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JEE Main 2025 January 22, Shift 2 Question Paper with Solutions

All 75 questions from the JEE Main 2025 (January 22, 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 correctElectronic Devices
To obtain the given truth table, following logic gate should be placed at G:
Logic gate circuit with NAND gates, inputs A and B, and gate G at output position to be identified
(A)
(B)
(C)
(D)
Q27Single correctProperties of Solids and Liquids
A small rigid spherical ball of mass MM is dropped in a long vertical tube containing glycerine. The velocity of the ball becomes constant after some time. If the density of glycerine is half of the density of the ball, then the viscous force acting on the ball will be (consider gg as acceleration due to gravity)
(A)
(B)
(C)
(D)
Q28Single correctVector Algebra
The torque due to the force (2i^+j^+2k^)(2\hat{i} + \hat{j} + 2\hat{k}) about the origin, acting on a particle whose position vector is (i^+j^+k^)(\hat{i} + \hat{j} + \hat{k}), would be
(A)
(B)
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(D)
Q29Single correctOptics
A symmetric thin biconvex lens is cut into four equal parts by two planes ABAB and CDCD as shown in figure. If the power of original lens is 4 D then the power of a part of the divided lens is
Symmetric biconvex lens with two cutting planes AB (perpendicular to axis) and CD (along optical axis) dividing lens into four equal parts
(A)
(B)
(C)
(D)
Q30Single correctElectrostatics
For a short dipole placed at origin O, the dipole moment P is along x-axis, as shown in the figure. If the electric potential and electric field at A are V0V_0 and E0E_0, respectively, then the correct combination of the electric potential and electric field, respectively, at point B on the y-axis is given by
Coordinate system with electric dipole at origin O, dipole moment P along x-axis, point A on x-axis at distance r, point B on y-axis at distance 2r
(A)
(B)
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Q31Single correctOptics
A transparent film of refractive index, 2.0 is coated on a glass slab of refractive index, 1.45. What is the minimum thickness of transparent film to be coated for the maximum transmission of Green light of wavelength 550 nm. [Assume that the light is incident nearly perpendicular to the glass surface.]
(A)
(B)
(C)
(D)
Q32Single correctThermodynamics
Given are statements for certain thermodynamic variables, (A) Internal energy, volume (V) and mass (M) are extensive variables. (B) Pressure (P), temperature (T) and density (ρ\rho) are intensive variables. (C) Volume (V), temperature (T) and density (ρ\rho) are intensive variables. (D) Mass (M), temperature (T) and internal energy are extensive variables. Choose the correct answer from the options given below:
(A)
(B)
(C)
(D)
Q33Single correctAtoms and Nuclei
An electron projected perpendicular to a uniform magnetic field B moves in a circle. If Bohr's quantization is applicable, then the radius of the electronic orbit in the first excited state is:
(A)
(B)
(C)
(D)
Q34Single correctOptics
Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A): In Young's double slit experiment, the fringes produced by red light are closer as compared to those produced by blue light. Reason (R): The fringe width is directly proportional to the wavelength of light. In the light of the above statements, choose the correct answer from the options given below:
(A)
(B)
(C)
(D)
Q35Single correctElectromagnetic Induction and Alternating Currents
A rectangular metallic loop is moving out of a uniform magnetic field region to a field free region with a constant speed. When the loop is partially inside the magnetic field, the plot of magnitude of induced emf (ε)(\varepsilon) with time (t) is given by
(A)
(B)
(C)
(D)
Q36Single correctWork, Energy and Power
A ball of mass 100 g is projected with velocity 20 m/s20\text{ m/s} at 60∘60^\circ with horizontal. The decrease in kinetic energy of the ball during the motion from point of projection to highest point is
(A)
(B)
(C)
(D)
Q37Single correctWork, Energy and Power
A body of mass 100 g is moving in circular path of radius 2 m on vertical plane as shown in figure. The velocity of the body at point A is 10 m/s10\text{ m/s}. The ratio of its kinetic energies at point B and C is: (Take acceleration due to gravity as 10 m/s210\text{ m/s}^2)
Circular path in vertical plane with center O. Point A at bottom, point B at 30° from vertical (bottom right), point C at 90° from point A (horizontal right from center). Radius is 2 m.
