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JEE Main 2025 January 29, Shift 2 Question Paper with Solutions

All 75 questions from the JEE Main 2025 (January 29, 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 correct
The difference of temperature in a material can convert heat energy into electrical energy. To harvest the heat energy, the material should have
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Q27Single correct
Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A): With the increase in the pressure of an ideal gas, the volume falls off more rapidly in an isothermal process in comparison to the adiabatic process.

Reason (R): In isothermal process, PV = constant, while in adiabatic process PV^γ = constant. Here γ is the ratio of specific heats, P is the pressure and V is the volume of the ideal gas.

In the light of the above statements, choose the correct answer from the options given below:
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Q28Single correct
An electric dipole is placed at a distance of 2 cm from an infinite plane sheet having positive charge density σ0_0. Choose the correct option from the following.
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Q29Single correct
In an experiment with photoelectric effect, the stopping potential:
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Q30Single correct
A point charge causes an electric flux of -2 × 1040^{4} Nm2m^{2}C−1C^{-1} to pass through a spherical Gaussian surface of 8.0 cm radius, centred on the charge. The value of the point charge is:

(Given ε0_0 = 8.85 × 10−120^{-12} C2C^{2}N−1m−2N^{-1}m^{-2})
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Q31Single correct
A poly-atomic molecule (CVC_V = 3R, CPC_P = 4R, where R is gas constant) goes from phase space point A(PAP_A = 1050^{5} Pa, VAV_A = 4 × 10−60^{-6} m3m^{3}) to point B (PBP_B = 5 × 1040^{4} Pa, VBV_B = 6 × 10−60^{-6} m3m^{3}) to point C (PCP_C = 1040^{4} Pa, VCV_C = 8 × 10−60^{-6} m3m^{3}). A to B is an adiabatic path and B to C is an isothermal path. The net heat absorbed per unit mole by the system is:
Figure for JEE Main 31
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Q32Single correct
Two identical symmetric double convex lenses of focal length f are cut into two equal parts L1L_1, L2L_2 by AB plane and L3L_3, L4L_4 by XY plane as shown in figure respectively. The ratio of focal lengths of lenses L1L_1 and L3L_3 is
Figure for JEE Main 32
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Q33Single correct
A plane electromagnetic wave propagates along the +x direction in free space. The components of the electric field, E⃗ and magnetic field, B⃗ vectors associated with the wave in Cartesian frame are:
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Q34Single correctOptics
Two concave refracting surfaces of equal radii of curvature and refractive index 1.5 face each other in air as shown in figure. A point object O is placed midway, between P and B. The separation between the images of O, formed by each refracting surface is:
Figure for JEE Main 34
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Q35Single correctSimple Harmonic Motion
Two bodies A and B of equal mass are suspended from two massless springs of spring constant k1k_1 and k2k_2, respectively. If the bodies oscillate vertically such that their amplitudes are equal, the ratio of the maximum velocity of A to the maximum velocity of B is
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Q36Single correctCollision
Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A): Three identical spheres of same mass undergo one dimensional motion as shown in figure with initial velocities vA = 5 m/s, vB = 2 m/s, vC = 4 m/s. If we wait sufficiently long for elastic collision to happen, then vA = 4 m/s, vB = 2 m/s, vC = 5 m/s will be the final velocities.

Reason (R): In an elastic collision between identical masses, two objects exchange their velocities.

In the light of the above statements, choose the correct answer from the options given below:
Figure for JEE Main 36
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Q37Single correctPower and Momentum
A sand dropper drops sand of mass m(t) on a conveyor belt at a rate proportional to the square root of speed (v) of the belt, i.e. dmdt\dfrac{dm}{dt} ∝\propto v\sqrt{v}. If P is the power delivered to run the belt at constant speed then which of the following relationship is true?
