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JEE Main 2025 January 23, Shift 2 Question Paper with Solutions

All 75 questions from the JEE Main 2025 (January 23, Shift 2) shift — Physics (25), Chemistry (25) and Mathematics (25) — with the correct answer and a step-by-step solution for every question.

Physics PYQs 2025Chemistry PYQs 2025Mathematics PYQs 2025

Physics25 questions

Q26Single correctMagnetic Effects of Current and Magnetism
A galvanometer having a coil of resistance 30Ω30\Omega need 20 mA of current for full-scale deflection. If a maximum current of 3 A is to be measured using this galvanometer, the resistance of the shunt to be added to the galvanometer should be 30XΩ\frac{30}{X}\Omega, where X is
(A)
(B)
(C)
(D)
Q27Single correctKinematics
A ball having kinetic energy KE, is projected at an angle of 60∘60^\circ from the horizontal. What will be the kinetic energy of ball at the highest point of its flight?
(A)
(B)
(C)
(D)
Q28Single correctElectrostatics
Two charges 7μC7\mu\text{C} and −4μC-4\mu\text{C} are placed at (−7 cm,0,0)(-7\text{ cm}, 0, 0) and (7 cm,0,0)(7\text{ cm}, 0, 0) respectively. Given, ϵ0=8.85×10−12C2 N−1 m−2\epsilon_0 = 8.85 \times 10^{-12}\text{C}^2\text{ N}^{-1}\text{ m}^{-2}, the electrostatic potential energy of the charge configuration is:
(A)
(B)
(C)
(D)
Q29Single correctElectrostatics
Two point charges −4μC-4\mu\text{C} and 4μC4\mu\text{C}, constituting an electric dipole, are placed at (−9,0,0) cm(-9, 0, 0)\text{ cm} and (9,0,0) cm(9, 0, 0)\text{ cm} in a uniform electric field of strength 104 NC−110^4 \text{ NC}^{-1}. The work done on the dipole in rotating it from the equilibrium through 180∘180^\circ is:
(A)
(B)
(C)
(D)
Q30Single correctProperties of Solids and Liquids
A massless spring gets elongated by amount x1x_1 under a tension of 5 N. Its elongation is x2x_2 under the tension of 7 N. For the elongation of (5x1−2x2)(5x_1 - 2x_2), the tension required is:
(A)
(B)
(C)
(D)
Q31Single correctThermodynamics
Water of mass m gram is slowly heated to increase the temperature from T1T_1 to T2T_2. The change in entropy of the water, given specific heat of water is 1 Jkg−1K−11\text{ Jkg}^{-1}\text{K}^{-1}, is:
(A)
(B)
(C)
(D)
Q32Single correctProperties of Solids and Liquids
Water flows in a horizontal pipe whose one end is closed with a valve. The reading of the pressure gauge attached to the pipe is P1P_1. The reading of the pressure gauge falls to P2P_2 when the valve is opened. The speed of water flowing in the pipe is proportional to
(A)
(B)
(C)
(D)
Q33Single correctOptics
A concave mirror of focal length f in air is dipped in a liquid of refractive index μ\mu. Its focal length in the liquid will be:
(A)
(B)
(C)
(D)
Q34Single correctCurrent Electricity
What is the current through the battery in the circuit shown below
Circuit diagram showing a 5V battery connected to two 20Ω resistors in parallel
(A)
(B)
(C)
(D)
Q35Single correctOptics
The refractive index of the material of a glass prism is 3\sqrt{3}. The angle of minimum deviation is equal to the angle of the prism. What is the angle of the prism?
(A)
(B)
(C)
(D)
Q36Single correctOptics
The width of one of the two slits in Young's double slit experiment is d while that of the other slit is xd. If the ratio of the maximum to the minimum intensity in the interference pattern on the screen is 9:4 then what is the value of x? (Assume that the field strength varies according to the slit width.)