(A)
(B)
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Q38Single correctGravitation
Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A): A simple pendulum is taken to a planet of mass and radius, 4 times and 2 times, respectively, than the Earth. The time period of the pendulum remains same on earth and the planet. Reason (R): The mass of the pendulum remains unchanged at Earth and the other planet. In the light of the above statements, choose the correct answer from the options given below:
(A)
(B)
(C)
(D)
Q39Single correctElectromagnetic Induction and Alternating Currents
A series LCR circuit is connected to an alternating source of emf E. The current amplitude at resonant frequency is I0I_0. If the value of resistance R becomes twice of its initial value then amplitude of current at resonance will be
(A)
(B)
(C)
(D)
Q40Single correctElectrostatics
Which one of the following is the correct dimensional formula for the capacitance in F? M, L, T and C stand for unit of mass, length, time and charge,
(A)
(B)
(C)
(D)
Q41Single correctProperties of Solids and Liquids
A tube of length L is shown in the figure. The radius of cross section at the point (1) is 2 cm and at the point (2) is 1 cm, respectively. If the velocity of water entering at point (1) is 2 m/s2\text{ m/s}, then velocity of water leaving the point (2) will be
A horizontal tube with narrowing cross-section. Point (1) has radius 2 cm, point (2) has radius 1 cm. Water flows from left to right through the tube.
(A)
(B)
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(D)
Q42Single correctDual Nature of Matter and Radiation
A light source of wavelength λ\lambda illuminates a metal surface and electrons are ejected with maximum kinetic energy of 2 eV. If the same surface is illuminated by a light source of wavelength λ2\frac{\lambda}{2}, then the maximum kinetic energy of ejected electrons will be (The work function of metal is 1 eV)
(A)
(B)
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(D)
Q43Single correctUnits and Measurements
The maximum percentage error in the measurement of density of a wire is [Given, mass of wire =(0.60±0.003)g= (0.60 \pm 0.003)\text{g}, radius of wire =(0.50±0.01)cm= (0.50 \pm 0.01)\text{cm}, length of wire =(10.00±0.05)cm= (10.00 \pm 0.05)\text{cm}]
(A)
(B)
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Q44Single correctKinetic Theory of Gases
For a diatomic gas, if γ1=(CpCv)\gamma_1 = \left(\frac{C_p}{C_v}\right) for rigid molecules and γ2=(CpCv)\gamma_2 = \left(\frac{C_p}{C_v}\right) for another diatomic molecules, but also having vibrational modes. Then, which one of the following options is correct? (CpC_p and CvC_v are specific heats of the gas at constant pressure and volume)
(A)
(B)
(C)
(D)
Q45Single correctWork, Energy and Power
A force F⃗=2i^+bj^+k^\vec{F} = 2\hat{i} + b\hat{j} + \hat{k} is applied on a particle and it undergoes a displacement i^−2j^−k^\hat{i} - 2\hat{j} - \hat{k}. What will be the value of b, if work done on the particle is zero.
(A)
(B)
(C)
(D)
Q46NumericalMagnetic Effects of Current and Magnetism
A proton is moving undeflected in a region of crossed electric and magnetic fields at a constant speed of 2×1052 \times 10^5 ms−1s^{-1}. When the electric field is switched off, the proton moves along a circular path of radius 2 cm. The magnitude of electric field is x×104x \times 10^4 N/C. The value of x is _______ Take the mass of the proton =1.6×10−27= 1.6 \times 10^{-27} kg.
Q47NumericalCurrent Electricity
The net current flowing in the given circuit is _______ A.