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Q38Single correctOptics - Lens
A convex lens made of glass (refractive index = 1.5) has focal length 24 cm in air. When it is totally immersed in water (refractive index = 1.33), its focal length changes to
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Q39Single correctCapacitors
A capacitor, C1C_1 = 66 μ\muF is charged to a potential difference of V0V_0 = 5V using a 5V battery. The battery is removed and another capacitor, C2C_2 = 1212 μ\muF is inserted in place of the battery. When the switch 'S' is closed, the charge flows between the capacitors for some time until equilibrium condition is reached. What are the charges (q1q_1 and q2q_2) on the capacitors C1C_1 and C2C_2 when equilibrium condition is reached?
Figure for JEE Main 39
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Q40Single correctAngular Momentum
Three equal masses m are kept at vertices (A, B, C) of an equilateral triangle of side a in free space. At t = 0, they are given an initial velocity V⃗A = V0V_0AC̅, V⃗B = V0V_0BA̅ and V⃗C = V0V_0CB̅. Here, AC̅, CB̅ and BA̅ are unit vectors along the edges of the triangle. If the three masses interact gravitationally, then the magnitude of the net angular momentum of the system at the point of collision is:
Figure for JEE Main 40
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Q41Single correctUnits and Measurements
Match List-I with List-II.
List-IList-II
A. Young's ModulusI. ML−1T−1ML^{-1}T^{-1}
B. TorqueII. ML−1T−2ML^{-1}T^{-2}
C. Coefficient of ViscosityIII. M−1L3T−2M^{-1}L^{3}T^{-2}
D. Gravitational ConstantIV. ML2T−2ML^{2}T^{-2}
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Q42Single correctMagnetic Effects of Current and Magnetism
Match List-I with List-II.
List-IList-II
A. Magnetic inductionI. Ampere meter2r^{2}
B. Magnetic intensityII. Weber
C. Magnetic fluxIII. Gauss
D. Magnetic momentIV. Ampere meter
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Q43Single correctElectronic Devices
The truth table for the circuit given below is :
Figure for JEE Main 43
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Q44Single correctHeat and Thermodynamics
A cup of coffee cools from 90°90°C to 80°80°C inttminutes when the room temperature is 20°20°C. The time taken by the similar cup of coffee to cool from 80°80°C to 60°60°C at the same room temperature is :
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Q45Single correctAtoms and Nuclei
The number of spectral lines emitted by atomic hydrogen that is in the 4th energy level, is
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Q46NumericalMagnetic Effects of Current and Magnetism
The magnetic field inside a 200 turns solenoid of radius 10 cm is2.9×10−42.9 \times 10^{-4}Tesla. If the solenoid carries a current of 0.29 A, then the length of the solenoid is __________π\picm.
Q47NumericalElectrostatics
A parallel plate capacitor consisting of two circular plates of radius 10 cm is being charged by a constant current of 0.15 A. If the rate of change of potential difference between the plates is7×1087 \times 10^{8}V/s then the integer value of the distance between the parallel plates is _________μ\mum. (Take,ϵ0=9×10−12Fm\epsilon_{0} = 9 \times 10^{-12} \dfrac{F}{m},π=227\pi = \dfrac{22}{7})
Q48NumericalUnits and Measurements
A physical quantity Q is related to four observables a, b, c, d as follows:Q=ab4cdQ = \dfrac{ab^{4}}{cd}where,a=(60±3)a = (60 \pm 3)Pa;b=(20±0.1)b = (20 \pm 0.1)m;c=(40±0.2)Nsm−2andd=(50±0.1)c = (40 \pm 0.2)Nsm^{-2}andd = (50 \pm 0.1)m, then the percentage error in Q isx1000\dfrac{x}{1000}, wherex= ______.
Q49NumericalGravitation
Two planets, A and B are orbiting a common star in circular orbits of radiiRAandRBR_{A}andR_{B}, respectively, withRB=2RAR_{B} = 2R_{A}. The planet B is424\sqrt{2}times more massive than planet A. The ratio(LBLA)\left(\dfrac{L_{B}}{L_{A}}\right)of angular momentum(LB)(L_{B})of planet B to that of planet A(LA)(L_{A})is closest to integer __________.