(A)
(B)
(C)
(D)
Q37Single correctThermodynamics
Using the given P-V diagram, the work done by an ideal gas along the path ABCD is:
P-V diagram showing cyclic path ABCD
(A)
(B)
(C)
(D)
Q38Single correctElectromagnetic Waves
A plane electromagnetic wave of frequency 20 MHz travels in free space along the +x direction. At a particular point in space and time, the electric field vector of the wave is Ey=9.3 Vm−1E_y = 9.3\text{ Vm}^{-1}. The magnetic field vector of the wave is:
(A)
(B)
(C)
(D)
Q39Single correctOscillations and Waves
The equation of a transverse wave travelling along a string is y(x,t)=4.0sin⁡[20×10−3x+600t] mmy(x,t) = 4.0\sin[20 \times 10^{-3}x + 600t]\text{ mm}, where x is in mm and t is in second. The velocity of the wave is:
(A)
(B)
(C)
(D)
Q40Single correctAtoms and Nuclei
Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A): The binding energy per nucleon is found to be practically independent of the atomic number A, for nuclei with mass numbers between 30 and 170. Reason (R): Nuclear force is long range. In the light of the above statements, choose the correct answer from the options given below:
(A)
(B)
(C)
(D)
Q41Single correctGravitation
If a satellite orbiting the Earth is 9 times closer to the Earth than the Moon, what is the time period of rotation of the satellite? Given rotational time period of Moon = 27 days and gravitational attraction between the satellite and the moon is neglected.
(A)
(B)
(C)
(D)
Q42Single correctRotational Motion
A circular disk of radius R meter and mass M kg is rotating around the axis perpendicular to the disk. An external torque is applied to the disk such that θ(t)=5t2−8t\theta(t) = 5t^2 - 8t, where θ(t)\theta(t) is the angular position of the rotating disc as a function of time t. How much power is delivered by the applied torque, when t=2 st = 2\text{ s}?
(A)
(B)
(C)
(D)
Q43Single correctUnits and Measurements
The energy of a system is given as E(t)=α3e−βtE(t) = \alpha^3 e^{-\beta t}, where t is the time and β=0.3 s−1\beta = 0.3\text{ s}^{-1}. The errors in the measurement of α\alpha and t are 1.21.2% and 1.6%, respectively. At t=5 st = 5\text{ s}, maximum percentage error in the energy is:
(A)
(B)
(C)
(D)
Q44Single correctUnits and Measurements
Match List - I with List - II.
List - IList - II
A. Permeability of free spaceI. [ML2T−2][M L^2 T^{-2}]
B. Magnetic fieldII. [MT−2A−1][M T^{-2} A^{-1}]
C. Magnetic momentIII. [MLT−2A−2][M L T^{-2} A^{-2}]
D. Torsional constantIV. [L2A][L^2 A]
(A)
(B)
(C)
(D)
Q45Single correctDual Nature of Matter and Radiation
In photoelectric effect an em-wave is incident on a metal surface and electrons are ejected from the surface. If the work function of the metal is 2.14 eV2.14 \, \text{eV} and stopping potential is 2 V2 \, \text{V}, what is the wavelength of the em-wave? (Given hc=1242 eV⋅nmhc = 1242 \, \text{eV} \cdot \text{nm} where h is the Planck's constant and c is the speed of light in vacuum.)
(A)
(B)
(C)
(D)
Q46NumericalElectromagnetic Induction and Alternating Currents
A time varying potential difference is applied between the plates of a parallel plate capacitor of capacitance 2.5μF2.5\mu\text{F}. The dielectric constant of the medium is 5. The current is:
Q47NumericalElectromagnetic Induction and Alternating Currents
In a series LCR circuit, a resistor of 300 Ω300\,\Omega, a capacitor of 25 nF25\,\text{nF} and an inductor of 100 mH100\,\text{mH} are used. The angular frequency at resonance is:
Q48NumericalProperties of Solids and Liquids
An air bubble of radius 1.0 mm1.0\,\text{mm} is observed at a depth of 20 cm20\,\text{cm} below the free surface of a liquid having surface tension 0.075 Nm−10.075\,\text{Nm}^{-1} and density 103 kgm−310^3\,\text{kgm}^{-3}. The difference in pressure is:
Q49NumericalGravitation
A satellite of mass M2\frac{M}{2} is revolving around earth in a circular orbit at a height of R3\frac{R}{3} from earth surface. The angular momentum of the satellite is MGMRxM\sqrt{\frac{GMR}{x}}. The value of x is ______, where M and R are the mass and radius of earth, respectively. (G is the gravitational constant)
Q50NumericalCurrent Electricity
At steady state the charge on the capacitor, as shown in the circuit below, is _____ μC\mu\text{C}.

