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JEE Mains 2025 (23 Jan, 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
    Regarding self-inductance : (A) The self-inductance of the coil depends on its geometry. (B) Self-inductance does not depend on the permeability of the medium. (C) Self-induced e.m.f. opposes any change in the current in a circuit. (D) Self-inductance is electromagnetic analogue of mass in mechanics. (E) Work needs to be done against self-induced e.m.f. in establishing the current.Choose the correct answer from the options given below:

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  2. 2
    A light hollow cube of side length 10 cm and mass 10 g is floating in water. It is pushed down and released to execute simple harmonic oscillations. The time period of oscillations is yπ×10−2 sy\pi \times10^{- 2}\text{ }s, where the value of yy is (Acceleration due to gravity, g=10 m/s2g = 10\text{ }m/s^{2}, density of water =103 kg/m3=10^{3}\text{ }kg/m^{3} )

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  3. 3
    Given below are two statements: Statement-I : The hot water flows faster than cold water. Statement-II : Soap water has higher surface tension as compared to fresh water. In the light above statements, choose the correct answer from the options given below

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  4. 4
    A sub-atomic particle of mass 10−30 kg10^{- 30}\text{ }kg is moving with a velocity 2.21×106 m/s2.21 \times10^{6}\text{ }m/s. Under the matter wave consideration, the particle will behave closely like _____\_\_\_\_\_ . (h=6.63×10−34 J.s)\left( h = 6.63 \times10^{- 34}\text{ }J.s \right)

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  5. 5
    A spherical surface of radius of curvature RR, separates air from glass (refractive index =1.5= 1.5 ). The centre of curvature is in the glass medium. A point object ' O ' placed in air on the optic axis of the surface, so that its real image is formed at ' I ' inside glass. The line OI intersects the spherical surface at PP and PO=PIPO = PI. The distance POPO equals to-

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  6. 6
    A radioactive nucleus n2n_{2} has 3 times the decay constant as compared to the decay constant of another radioactive nucleus n1n_{1}. If initial number of both nuclei are the same, what is the ratio of number of nuclei of n2n_{2} to the number of nuclei of n1n_{1}, after one half-life of n1n_{1} ?

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  7. 7
    Identify the valid statements relevant to the given circuit at the instant when the key is closed. (A) There will be no current through resistor RR. (B) There will be maximum current in the connecting wires. (C) Potential difference between the capacitor plates A and B is minimum. (D) Charge on the capacitor plates is minimum. Choose the correct answer from the options given below :

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  8. 8
    The position of a particle moving on xx-axis is given by x(t)=Asin⁡t+Bcos⁡2t+Ct2+Dx(t) = A\sin t + B\cos^{2}t + Ct^{2}+ D, where tt is time. The dimension of ABCD\frac{ABC}{D} is-

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  9. 9
    Match the List-I with List-II
    List-IList-II
    (A) Pressure varies inversely with volume of an ideal gas.(I) Adiabatic process
    (B) Heat absorbed goes partly to increase internal energy and partly to do work.(II) Isochoric process
    (C) Heat is neither absorbed nor released by a system(III) Isothermal process
    (D) No work is done on or by a gas(IV) Isobaric process
    Choose the correct answer from the options given below :

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  10. 10
    Consider a moving coil galvanometer (MCG) : (A) The torsional constant in moving coil galvanometer has dimensions [ML2 T−2]\left\lbrack ML^{2}{\text{ }T}^{- 2} \right\rbrack (BB) Increasing the current sensitivity may not necessarily increase the voltage sensitivity. (C) If we increase number of turns (N)(N) to its double (2 N)(2\text{ }N), then the voltage sensitivity doubles. (D) MCG can be converted into an ammeter by introducing a shunt resistance of large value in parallel with galvanometer. (E) Current sensitivity of MCG depends inversely on number of turns of coil. Choose the correct answer from the options given below :

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  11. 11
    A point particle of charge QQ is located at PP along the axis of an electric dipole 1 at a distance rr as shown in the figure. The point P is also on the equatorial plane of a second electric dipole 2 at a distance rr. The dipoles are made of opposite charge qq separated by a distance 2a2a. For the charge particle at P not to experience any net force, which of the following correctly describes the situation?

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  12. 12
    A gun fires a lead bullet of temperature 300 K into a wooden block. The bullet having melting temperature of 600 K penetrates into the block and melts down. If the total heat required for the process is 625 J , then the mass of the bullet is _____\_\_\_\_\_ grams. (Latent heat of fusion of lead =2.5×104JKg−1= 2.5 \times10^{4}JKg^{- 1} and specific heat capacity of lead =125JKg−1 K−1= 125JKg^{- 1}{\text{ }K}^{- 1} )

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  13. 13
    What is the lateral shift of a ray refracted through a parallel-sided glass slab of thickness ' hh ' in terms of the angle of incidence ' ii ' and angle of refraction ' rr ', if the glass slab is placed in air medium?

