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JEE Mains 2025 (2 Apr, Shift 1), 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
    A light wave is propagating with plane wave fronts of the type x+y+z=x+y+z= constant. The angle made by the direction of wave propagation with the x -axis is :

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  2. 2
    The equation for real gas is given by (P+aV2)(V−b)=RT\left( P +\frac{a}{V^{2}} \right)(V - b) = RT, where P,V,TP,V,T and R are the pressure, volume, temperature and gas constant, respectively. The dimension of ab−2ab^{- 2} is equivalent to that of :

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  3. 3
    A cord of negligible mass is wound around the rim of a wheel supported by spokes with negligible mass. The mass of wheel is 10 kg and radius is 10 cm and it can freely rotate without any friction. Initially the wheel is at rest. If a steady pull of 20 N is applied on the cord, the angular velocity of the wheel, after the cord is unwound by 1 m , would be :

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  4. 4
    A slanted object ABAB is placed on one side of convex lens as shown in the diagram. The image is formed on the opposite side. Angle made by the image with principal axis is:

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  5. 5
    Consider two infinitely large plane parallel conducting plates as shown below. The plates are uniformly charged with a surface charge density +σ+\sigma and −2σ- 2\sigma. The force experienced by a point charge +q placed at the mid point between two plates will be :

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  6. 6
    A river is flowing from west to east direction with speed of 9 km h−19\text{ }km{\text{ }h}^{- 1}. If a boat capable of moving at a maximum speed of 27 km h−127\text{ }km{\text{ }h}^{- 1} in still water, crosses the river in half a minute, while moving with maximum speed at an angle of 150∘150^{\circ} to direction of river flow, then the width of the river is :

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  7. 7
    A point charge +q is placed at the origin. A second point charge +9 q is placed at (d,0,0)(d,0,0) in Cartesian coordinate system. The point in between them where the electric field vanishes is :

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  8. 8
    The battery of a mobile phone is rated as 4.2 V , 5800 mAh . How much energy is stored in it when fully charged?

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  9. 9
    A particle is subjected two simple harmonic motions as : x1=7sin⁡5t cmx_{1}=\sqrt{7}\sin 5t\ cm and x2=27sin⁡(5t+π3)cmx_{2}= 2\sqrt{7}\sin\left( 5t+\frac{\pi}{3} \right)cm where x is displacement and tt is time in seconds. The maximum acceleration of the particle is x×10−2 ms−2x \times10^{- 2}{\text{ }ms}^{- 2}. The value of x is :

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  10. 10
    The relationship between the magnetic susceptibility (χ)(\chi) and the magnetic permeability (μ)(\mu) is given by : ( μ0\mu_{0} is the permeability of free space and μr\mu_{r} is relative permeability)

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  11. 11
    A Zener diode with 5 V Zener voltage is used to regulate an unregulated dc voltage input of 25 V . For a 400Ω400\Omega resistor connected in series, the zener current is found to be 4 times load current. The load current (IL)\left( I_{L} \right) and load resistance (RL)\left( R_{L} \right) are :

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  12. 12
    In an adiabatic process, which of the following statements is true ?

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  13. 13
    A square Lamina OABCOABC of length 10 cm is pivoted at ' O '. Forces act at Lamina as shown in figure. If Lamina remains stationary, then the magnitude of FF is :

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  14. 14
    Let B1B_{1} be the magnitude of magnetic field at center of a circular coil of radius RR carrying current II. Let B2B_{2} be the magnitude of magnetic field at an axial distance ' xx ' from the center. For x:R=3:4,B2B1x:R= 3:4,\frac{B_{2}}{B_{1}} is :

