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JEE Mains 2025 (29 Jan, Shift 2), Paper 1 (B.E./B.Tech) question paper

75 questions · 300 marks · 180 minutes · +4 per correct answer, -1 per wrong one

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Physics

25 questions

  1. 1
    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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  2. 2
    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=PV = constant, while in adiabatic process PVγ=PV^{\gamma}= constant. Here γ\gamma 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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  3. 3
    An electric dipole is placed at a distance of 2 cm from an infinite plane sheet having positive charge density σ0\sigma_{0}. Choose the correct option from the following.

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  4. 4
    In an experiment with photoelectric effect, the stopping potential.

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  5. 5
    A point charge causes an electric flux of −2×104- 2 \times10^{4} Nm2C−1Nm^{2}C^{- 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=8.85×10−12C2 N−1 m−2\epsilon_{0}= 8.85 \times10^{- 12}C^{2}{\text{ }N}^{- 1}{\text{ }m}^{- 2} )

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  6. 6
    A poly-atomic molecule (CV=3R,Cp=4R\left( C_{V}= 3R,C_{p}= 4R \right., where RR is gas constant) goes from phase space point A(PA=105 Pa, VA=4×10−6 m3)A\left( P_{A}=10^{5}\text{ }Pa,{\text{ }V}_{A}= 4 \times10^{- 6}{\text{ }m}^{3} \right) to point B(PB=5B\left( P_{B}= 5 \right.  ×104 Pa, VB=6×10−6 m3)\left. \ \times10^{4}\text{ }Pa,{\text{ }V}_{B}= 6 \times10^{- 6}{\text{ }m}^{3} \right) to point C(PC=104 PaC\left( P_{C}=10^{4}\text{ }Pa \right., VC=8×10−6 m3V_{C}= 8 \times10^{- 6}{\text{ }m}^{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 :

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  7. 7
    Two identical symmetric double convex lenses of focal length ff are cut into two equal parts L1, L2L_{1},{\text{ }L}_{2} by ABAB plane (containing the principal axis) and L3,L4L_{3},L_{4} by XYXY plane (perpendicular to the principal axis, through the centre) as shown in figure respectively. The ratio of focal lengths of lenses L1L_{1} and L3L_{3} is

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  8. 8
    A plane electromagnetic wave propagates along the +x+ x direction in free space. The components of the electric field, E→\overrightarrow{E} and magnetic field, B→\overrightarrow{B} vectors associated with the wave in Cartesian frame are :

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  9. 9
    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 :

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  10. 10
    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 AA to the maximum velocity of BB is

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  11. 11
    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/sv_{A}= 5\text{ }m/s,v_{B}= 2\text{ }m/s,v_{c}= 4\text{ }m/s If we wait sufficiently long for elastic collision to happen, then vA=4 m/s,vB=2 m/s,vc=5 m/sv_{A}= 4\text{ }m/s,v_{B}= 2\text{ }m/s,v_{c}= 5\text{ }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 :

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  12. 12
    A sand dropper drops sand of mass m(t)m(t) on a conveyer belt at a rate proportional to the square root of speed (v) of the belt, i.e. dmdt∝v\frac{dm}{dt}\propto\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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  13. 13
    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= 1.33 ), its focal length changes to

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  14. 14
    A capacitor, C1=6μ FC_{1}= 6\mu\text{ }F is charged to a potential difference of V0=5 VV_{0}= 5\text{ }V using a 5 V battery. The battery is removed and another capacitor, C2=12C_{2}= 12 μF\mu F 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 (q1\left( q_{1} \right. and q2q_{2} ) on the capacitors C1C_{1} and C2C_{2} when equilibrium condition is reached.