Q50NumericalMotion in a Plane
Two cars P and Q are moving on a road in the same direction. Acceleration of car P increases linearly with time whereas car Q moves with a constant acceleration. Both cars cross each other at timet=0t = 0, for the first time. The maximum possible number of crossing(s) (including the crossing att=0t = 0) is ________.

Chemistry25 questions

Q51Single correctCoordination Compounds
The calculated spin-only magnetic moments ofK3[Fe(OH)6]andK4[Fe(OH)6]\text{K}_3[\text{Fe}(\text{OH})_6]and\text{K}_4[\text{Fe}(\text{OH})_6]respectively are :
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Q52Single correctAtomic Structure
For hydrogen like species, which of the following graphs provides the most appropriate representation of E vs Z plot for a constant n? [E : Energy of the stationary state, Z : atomic number, n = principal quantum number]
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Q53Single correctSurface Chemistry
Given below are two statements: Statement (I): In partition chromatography, stationary phase is thin film of liquid present in the inert support. Statement (II): In paper chromatography, the material of paper acts as a stationary phase. In the light of the above statements, choose the correct answer from the options given below:
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Q54Single correctBiomolecules
Identify the essential amino acids from below: (A) Valine (B) Proline (C) Lysine (D) Threonine (E) Tyrosine. Choose the correct answer from the options given below:
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Q55Single correctHaloalkanes and Haloarenes
Which among the following halides will generate the most stable carbocation in Nucleophilic substitution reaction?
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Q56Single correctChemical Equilibrium
Consider the equilibriumCO(g)+3H2(g)⇌CH4(g)+H2O(g)\text{CO}(g) + 3\text{H}_2(g) \rightleftharpoons \text{CH}_4(g) + \text{H}_2\text{O}(g). If the pressure applied over the system increases by two fold at constant temperature then (A) Concentration of reactants and products increases. (B) Equilibrium will shift in forward direction. (C) Equilibrium constant increases since concentration of products increases. (D) Equilibrium constant remains unchanged as concentration of reactants and products remain same. Choose the correct answer from the options given below:
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Q57Single correctSolutions
Given below are two statements: Statement (I): NaCl is added to the ice at 0°0°C, present in the ice cream box to prevent the melting of ice cream. Statement (II): On addition of NaCl to ice at 0°0°C, there is a depression in freezing point. In the light of the above statements, choose the correct answer from the options given below:
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Q58Single correctHydrocarbons
Given below are two statements: Statement (I): On nitration of m-xylene withHNO3\text{HNO}_3,H2SO4\text{H}_2\text{SO}_4followed by oxidation, 4-nitrobenzene-1,3-dicarboxylic acid is obtained as the major product. Statement (II):CH3\text{CH}_3group is o/p-directing whileNO2\text{NO}_2group is m-directing group. In the light of the above statements, choose the correct answer from the options given below:
Figure for JEE Main 58
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Q59Single correctChemical Kinetics and Equilibrium
0.1 M solution of KI reacts with excess of H2SO4H_2SO_4and KIO3O_3solution. According to equation5I−+IO3−+6H+→3I2+3H2O5\text{I}^- + \text{IO}_3^- + 6\text{H}^+ \rightarrow 3\text{I}_2 + 3\text{H}_2\text{O}Identify the correct statements: (A) 200 mL of KI solution reacts with 0.004 mol of KIO3O_3(B) 200 mL of KI solution reacts with 0.006 mol of H2SO4H_2SO_4(C) 0.5 L of KI solution produced 0.005 mol of I2I_2(D) Equivalent weight of KIO3O_3is equal toMolecular weight5\dfrac{\text{Molecular weight}}{5}
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Q60Single correctElectrochemistry
Match List-I with List-II:
List-I (Applications)List-II (Batteries/Cell)
A. TransistorsI. Anode - Zn/Hg; Cathode - HgO + C
B. Hearing aidsII. Hydrogen fuel cell