Circuit with 2V battery, resistors 2Ω, 4Ω, 2.5Ω, 6Ω, 3Ω, 5Ω, 8Ω, 4Ω, 1Ω and capacitor 1μF arranged in complex network
Q48NumericalElectromagnetic Induction and Alternating Currents
A parallel plate capacitor of area A=16A = 16 cm2m^2 and separation between the plates 10 cm, is charged by a DC current. Consider a hypothetical plane surface of area A0=3.2A_0 = 3.2 cm2m^2 inside the capacitor and parallel to the plates. At an instant, the current through the circuit is 6A. At the same instant the displacement current through A0A_0 is ________ mA.
Q49NumericalRotational Motion
A tube of length 1 m is filled completely with an ideal liquid of mass 2 M, and closed at both ends. The tube is rotated uniformly in horizontal plane about one of its ends. If the force exerted by the liquid at the other end is F then angular velocity of the tube is FαM\sqrt{\frac{F}{\alpha M}} in SI unit. The value of α\alpha is __________.
Q50NumericalMagnetic Effects of Current and Magnetism
Two long parallel wires X and Y, separated by a distance of 6 cm, carry currents of 5A and 4A, respectively, in opposite directions as shown in the figure. Magnitude of the resultant magnetic field at point P at a distance of 4 cm from wire Y is x×10−5x \times 10^{-5} T. The value of x is __________. Take permeability of free space as μ0=4π×10−7\mu_0 = 4\pi \times 10^{-7} SI units.
Two parallel vertical wires X and Y separated by 6 cm. Wire X carries 5A upward, wire Y carries 4A downward. Point P is 4 cm to the right of wire Y.

Chemistry25 questions

Q51Single correctPurification and Characterisation of Organic Compounds
Given below are two statements : Statement (I) : Nitrogen, sulphur, halogen and phosphorus present in an organic compound are detected by Lassaigne's Test. Statement (II) : The elements present in the compound are converted from covalent form into ionic form by fusing the compound with Magnesium in Lassaigne's test. In the light of the above statements, choose the correct answer from the options given below :
(A)
(B)
(C)
(D)
Q52Single correctSolutions
Density of 3 M NaCl solution is 1.25 g/mL. The molality of the solution is :
(A)
(B)
(C)
(D)
Q53Single correctCoordination Compounds
The correct order of the following complexes in terms of their crystal field stabilization energies is:
(A)
(B)
(C)
(D)
Q54Single correctRedox Reactions and Electrochemistry
Given below are two statements: Statement (I): Corrosion is an electrochemical phenomenon in which pure metal acts as an anode and impure metal as a cathode. Statement (II): The rate of corrosion is more in alkaline medium than in acidic medium. In the light of the above statements, choose the correct answer from the options given below:
(A)
(B)
(C)
(D)
Q55Single correctChemical Kinetics
Consider the given figure and choose the correct option:
Energy diagram showing reactant, activated complex at peak with E1 marked, product at lower energy than reactant with E2 marked from product to reactant level, and reaction coordinate on x-axis
(A)
(B)
(C)
(D)
Q56Single correctHydrocarbons
The maximum number of RBr producing 2-methylbutane by above sequence of reactions is ______ (Consider the structural isomers only)
Reaction scheme showing RBr reacting with (i) Mg, dry ether then (ii) H2O to produce 2-Methylbutane structure
(A)
(B)
(C)
(D)
Q57Single correctp-Block Elements
The species which does not undergo disproportionation reaction is:
(A)
(B)
(C)
(D)
Q58Single correctEquilibrium
The molar solubility(s) of zirconium phosphate with molecular formula (Zr4+)3(PO43−)4(Zr^{4+})_3(PO_4^{3-})_4 is given by relation:
(A)
(B)
(C)
(D)
Q59Single correctCoordination Compounds
Identify the homoleptic complex(es) that is/are low spin. (A) [Fe(CN)5NO]2−[Fe(CN)_5NO]^{2-} (B) [CoF6]3−[\text{CoF}_6]^{3-} (C) [Fe(CN)6]4−[Fe(CN)_6]^{4-} (D) [Co(NH3)6]3+[Co(NH_3)_6]^{3+} (E) [Cr(H2O)6]2+[Cr(H_2O)_6]^{2+} Choose the correct answer from the options given below:
(A)
(B)
(C)
(D)
Q60Single correctChemical Thermodynamics
Match List - I with List - II.