Chemistry25 questions

Q51Single correctOrganic Compounds Containing Oxygen
Identify the products [A] and [B], respectively in the following reaction:
Chlorobenzene reacting with (i) NaOH, 623 K, 300 atm then Na₂Cr₂O₇/H₂SO₄ to form [A], then (ii) H⁺ to form [B]
(A)
(B)
(C)
(D)
Q52Single correctd- and f-Block Elements
Consider the following reactions: K2Cr2O7→KOH,−H2O[A]→H2SO4,−H2OCrO3\text{K}_2\text{Cr}_2\text{O}_7 \xrightarrow{\text{KOH}, -\text{H}_2\text{O}} [\text{A}] \xrightarrow{\text{H}_2\text{SO}_4, -\text{H}_2\text{O}} \text{CrO}_3. The sum of spin only magnetic moments of the compounds A and B is:
(A)
(B)
(C)
(D)
Q53Single correctChemical Thermodynamics
The effect of temperature on spontaneity of reactions are represented as:
Table showing ΔH, ΔS, Temperature, and Spontaneity columns with four rows (A), (B), (C), (D)
(A)
(B)
(C)
(D)
Q54Single correctChemical Kinetics
Which of the following graphs most appropriately represents a zero order reaction?
(A)
(B)
(C)
(D)
Q55Single correctEquilibrium
Consider the reaction X2Y(g)⇌X2(g)+12Y2(g)\text{X}_2\text{Y}(\text{g}) \rightleftharpoons \text{X}_2(\text{g}) + \frac{1}{2}\text{Y}_2(\text{g}). The equation representing correct relationship between the degree of dissociation (x) of X2Y\text{X}_2\text{Y} and equilibrium constant (KpK_p) is:
(A)
(B)
(C)
(D)
Q56Single correctOrganic Compounds Containing Oxygen
Given below are two statements: Consider the following reaction for hydration of aldehydes and ketones.
Carbonyl compound (R-CO-R) reacting with water in equilibrium to form geminal diol (R-C(OH)₂-R) with equilibrium constant K
(A)
(B)
(C)
(D)
Q57Single correctAtomic Structure
Given below are two statements: Statement (I): For a given shell, the total number of allowed orbitals is given by n². Statement (II): For any subshell, the spatial orientation of the orbitals is given by -l to +l values including zero. In the light of the above statements, choose the correct answer from the options given below:
(A)
(B)
(C)
(D)
Q58Single correctRedox Reactions and Electrochemistry
Standard electrode potentials for a few half cells are mentioned below: ECu2+/Cu∘=0.34 VE^\circ_{\text{Cu}^{2+}/\text{Cu}} = 0.34\text{ V}, EZn2+/Zn∘=−0.76 VE^\circ_{\text{Zn}^{2+}/\text{Zn}} = -0.76\text{ V}, EAg+/Ag∘=0.80 VE^\circ_{\text{Ag}^{+}/\text{Ag}} = 0.80\text{ V}. The galvanic cell that gives the highest voltage is:
(A)
(B)
(C)
(D)
Q59Single correctBiomolecules
The α\alpha-Helix and β\beta-Pleated sheet structures of protein are associated with its:
(A)
(B)
(C)
(D)
Q60Single correctp-Block Elements
Given below are the atomic numbers of some group 14 elements. The atomic number of the element with lowest melting point is:
(A)
(B)
(C)
(D)
Q61Single correctAtomic Structure
Given below are two statements about X-ray spectra of elements: Statement (I): A plot of v\sqrt{v} (v = frequency of X-rays emitted\text{frequency of X-rays emitted}) vs atomic mass is a straight line. Statement (II): A plot of v (v = frequency of X-rays emitted\text{frequency of X-rays emitted}) vs atomic number is a straight line. In the light of the above statements, choose the correct answer from the options given below:
(A)
(B)
(C)
(D)
Q62Single correctPrinciples Related to Practical Chemistry
Identify A, B and C in the given below reaction sequence: A→HNO3Pb(NO3)2→H2SO4B→ΔC\text{A} \xrightarrow{\text{HNO}_3} \text{Pb(NO}_3\text{)}_2 \xrightarrow{\text{H}_2\text{SO}_4} \text{B} \xrightarrow{\Delta} \text{C}.
