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JEE Mains 2025 (2 Apr, 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
    Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : Net dipole moment of a polar linear isotropic dielectric substance is not zero even in the absence of an external electric field. Reason (R) : In absence of an external electric field, the different permanent dipoles of a polar dielectric substance are oriented in random directions. In the light of the above statements, choose the most appropriate answer from the options given below:

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  2. 2
    In a moving coil galvanometer, two moving coils M1M_{1} and M2M_{2} have the following particulars:
    R1=5Ω, N1=15, A1=3.6×10−3 m2, B1=0.25 TR_{1}= 5\Omega,{\text{ }N}_{1}= 15,{\text{ }A}_{1}= 3.6 \times10^{- 3}{\text{ }m}^{2},{\text{ }B}_{1}= 0.25\text{ }T
    R2=7Ω, N2=21, A2=1.8×10−3 m2, B2=0.50 TR_{2}= 7\Omega,{\text{ }N}_{2}= 21,{\text{ }A}_{2}= 1.8 \times10^{- 3}{\text{ }m}^{2},{\text{ }B}_{2}= 0.50\text{ }T
    Assuming that torsional constant of the springs are same for both coils, what will be the ratio of voltage sensitivity of M1M_{1} and M2M_{2} ?

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  3. 3
    The moment of inertia of a circular ring of mass MM and diameter r about a tangential axis lying in the plane of the ring is :

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  4. 4
    Two water drops each of radius ' rr ' coalesce to form a bigger drop. If ' T ' is the surface tension, the surface energy released in this process is :

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  5. 5
    An electron with mass ' m ' with an initial velocity (t=0)v→=v0i^ (v0>0)(t = 0)\overrightarrow{v}=v_{0}\widehat{i}\ \left( v_{0}> 0 \right) enters a magnetic field B→=B0j^\overrightarrow{B}=B_{0}\widehat{j}. If the initial de-Broglie wavelength at t=0t= 0 is λ0\lambda_{0} then its value after time ' tt ' would be :

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  6. 6
    A sinusoidal wave of wavelength 7.5 cm travels a distance of 1.2 cm along the xx- direction in 0.3 sec . The crest PP is at x=0x= 0 at t=0sect= 0sec and maximum displacement of the wave is 2 cm . Which equation correctly represents this wave ?

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  7. 7
    Given a charge qq, current II and permeability of vacuum μ0\mu_{0}. Which of the following quantity has the dimension of momentum?

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  8. 8
    A solenoid having area A and length ' ll ' is filled with a material having relative permeability 2 . The magnetic energy stored in the solenoid is :

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  9. 9
    Two large plane parallel conducting plates are kept 10 cm apart as shown in figure. The potential difference between them is V. The potential difference between the points A and B (shown in the figure) is :

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  10. 10
    Identify the characteristics of an adiabatic process in a monoatomic gas. (A) Internal energy is constant. (B) Work done in the process is equal to the change in internal energy. (C) The product of temperature and volume is a constant. (D) The product of pressure and volume is a constant. (E) The work done to change the temperature from T1T_{1} to T2T_{2} is proportional to (T2−T1)\left( T_{2} - T_{1} \right) Choose the correct answer from the options given below :

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  11. 11
    Assuming the validity of Bohr's atomic model for hydrogen like ions the radius of Li++Li^{+ +}ion in its ground state is given by 1Xa0\frac{1}{X}a_{0}, where X=X= . (Where a0a_{0} is the first Bohr's radius.)

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  12. 12
    Energy released when two deuterons ( 1H2)\left( \ _{1}H^{2} \right) fuse to form a helium nucleus ( 2He4)\left( \ _{2}He^{4} \right) is : (Given : Binding energy per nucleon of  1H2=1.1MeV\ _{1}H^{2}= 1.1MeV and binding energy per nucleon of  2He4=7.0MeV\ _{2}He^{4}= 7.0MeV )

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  13. 13
    In the digital circuit shown in the figure, for the given inputs the P and Q values are.