List - I (Partial Derivatives)List - II (Thermodynamic Quantity)
A. (∂G∂T)P\left(\frac{\partial G}{\partial T}\right)_PI. Cp
B. (∂H∂T)P\left(\frac{\partial H}{\partial T}\right)_PII. -S
C. (∂G∂P)T\left(\frac{\partial G}{\partial P}\right)_TIII. Cv
D. (∂U∂T)V\left(\frac{\partial U}{\partial T}\right)_VIV. V
(A)
(B)
(C)
(D)
Q61Single correctBiomolecules
Identify the number of structure/s from the following which can be correlated to D-glyceraldehyde.
Four Fischer projections labeled A, B, C, D showing different arrangements of CHO, OH, and H groups on vertical carbon chains
(A)
(B)
(C)
(D)
Q62Single correctAtomic Structure
Given below are two statements: Statement (I): A spectral line will be observed for a 2px→2py2p_x \rightarrow 2p_y transition. Statement (II): 2Px2P_x and 2py2p_y are degenerate orbitals. In the light of the above statements, choose the correct answer from the options given below:
(A)
(B)
(C)
(D)
Q63Single correctClassification of Elements and Periodicity in Properties
Given below are two statements : Statement (I) : An element in the extreme left of the periodic table forms acidic oxides. Statement (II) : Acid is formed during the reaction between water and oxide of a reactive element present in the extreme right of the periodic table. In the light of the above statements, choose the correct answer from the options given below :
(A)
(B)
(C)
(D)
Q64Single correctHydrocarbons
Toluene (excess)→(ii)H3O+(i)CrO2Cl2,CS2Filter⟶Residue (A)→(iii)NaHSO3+HCl (dil)Compound (B)\text{Toluene (excess)} \xrightarrow[(ii) H_3O^+]{(i) CrO_2Cl_2, CS_2} \text{Filter} \longrightarrow \text{Residue (A)} \xrightarrow[(iii) NaHSO_3]{+ HCl \text{ (dil)}} \text{Compound (B)}. Structure of residue (A) and compound (B) formed respectively is:
(A)
(B)
(C)
(D)
Q65Single correctHydrocarbons
The alkane from below having two secondary hydrogens is :
(A)
(B)
(C)
(D)
Q66Single correctHydrocarbons
When sec-butylcyclohexane reacts with bromine in the presence of sunlight, the major product is :
(A)
(B)
(C)
(D)
Q67Single correctSome Basic Principles of Organic Chemistry
The most stable carbocation from the following is :
(A)
(B)
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(D)
Q68Single correctOrganic Compounds Containing Nitrogen
Match the Compounds (List - I) with the appropriate Catalyst/Reagents (List - II) for their reduction into corresponding amines. List-I (Compounds): (A) R-C(=O)-NH2NH_2, (B) Nitrobenzene, (C) R-C≡N, (D) Quinoline derivative with N-R. List-II (Catalyst/Reagents): (I) NaOH (aqueous), (II) H2H_2/Ni, (III) LiAlH4LiAlH_4, H2H_2O, (IV) Sn, HCl. Choose the correct answer from the options given below:
Matching table with List-I showing drawn structures: (A) Amide R-CONH2, (B) Nitrobenzene, (C) Nitrile R-CN, (D) Quinoline N-oxide derivative; and List-II showing reagents: (I) NaOH, (II) H2/Ni, (III) LiAlH4, (IV) Sn/HCl
(A)
(B)
(C)
(D)
Q69Single correctChemical Bonding and Molecular Structure
Arrange the following compounds in increasing order of their dipole moment: HBr, H2H_2S, NF3NF_3 and CHCl3CHCl_3
(A)
(B)
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Q70Single correctp-Block Elements
The maximum covalency of a non-metallic group 15 element 'E' with weakest E-E bond is:
(A)
(B)
(C)
(D)
Q71NumericalHydrocarbons
The compound with molecular formula C6H6C_6H_6, which gives only one monobromo derivative and takes up four moles of hydrogen per mole for complete hydrogenation has _____ π\pi electrons.