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  15. 15
    Three equal masses mm are kept at vertices (A, B, C) of an equilateral triangle of side aa in free space. At t=0t = 0, they are given an initial velocity V→A=V0AC→, V→B=V0BA→{\overrightarrow{V}}_{A}=V_{0}\overrightarrow{AC},\ {\overrightarrow{V}}_{B}=V_{0}\overrightarrow{BA} and  V→C=V0CB→\ {\overrightarrow{V}}_{C}=V_{0}\overrightarrow{CB}. Here, AC→,CB→\overrightarrow{AC},\overrightarrow{CB} and BA→\overrightarrow{BA} 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 :

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  16. 16
    Match List-I with List-II.
    List-IList-II
    (A) Young's Modulus(I) ML−1T−1ML^{-1}T^{-1}
    (B) Torque(II) ML−1T−2ML^{-1}T^{-2}
    (C) Coefficient of Viscosity(III) M−1L3T−2M^{-1}L^{3}T^{-2}
    (D) Gravitational Constant(IV) ML2T−2ML^{2}T^{-2}
    Choose the correct answer from the options given below :

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  17. 17
    Match List-I with List-II.
    List-IList-II
    (A) Magnetic induction(I) Ampere meter2^{2}
    (B) Magnetic intensity(II) Weber
    (C) Magnetic flux(III) Gauss
    (D) Magnetic moment(IV) Ampere/meter
    Choose the correct answer from the options given below :

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  18. 18
    The truth table for the circuit given below is :

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  19. 19
    A cup of coffee cools from 90∘C90^{\circ}C to 80∘C80^{\circ}C in t minutes when the room temperature is 20∘C20^{\circ}C. The time taken by the similar cup of coffee to cool from 80∘C80^{\circ}C to 60∘C60^{\circ}C at the same room temperature is :

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  20. 20
    The number of spectral lines emitted by atomic hydrogen that is in the 4th 4^{\text{th~}} energy level, is

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  21. 21
    The magnetic field inside a 200 turns solenoid of radius 10 cm is 2.9×10−4Tesla2.9 \times10^{- 4}Tesla. If the solenoid carries a current of 0.29 A , then the length of the solenoid is ______\_\_\_\_\_\_ πcm\pi cm.
  22. 22
    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 is 7×1087 \times10^{8} V/sV/s then the integer value of the distance between the parallel plates is - (Take \left( Take\ \right.  ,ϵ0=9×10−12 F m,π=227)\left. \ ,\epsilon_{0}= 9 \times10^{- 12}\frac{\text{ }F}{\text{ }m},\pi =\frac{22}{7} \right) _____\_\_\_\_\_ μm\mu m.
  23. 23
    A physical quantity Q is related to four observables a, b,c,db,c,d as follows : Q=ab4cdQ = \frac{ab^{4}}{cd} where, a=(60±3)Pa;b=(20±0.1)ma = (60 \pm 3)Pa;b = (20 \pm 0.1)m; c=(40±0.2)Nsm−2c = (40 \pm 0.2)Nsm^{- 2} and d=(50±0.1)md = (50 \pm 0.1)m, then the percentage error in Q is x1000\frac{x}{1000}, where x=x =  ____\ \_\_\_\_ .
  24. 24
    Two planets, AA and BB are orbiting a common star in circular orbits of radii RAR_{A} and RBR_{B}, respectively, with RB=2RAR_{B}= 2R_{A}. The planet BB is 424\sqrt{2} times more massive than planet AA. The ratio (LBLA)\left( \frac{L_{B}}{L_{A}} \right) of angular momentum (LB)\left( L_{B} \right) of planet BB to that of planet A(LA)A\left( L_{A} \right) is closest to integer ____\_\_\_\_
  25. 25
    Two cars P and Q are moving on a road in the same direction. Acceleration of car PP increases linearly with time whereas car QQ moves with a constant acceleration. Both cars cross each other at time t=0t = 0, for the first time. The maximum possible number of crossing(s) (including the crossing at t=0t = 0 ) is  ____\ \_\_\_\_

Chemistry

25 questions

  1. 26
    The calculated spin-only magnetic moments of K3[Fe(OH)6]K_{3}\left\lbrack Fe(OH)_{6} \right\rbrack and K4[Fe(OH)6]K_{4}\left\lbrack Fe(OH)_{6} \right\rbrack respectively are :

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  2. 27
    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=n = principal quantum number]

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  3. 28
    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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  4. 29
    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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  5. 30
    Which among the following halides will generate the most stable carbocation in Nucleophillic substitution reaction?