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  14. 14
    Two identical objects are placed in front of convex mirror and concave mirror having same radii of curvature of 12 cm , at same distance of 18 cm from the respective mirrors. The ratio of sizes of the images formed by convex mirror and by concave mirror is :

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  15. 15
    A sportsman runs around a circular track of radius rr such that he traverses the path ABABABAB. The distance travelled and displacement, respectively, are

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  16. 16
    A body of mass 1 kg is suspended with the help of two strings making angles as shown in figure. Magnitude of tensions T1T_{1} and T2T_{2}, respectively, are (in N ) :

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  17. 17
    A bi-convex lens has radius of curvature of both the surfaces same as 16 cm\frac{1}{6}\text{ }cm. If this lens is required to be replaced by another convex lens having different radii of curvatures on both sides (R1≠R2)\left( R_{1}\neq R_{2} \right), without any change in lens power then possible combination of R1R_{1} and R2R_{2} is :

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  18. 18
    If μ0\mu_{0} and ε0\varepsilon_{0} are the permeability and permittivity of free space, respectively, then the dimension of (1μ0ε0)\left( \frac{1}{\mu_{0}\varepsilon_{0}} \right) is :

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  19. 19
    Match List-I with List-II.
    List-IList-II
    (A) Heat capacity of body(I) Jkg−1J{kg}^{-1}
    (B) Specific heat capacity of body(II) JK−1{JK}^{-1}
    (C) Latent heat(III) Jkg−1K−1J{kg}^{-1}K^{-1}
    (D) Thermal conductivity(IV) Jm−1K−1s−1{Jm}^{-1}K^{-1}s^{-1}
    Choose the correct answer from the options given below :

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  20. 20
    Consider a circular loop that is uniformly charged and has a radius a2a\sqrt{2}. Find the position along the positive zz-axis of the cartesian coordinate system where the electric field is maximum if the ring was assumed to be placed in xy-plane at the origin :

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  21. 21
    A wheel of radius 0.2 m rotates freely about its center when a string that is wrapped over its rim is pulled by force of 10 N as shown in figure. The established torque produces an angular acceleration of 2rad/s22rad/s^{2}. Moment of inertia of the wheel is ____\_\_\_\_ kgm2kgm^{2}. (Acceleration due to gravity =10 m/s2= 10\text{ }m/s^{2} )
  22. 22
    The internal energy of air i n 4 m×4 m×3 m4\text{ }m \times 4\text{ }m \times 3\text{ }m sized room at 1 atmospheric pressure will be ____\_\_\_\_ ×106 J\times 10^{6}\text{ }J. (Consider air as diatomic molecule)
  23. 23
    A ray of light suffers minimum deviation when incident on a prism having angle of the prism equal to 60∘60^{\circ}. The refractive index of the prism material is 2\sqrt{2}. The angle of incidence (in degrees) is ____\_\_\_\_.
  24. 24
    The length of a light string is 1.4 m when the tension on it is 5 N . If the tension increases to 7 N , the length of the string is 1.56 m . The original length of the string is ____\_\_\_\_ m.
  25. 25
    A satellite of mass 1000 kg is launched to revolve around the earth in an orbit at a height of 270 km from the earth's surface. Kinetic energy of the satellite in this orbit is ____\_\_\_\_ ×1010 J\times 10^{10}\text{ }J. (Mass of earth =6×1024 kg= 6 \times 10^{24}\text{ }kg, Radius of earth == 6.4×106 m6.4 \times 10^{6}\text{ }m, Gravitational constant == 6.67×10−11Nm2 kg−26.67 \times 10^{- 11}{Nm}^{2}{\text{ }kg}^{- 2} )

Chemistry

25 questions

  1. 26
    When a concentrated solution of sulphanilic acid and 1-naphthylamine is treated with nitrous acid ( 273 K ) and acidified with acetic acid, the mass (g) of 0.1 mole of product formed is : (Given molar mass in gmol−1H:1,C:12, N:14gmol^{- 1}H:1,C:12,\text{ }N:14, O : 16, S : 32)