(A)
(B)
(C)
(D)
Q63Single correctOrganic Compounds Containing Oxygen
Given below are two statements: Statement (I): The boiling points of alcohols and phenols increase with increase in the number of C-atoms. Statement (II): The boiling points of alcohols and phenols are higher in comparison to other class of compounds such as ethers, haloalkanes. In the light of the above statements, choose the correct answer from the options given below:
(A)
(B)
(C)
(D)
Q64Single correctSolutions
Consider a binary solution of two volatile liquid components 1 and 2. x1x_1 and y1y_1 are the mole fractions of component 1 in liquid and vapour phase, respectively. The slope and intercept of the graph are:
(A)
(B)
(C)
(D)
Q65Single correctOrganic Compounds Containing Halogens
The ascending order of relative rate of solvolysis of following compounds is:
Four organic bromide structures (A), (B), (C), (D) showing different positions and environments of Br atom
(A)
(B)
(C)
(D)
Q66Single correctd- and f-Block Elements
Match List - I with List - II.
(A)
(B)
(C)
(D)
Q67Single correctHydrocarbons
Match List - I with List - II.
Isomers of C₁₀H₁₄Ozonolysis product
(A)(I)
(B)(II)
(C)(III)
(D)(IV)
Cyclic aromatic isomers with methyl substituents and their ozonolysis products showing dicarbonyl compounds
(A)
(B)
(C)
(D)
Q68Single correctEquilibrium
pH of water is 7 at 25∘C25^\circ\text{C}. If water is heated to 80∘C80^\circ\text{C}, its pH will:
(A)
(B)
(C)
(D)
Q69Single correctCoordination Compounds
Identify the coordination complexes in which the central metal ion has d4d^4 configuration.
Coordination complexes with different metal centers and ligands
(A)
(B)
(C)
(D)
Q70Single correctSolutions
When a non-volatile solute is added to the solvent, the vapour pressure of the solvent decreases by 10 mm of Hg. The mole fraction of the solute in the solution is 0.2. What would be the mole fraction of the solvent if decrease in vapour pressure is 20 mm of Hg?\text{When a non-volatile solute is added to the solvent, the vapour pressure of the} \ \text{solvent decreases by 10 mm of Hg. The mole fraction of the solute in the solution} \ \text{is 0.2. What would be the mole fraction of the solvent if decrease in vapour} \ \text{pressure is 20 mm of Hg?}
(A)
(B)
(C)
(D)
Q71NumericalSome Basic Concepts in Chemistry
0.010.01 mole of an organic compound (X) containing 1010% hydrogen, on complete combustion produced 0.9 g H2O0.9\,\text{g}\,\text{H}_2\text{O}. Molar mass of X is:
Q72NumericalHydrocarbons
A compound 'X' absorbs 2 moles of hydrogen and 'X' upon oxidation with KMnO4∣H+\text{KMnO}_4|\text{H}^+ gives CH3-C-CH3\text{CH}_3\text{-C-CH}_3, CH3-C-OH\text{CH}_3\text{-C-OH}. The compound X is:
Ketone and carboxylic acid products from oxidation
Q73NumericalSome Basic Concepts in Chemistry
When 81.0 g81.0\,\text{g} of aluminium is allowed to react with 128.0 g128.0\,\text{g} of oxygen gas, the mass of aluminium oxide produced in grams is _____.