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  6. 31
    Consider the equilibrium
    CO( g)+3H2( g)⇌CH4( g)+H2O( g)CO(\text{ }g) + 3H_{2}(\text{ }g) \rightleftharpoons CH_{4}(\text{ }g) +H_{2}O(\text{ }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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  7. 32
    Given below are two statements : Statement (I): NaCl is added to the ice at 0∘C0^{\circ}C, present in the ice cream box to prevent the melting of ice cream. Statement (II) : On addition of NaCl to ice at 0∘C0^{\circ}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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  8. 33
    Given below are two statements : Statement (I) : On nitration of m-xylene with HNO3,H2SO4HNO_{3},H_{2}SO_{4} followed by oxidation, 4-nitrobenzene-1, 3-dicarboxylic acid is obtained as the major product. Statement (II) : CH3CH_{3} group is o/p-directing whileNO2NO_{2} group is m-directing group. In the light of the above statements, choose the correct answer from the options given below :

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  9. 34
    0.1 M solution of KI reacts with excess of H2SO4H_{2}SO_{4} and KIO3KIO_{3} solution. According to equation
    5I−+IO3−+6H+→3I2+3H2O5I^{-}+IO_{3}^{-}+ 6H^{+}\rightarrow 3I_{2}+ 3H_{2}O
    Identify the correct statements : (A) 200 mL of KI solution reacts with 0.004 mol of KIO3KIO_{3} (B) 200 mL of KI solution reacts with 0.006 mol of H2SO4H_{2}SO_{4} (C) 0.5 L of KI solution produced 0.005 mol of I2I_{2} (D) Equivalent weight of KIO3KIO_{3} is equal to
    ( Molecular weight 5)\left( \frac{\text{~Molecular weight~}}{5} \right)
    Choose the correct answer from the options given below:

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  10. 35
    Match List-I with List-II :
    List-I (Application)List-II (Battery/Cell)
    (A)Transistors(I)Anode -Zn/Hg; Cathode -HgO + C
    (B)Hearing aids(II)Hydrogen fuel cell
    (C)Invertors(III)Anode - Zn ; Cathode - Carbon
    (D)Apollo space ship(IV)Anode - Pb ; Cathode - PbO2PbO_{2} on a lead grid
    Choose the correct answer from the options given below :

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  11. 36
    O2O_{2} gas will be evolved as a product of electrolysis of : (a) an aqueous solution of AgNO3AgNO_{3} using silver electrodes. (b) an aqueous solution of AgNO3AgNO_{3} using platinum electrodes. (c) a dilute solution of H2SO4H_{2}SO_{4} using platinum electrodes. (d) a high concentration solution of H2SO4H_{2}SO_{4} using platinum electrodes. Choose the correct answer from the options given below :

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  12. 37
    Identify the homoleptic complexes with odd number of d electrons in the central metal. (A) [FeO4]2−\left\lbrack FeO_{4} \right\rbrack^{2 -} (B) [Fe(CN)6]3−\left\lbrack Fe(CN)_{6} \right\rbrack^{3 -} (C) [Fe(CN)5NO]2−\left\lbrack Fe(CN)_{5}NO \right\rbrack^{2 -} (D) [CoCl4]2−\left\lbrack CoCl_{4} \right\rbrack^{2 -} (E) [Co(H2O)3 F3]\left\lbrack Co\left( H_{2}O \right)_{3}{\text{ }F}_{3} \right\rbrack Choose the correct answer from the options given below :

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  13. 38
    Total number of sigma (σ)(\sigma) and pi (π)(\pi) bonds respectively present in hex-1-en-4-yne are :

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  14. 39
    If C (diamond) →C\rightarrow C (graphite) +XkJmol−1+ XkJmol^{- 1}
    C( diamond )+O2( g)→CO2( g)+Y kJ mol−1C( graphite )+O2( g)→CO2( g)+Z kJ mol−1\begin{matrix} & C(\text{~diamond~}) +O_{2}(\text{ }g) \rightarrow CO_{2}(\text{ }g) + Y\text{ }kJ{\text{ }mol}^{- 1} \\ & C(\text{~graphite~}) +O_{2}(\text{ }g) \rightarrow CO_{2}(\text{ }g) + Z\text{ }kJ{\text{ }mol}^{- 1} \end{matrix}
    At constant temperature. Then

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  15. 40
    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×12 m\geq\sqrt{\frac{h}{\pi}}\times\frac{1}{2\text{ }m}. In the light of the above statements, choose the correct answer from the options given below :

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  16. 41
    Which one of the following reaction sequences will give an azo dye?

