. Let the sets {aR:e1 and e2 are the eccentricities of hyperbolas}=(α,β)\{a\in\mathbb{R}:e_{1}\ \text{and}\ e_{2}\ \text{are the eccentricities of hyperbolas}\}=(\alpha,\beta), and {aR:e1 and e2 are the eccentricities of an ellipse and a hyperbola, respectively}=(γ,)\{a\in\mathbb{R}:e_{1}\ \text{and}\ e_{2}\ \text{are the eccentricities of an ellipse and a hyperbola, respectively}\}=(\gamma,\infty). Then α2+β2+γ2\alpha^{2}+\beta^{2}+\gamma^{2} is equal to:
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Q4Single correctComplex Numbers and Quadratic Equations
Let the set of all values of kRk\in\mathbb{R} such that the equation z(zˉ+2+i)+k(2+3i)=0, zCz(\bar{z}+2+i)+k(2+3i)=0,\ z\in\mathbb{C}, has at least one solution, be the interval [α,β][\alpha,\beta]. Then 9(α+β)9(\alpha+\beta) is equal to :
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Q5Single correctSequence and Series
The value of 1323+33+1531^{3}-2^{3}+3^{3}-\ldots+15^{3} is:
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Q6Single correctSequence and Series
The sum of the first ten terms of an A.P. is 160160 and the sum of the first two terms of a G.P. is 88. 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:
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Q7Single correctPermutations and Combinations
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:
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Q8Single correctBinomial Theorem and its Simple Applications
If the coefficients of the middle terms in the binomial expansions of (1+αx)26(1+\alpha x)^{26} and (1αx)28, α0(1-\alpha x)^{28},\ \alpha\neq 0, are equal, then the value of α\alpha is:
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Q9Single correctStatistics and Probability
A data consists of 2020 observations x1,x2,,x20x_{1},x_{2},\ldots,x_{20}. If i=120(xi+5)2=2500\displaystyle\sum_{i=1}^{20}(x_{i}+5)^{2}=2500 and i=120(xi5)2=100\displaystyle\sum_{i=1}^{20}(x_{i}-5)^{2}=100, then the ratio of mean to standard deviation of this data is:
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Q10Single correctStatistics and Probability
A bag contains (N+1)(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 916\dfrac{9}{16}, then N is equal to:
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(D)
Q11Single correctCo-ordinate Geometry
If the eccentricity e of the hyperbola x2a2y2b2=1\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1, passing through (6,43)(6,\,4\sqrt{3}), satisfies 15(e2+1)=34e15(e^{2}+1)=34e, then the length of the latus rectum of the hyperbola x2b2y22(a2+1)=1\dfrac{x^{2}}{b^{2}}-\dfrac{y^{2}}{2(a^{2}+1)}=1 is:
(A)
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(D)
Q12Single correctCo-ordinate Geometry
Let chord PQ of length 3133\sqrt{13} of the parabola y2=12xy^{2}=12x be such that the ordinates of points P and Q are in the ratio 1:21:2. If the chord PQ subtends an angle α\alpha at the focus of the parabola, then sinα\sin\alpha is equal to:
(A)
(B)
(C)
(D)
Q13Single correctTrigonometry
Let 0<α<1, β=13α0<\alpha<1,\ \beta=\dfrac{1}{3\alpha} and tan1(1α)+tan1(1β)=π4\tan^{-1}(1-\alpha)+\tan^{-1}(1-\beta)=\dfrac{\pi}{4}. Then 6(α+β)6(\alpha+\beta) is equal to:
(A)
(B)
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(D)
Q14Single correctTrigonometry
Let S={θ(2π,2π):cosθ+1=3sinθ}S = \{\theta \in (-2\pi, 2\pi) : \cos\theta + 1 = \sqrt{3}\sin\theta\}. Then θSθ\displaystyle\sum_{\theta \in S}\theta is equal to:
(A)
(B)
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(D)
Q15Single correctThree Dimensional Geometry
Let the image of the point P(1,6,a)P(1, 6, a) in the line L:x1=y12=za+1b,b>0L : \dfrac{x}{1} = \dfrac{y-1}{2} = \dfrac{z-a+1}{b}, b > 0, be (a3,0,a+c)\left(\dfrac{a}{3}, 0, a+c\right). If S(α,β,γ),α>0S(\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 2142\sqrt{14}, then α+β+γ\alpha + \beta + \gamma is equal to:
(A)
(B)
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(D)
Q16Single correctThree Dimensional Geometry
Let a line L be perpendicular to both the lines L1:x+13=y+35=z+57L_1 : \dfrac{x+1}{3} = \dfrac{y+3}{5} = \dfrac{z+5}{7} and L2:x21=y44=z67L_2 : \dfrac{x-2}{1} = \dfrac{y-4}{4} = \dfrac{z-6}{7}. If θ\theta is the acute angle between the lines L and L3:x872=y471=z2L_3 : \dfrac{x-\tfrac{8}{7}}{2} = \dfrac{y-\tfrac{4}{7}}{1} = \dfrac{z}{2}, then tanθ\tan\theta is equal to:
(A)
(B)
(C)
(D)
Q17Single correctLimit, Continuity and Differentiability
The value of limx0(x2sin2xx2sin2x)\displaystyle\lim_{x\to 0}\left(\dfrac{x^2 \sin^2 x}{x^2 - \sin^2 x}\right) is:
(A)
(B)
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(D)
Q18Single correctIntegral Calculus
The value of the integral π/4π/4(32cos4x1+esinx)dx\displaystyle\int_{-\pi/4}^{\pi/4}\left(\dfrac{32\cos^4 x}{1 + e^{\sin x}}\right)dx is:
(A)
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(D)
Q19Single correctIntegral Calculus
The area of the region {(x,y):0y6x,  y24x3,  x0}\{(x, y) : 0 \le y \le 6 - x,\; y^2 \ge 4x - 3,\; x \ge 0\} is:
(A)
(B)
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(D)
Q20Single correctIntegral Calculus
Let e be the base of natural logarithm and let f:{1,2,3,4}{1,e,e2,e3}f : \{1, 2, 3, 4\} \to \{1, e, e^2, e^3\} and g:{1,e,e2,e3}{1,12,13,14}g : \{1, e, e^2, e^3\} \to \left\{1, \dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}\right\} be two bijective functions such that f is strictly decreasing and g is strictly increasing. If ϕ(x)=[f1{g1(12)}]x\phi(x) = \left[f^{-1}\left\{g^{-1}\left(\dfrac{1}{2}\right)\right\}\right]^{x}, then the area of the region R={(x,y):x2yϕ(x),  0x1}R = \{(x, y) : x^2 \le y \le \phi(x),\; 0 \le x \le 1\} is:
(A)
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Q21NumericalMatrices and Determinants
Let A=[111101001]A = \begin{bmatrix} -1 & 1 & -1 \\ 1 & 0 & 1 \\ 0 & 0 & 1 \end{bmatrix} satisfy A2+αadj(adj(A))+βadj(A)adj(adj(A))=[222201001]A^2 + \alpha\,\mathrm{adj}(\mathrm{adj}(A)) + \beta\,\mathrm{adj}(A)\cdot\mathrm{adj}(\mathrm{adj}(A)) = \begin{bmatrix} 2 & -2 & 2 \\ -2 & 0 & -1 \\ 0 & 0 & -1 \end{bmatrix} for some α,βR\alpha, \beta \in \mathbb{R}. Then (αβ)2(\alpha - \beta)^2 is equal to ____.
Q22NumericalCo-ordinate Geometry
Let the centre of the circle x2+y2+2gx+2fy+25=0x^2 + y^2 + 2gx + 2fy + 25 = 0 be in the first quadrant and lie on the line 2xy=42x - y = 4. Let the area of an equilateral triangle inscribed in the circle be 27327\sqrt{3}. Then the square of the length of the chord of the circle on the line x=1x = 1 is ____ .
Q23NumericalVector Algebra
If a=i^+j^+k^,  b=j^k^\vec{a} = \hat{i} + \hat{j} + \hat{k},\; \vec{b} = \hat{j} - \hat{k} and c\vec{c} be three vectors such that a×c=b\vec{a} \times \vec{c} = \vec{b} and ac=3\vec{a} \cdot \vec{c} = 3, then c(a2b)\vec{c} \cdot (\vec{a} - 2\vec{b}) is equal to ____ .
Q24NumericalSequence and Series
For the functions f(θ)=αtan2θ+βcot2θf(\theta) = \alpha\tan^2\theta + \beta\cot^2\theta and g(θ)=αsin2θ+βcos2θ,  α>β>0g(\theta) = \alpha\sin^2\theta + \beta\cos^2\theta,\; \alpha > \beta > 0, let min0<θ<π/2f(θ)=max0<θ<πg(θ)\min_{0 < \theta < \pi/2} f(\theta) = \max_{0 < \theta < \pi} g(\theta). If the first term of a G.P. is (α2β)\left(\dfrac{\alpha}{2\beta}\right), its common ratio is (2βα)\left(\dfrac{2\beta}{\alpha}\right) and the sum of its first 1010 terms is mn,  gcd(m,n)=1\dfrac{m}{n},\; \gcd(m, n) = 1, then m+nm + n is equal to ____ .
Q25NumericalDifferential Equations
Let y=y(x)y = y(x) be the solution of the differential equation (x2xx21)dy+(y(xx21)x)dx=0,  x1\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)=1y(1) = 1, then the greatest integer less than y(5)y(\sqrt{5}) is ____ .

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