C. InvertorsIII. Anode - Zn; Cathode - Carbon
D. Apollo space shipIV. Anode - Pb; Cathode - Pb | PbO2O_2
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Q61Single correctElectrochemistry
O2O_2gas will be evolved as a product of electrolysis of: (A) an aqueous solution of AgNO3O_3using silver electrodes. (B) an aqueous solution of AgNO3O_3using platinum electrodes. (C) a dilute solution of H2SO4H_2SO_4using platinum electrodes. (D) a high concentration solution of H2SO4H_2SO_4using platinum electrodes. Choose the correct answer from the options given below:
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Q62Single correctCoordination Compounds
Identify the homoleptic complexes with odd number of d electrons in the central metal. (A) [FeO4]2−[\text{FeO}_4]^{2-} (B) [Fe(CN)6]3−[\text{Fe(CN)}_6]^{3-} (C) [Fe(CN)5NO]2−[\text{Fe(CN)}_5\text{NO}]^{2-} (D) [CoCl4]2−[\text{CoCl}_4]^{2-} (E) [Co(H2O)3F3][\text{Co(H}_2\text{O)}_3\text{F}_3] Choose the correct answer from the options given below:
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Q63Single correctOrganic Chemistry
Total number of sigma (σ\sigma) _____ and pi (π\pi) _____ bonds respectively present in hex-1-en-4-yne are:
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Q64Single correctThermodynamics
If C(diamond) →\rightarrow C(graphite) + X kJ mol−1l^{-1}, C(diamond) + O2O_2(g) →\rightarrow CO2O_2(g) + Y kJ mol−1l^{-1}, C(graphite) + O2O_2(g) →\rightarrow CO2O_2(g) + Z kJ mol−1l^{-1} at constant temperature. Then
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Q65Single correctAtomic Structure
Given below are two statements: Statement (I): It is impossible to specify simultaneously with arbitrary precision, both the linear momentum and the position of a particle. Statement (II): If the uncertainty in the measurement of position and uncertainty in measurement of momentum are equal for an electron, then the uncertainty in the measurement of velocity is≥hπ×12m\geq \sqrt{\dfrac{h}{\pi}} \times \dfrac{1}{2m}. In the light of the above statements, choose the correct answer from the options given below:
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Q66Single correctOrganic Chemistry - Some Basic Principles and Techniques
Which one of the following reaction sequences will give an azo dye?
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Q67Single correctPhysical Chemistry - Chemical Kinetics
Drug X becomes ineffective after 50% decomposition. The original concentration of drug in a bottle was 16 mg/mL which becomes 4 mg/mL in 12 months. The expiry time of the drug in months is _____. Assume that the decomposition of the drug follows first order kinetics.
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Q68Single correctPeriodic Table and Properties
The type of oxide formed by the element among Li, Na, Be, Mg, B and Al that has the least atomic radius is:
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Q69Single correctPeriodic Table and Properties
First ionisation enthalpy values of first four group 15 elements are given below. Choose the correct value for the element that is a main component of apatite family:
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Q70Single correctOrganic Chemistry - Alcohols, Phenols and Ethers
Which one of the following, with HBr will give a phenol?
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Q71NumericalCoordination Chemistry
Consider the following low-spin complexes K3[Co(NO2)6K_3[Co(NO_2)_6], K4[Fe(CN)6K_4[Fe(CN)_6], K3[Fe(CN)6K_3[Fe(CN)_6], Cu2[Fe(CN)6u_2[Fe(CN)_6] and Zn2[Fe(CN)6n_2[Fe(CN)_6]. The sum of the spin-only magnetic moment values of complexes having yellow colour is _______ B.M. (answer is nearest integer)
Q72NumericalOrganic Chemistry - Hydrocarbons
Isomeric hydrocarbons→\rightarrownegative Baeyer's test (Molecular formula C9H12C_9H_{12}). The total number of isomers from above with four different non-aliphatic substitution sites is -
Q73NumericalOrganic Chemistry - Aldehydes, Ketones and Carboxylic Acids
In the Claisen-Schmidt reaction to prepare dibenzalacetone from 5.3 g benzaldehyde, a total of 3.51 g of product was obtained. The percentage yield in this reaction was ______ %.