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  15. 15
    Considering Bohr's atomic model for hydrogen atom : (A) the energy of H atom in ground state is same as energy of He+{He}^{+}ion in its first excited state. (B) the energy of H atom in ground state is same as that for Li++{Li}^{+ +}ion in its second excited state. (C) the energy of H atom in its ground state is same as that of He+{He}^{+}ion for its ground state. (D) the energy of He+{He}^{+}ion in its first excited state is same as that for Li++{Li}^{+ +}ion in its ground state Choose the correct answer from the options given below :

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  16. 16
    Moment of inertia of a rod of mass ' M ' and length 'L' about an axis passing through its center and normal to its length is ' α\alpha '. Now the rod is cut into two equal parts and these parts are joined symmetrically to form a cross shape. Moment of inertia of cross about an axis passing through its center and normal to plane containing cross is :

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  17. 17
    A spherical surface separates two media of refractive indices 1 and 1.5 as shown in figure Distance of the image of an object ' O ', is :( C is the center of curvature of the spherical surface and R is the radius of curvature)

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  18. 18
    Match List-I with List-II.
    List-IList-II
    (A) Coefficient of viscosity(I) [ML0T−3]\left\lbrack ML^{0}T^{-3} \right\rbrack
    (B) Intensity of wave(II) [ML−2T−2]\left\lbrack ML^{-2}T^{-2} \right\rbrack
    (C) Pressure gradient(III) [M−1LT2]\left\lbrack M^{-1}LT^{2} \right\rbrack
    (D) Compressibility(IV) [ML−1T−1]\left\lbrack ML^{-1}T^{-1} \right\rbrack
    Choose the correct answer from the options given below :

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  19. 19
    A small bob of mass 100 mg and charge +10μC+ 10\mu C is connected to an insulating string of length 1 m . It is brought near to an infinitely long nonconducting sheet of charge density ' σ\sigma ' as shown in figure. If string subtends an angle of 45∘45^{\circ} with the sheet at equilibrium the charge density of sheet will be : (Given, ε0=8.85×10−12 F m\varepsilon_{0}= 8.85 \times10^{- 12}\frac{\text{ }F}{\text{ }m} and acceleration due to gravity, g=10 m/s2g = 10\text{ }m/s^{2} )

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  20. 20
    A monochromatic light is incident on a metallic plate having work function ϕ\phi. An electron, emitted normally to the plate from a point AA with maximum kinetic energy, enters a constant magnetic field, perpendicular to the initial velocity of electron. The electron passes through a curve and hits back the plate at a point B . The distance between A and B is : (Given : The magnitude of charge of an electron is e and mass is m,hm,h is Planck's constant and c is velocity of light. Take the magnetic field exists throughout the path of electron)

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  21. 21
    A vessel with square cross-section and height of 6 m is vertically partitioned. A small window of 100 cm2100\text{ }cm^{2} with a hinged door is fitted at a depth of 3 m in the partition wall. One part of the vessel is filled completely with water and the other side is filled with the liquid having density 1.5×103 kg/m31.5 \times 10^{3}\text{ }kg/m^{3}. What force (in N) one needs to apply on the hinged door so that it does not get opened?(Acceleration due to gravity =10 m/s2= 10\text{ }m/s^{2} )
  22. 22
    A steel wire of length 2 m and Young's modulus 2.0×1011Nm−22.0 \times 10^{11}{Nm}^{- 2} is stretched by a force. If Poisson ratio and transverse strain for the wire are 0.2 and 10−310^{- 3} respectively, then the elastic potential energy density of the wire is α×105\alpha \times 10^{5} (in SI units). The value of α\alpha is ____\_\_\_\_.
  23. 23
    If the measured angular separation between the second minimum to the left of the central maximum and the third minimum to the right of the central maximum is 30∘30^{\circ} in a single slit diffraction pattern recorded using 628 nm light, then the width of the slit is ____\_\_\_\_ μm\mu m.
  24. 24
    γA\gamma_{A} is the specific heat ratio of monoatomic gas A having 3 translational degrees of freedom. γB\gamma_{B} is the specific heat ratio of polyatomic gas BB having 3 translational, 3 rotational degrees of freedom and 1 vibrational mode. If γAγB=(1+1n)\frac{\gamma_{A}}{\gamma_{B}} = \left( 1 + \frac{1}{n} \right), then the value of nn is .
  25. 25
    A person travelling on a straight line moves with a uniform velocity v1v_{1} for a distance x and with a uniform velocity v2v_{2} for the next 32x\frac{3}{2}x distance. The average velocity in this motion is 507 m/s\frac{50}{7}\text{ }m/s. If v1v_{1} is 5 m/s5\text{ }m/s then v2=v_{2} = ____\_\_\_\_ m/sm/s.