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  17. 42
    Drug X becomes ineffective after 50%50\% decomposition. The original concentration of drug in a bottle was 16mg/mL16mg/mL which becomes 4 mg/mLmg/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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  18. 43
    The type of oxide formed by the element among Li,Na,Be,Mg,BLi,Na,Be,Mg,B and Al that has the least atomic radius is :

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  19. 44
    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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  20. 45
    Which one of the following, with HBr will give a phenol?

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  21. 46
    Consider the following low-spin complexes K3[Co(NO2)6],K4[Fe(CN)6],K3[Fe(CN)6]K_{3}\left\lbrack Co\left( NO_{2} \right)_{6} \right\rbrack,K_{4}\left\lbrack Fe(CN)_{6} \right\rbrack,K_{3}\left\lbrack Fe(CN)_{6} \right\rbrack, Cu2[Fe(CN)6]Cu_{2}\left\lbrack Fe(CN)_{6} \right\rbrack and Zn2[Fe(CN)6]Zn_{2}\left\lbrack Fe(CN)_{6} \right\rbrack. The sum of the spin-only magnetic moment values of complexes having yellow colour is _____\_\_\_\_\_ B.M. (answer is nearest integer)
  22. 47
    Isomeric hydrocarbons →\rightarrow negative Baeyer's test (Molecular formula C9H12C_{9}H_{12} ). The total number of isomers from above with four different non-aliphatic substitution sites is -
  23. 48
    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 ______\_\_\_\_\_\_ %.
  24. 49
    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=137ingmol −1O = 16,\text{ }S = 32,Ba = 137in_{gmol}\ ^{- 1} )
  25. 50
    Total number of non -bonded electrons present in NO2−NO_{2}^{-}ion based on Lewis theory is ______\_\_\_\_\_\_ .

Mathematics

25 questions

  1. 51
    If the set of all a∈Ra \in R, for which the equation 2x2+(a−5)x+15=3a2x^{2}+ (a - 5)x + 15 = 3a has no real root, is the interval (α,β)(\alpha,\beta), and X={x∈Z:α<x<β}X = \{ x \in Z:\alpha < x < \beta\}, then ∑x∈Xx2\sum_{x \in X} x^{2} is equal to

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  2. 52
    If sin⁡x+sin⁡2x=1,x∈(0,π2)\sin x +\sin^{2}x = 1,x \in\left( 0,\frac{\pi}{2} \right), then (cos⁡12x+tan⁡12x)+3(cos⁡10x+tan⁡10x+cos⁡8x+tan⁡8x)\left( \cos^{12}x +\tan^{12}x \right)+ 3\left( \cos^{10}x +\tan^{10}x +\cos^{8}x +\tan^{8}x \right) +(cos⁡6x+tan⁡6x)+\left( \cos^{6}x +\tan^{6}x \right) is equal to

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  3. 53
    Let the area enclosed between the curves ∣y∣=1−|y| = 1 - x2x^{2} and x2+y2=1x^{2}+y^{2}= 1 be α\alpha. If 9α=βπ+γ;β,γ9\alpha = \beta\pi + \gamma;\beta,\gamma are integers, then the value of ∣β−γ∣|\beta - \gamma| equals

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  4. 54
    If the domain of the function log⁡5(18x−x2−77)\log_{5}\left( 18x -x^{2}- 77 \right) is (α,β)(\alpha,\beta) and the domain of the functionlog⁡(x−1)(2x2+3x−2x2−3x−4)\log_{(x - 1)}\left( \frac{2x^{2}+ 3x - 2}{x^{2}- 3x - 4} \right) is (γ,δ)(\gamma,\delta), then α2+β2+γ2\alpha^{2}+\beta^{2}+\gamma^{2} is equal to :

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  5. 55
    Let the function f(x)=(x2−1)∣x2−ax+2∣+cos⁡∣x∣f(x) =\left( x^{2}- 1 \right)\left| x^{2}- ax + 2 \right|+ \cos|x| be not differentiable at the two points x=α=2x = \alpha = 2 and x=βx = \beta. Then the distance of the point (α,β)(\alpha,\beta) from the line 12x+5y+10=012x + 5y + 10 = 0 is equal to :