Q72NumericalSolutions
20 mL of 2 M NaOH solution is added to 400 mL of 0.5 M NaOH solution. The final concentration of the solution is _____ ×10−2\times 10^{-2} M. (Nearest integer)
Q73NumericalChemical Thermodynamics
Consider the following cases of standard enthalpy of reaction (ΔHr∘\Delta H^\circ_r in kJ mol−1l^{-1}): C2H6(g)+72O2(g)→2CO2(g)+3H2O(l)C_2H_6(g) + \frac{7}{2}O_2(g) \rightarrow 2CO_2(g) + 3H_2O(l), ΔH1∘=−1550\Delta H^\circ_1 = -1550. C(graphite)+O2(g)→CO2(g)C(\text{graphite}) + O_2(g) \rightarrow CO_2(g), ΔH2∘=−393.5\Delta H^\circ_2 = -393.5. H2(g)+12O2(g)→H2O(l)H_2(g) + \frac{1}{2}O_2(g) \rightarrow H_2O(l), ΔH3∘=−286\Delta H^\circ_3 = -286. The magnitude of ΔHf∘(C2H6(g))\Delta H^\circ_f(C_2H_6(g)) is _____ kJ mol−1l^{-1} (Nearest integer).
Q74Numericald- and f-Block Elements
Niobium (Nb) and ruthenium (Ru) have 'x' and 'y' number of electrons in their respective 4d orbitals. The value of x+yx + y is _____.
Q75NumericalCoordination Compounds
The complex of Ni2+Ni^{2+} ion and dimethyl glyoxime contains _____ number of Hydrogen (H) atoms.

Mathematics25 questions

Q1Single correctMatrices and Determinants
For a 3×33 \times 3 matrix M, let trace (M) denote the sum of all the diagonal elements of M. Let A be a 3×33 \times 3 matrix such that ∣A∣=12|A| = \frac{1}{2} and trace (A)=3(A) = 3. If B=adj(adj(2A))B = \text{adj}(\text{adj}(2A)), then the value of ∣B∣|B| + trace (B) equals :
(A)
(B)
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(D)
Q2Single correctPermutations and Combinations
In a group of 3 girls and 4 boys, there are two boys B1B_1 and B2B_2. The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but B1B_1 and B2B_2 are not adjacent to each other, is :
(A)
(B)
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Q3Single correctBinomial Theorem and its Simple Applications
Let α,β,γ\alpha, \beta, \gamma and δ\delta be the coefficients of x7,x5,x3x^7, x^5, x^3 and x respectively in the expansion of (x+x3−1)5+(x−x3−1)5(x + \sqrt{x^3 - 1})^5 + (x - \sqrt{x^3 - 1})^5, x>1x > 1. If u and v satisfy the equations αu+βv=18\alpha u + \beta v = 18 and γu+δv=20\gamma u + \delta v = 20 then u+vu + v equals :
(A)
(B)
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(D)
Q4Single correctThree Dimensional Geometry
Let a line pass through two distinct points P(−2,−1,3)P(-2, -1, 3) and Q, and be parallel to the vector 3i^+2j^+2k^3\hat{i} + 2\hat{j} + 2\hat{k}. If the distance of the point Q from the point R(1,3,3)R(1, 3, 3) is 5, then the square of the area of △PQR\triangle \text{PQR} is equal to :
(A)
(B)
(C)
(D)
Q5Single correctStatistics and Probability
If A and B are two events such that P(A∩B)=0.1P(A \cap B) = 0.1, and P(A∣B)P(A | B) and P(B∣A)P(B | A) are the roots of the equation 12x2−7x+1=012x^2 - 7x + 1 = 0, then the value of P(A‾∪B‾)P(A‾∩B‾)\frac{P(\overline{A} \cup \overline{B})}{P(\overline{A} \cap \overline{B})} is :