Chemistry

25 questions

  1. 26
    Designate whether each of the following compounds is aromatic or not aromatic.

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  2. 27
    An optically active alkyl halide C4H9Br[A]C_{4}H_{9}Br\lbrack A\rbrack reacts with hot KOH dissolved in ethanol and forms alkene [B] as major product which reacts with bromine to give dibromide [C]. The compound [C] is converted into a gas [D] upon reacting with alcoholic NaNH2NaNH_{2}. During hydration 18 gram of water is added to 1 mole of gas [D] on warming with mercuric sulphate and dilute acid at 333 K to form compound [E]. The IUPAC name of compound [E] is :

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  3. 28
    The property/properties that show irregularity in first four elements of group-17 is/are : (A) Covalent radius (B) Electron affinity (C) Ionic radius (D) First ionization energy Choose the correct answer from the options given below:

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  4. 29
    Which of the following graph correctly represents the plots of KHK_{H} at 1 bar gases in water versus temperature?

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  5. 30
    According to Bohr's model of hydrogen atom, which of the following statement is incorrect?

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  6. 31
    Two vessels A and B are connected via stopcock. The vessel A is filled with a gas at a certain pressure. The entire assembly is immersed in water and is allowed to come to thermal equilibrium with water. After opening the stopcock the gas from vessel A expands into vessel BB and no change in temperature is observed in the thermometer. Which of the following statement is true?

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  7. 32
    A solution is made by mixing one mole of volatile liquid AA with 3 moles of volatile liquid BB. The vapour pressure of pure A is 200 mm Hg and that of the solution is 500 mm Hg . The vapour pressure of pure B and the least volatile component of the solution, respectively, are :

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  8. 33
    CaCO3( s)+2HCl(aq)→CaCl2(aq)+CO2( g)+H2O(l)CaCO_{3}(\text{ }s) + 2HCl(aq) \rightarrow CaCl_{2}(aq) +CO_{2}(\text{ }g)+H_{2}O(l) Consider the above reaction, what mass of CaCl2CaCl_{2} will be formed if 250 mL of 0.76 M HCl reacts with 1000 g of CaCO3CaCO_{3} ? (Given : Molar mass of Ca,C,O,HCa,C,O,H and Cl are 40, 12,16,112,16,1 and 35.5 g mol−135.5\text{ }g{\text{ }mol}^{- 1}, respectively)

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  9. 34
    If equal volumes of AB2AB_{2} and XY (both are salts) aqueous solutions are mixed, which of the following combination will give a precipitate of AY2AY_{2} at 300 K ?(Given Ksp(K_{sp}( at 300 K)) for AY2=5.2×10−7AY_{2}= 5.2 \times10^{- 7} )

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  10. 35
    Among SO2,NF3,NH3,XeF2,ClF3SO_{2},NF_{3},NH_{3},XeF_{2},ClF_{3} and SF4SF_{4}, the hybridization of the molecule with non-zero dipole moment and highest number of lone-pairs of electrons on the central atom is.