Q74NumericalOrganic Chemistry - Some Basic Principles and Techniques
In the sulphur estimation, 0.20 g of a pure organic compound gave 0.40 g of barium sulphate. The percentage of sulphur in the compound is ____×10−1\times10^{-1}%. (Molar mass: O = 16, S = 32, Ba = 137 in g mol−1l^{-1})
Q75NumericalChemical Bonding and Molecular Structure
Total number of non bonded electrons present in NO2−_2^-ion based on Lewis theory is _____.

Mathematics25 questions

Q1Single correctComplex Numbers and Quadratic Equations
If the set of alla∈Ra \in \mathbb{R}, for which the equation2x2+(a−5)x+15=3a2x^2 + (a - 5)x + 15 = 3ahas no real root, is the interval(α,β)(\alpha, \beta), andX={x∈Z:α<x<β}X = \{x \in \mathbb{Z} : \alpha < x < \beta\}, then∑x∈Xx2\sum_{x \in X} x^2is equal to
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Q2Single correctTrigonometry
Ifsin⁡x+sin⁡2x=1\sin x + \sin^2 x = 1,x∈(0,π2)x \in (0, \dfrac{\pi}{2}), then(cos⁡12x+tan⁡12x)+3(cos⁡10x+tan⁡10x+cos⁡8x+tan⁡8x)+(cos⁡6x+tan⁡6x)(\cos^{12}x + \tan^{12}x) + 3(\cos^{10}x + \tan^{10}x + \cos^8x + \tan^8x) + (\cos^6x + \tan^6x)is equal to
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Q3Single correctIntegral Calculus
Let the area enclosed between the curves∣y∣=1−x2andx2+y2=1beα|y| = 1 - x^2andx^2 + y^2 = 1be\alpha. If9α=βπ+γ9\alpha = \beta\pi + \gamma;β,γ\beta, \gammaare integers, then the value of∣β−γ∣|\beta - \gamma|equals
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Q4Single correctLimit, Continuity and Differentiability
If the domain of the functionlog⁡5(18x−x2−77)is(α,β)\log_5(18x - x^2 - 77)is(\alpha, \beta)and the domain of the functionlog⁡(x−1)(2x2+3x−2x2−3x−4)is(γ,δ)\log_{(x-1)}\left(\dfrac{2x^2 + 3x - 2}{x^2 - 3x - 4}\right)is(\gamma, \delta), thenα2+β2+γ2\alpha^2 + \beta^2 + \gamma^2is equal to:
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Q5Single correctLimit, Continuity and Differentiability
Let the functionf(x)=(x2−1)∣x2−ax+2∣+cos⁡∣x∣f(x) = (x^2 - 1)|x^2 - ax + 2| + \cos|x|be not differentiable at the two pointsx=α=2andx=βx = \alpha = 2andx = \beta. Then the distance of the point(α,β)(\alpha, \beta)from the line12x+5y+10=012x + 5y + 10 = 0is equal to:
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Q6Single correctThree Dimensional Geometry
Let a straight line L pass through the point P(2, -1, 3) and be perpendicular to the linesx−12=y+1−1=z−32andx−31=y−23=z+24\dfrac{x-1}{2} = \dfrac{y+1}{-1} = \dfrac{z-3}{2}and\dfrac{x-3}{1} = \dfrac{y-2}{3} = \dfrac{z+2}{4}. If the line L intersects the yz-plane at the point Q, then the distance between the points P and Q is:
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Q7Single correctSets, Relations and Functions
LetS=N∪{0}S = \mathbb{N} \cup \{0\}. Define a relation R from S toR\mathbb{R}by:R={(x,y):log⁡ey=xlog⁡e(25),x∈S,y∈R}\mathbf{R} = \{(x, y) : \log_e y = x \log_e \left(\dfrac{2}{5}\right), x \in S, y \in \mathbb{R}\}. Then, the sum of all the elements in the range of R is equal to
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Q8Single correctTrigonometry