Q74NumericalChemical Thermodynamics
The bond dissociation enthalpy of X2X_2, ΔHbond\Delta H_{\text{bond}}, calculated from the given data is _____ kJ mol−1\text{kJ mol}^{-1}. (Nearest integer). M+X−(s)→M+(g)+X−(g)\text{M}^+\text{X}^-(s) \rightarrow \text{M}^+(g) + \text{X}^-(g), ΔHlattice∗=800 kJ mol−1\Delta H^*_{\text{lattice}} = 800\text{ kJ mol}^{-1}. M(s)→M(g)\text{M}(s) \rightarrow \text{M}(g), ΔHsub∘=100 kJ mol−1\Delta H^\circ_{\text{sub}} = 100\text{ kJ mol}^{-1}. M(g)→M+(g)+e−(g)\text{M}(g) \rightarrow \text{M}^+(g) + e^-(g), ΔHi=500 kJ mol−1\Delta H_i = 500\text{ kJ mol}^{-1}. X(g)+e−(g)→X−(g)\text{X}(g) + e^-(g) \rightarrow \text{X}^-(g), ΔHeg∗=−300 kJ mol−1\Delta H^*_{\text{eg}} = -300\text{ kJ mol}^{-1}. M(s)+12X2(g)→M+X−(s)\text{M}(s) + \frac{1}{2}\text{X}_2(g) \rightarrow \text{M}^+\text{X}^-(s), ΔHf∘=−400 kJ mol−1\Delta H^\circ_f = -400\text{ kJ mol}^{-1}. [Given: M+X−\text{M}^+\text{X}^- is a pure ionic compound and X forms a diatomic molecule X2\text{X}_2 in gaseous state]
Q75NumericalOrganic Compounds Containing Nitrogen
Consider the following sequence of reactions. Total number of sp3\text{sp}^3 hybridised carbon atoms in the major product C formed is _____.
p-ethoxyaniline converted to diazonium salt, then coupled with phenol, followed by Williamson ether synthesis with propyl bromide, forming C₁₆H₁₈N₂O₂

Mathematics25 questions

Q1Single correctThree Dimensional Geometry
The distance of the line x−22=y−63=z−34\frac{x-2}{2} = \frac{y-6}{3} = \frac{z-3}{4} from the point (1,4,0)(1, 4, 0) along the line x1=y−22=z+33\frac{x}{1} = \frac{y-2}{2} = \frac{z+3}{3} is:
(A)
(B)
(C)
(D)
Q2Single correctSets, Relations and Functions
Let A={(x,y)∈R×R:∣x+y∣≥3}A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + y| \geq 3\} and B={(x,y)∈R×R:∣x∣+∣y∣≤3}B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x| + |y| \leq 3\}. If C={(x,y)∈A∩B:x=0 or y=0}C = \{(x, y) \in A \cap B : x = 0 \text{ or } y = 0\}, then ∑(x,y)∈C∣x+y∣\sum_{(x,y)\in C}|x + y| is:
(A)
(B)
(C)
(D)
Q3Single correctSets, Relations and Functions
Let X=R×RX = \mathbb{R} \times \mathbb{R}. Define a relation R on X as: (a1,b1)R(a2,b2)⇔b1=b2(a_1, b_1)R(a_2, b_2) \Leftrightarrow b_1 = b_2. Statement I: R is an equivalence relation. Statement II: For some (a,b)∈X(a, b) \in X, the set S={(x,y)∈X:(x,y)R(a,b)}S = \{(x, y) \in X : (x, y)R(a, b)\} represents a line parallel to y=xy = x. In the light of the above statements, choose the correct answer from the options given below:
(A)
(B)
(C)
(D)
Q4Single correctIntegral Calculus
Let ∫x3sin⁡x dx=g(x)+C\int x^3 \sin x \, dx = g(x) + C, where C is the constant of integration. If 8(g(π2)+g′(π2))=απ3+βπ2+γ8\left(g\left(\frac{\pi}{2}\right) + g'\left(\frac{\pi}{2}\right)\right) = \alpha\pi^3 + \beta\pi^2 + \gamma, α,β,γ∈Z\alpha, \beta, \gamma \in \mathbb{Z}, then α+β−γ\alpha + \beta - \gamma equals:
(A)
(B)
(C)
(D)
Q5Single correctCo-ordinate Geometry