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  2. 27
    The d-orbital electronic configuration of the complex among [Co(en)3]3+, [CoF6]3−\left\lbrack Co(en)_{3} \right\rbrack^{3 +},\ \left\lbrack CoF_{6} \right\rbrack^{3 -}, [Mn(H2O)6]2+\left\lbrack Mn\left( H_{2}O \right)_{6} \right\rbrack^{2 +} and [Zn(H2O)6]2+\left\lbrack Zn\left( H_{2}O \right)_{6} \right\rbrack^{2 +} that has the highest CFSE is :

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  3. 28
    Given below are two statements: Statement (I) : Neopentane forms only one monosubstituted derivative. Statement (II) : Melting point of neopentane is higher than n-pentane In the light of the above statements, choose the most appropriate answer from the options given below :

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  4. 29
    Which among the following molecules is (a) involved in sp3 dsp^{3}\text{ }d hybridization, (b) has different bond lengths and (c) has lone pair of electrons on the central atom?

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  5. 30
    Formation of Na4[Fe(CN)5NOS]Na_{4}\left\lbrack Fe(CN)_{5}NOS \right\rbrack, a purple coloured complex formed by addition of sodium nitroprusside in sodium carbonate extract of salt indicates the presence of :

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  6. 31
    In 3,3-dimethylhex-1-en-4-yne, there are ___ sp3sp^{3}, ___ sp2sp^{2} and ___ spsp hybridised carbon atoms respectively:

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  7. 32
    Which of the following statements are true ? (A) The subsidiary quantum number ll describes the shape of the orbital occupied by the electron. (B) is the boundary surface diagram of the 2px2p_{x} orbital. (C) The + and - signs in the wave function of the 2px2p_{x} orbital refer to charge. (D) The wave function of 2px2p_{x} orbital is zero everywhere in the xy plane.

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  8. 33
    The type of hybridization and the magnetic property of [MnCl6]3−\left\lbrack MnCl_{6} \right\rbrack^{3 -} are :

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  9. 34
    Consider the following reactions. From these reactions which reaction will give carboxylic acid as a major product ? (A) R−C≡NR - C \equiv N →(i)H+/H2Omild Condition\underset{mild\ Condition}{\overset{{(i)H}^{+}/H_{2}O}{\rightarrow}} (B) R−MgX→(i)CO2(ii)H3O+R - \underset{(ii)H_{3}O^{+}}{MgX\overset{(i){CO}_{2}}{\rightarrow}} (C) R−C≡N→(i) SnCl2/HCl  (ii) H3O+ R - C \equiv N\underset{\text{~(ii)~}H_{3}O^{+}\ }{\overset{\text{(i)~}{SnCl}_{2}/HCl\ }{\rightarrow}} (D) R−CH2−OH→PCC R - {CH}_{2} - OH\overset{PCC\ }{\rightarrow} (E) R−COCl→(i) H2/Pd−BaSO4(ii) Br2 waterR - COCl\underset{\text{(ii)~}{Br}_{2}\text{~water}}{\overset{\text{(i)~}H_{2}/Pd - {BaSO}_{4}}{\rightarrow}} Choose the correct answer from the options given below :

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  10. 35
    Electronic configuration of four elements A,B,CA,B,C and D are given below (A) 1 s22 s22p31{\text{ }s}^{2}2{\text{ }s}^{2}2p^{3} (B) 1 s22 s22p41{\text{ }s}^{2}2{\text{ }s}^{2}2p^{4} (C) 1 s22 s22p51{\text{ }s}^{2}2{\text{ }s}^{2}2p^{5} (D) 1 s22 s22p21{\text{ }s}^{2}2{\text{ }s}^{2}2p^{2} Which of the following is the correct order of increasing electronegativity (Pauling's scale) ?