Let the linex+y=1x + y = 1meet the axes of x and y at A and B, respectively. A right-angled triangle AMN is inscribed in the triangle OAB, where O is the origin and the points M and N lie on the lines OB and AB, respectively. If the area of the triangle AMN is49\dfrac{4}{9}of the area of the triangle OAB and AN:NB =λ\lambda:1, then the sum of all possible value(s) ofλ\lambdais:
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Q9Single correctCo-ordinate Geometry
Ifαx+βy=109\alpha x + \beta y = 109is the equation of the chord of the ellipsex29+y24=1\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1, whose mid point is(52,12)\left(\dfrac{5}{2}, \dfrac{1}{2}\right), thenα+β\alpha + \betais equal to
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Q10Single correctPermutations and Combinations
If all the words with or without meaning made using all the letters of the word "KANPUR" are arranged as in a dictionary, then the word at 440th position in this arrangement, is:
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Q11Single correctMatrices and Determinants
Letα(α≠β\alpha(\alpha \neq \beta) be the values of m, for which the equationsx+y+z=1x + y + z = 1;x+2y+4z=mandx+4y+10z=m2x + 2y + 4z = mandx + 4y + 10z = m^2have infinitely many solutions. Then the value of∑n=110(nα+nβ)\sum_{n=1}^{10} (n^\alpha + n^\beta)is equal to:
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Q12Single correctMatrices and Determinants
LetA=[aij]A = [a_{ij}]be a matrix of order3×33 \times 3, withaij=(2)i+ja_{ij} = (\sqrt{2})^{i+j}. If the sum of all the elements in the third row ofA2isα+β2A^2is\alpha + \beta\sqrt{2},α,β∈Z\alpha, \beta \in \mathbb{Z}, thenα+β\alpha + \betais equal to
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Q13Single correctThree Dimensional Geometry
Let P be the foot of the perpendicular from the point (1, 2, 2) on the line L:x−11=y+1−1=z−22\dfrac{x-1}{1} = \dfrac{y+1}{-1} = \dfrac{z-2}{2}. Let the liner⃗=(i^+j^−2k^)+λ(i^−j^+k^)\vec{r} = (\hat{i} + \hat{j} - 2\hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}),λ∈R\lambda \in \mathbb{R}, intersect the line L at Q. Then2(PQ)22(PQ)^2is equal to:
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Q14Single correctCo-ordinate Geometry
Let a circle C pass through the points (4, 2) and (0, 2), and its centre lie on3x+2y+2=03x + 2y + 2 = 0. Then the length of the chord, of the circle C, whose midpoint is (1, 2), is:
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Q15Single correctStatistics and Probability
Let A =[aij][a_{ij}]be a2×22 \times 2matrix such thataij∈{0,1}a_{ij} \in \{0, 1\}for all i and j. Let the random variable X denote the possible values of the determinant of the matrix A. Then, the variance of X is:
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Q16Single correctProbability
Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains n white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability that the ball drawn is white is2945is \dfrac{29}{45}, then n is equal to:
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Q17Single correctNumber Theory
The remainder, when 71037^{103} is divided by 23, is equal to:
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Q18Single correctCalculus - Integration and Maxima-Minima
Letf(x)=∫0xt(t2−9t+20)dtf(x) = \int_0^x t(t^2 - 9t + 20)dt,1≤x≤51 \leq x \leq 5. If the range offis[α,β][\alpha, \beta], then4(α+β)4(\alpha + \beta)equals:
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Q19Single correctVector Algebra
Leta^\hat{a}be a unit vector perpendicular to the vectorsb⃗=i^−2j^+3k^andc⃗=2i^+3j^−k^\vec{b} = \hat{i} - 2\hat{j} + 3\hat{k}and\vec{c} = 2\hat{i} + 3\hat{j} - \hat{k}, and makes an angle ofcos⁡−1(−13)\cos^{-1}\left(-\dfrac{1}{3}\right)with the vectori^+j^+k^\hat{i} + \hat{j} + \hat{k}. Ifa^\hat{a}makes an angle ofπ3\dfrac{\pi}{3}with the vectori^+αj^+k^\hat{i} + \alpha\hat{j} + \hat{k}, then the value ofα\alphais:
(A)
(B)
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(D)
Q20Single correctDifferential Equations
If for the solution curvey=f(x)y = f(x)of the differential equationdydx+(tan⁡x)y=2+sec⁡x(1+2sec⁡x)2\dfrac{dy}{dx} + (\tan x)y = \dfrac{2 + \sec x}{(1 + 2\sec x)^2},x∈(−π2,π2)x \in \left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right),f(π3)=310f\left(\dfrac{\pi}{3}\right) = \dfrac{\sqrt{3}}{10}, thenf(π4)f\left(\dfrac{\pi}{4}\right)is equal to:
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Q21NumericalDefinite Integration
If∫0π/424[∣sin⁡(4x−π12)∣+[2sin⁡x]]dx=2π+α\int_0^{\pi/4} 24\left[\left|\sin\left(4x - \dfrac{\pi}{12}\right)\right| + [2\sin x]\right]dx = 2\pi + \alpha, where[⋅][\cdot]denotes the greatest integer function, thenα\alphais equal to _______.
Q22NumericalLimits and Integration
Iflim⁡t→0(∫01(3x+5)tdx)1/t=α5e(85)2/3\lim_{t \to 0} \left(\int_0^1 (3x+5)^t dx\right)^{1/t} = \dfrac{\alpha}{5e}\left(\dfrac{8}{5}\right)^{2/3}, thenα\alphais equal to _______.
Q23NumericalArithmetic Progression
Leta1,a2,…,a2024a_1, a_2, \ldots, a_{2024}be an Arithmetic Progression such thata1+(a5+a10+a15+…+a2020)+a2024=2233a_1 + (a_5 + a_{10} + a_{15} + \ldots + a_{2020}) + a_{2024} = 2233. Thena1+a2+a3+…+a2024a_1 + a_2 + a_3 + \ldots + a_{2024}is equal to _______.
Q24NumericalComplex Numbers
Let integers a,b∈[−3,3]a, b \in [-3, 3] be such that a+b≠0a + b \neq 0. Then the number of all possible ordered pairs (a, b), for which ∣z−a∣∣z+b∣=1\dfrac{|z-a|}{|z+b|} = 1 and ∣z+1ωω2ωz+ω21ω21z+ω∣=1\begin{vmatrix} z+1 & \omega & \omega^2 \\ \omega & z+\omega^2 & 1 \\ \omega^2 & 1 & z+\omega \end{vmatrix} = 1, z∈Cz \in \mathbb{C}, where ω\omega and ω2\omega^2 are the roots of x2+x+1=0x^2 + x + 1 = 0, is equal to _______.
Q25NumericalConic Sections - Parabola
Lety2=12xy^2 = 12xbe the parabola andSbe its focus. LetPQbe a focal chord of the parabola such that(SP)(SQ)=1474(SP)(SQ) = \dfrac{147}{4}. LetCbe the circle described takingPQas a diameter. If the equation of circleCis64x2+64y2−αx−643y=β64x^2 + 64y^2 - \alpha x - 64\sqrt{3}y = \beta, thenβ−α\beta - \alphais equal to _______.

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