A rod of length eight units moves such that its ends A and B always lie on the lines x−y+2=0x - y + 2 = 0 and y+2=0y + 2 = 0, respectively. If the locus of the point P, that divides the rod AB internally in the ratio 2:12:1 is 9(x2+αy2+βxy+γx+28y)−76=09(x^2 + \alpha y^2 + \beta xy + \gamma x + 28y) - 76 = 0, then α−β−γ\alpha - \beta - \gamma is equal to:
(A)
(B)
(C)
(D)
Q6Single correctThree Dimensional Geometry
If the square of the shortest distance between the lines x−21=y−12=z+3−3\frac{x-2}{1} = \frac{y-1}{2} = \frac{z+3}{-3} and x+12=y+34=z+5−5\frac{x+1}{2} = \frac{y+3}{4} = \frac{z+5}{-5} is mn\frac{m}{n}, where m, n are coprime numbers, then m+nm + n is equal to:
(A)
(B)
(C)
(D)
Q7Single correctLimit, Continuity and Differentiability
lim⁡x→∞(2x2−3x+5)(3x−1)x2(3x2+5x+4)(3x+2)x\lim_{x\to\infty} \frac{(2x^2-3x+5)(3x-1)^{\frac{x}{2}}}{(3x^2+5x+4)\sqrt{(3x+2)^x}} is equal to:
(A)
(B)
(C)
(D)
Q8Single correctVector Algebra
Let the point A divide the line segment joining the points P(−1,−1,2)P(-1, -1, 2) and Q(5,5,10)Q(5, 5, 10) internally in the ratio r:1(r>0)r:1(r > 0). If O is the origin and (OQ⃗⋅OA⃗)−15∣OP⃗×OA⃗∣2=10(\vec{OQ} \cdot \vec{OA}) - \frac{1}{5}|\vec{OP} \times \vec{OA}|^2 = 10, then the value of r is:
(A)
(B)
(C)
(D)
Q9Single correctCo-ordinate Geometry
The length of the chord of the ellipse x24+y22=1\frac{x^2}{4} + \frac{y^2}{2} = 1, whose mid-point is (1,12)\left(1, \frac{1}{2}\right), is:
(A)
(B)
(C)
(D)
Q10Single correctMatrices and Determinants
The system of equations x+y+z=6,x+2y+5z=9,x+5y+λz=μx + y + z = 6 , x + 2y + 5z = 9 , x + 5y + \lambda z = \mu, has no solution if
(A)
(B)
(C)
(D)
Q11Single correctTrigonometry
Let the range of the function f(x)=6+16cos⁡x⋅cos⁡(π3−x)⋅cos⁡(π3+x)⋅sin⁡3x⋅cos⁡6x,x∈Rf(x) = 6 + 16\cos x \cdot \cos\left(\frac{\pi}{3} - x\right) \cdot \cos\left(\frac{\pi}{3} + x\right) \cdot \sin 3x \cdot \cos 6x , x \in \mathbb{R} be [α,β][\alpha, \beta]. Then the distance of the point (α,β)(\alpha, \beta) from the line 3x+4y+12=03x + 4y + 12 = 0 is:
(A)
(B)
(C)
(D)
Q12Single correctDifferential Equations
Let x=x(y)x = x(y) be the solution of the differential equation y=(x−ydxdy)sin⁡(xy),y>0y = \left(x - y\frac{dx}{dy}\right)\sin\left(\frac{x}{y}\right) , y > 0 and x(1)=π2x(1) = \frac{\pi}{2}. Then cos⁡(x(2))\cos(x(2)) is equal to:
(A)
(B)
(C)
(D)
Q13Single correctLimit, Continuity and Differentiability
A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at the rate of 81 cm³/min and the thickness of the ice-cream layer decreases at the rate of 14π\frac{1}{4\pi} cm/min. The surface area (in cm²) of the chocolate ball (without the ice-cream layer) is:
(A)
(B)
(C)
(D)
Q14Single correctComplex Numbers and Quadratic Equations
The number of complex numbers z, satisfying ∣z∣=1|z| = 1 and ∣zzˉ+zˉz∣=1\left|\frac{z}{\bar{z}} + \frac{\bar{z}}{z}\right| = 1, is:
(A)
(B)
(C)
(D)
Q15Single correctMatrices and Determinants