(A)
(B)
(C)
(D)
Q6Single correctIntegral Calculus
If ∫ex(xsin⁡−1x1−x2+sin⁡−1x(1−x2)3/2+x1−x2)dx=g(x)+C\int e^x \left(\frac{x \sin^{-1} x}{\sqrt{1-x^2}} + \frac{\sin^{-1} x}{(1-x^2)^{3/2}} + \frac{x}{1-x^2}\right)dx = g(x) + C, where C is the constant of integration, then g(12)g\left(\frac{1}{2}\right) equals :
(A)
(B)
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(D)
Q7Single correctIntegral Calculus
The area of the region enclosed by the curves y=x2−4x+4y = x^2 - 4x + 4 and y2=16−8xy^2 = 16 - 8x is :
(A)
(B)
(C)
(D)
Q8Single correctLimit, Continuity and Differentiability
Let f(x)=∫0x2t2−8t+15etdtf(x) = \int_0^{x^2} \frac{t^2 - 8t + 15}{e^t} dt, x∈Rx \in \mathbb{R}. Then the numbers of local maximum and local minimum points of f, respectively, are :
(A)
(B)
(C)
(D)
Q9Single correctCo-ordinate Geometry
Let P(4,43)P(4, 4\sqrt{3}) be a point on the parabola y2=4axy^2 = 4ax and PQ be a focal chord of the parabola. If M and N are the foot of perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to :
(A)
(B)
(C)
(D)
Q10Single correctVector Algebra
Let a⃗\vec{a} and b⃗\vec{b} be two unit vectors such that the angle between them is π3\frac{\pi}{3}. If λa⃗+2b⃗\lambda\vec{a} + 2\vec{b} and 3a⃗−λb⃗3\vec{a} - \lambda\vec{b} are perpendicular to each other, then the number of values of λ\lambda in [−1,3][-1, 3] is :
(A)
(B)
(C)
(D)
Q11Single correctLimit, Continuity and Differentiability
If lim⁡x→∞((e1−e)(1e−x1+x))x=α\lim_{x \to \infty} \left( \left(\frac{e}{1-e}\right) \left(\frac{1}{e} - \frac{x}{1+x}\right) \right)^x = \alpha, then the value of log⁡eα1+log⁡eα\frac{\log_e \alpha}{1 + \log_e \alpha} equals:
(A)
(B)
(C)
(D)
Q12Single correctSets, Relations and Functions
Let A={1,2,3,4}A = \{1, 2, 3, 4\} and B={1,4,9,16}B = \{1, 4, 9, 16\}. Then the number of many-one functions f:A→Bf : A \to B such that 1∈f(A)1 \in f(A) is equal to:
(A)
(B)
(C)
(D)
Q13Single correctSequence and Series
Suppose that the number of terms in an A.P. is 2k,k∈N2k, k \in \mathbb{N}. If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 5555 and the last term of the A.P. exceeds the first term by 2727, then k is equal to:
(A)
(B)
(C)
(D)
Q14Single correctThree Dimensional Geometry
The perpendicular distance of the line x−12=y+2−1=z+32\frac{x-1}{2} = \frac{y+2}{-1} = \frac{z+3}{2} from the point P(2,−10,1)P(2, -10, 1) is:
(A)
(B)
(C)
(D)
Q15Single correctMatrices and Determinants
If the system of linear equations: x+y+2z=6x + y + 2z = 6, 2x+3y+az=a+12x + 3y + az = a + 1, −x−3y+bz=2b-x - 3y + bz = 2b where a,b∈Ra, b \in \mathbb{R}, has infinitely many solutions, then 7a+3b7a + 3b is equal to:
(A)
(B)
(C)
(D)
Q16Single correctDifferential Equations