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  11. 36
    Match List-I with List-II
    List-I (Purification technique)List-II (Mixture of organic compounds)
    (A)Distillation (simple)(I)Diesel + Petrol
    (B)Fractional distillation(II)Aniline + Water
    (C)Distillation under reduced pressure(III)Chloroform + Aniline
    (D)Steam distillation(IV)Glycerol + Spent-lye
    Choose the correct answer from the options given below:

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  12. 37
    ' x ' g of NaCl is added to water in a beaker with a lid. The temperature of the system is raised from 1∘C1^{\circ}C to 25∘C25^{\circ}C. Which out of the following plots, is best suited for the change in the molarity (M) of the solution with respect to temperature ?[Consider the solubility of NaCl remains unchanged over the temperature range]

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  13. 38
    Arrange the following in order of magnitude of work done by the system / on the system at constant temperature : (a) ∣wreversible ∣\left| w_{\text{reversible~}} \right| for expansion in infinite stage. (b) ∣wirreversible ∣\left| w_{\text{irreversible~}} \right| for expansion in single stage. (c) ∣wreversible ∣\left| w_{\text{reversible~}} \right| for compression in infinite stage. (d) ∣wirreversible ∣\left| w_{\text{irreversible~}} \right| for compression in single stage. Choose the correct answer from the options given below:

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  14. 39
    Reactant A converts to product D through the given mechanism (with the net evolution of heat) : A→BA \rightarrow B slow; ΔH=+ve\Delta H = + ve B→CB \rightarrow C fast; ΔH=−ve\Delta H = - ve C→DC \rightarrow D fast ;ΔH=−ve;\Delta H = - ve Which of the following represents the above reaction mechanism?

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  15. 40
    The nature of oxide (TeO2)\left( TeO_{2} \right) and hydride (TeH2)\left( TeH_{2} \right) formed by Te , respectively are :

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  16. 41
    Match List-I With List-II.
    List-I (Reaction)List-II (Name of reaction)
    (A)2C6H5X+2Na→dry etherC6H5−C6H5+2NaX2C_{6}H_{5}X + 2Na \xrightarrow{\text{dry ether}} C_{6}H_{5}-C_{6}H_{5} + 2NaX(I)Lucas reaction
    (B)ArN2+X−→Cu, HClArCl+N2↑+CuXArN_{2}^{+}X^{-} \xrightarrow{Cu,\ HCl} ArCl + N_{2} \uparrow + CuX(II)Finkelstein reaction
    (C)C2H5Br+NaI→dry acetoneC2H5I+NaBrC_{2}H_{5}Br + NaI \xrightarrow{\text{dry acetone}} C_{2}H_{5}I + NaBr(III)Fittig reaction
    (D)CH3C(OH)(CH3)CH3→HCl, ZnCl2CH3C(Cl)(CH3)CH3CH_{3}C(OH)(CH_{3})CH_{3} \xrightarrow{HCl,\ ZnCl_{2}} CH_{3}C(Cl)(CH_{3})CH_{3}(IV)Gattermann reaction
    Choose the correct answer from the options given below:

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  17. 42
    Consider the following chemical equilibrium of the gas phase reaction at a constant temperature : A(g)⇌B(g)+C(g)A(g) \rightleftharpoons B(g) + C(g).If p being the total pressure, KpK_{p} is the pressure equilibrium constant and α\alpha is the degree of dissociation, then which of the following is true at equilibrium ?

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  18. 43
    Which of the following graphs correctly represents the variation of thermodynamic properties of Haber's process?

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  19. 44
    A tetrapeptide " xx " on complete hydrolysis produced glycine (Gly), alanine (Ala), valine (Val), leucine (Leu) in equimolar proportion each. The number of tetrapeptides (sequences) possible involving each of these amino acids is.

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  20. 45
    In Dumas' method for estimation of nitrogen, 0.5 gram of an organic compound gave 60 mL of nitrogen collected at 300 K temperature and 715 mm Hg pressure. The percentage composition of nitrogen in the compound (Aqueous tension at 300 K=15 mmHg300\text{ }K = 15\text{ }mmHg ) is.