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  6. 56
    Let a straight line LL pass through the point P(2,−1,3)P(2, - 1,3) and be perpendicular to the lines x−12=y+11=z−3−2\frac{x - 1}{2}=\frac{y + 1}{1}=\frac{z - 3}{- 2} and x−31=y−23=z+24\frac{x - 3}{1}=\frac{y - 2}{3}=\frac{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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  7. 57
    Let S=N∪{0}\mathbf{S = N \cup \{ 0\}}. Define a relation RR from S to RR by :
    R={(x,y):log⁡ey=xlog⁡e(25),x∈ S,y∈R}.R =\left\{ (x,y):\log_{e}y = x\log_{e}\left( \frac{2}{5} \right),x \in\text{ }S,y \in R \right\}.
    Then, the sum of all the elements in the range of RR is equal to

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  8. 58
    Let the line x+y=1x + y = 1 meet the axes of xx and yy at AA and BB, respectively. A right angled triangle AMN is inscribed in the triangle OABOAB, where OO is the origin and the points M and N lie on the lines OB and ABAB, respectively. If the area of the triangle AMN is 49\frac{4}{9} of the area of the triangle OABOAB and AN:NB=λ:1AN:NB = \lambda:1, then the sum of all possible value(s) of λ\lambda is :

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  9. 59
    If αx+βy=109\alpha x + \beta y = 109 is the equation of the chord of the ellipse x29+y24=1\frac{x^{2}}{9}+\frac{y^{2}}{4}= 1, whose mid point is (52,12)\left( \frac{5}{2},\frac{1}{2} \right), then α+β\alpha + \beta is equal to

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  10. 60
    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 440^{\text{th~}} position in this arrangement, is :

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  11. 61
    Let α,β(α≠β)\alpha,\beta(\alpha \neq \beta) be the values of m , for which the equations x+y+z=1;x+2y+4z=mx + y + z = 1;x + 2y + 4z = m and x+4y+10z=m2x + 4y + 10z =m^{2} have infinitely many solutions. Then the value of ∑n=110(nα+nβ)\sum_{n = 1}^{10} \left( n^{\alpha}+n^{\beta} \right) is equal to :

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  12. 62
    Let A=[aij]A =\left\lbrack a_{ij} \right\rbrack be a matrix of order 3×33 \times 3, with aij=(2)i+ja_{ij}= (\sqrt{2})^{i + j}. If the sum of all the elements in the third row of A2A^{2} is α+β2,α,β∈Z\alpha + \beta\sqrt{2},\alpha,\beta \in Z, then α+β\alpha + \beta is equal to

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  13. 63
    Let PP be the foot of the perpendicular from the point (1,2,2)(1,2,2) on the line L:x−11=y+1−1=z−22L:\frac{x - 1}{1}=\frac{y + 1}{- 1}=\frac{z - 2}{2}. Let the line r→=(−i^+j^−2k^)+λ(i^−j^+k^),λ∈R\overrightarrow{r}= ( -\widehat{i}+\widehat{j}- 2\widehat{k}) + \lambda(\widehat{i}-\widehat{j}+\widehat{k}),\lambda \in R, intersect the line L at Q . Then 2(PQ)22(PQ)^{2} is equal to:

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  14. 64
    Let a circle CC pass through the points (4,2)(4,2) and (0(0, 2 ), and its centre lie on 3x+2y+2=03x + 2y + 2 = 0. Then the length of the chord, of the circle CC, whose midpoint is (1,2)(1,2), is:

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  15. 65
    Let A=[aij]A =\left\lbrack a_{ij} \right\rbrack be a 2×22 \times 2 matrix such that aij∈{0,1}a_{ij}\mathbf{\in \{ 0,1\}} for all i and j . Let the random variable X denote the possible values of the determinant of the matrix AA. Then, the variance of X is:

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  16. 66
    Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains nn 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, is 29/4529/45, then n is equal to:

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  17. 67
    The remainder, when 71037^{103} is divided by 23 , is equal to:

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  18. 68
    Let f(x)=∫0xt(t2−9t+20)dt, 1≤x≤5f(x) =\int_{0}^{x} t\left( t^{2}- 9t + 20 \right)dt,\ 1 \leq x \leq 5. If the range of ff is [α,β]\lbrack\alpha,\beta\rbrack, then 4(α+β)4(\alpha + \beta) equals:

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  19. 69
    Let â be a unit vector perpendicular to the vectors b→=i^−2j^+3k^\overrightarrow{b}=\widehat{i}- 2\widehat{j}+ 3\widehat{k} and c→=2i^+3j^−k^\overrightarrow{c}= 2\widehat{i}+ 3\widehat{j}-\widehat{k}, and makes an angle of cos⁡−1(−13)\cos^{- 1}\left( -\frac{1}{3} \right) with the vector i^+j^+k^\widehat{i}+\widehat{j}+\widehat{k}. If a^\widehat{a} makes an angle of π3\frac{\pi}{3} with the vector i^+αj^+k^\widehat{i}+ \alpha\widehat{j}+\widehat{k}, then the value of α\alpha is :

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  20. 70
    If for the solution curve y=f(x)y = f(x) of the differential equation dydx+(tan⁡x)y=2+secx(1+2secx)2\frac{dy}{dx}+ (\tan x)y =\frac{2 + secx}{(1 + 2secx)^{2}}, x∈(−π2,π2),f(π3)=310x \in\left( \frac{- \pi}{2},\frac{\pi}{2} \right),f\left( \frac{\pi}{3} \right)=\frac{\sqrt{3}}{10}, then f(π4)f\left( \frac{\pi}{4} \right) is equal to:

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  21. 71
    If 24∫0π4(sin⁡∣4x−π12∣+[2sin⁡x])dx=2π+α24\int_{0}^{\frac{\pi}{4}} \left( \sin\left| 4x-\frac{\pi}{12} \right|+ \lbrack 2\sin x\rbrack \right)dx= 2\pi+\alpha, where [.] denotes the greatest integer function, then α\alpha is equal to  _____\ \_\_\_\_\_ .
  22. 72
    If lim⁡t→0(∫01(3x+5)tdx)1t=α5e(85)23\lim_{t \rightarrow 0} \left( \int_{0}^{1} (3x + 5)^{t}dx \right)^{\frac{1}{t}}=\frac{\alpha}{5e}\left( \frac{8}{5} \right)^{\frac{2}{3}}, then α\alpha is equal to ____\_\_\_\_ .
  23. 73
    Let a1,a2,…,a2024a_{1},a_{2},\ldots,a_{2024} be an Arithmetic Progression such that a1+(a5+a10+a15+…+a2020)+a2024=a_{1}+\left( a_{5}+a_{10}+a_{15}+ \ldots +a_{2020} \right)+a_{2024}= 2233. Then a1+a2+a3+…+a2024a_{1}+a_{2}+a_{3}+ \ldots +a_{2024} is equal to
  24. 74
    Let integers a,b∈[−3,3]a,b \in \lbrack - 3,3\rbrack be such that a+b≠0a + b \neq 0. Then the number of all possible ordered pairs (a, b), for which ∣z−az+b∣=1\left| \frac{z - a}{z + b} \right|= 1 and ∣z+1ωω2ωz+ω21ω21z+ω∣=1,z∈C\left| \begin{matrix} z + 1 & \omega & \omega^{2} \\ \omega & z +\omega^{2} & 1 \\ \omega^{2} & 1 & z + \omega \end{matrix} \right|= 1,z \in C, where ω\omega and ω2\omega^{2} are the roots of x2+x+x^{2}+ x + 1=01 = 0, is equal to  ____\ \_\_\_\_
  25. 75
    Let y2=12xy^{2}= 12x the parabola and S be its focus. Let PQ be a focal chord of the parabola such that (SP) (SQ)=1474(SQ) =\frac{147}{4}. Let CC be the circle described taking PQPQ as a diameter. If the equation of a circle CC is 64x2+64y2−αx−643y=β64x^{2}+ 64y^{2}- \alpha x - 64\sqrt{3}y = \beta, then β−α\beta - \alpha is equal to  _____\ \_\_\_\_\_ -

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