If x=f(y)x = f(y) is the solution of the differential equation (1+y2)+(x−2etan⁡−1y)dydx=0(1 + y^2) + (x - 2e^{\tan^{-1} y}) \frac{dy}{dx} = 0, y∈(−π2,π2)y \in (-\frac{\pi}{2}, \frac{\pi}{2}) with f(0)=1f(0) = 1, then f(13)f\left(\frac{1}{\sqrt{3}}\right) is equal to:
(A)
(B)
(C)
(D)
Q17Single correctComplex Numbers and Quadratic Equations
Let αθ\alpha_\theta and βθ\beta_\theta be the distinct roots of 2x2+(cos⁡θ)x−1=02x^2 + (\cos \theta)x - 1 = 0, θ∈(0,2π)\theta \in (0, 2\pi). If m and M are the minimum and the maximum values of αθ4+βθ4\alpha_\theta^4 + \beta_\theta^4, then 16(M+m)16(M + m) equals:
(A)
(B)
(C)
(D)
Q18Single correctTrigonometry
The sum of all values of θ∈[0,2π]\theta \in [0, 2\pi] satisfying 2sin⁡2θ=cos⁡2θ2\sin^2 \theta = \cos 2\theta and 2cos⁡2θ=3sin⁡θ2\cos^2 \theta = 3\sin \theta is:
(A)
(B)
(C)
(D)
Q19Single correctComplex Numbers and Quadratic Equations
Let the curve z(1+i)+zˉ(1−i)=4z(1 + i) + \bar{z}(1 - i) = 4, z∈Cz \in \mathbb{C}, divide the region ∣z−3∣≤1|z - 3| \leq 1 into two parts of areas α\alpha and β\beta. Then ∣α−β∣|\alpha - \beta| equals:
(A)
(B)
(C)
(D)
Q20Single correctCo-ordinate Geometry
Let E: x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a>ba > b and H: x2A2−y2B2=1\frac{x^2}{A^2} - \frac{y^2}{B^2} = 1. Let the distance between the foci of E and the foci of H be 232\sqrt{3}. If a−A=2a - A = 2, and the ratio of the eccentricities of E and H is 13\frac{1}{3}, then the sum of the lengths of their latus rectums is equal to:
(A)
(B)
(C)
(D)
Q21NumericalBinomial Theorem and its Simple Applications
If ∑r=130r2(30Cr)230Cr−1=α×229\sum_{r=1}^{30} \frac{r^2(^{30}C_r)^2}{^{30}C_{r-1}} = \alpha \times 2^{29}, then α\alpha is equal to ______
Q22NumericalSets, Relations and Functions
Let A={1,2,3}A = \{1, 2, 3\}. The number of relations on A, containing (1,2)(1, 2) and (2,3)(2, 3), which are reflexive and transitive but not symmetric, is ______
Q23NumericalCo-ordinate Geometry
Let A(6,8)A(6, 8), B(10cos⁡α,−10sin⁡α)B(10\cos\alpha, -10\sin\alpha) and C(−10sin⁡α,10cos⁡α)C(-10\sin\alpha, 10\cos\alpha), be the vertices of a triangle. If L(a,9)L(a, 9) and G(h, k) be its orthocenter and centroid respectively, then (5a−3h+6k+100sin⁡2α)(5a - 3h + 6k + 100\sin 2\alpha) is equal to ______
Q24NumericalDifferential Equations
Let y=f(x)y = f(x) be the solution of the differential equation dydx+xyx2−1=x6+4x1−x2\frac{dy}{dx} + \frac{xy}{x^2-1} = \frac{x^6+4x}{\sqrt{1-x^2}}, −1<x<1-1 < x < 1 such that f(0)=0f(0) = 0. If 6∫−1/21/2f(x)dx=2π−α6\int_{-1/2}^{1/2} f(x)dx = 2\pi - \alpha then α2\alpha^2 is equal to _______
Q25NumericalCo-ordinate Geometry
Let the distance between two parallel lines be 5 units and a point P lie between the lines at a unit distance from one of them. An equilateral triangle PQR is formed such that Q lies on one of the parallel lines, while R lies on the other. Then (QR)2(QR)^2 is equal to ______

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