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  21. 46
    For the reaction A→BA \rightarrow B the following graph was obtained. The time required (in seconds) for the concentration of A to reduce to 2.5 g L−12.5\text{ }g{\text{ }L}^{- 1} (if the initial concentration of AA was 50 g L−150\text{ }g{\text{ }L}^{- 1} ) is ____ . (Nearest integer) Given : log⁡2=0.3010\log2 = 0.3010
  22. 47
    0.2%(w/v)0.2\%(w/v) solution of NaOH is measured to have resistivity 870.0 mΩ m870.0\text{ }m\Omega\text{ }m. The molar conductivity of the solution will be ____\_\_\_\_ ×102 mS dm2 mol−1\times 10^{2}\ mS\ dm^{2}\ mol^{-1}. (Nearest integer)
  23. 48
    Consider the above sequence of reactions. 151 g of 2-bromopentane is made to react. Yield of major product P is 80%80\% whereas Q is 100%100\%. Mass of product Q obtained is ____ g. (Given molar mass in g mol−1g\ mol^{-1}: H: 1, C: 12, O: 16, Br: 80)
  24. 49
    When 1 g each of compounds AB and AB2{AB}_{2} are dissolved in 15 g of water separately, they increased the boiling point of water by 2.7 K and 1.5 K respectively. The atomic mass of A (in amu) is ____ ×10−1\times 10^{- 1} (Nearest integer).(Given : Molal boiling point elevation constant is 0.5 K kg mol−10.5\text{ }K\text{ }kg{\text{ }mol}^{- 1} )
  25. 50
    The spin-only magnetic moment value of Mn+M^{n +} ion formed by the metal among Ni,Zn,MnNi, Zn, Mn and Cu that has the least enthalpy of atomisation is ____ . (in nearest integer) Here n is equal to the number of diamagnetic complexes among K2[NiCl4]K_{2}\left\lbrack {NiCl}_{4} \right\rbrack, [Zn(H2O)6]Cl2\left\lbrack Zn\left( H_{2}O \right)_{6} \right\rbrack{Cl}_{2}, K3[Mn(CN)6]K_{3}\left\lbrack Mn(CN)_{6} \right\rbrack and [Cu(PPh3)3I]\left\lbrack Cu\left( {PPh}_{3} \right)_{3}I \right\rbrack

Mathematics

25 questions

  1. 51
    If the image of the point P(1,0,3)P(1,0,3) in the line joining the points A(4,7,1)A(4,7,1) and B(3,5,3)B(3,5,3) is Q(α,β,γ)Q(\alpha,\beta,\gamma), then α+β+γ\alpha+\beta+\gamma is equal to.

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  2. 52
    Let f:[1,∞)→[2,∞)f:\lbrack 1,\infty) \rightarrow \lbrack 2,\infty) be a differentiable function, If 10∫1xf(t)dt=5xf(x)−x5−910\int_{1}^{x} f(t)dt = 5xf(x) -x^{5}- 9 for all x≥1x \geq 1, then the value of f(3)f(3) is :

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  3. 53
    The number of terms of an A.P. is even; the sum of all the odd terms is 24 , the sum of all the even terms is 30 and the last term exceeds the first by 212\frac{21}{2}. Then the number of terms which are integers in the A.P. is :

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  4. 54
    Let A={1,2,3,…,100}\mathbf{A = \{ 1,2,3,\ldots,100\}} and R be a relation on A such that R={(a,b):a=2b+1}R = \{(a,b):a = 2b + 1\}. Let (a1,a2)\left( a_{1},a_{2} \right), (a2,a3),(a3,a4),…,(ak,ak+1)\left( a_{2},a_{3} \right),\left( a_{3},a_{4} \right),\ldots,\left( a_{k},a_{k+ 1} \right) be a sequence of kk elements of R such that the second entry of an ordered pair is equal to the first entry of the next ordered pair. Then the largest integer k , for which such a sequence exists, is equal to :