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  11. 36
    Given below are two statements : Statement (I) : Vanillin will react with NaOH and also with Tollen's reagent. Statement (II) : Vanillin will undergo self aldol condensation very easily. In the light of the above statements, choose the most appropriate answer from the options given below :

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  12. 37
    Identify the correct statement among the following:

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  13. 38
    The correct order of basic nature on aqueous solution for the bases NH3,H2 N−NH2,CH3CH2NH2NH_{3},H_{2}\text{ }N -NH_{2},CH_{3}CH_{2}NH_{2}, (CH3CH2)2NH\left( CH_{3}CH_{2} \right)_{2}NH and (CH3CH2)3 N\left( CH_{3}CH_{2} \right)_{3}\text{ }N is :

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  14. 39
    Given below are two statements : Statement (I) : The metallic radius of Al is less than that of Ga . Statement (II) : The ionic radius of Al3+{Al}^{3 +} is less than that of Ga3+{Ga}^{3 +}. In the light of the above statements, choose the most appropriate answer from the options given below :

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  15. 40
    Given below are two statements : Statement (I) : In octahedral complexes, when Δo<P\Delta_{o} < P high spin complexes are formed. When Δ0>P\Delta_{0} > P low spin complexes are formed. Statement (II) : In tetrahedral complexes because of Δt<P\Delta_{t} < P, low spin complexes are rarely formed. In the light of the above statements, choose the most appropriate answer from the options given below :

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  16. 41
    Choose the correct tests with respective observations. (A) CuSO4{CuSO}_{4} (acidified with acetic acid) + K4[Fe(CN)6]→K_{4}\left\lbrack Fe(CN)_{6} \right\rbrack \rightarrow Chocolate brown precipitate. (B) FeCl3+K4[Fe(CN)6]→{FeCl}_{3} + K_{4}\left\lbrack Fe(CN)_{6} \right\rbrack \rightarrow Prussian blue precipitate. (C) ZnCl2+K4[Fe(CN)6]{ZnCl}_{2} + K_{4}\left\lbrack Fe(CN)_{6} \right\rbrack, neutralised with NH4OH{NH}_{4}OH →\rightarrow White or bluish white precipitate. (D) MgCl2+K4[Fe(CN)6]→{MgCl}_{2} + K_{4}\left\lbrack Fe(CN)_{6} \right\rbrack \rightarrow Blue precipitate. (E) BaCl2+K4[Fe(CN)6]{BaCl}_{2} + K_{4}\left\lbrack Fe(CN)_{6} \right\rbrack, neutralised with NaOH →\rightarrow White precipitate. Choose the correct answer from the options given below :

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  17. 42
    On complete combustion 1.0 g of an organic compound ( X ) gave 1.46 g of CO2CO_{2} and 0.567 g of H2OH_{2}O. The empirical formula mass of compound (X) is ___ g .(Given molar mass in gmol−1C:12,H:1,O:16gmol^{- 1}C:12,H:1,O:16 )

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  18. 43
    Consider the following compound (X) The most stable and least stable carbon radicals, respectively, produced by homolytic cleavage of corresponding C-H bond are :

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  19. 44
    Consider the following molecules : (p) (q) (r) (s) The correct order of rate of hydrolysis is :

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  20. 45
    A molecule with the formula AX4YAX_{4}Y has all its elements from p-block. Element A is rarest, monoatomic, non-radioactive from its group and has the lowest ionization enthalpy value among AA, XX and YY. Elements XX and YY have first and second highest electronegativity values respectively among all the known elements. The shape of the molecule is :