Let A=[aij]A = [a_{ij}] be 3×33 \times 3 matrix such that A[010]=[001]A\begin{bmatrix} 0 1 0 \end{bmatrix} = \begin{bmatrix} 0 0 1 \end{bmatrix}, A[413]=[110]A\begin{bmatrix} 4 1 3 \end{bmatrix} = \begin{bmatrix} 1 1 0 \end{bmatrix} and A[212]=[100]A\begin{bmatrix} 2 1 2 \end{bmatrix} = \begin{bmatrix} 1 0 0 \end{bmatrix}, then a23a_{23} equals:
(A)
(B)
(C)
(D)
Q16Single correctIntegral Calculus
If I=∫0π2sin⁡32xsin⁡32x+cos⁡32xdxI = \int_0^{\frac{\pi}{2}} \frac{\sin^{\frac{3}{2}}x}{\sin^{\frac{3}{2}}x + \cos^{\frac{3}{2}}x} dx, then ∫021xsin⁡xcos⁡xsin⁡4x+cos⁡4xdx\int_0^{21} \frac{x\sin x\cos x}{\sin^4 x + \cos^4 x} dx equals:
(A)
(B)
(C)
(D)
Q17Single correctStatistics and Probability
A board has 16 squares as shown in the figure: Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is :
4x4 grid of 16 squares
(A)
(B)
(C)
(D)
Q18Single correctCo-ordinate Geometry
Let the shortest distance from (a, 0), a > 0, to the parabola y2y^2 = 4x be 4. Then the equation of the circle passing through the point (a, 0) and the focus of the parabola, and having its centre on the axis of the parabola is :
(A)
(B)
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(D)
Q19Single correctBinomial Theorem and its Simple Applications
If in the expansion of (1+x)p(1−x)q(1 + x)^p(1 - x)^q, the coefficients of x and x2x^2 are 1 and -2, respectively, then p2+q2p^2 + q^2 is equal to :
(A)
(B)
(C)
(D)
Q20Single correctIntegral Calculus
If the area of the region {(x,y):−1≤x≤1,0≤y≤a+e∣x∣−e−x,a>0}\{(x, y) : -1 \leq x \leq 1, 0 \leq y \leq a + e^{|x|} - e^{-x}, a > 0\} is e2+8e+1e\frac{e^2 + 8e + 1}{e}, then the value of a is:
(A)
(B)
(C)
(D)
Q21NumericalStatistics and Probability
The variance of the numbers 8,21,34,47,…,320is8, 21, 34, 47, \ldots, 320 is
Q22NumericalSequence and Series
The roots of the quadratic equation 3x23x^2 - px + q = 0 are 10th10^{th} and 11th11^{th} terms of an arithmetic progression with common difference 32\frac{3}{2}. If the sum of the first 11 terms of this arithmetic progression is 88, then q - 2p is equal to
Q23NumericalPermutations and Combinations
The number of ways 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is
Q24NumericalCo-ordinate Geometry
The focus of the parabola y2=4x+16y^2 = 4x + 16 is the centre of the circle C of radius 5. If the values of λ\lambda, for which C passes through the point of intersection of the lines 3x−y=03x - y = 0 and x+λy=4x + \lambda y = 4, are λ1\lambda_1 and λ2\lambda_2, λ1<λ2\lambda_1 < \lambda_2, then 12λ1+29λ212\lambda_1 + 29\lambda_2 is equal to
Q25NumericalComplex Numbers and Quadratic Equations
Let α,β\alpha, \beta be the roots of the equation x2−ax−b=0x^2 - ax - b = 0 with Im(α)<Im(β)\text{Im}(\alpha) < \text{Im}(\beta). Let Pn=αn−βnP_n = \alpha^n - \beta^n. If P3=−57iP_3 = -5\sqrt{7}i, P4=−37iP_4 = -3\sqrt{7}i, P5=117iP_5 = 11\sqrt{7}i and P6=457iP_6 = 45\sqrt{7}i, then ∣α4+β4∣|\alpha^4 + \beta^4| is equal to

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