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  5. 55
    If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is :

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  6. 56
    The line LlL_{l} is parallel to the vector a→=−3i^+2j^+4k^\overrightarrow{a}= - 3\widehat{i}+ 2\widehat{j}+ 4\widehat{k} and passes through the point (7,6,2)(7,6,2) and the line L2L_{2} is parallel to the vector b→=2i^+j^+3k^\overrightarrow{b}= 2\widehat{i}+\widehat{j}+ 3\widehat{k} and passes through the point (5, 3, 4). The shortest distance between the lines L1L_{1} and L2L_{2} is :

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  7. 57
    Let (a,b)\left( a,b \right) be the point of intersection of the curve x2=2yx^{2}= 2y and the straight line y−y- 2x−6=02x- 6 = 0 in the second quadrant. Then the integral I=∫ab9x21+5xdxI=\int_{a}^{b} \frac{9x^{2}}{1 +5^{x}}dx is equal to:

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  8. 58
    If the system of equation
    2x+λy+3z=52x+\lambda y+ 3z= 5
    3x+2y−z=73x+ 2y-z= 7
    4x+5y+μz=94x+ 5y+\mu z= 9
    has infinitely many solutions, then (λ2+μ2)\left( \lambda^{2}+\mu^{2} \right) is equal to :

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  9. 59
    If θ∈[−7π6,4π3]\theta\in\left\lbrack -\frac{7\pi}{6},\frac{4\pi}{3} \right\rbrack, then the number of solutions of 3csc⁡2θ−2(3−1)csc⁡θ−4=0\sqrt{3}\csc^{2}\theta- 2(\sqrt{3}- 1)\csc\theta- 4 = 0, is equal to.

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  10. 60
    Given three identical bags each containing 10 balls, whose colours are as follows :
    BagRedBlueGreen
    Bag I325
    Bag II433
    Bag III514
    A person chooses a bag at random and takes out a ball. If the ball is Red, the probability that it is from bag I is p and if the ball is Green, the probability that it is from bag III is q , then the value of (1p+1q)\left( \frac{1}{p}+\frac{1}{q} \right) is :

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  11. 61
    If the mean and the variance of 6,4,a,8,b,12,106,4,a,8,b,12,10, 13 are 9 and 9.25 respectively, then a+b+aba+b+ab is equal to :

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  12. 62
    If the domain of the function f(x)=110+3x−x2+1x+∣x∣f(x) =\frac{1}{\sqrt{10 + 3x -x^{2}}}+\frac{1}{\sqrt{x + |x|}} is (a,b)(a,b), then (1+a)2+b2(1 +a)^{2}+b^{2} is equal to :

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  13. 63
    4∫01(13+x2+1+x2)dx−3log⁡e(3)4\int_{0}^{1} \left( \frac{1}{\sqrt{3 +x^{2}}+\sqrt{1 +x^{2}}} \right)dx - 3\log_{e}(\sqrt{3}) is equal to:

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  14. 64
    If lim⁡x→0cos⁡(2x)+acos(4x)−bx4\lim_{x\rightarrow 0} \frac{\cos(2x) +acos(4x) -b}{x^{4}} is finite, then (a+b)(a+b) is equal to:

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  15. 65
    If ∑r=010(10r+1−110r)⋅ 11Cr+1=α11−11111010\sum_{r = 0}^{10} \left( \frac{10^{r + 1}- 1}{10^{r}} \right)\cdot\ ^{11}C_{r + 1}=\frac{\alpha^{11}-11^{11}}{10^{10}}, then α\alpha is equal to :

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  16. 66
    The number of ways, in which the letters A,B,CA,B,C, D , E can be placed in the 8 boxes of the figure below so that no row remains empty and at most one letter can be placed in a box, is :