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  21. 46
    A transition metal (M) among Mn,Cr,CoMn,Cr,Co and Fe has the highest standard electrode potential (M3+/M2+)\left( M^{3 +}/M^{2 +} \right). It forms a metal complex of the type [M(CN)6]4−\left\lbrack M(CN)_{6} \right\rbrack^{4 -}. The number of electrons present in the ege_{g} orbital of the complex is ____ .
  22. 47
    Consider the following electrochemical cell at standard condition.
    Au( s)∣QH2,Q∣NH4X(0.01M)∣∣Ag+(1M)∣Ag( s)Ecell =+0.4 V\begin{matrix} & Au(\text{ }s)\left| {QH}_{2},Q \right|{NH}_{4}X(0.01M)||{Ag}^{+}(1M) \mid Ag(\text{ }s) \\ & E_{\text{cell~}} = + 0.4\text{ }V \end{matrix}
    The couple QH2/Q{QH}_{2}/Q represents quinhydrone electrode, the half cell reaction is given below
    [ Given : EAg+/Ago=+0.8 V and 2.303RT F=0.06 V]\left\lbrack \text{~Given :~}E_{{Ag}^{+}/Ag}^{o} = + 0.8\text{ }V\text{~and~}\frac{2.303RT}{\text{ }F} = 0.06\text{ }V \right\rbrack
    The pKb{pK}_{b} value of the ammonium halide salt (NH4X)\left( {NH}_{4}X \right) used here is ____ . (nearest integer)
  23. 48
    0.1 mol of the following given antiviral compound (P) will weigh____ ×10−1 g\times 10^{- 1}\text{ }g (Given : molar mass in gmol−1H:1,C:12, N:14g{mol}^{- 1}H:1,C:12,\text{ }N:14, O:16, F:19,I:127)O:16,\text{ }F:19,I:127)
  24. 49
    Consider the following equilibrium,
    CO(g)+2H2( g)⇌CH3OH(g)CO(g) + 2H_{2}(\text{ }g) \rightleftharpoons {CH}_{3}OH(g)
    0.1 mol of CO along with a catalyst is present in a 2dm32{dm}^{3} flask maintained at 500 K . Hydrogen is introduced into the flask until the pressure is 5 bar and 0.04 mol of CH3OH{CH}_{3}OH is formed. The Kp∘K_{p}^{\circ} is ____\_\_\_\_ ×10−3\times 10^{- 3} (nearest integer). Given : R=0.08dm3R = 0.08{dm}^{3} bar K−1 mol−1K^{- 1}{\text{ }mol}^{- 1} Assume only methanol is formed as the product and the system follows ideal gas behaviour.
  25. 50
    For the reaction A→A \rightarrow products, the plot of t1/2t_{1/2} against [A]0[A]_{0} is shown in the figure. The concentration of A at 10 minutes is ____×10−3 mol L−1\_\_\_\_ \times 10^{- 3}\text{ }mol{\text{ }L}^{- 1} (nearest integer). The reaction was started with 2.5 mol L−12.5\text{ }mol{\text{ }L}^{- 1} of A .

Mathematics

25 questions

  1. 51
    The largest n∈Nn \in N such that 3n3^{n} divides 50 ! is:

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  2. 52
    Let one focus of the hyperbola H:x2a2−y2b2=1H:\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}= 1 be at (10,0)(\sqrt{10},0) and the corresponding directrix be  x=910\ ^{x=\frac{9}{\sqrt{10}}}. If e and  l\ ^{l} respectively are the eccentricity and the length of the latus rectum of HH, then 9(e2+l)9\left( e^{2}+l \right) is equal to:

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  3. 53
    The number of sequences of ten terms, whose terms are either 0 or 1 or 2 , that contain exactly five 1 s and exactly three 2 s , is equal to

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  4. 54
    Let f:R→Rf:R \rightarrow R be a twice differentiable function such that (sin⁡xcos⁡y)(f(2x+2y)−f(2x−2y))=(cos⁡xsin⁡y)(f(2x+2y)+f(2x−2y))(\sin x\cos y)(f(2x+ 2y) -f(2x- 2y)) = (\cos x\sin y)(f(2x + 2y) +f(2x - 2y)), for all x,y∈Rx,y \in R. If f′(0)=12f^{'}(0) =\frac{1}{2}, then the value of 24f′′(5π3)24f^{''}\left( \frac{5\pi}{3} \right) is:

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  5. 55
    Let A=[α−16β],α>0A=\begin{bmatrix} \alpha & - 1 \\ 6 & \beta \end{bmatrix},\alpha> 0, such that det(A)=0det(A) = 0 and α+β=1\alpha+\beta= 1. If I denotes 2×22 \times 2 identity matrix, then the matrix (1+A)8(1 +A)^{8} is:

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  6. 56
    The term independent of xx in the expansion of ((x+1)(x23+1−x13)−(x−1)(x−x12))10,x>\left( \frac{\left( x+ 1 \right)}{\left( x^{\frac{2}{3}}+ 1 -x^{\frac{1}{3}} \right)}-\frac{\left( x- 1 \right)}{\left( x-x^{\frac{1}{2}} \right)} \right)^{10},x> 11 is:

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  7. 57
    If θ∈[−2π,2π]\theta\in \lbrack - 2\pi,2\pi\rbrack, then the number of solutions of 22cos⁡2θ+(2−6)cos⁡θ−3=02\sqrt{2}\cos^{2}\theta+ (2 -\sqrt{6})\cos\theta-\sqrt{3}= 0 is equal to:

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  8. 58
    Let a1,a2,a3…a_{1},a_{2},a_{3}\ldots be in an A.P. such that ∑k=112a2k−1=−725a1,a1≠0\sum_{k= 1}^{12} a_{2k- 1}= -\frac{72}{5}a_{1},a_{1}\neq 0. If ∑k=1nak=0\sum_{k= 1}^{n} a_{k}= 0, then nn is:

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  9. 59
    If the function f(x)=2x3−9ax2+12a2x+1f(x) = 2x^{3}- 9ax^{2}+ 12a^{2}x+ 1, where a >0> 0, attains its local maximum and local minimum values at pp and qq, respectively, such that p2=qp^{2}= q, then f(3)f(3) is equal to:

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  10. 60
    Let zz be a complex number such that ∣z∣=1|z| = 1. If 2+k2zk+z‾=kz,k∈R\frac{2 +k^{2}z}{k +\overline{z}}= kz,k \in R, then the maximum distance of k+ik2k+ik^{2} from the circle ∣z−(1+2i)∣=1|z- (1 + 2i)| = 1 is:

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  11. 61
    If a→\overrightarrow{a} is nonzero vector such that its projections on the vectors 2i^−j^+2k^,i^+2j^−2k^2\widehat{i}-\widehat{j}+ 2\widehat{k},\widehat{i}+ 2\widehat{j}- 2\widehat{k} and k^\widehat{k} are equal, then a unit vector along a→\overrightarrow{a} is:

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  12. 62
    Let A be the set of all functions f:Z→Zf:Z \rightarrow Z and R be a relation on A such that R={(f,g):f(0)=g(1)\mathbf{R = \{(f,g):}f(0) = g(1) and f(1)=g(0)}f(1) = g(0)\}. Then R is:

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  13. 63
    For α,β,γ∈R\alpha,\beta,\gamma \in R, if lim⁡x→0x2sin⁡αx+(γ−1)ex2sin⁡2x−βx=3\lim_{x \rightarrow 0} \frac{x^{2}\sin\alpha x + (\gamma- 1)e^{x^{2}}}{\sin2x -\beta x}= 3, then β+γ−α\beta+\gamma-\alpha is equal to:

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  14. 64
    If the system of linear equations
    3x+y+βz=32x+αy−z=−3x+2y+z=4\begin{matrix} & 3x+y+\beta z= 3 \\ & 2x+\alpha y-z= - 3 \\ & x+ 2y+z= 4 \end{matrix}
    has infinitely many solutions, then the value of 22β−9α22\beta- 9\alpha is :

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  15. 65
    Let Pn=αn+βn,n∈NP_{n}=\alpha^{n}+\beta^{n},n\in N. If P10=123,P9=76P_{10}= 123,P_{9}= 76, P8=47P_{8}= 47 and P1=1P_{1}= 1, then the quadratic equation having roots 1α\frac{1}{\alpha} and 1β\frac{1}{\beta} is:

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  16. 66
    If SS and S′S^{'} are the foci of the ellipse x218+y29=1\frac{x^{2}}{18}+\frac{y^{2}}{9}= 1 and P be a point on the ellipse, then min⁡(SP.S′P)+\min\left( SP.S^{'}P \right)+ max⁡(SP.S′P)\max\left( SP.S^{'}P \right) is equal to :

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  17. 67
    Let the vertices QQ and RR of the triangle PQRPQR lie on the line x+35=y−12=\frac{x+ 3}{5}=\frac{y- 1}{2}= z+43,QR=5\frac{z+ 4}{3},QR= 5 and the coordinates of the point P be (0,2,3)(0,2,3). If the area of the triangle PQRPQR is mn\frac{m}{n} then :

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  18. 68
    Let ABCDABCD be a tetrahedron such that the edges AB,ACAB,AC and AD are mutually perpendicular. Let the areas of the triangles ABC,ACDABC,ACD and ADB be 5,6 and 7 square units respectively. Then the area (in square units) of the △BCD\bigtriangleup BCD is equal to :

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  19. 69
    Let a∈Ra \in R and A be a matrix of order 3×33 \times 3 such that det(A)=−4det(A) = - 4 and A+I=[1a1210a12]A+I=\begin{bmatrix} 1 & a & 1 \\ 2 & 1 & 0 \\ a & 1 & 2 \end{bmatrix}, where II is the identity matrix of order 3×33 \times 3.If det((a+1)adj((a−1)A))det((a+ 1)adj((a- 1)A)) is 2m3n,m,n∈2^{m}3^{n},m,n\in {0,1,2,….20}\mathbf{\{ 0,1,2,\ldots.20\}}, then m+nm + n is equal to :

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  20. 70
    Let the focal chord PQPQ of the parabola y2=4xy^{2}= 4x make an angle of 60∘60^{\circ} with the positive xx-axis, where PP lies in the first quadrant. If the circle, whose one diameter is PS, S being the focus of the parabola, touches the y -axis at the point (0,α)(0,\alpha), then 5α25\alpha^{2} is equal to :

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  21. 71
    Let [•] denote the greatest integer function. If ∫0e3[1ex−1]dx=α−log⁡e2\int_{0}^{e^{3}} \left\lbrack \frac{1}{e^{x - 1}} \right\rbrack dx = \alpha - \log_{e}2, then α3\alpha^{3} is equal to
  22. 72
    Let f:R→Rf:R \rightarrow R be a thrice differentiable odd function satisfying f′(x)≥0,f′(x)=f(x),f(0)=0,f′(0)=3f'(x) \geq 0,f'(x) = f(x),f(0) = 0,f'(0) = 3. Then 9f(log⁡e3)9f\left( \log_{e}3 \right) is equal to .
  23. 73
    If the area of the region {(x,y):∣4−x2∣≤y≤x2,y≤4,x≥0}\left\{ (x,y):\left| 4 - x^{2} \right| \leq y \leq x^{2},y \leq 4,x \geq 0 \right\} is (802α−β),α,β∈N\left( \frac{80\sqrt{2}}{\alpha} - \beta \right),\alpha,\beta \in N, then α+β\alpha + \beta is equal to.
  24. 74
    Three distinct numbers are selected randomly from the set {1,2,3,……,40}\{ 1,2,3,\ldots\ldots,40\}. If the probability, that the selected numbers are in an increasing G.P. is mn\frac{m}{n}, gcd(m,n)=1gcd(m,n) = 1, then m+nm + n is equal to
  25. 75
    The absolute difference between the squares of the radii of the two circles passing through the point (−9,4)(-9,4) and touching the lines x+y=3x+y=3 and x−y=3x-y=3, is equal to

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