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  17. 67
    Let the point PP of the focal chord PQPQ of the parabola y2=16xy^{2}= 16x be (1,−4)(1, - 4). If the focus of the parabola divides the chord PQ in the ratio m:nm:n, gcd(m,n)=1gcd(m,n) = 1, then m2+n2m^{2}+n^{2} is equal to :

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  18. 68
    Let a→=2i^−3j^+k^,b→=3i^+2j^+5k^\overrightarrow{a}= 2\widehat{i}- 3\widehat{j}+\widehat{k},\overrightarrow{b}= 3\widehat{i}+ 2\widehat{j}+ 5\widehat{k} and a vector c→\overrightarrow{c} be such that (a→−c→)×b→=−18i^−3j^+12k^(\overrightarrow{a}-\overrightarrow{c}) \times\overrightarrow{b}= - 18\widehat{i}- 3\widehat{j}+ 12\widehat{k} and a→⋅c→=3\overrightarrow{a}\cdot\overrightarrow{c}= 3. If b→×c→=d→\overrightarrow{b}\times\overrightarrow{c}=\overrightarrow{d}, then ∣a→⋅d→∣|\overrightarrow{a}\cdot\overrightarrow{d}| is equal to :

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  19. 69
    Let the area of the triangle formed by a straight Line L:x+by+c=0L:x+by+c= 0 with co-ordinate axes be 48 square units. If the perpendicular drawn from the origin to the line LL makes an angle of 45∘45^{\circ} with the positive xx-axis, then the value of b2+c2b^{2}+c^{2} is:

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  20. 70
    Let AA be a 3×33 \times 3 real matrix such that A2(A−2I)−A^{2}(A- 2I) - 4( A−I)=O4(\text{ }A - I) = O, where I and O are the identity and null matrices, respectively. If A5=αA2+βA+γIA^{5}=\alpha A^{2}+\beta A+\gamma I, where α,β\alpha,\beta and γ\gamma are real constants, then α+β+γ\alpha+\beta+\gamma is equal to:

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  21. 71
    Let y=y(x)y = y(x) be the solution of the differential equation dydx+2ysec⁡2x=2sec⁡2x+3tan⁡x⋅sec⁡2x\frac{dy}{dx} + 2y\sec^{2}x = 2\sec^{2}x + 3\tan x \cdot \sec^{2}x such that y(0)=54y(0) = \frac{5}{4}. Then 12(y(π4)−e−2)12\left( y\left( \frac{\pi}{4} \right) - e^{- 2} \right) is equal to
  22. 72
    If the sum of the first 10 terms of the series 4.11+4.14+4.21+4.24+4.31+4.34+…\frac{4.1}{1 + {4.1}^{4}} + \frac{4.2}{1 + {4.2}^{4}} + \frac{4.3}{1 + {4.3}^{4}} + \ldots is mn\frac{m}{n}, where gcd(m,n)=1gcd(m,n) = 1, then m+nm + n is equal to.
  23. 73
    If y=cos⁡(π3+cos⁡−1x2)y = \cos\left( \frac{\pi}{3} + \cos^{- 1}\frac{x}{2} \right), then (x−y)2+3y2(x - y)^{2} + 3y^{2} is equal to
  24. 74
    Let A(4,−2),B(1,1)A(4, - 2),B(1,1) and C(9,−3)C(9, - 3) be the vertices of a triangle ABCABC. Then the maximum area of the parallelogram AFDE , formed with vertices D,ED,E and FF on the sides BC,CABC,CA and ABAB of the triangle ABC respectively, is
  25. 75
    If the set of all a∈R−{1}a \in R - \left\{ 1 \right\}, for which the roots of the equation (1−a)x2+2(a−3)x+9=0(1 - a)x^{2} + 2(a - 3)x + 9 = 0 are positive is (−∞,−α]∪[β,γ)( - \infty, - \alpha\rbrack \cup \lbrack\beta,\gamma), then 2α+β+γ2\alpha + \beta + \gamma is equal to

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