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  14. 14
    A solid sphere of mass ' m ' and radius ' rr ' is allowed to roll without slipping from the highest point of an inclined plane of length ' LL ' and makes an angle 30∘30^{\circ} with the horizontal. The speed of the particle at the bottom of the plane is v1v_{1}. If the angle of inclination is increased to 45∘45^{\circ} while keeping L constant. Then the new speed of the sphere at the bottom of the plane is v2v_{2}. The ratio of v1 2:v2 2v_{1}\ ^{2}:v_{2}\ ^{2} is

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  15. 15
    Refer to the circuit diagram given in the figure, which of the following observation are correct? (A) Total resistance of circuit is 6Ω6\Omega. (B) Current in Ammeter is 1 A (C) Potential across ABAB is 4 Volts. (D) Potential across CD is 4 Volts. (E) Total resistance of the circuit is 8Ω8\Omega. Choose the correct answer from the options given below:

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  16. 16
    The electric flux is ϕ=ασ+βλ\phi = \alpha\sigma + \beta\lambda where λ\lambda and σ\sigma are linear and surface charge density, respectively, (αβ)\left( \frac{\alpha}{\beta} \right) represents

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  17. 17
    Given a thin convex lens (refractive index μ2\mu_{2} ), kept in a liquid (refractive index μ1,μ1<μ2\mu_{1},\mu_{1}<\mu_{2} ) having radii of curvature ∣R1∣\left| R_{1} \right| and ∣R2∣\left| R_{2} \right|. Its second surface is silver polished. Where should an object be placed on the optic axis so that a real and inverted image is formed at the same place?

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  18. 18
    The electric field of an electromagnetic wave in free space is
    E→=57cos⁡[7.5×106t−5×10−3(3x+4y)] (4i^−3j^)N/C\begin{matrix} & \overrightarrow{E}= 57\cos\left\lbrack 7.5 \times10^{6}t - 5 \times10^{- 3}(3x + 4y) \right\rbrack \\ & \ (4\widehat{i}- 3\widehat{j})N/C \end{matrix}
    The associated magnetic field in Tesla is-

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  19. 19
    The motion of an airplane is represented by velocity-time graph as shown below. The distance covered by airplane in the first 30.5 second is _____\_\_\_\_\_ km.

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  20. 20
    Consider a circular disc of radius 20 cm with centre located at the origin. A circular hole of a radius 5 cm is cut from this disc in such a way that the edge of the hole touches the edge of the disc. The distance of centre of mass of residual or remaining disc from the origin will be-

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  21. 21
    A positive ion AA and a negative ion BB has charges 6.67×10−19C6.67 \times10^{- 19}C and 9.6×10−10C9.6 \times10^{- 10}C, and masses 19.2×10−27 kg19.2 \times10^{- 27}\text{ }kg and 9×10−27 kg9 \times10^{- 27}\text{ }kg respectively. At an instant, the ions are separated by a certain distance r. At that instant the ratio of the magnitudes of electrostatic force to gravitational force is P×1044P \times10^{44}, where the value of P is  ______\ \_\_\_\_\_\_ . (Take 14πε0=9×109Nm2C−2\frac{1}{4\pi\varepsilon_{0}}= 9 \times10^{9}Nm^{2}C^{- 2} and universal gravitational constant as 6.67×10−11Nm2 kg−26.67 \times10^{- 11}Nm^{2}{\text{ }kg}^{- 2} )
  22. 22
    Two particles are located at equal distance from origin. The position vectors of those are represented by A→=2i^+3nj^+2k^\overrightarrow{A}= 2\widehat{i}+ 3n\widehat{j}+ 2\widehat{k} and B→=2i^−2j^+4pk^\overrightarrow{B}= 2\widehat{i}- 2\widehat{j}+ 4p\widehat{k}, respectively. If both the vectors are at right angle to each other, the value of n−1n^{- 1} is ____\_\_\_\_
  23. 23
    An ideal gas initially at 0∘C0^{\circ}C temperature, is compressed suddenly to one fourth of its volume. If the ratio of specific heat at constant pressure to that at constant volume is 3/23/2, the change in temperature due to the thermodynamics process is ______\_\_\_\_\_\_ K.
  24. 24
    A force f=x2yi^+y2j^f =x^{2}y\widehat{i}+y^{2}\widehat{j} acts on a particle in a plane x+y=10x + y = 10. The work done by this force during a displacement from (0,0)(0,0) to (4 m,2 m)(4\text{ }m,2\text{ }m) is ______\_\_\_\_\_\_ Joule (round off to the nearest integer)
  25. 25
    In the given circuit the sliding contact is pulled outwards such that electric current in the circuit changes at the rate of 8 A/s8\text{ }A/s. At an instant when RR is 12Ω12\Omega, the value of the current in the circuit will be ___\_\_\_ A.

Chemistry

25 questions

  1. 26
    The element that does not belong to the same period of the remaining elements (modern periodic table) is:

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  2. 27
    Heat treatment of muscular pain involves radiation of wavelength of about 900 nm . Which spectral line of H atom is suitable for this ? Given: Rydberg constant
     RH=105 cm−1, h=6.6×10−34 J s,c=3×108 m/s)\left. \ R_{H}=10^{5}{\text{ }cm}^{- 1},\text{ }h = 6.6 \times10^{- 34}\text{ }J\text{ }s,c = 3 \times10^{8}\text{ }m/s \right)

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  3. 28
    The incorrect statements among the following is

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  4. 29
    CrCl3⋅xNH3CrCl_{3} \cdot xNH_{3} can exist as a complex. 0.1 molal aqueous solution of this complex shows a depression in freezing point of 0.558∘C0.558^{\circ}C. Assuming 100%100\% ionisation of this complex and coordination number of Cr is 6 , the complex will be (Given Kf=1.86 K kg mol−1K_{f}= 1.86\text{ }K\text{ }kg{\text{ }mol}^{- 1} )

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  5. 30
    FeO42−→+2.0VFe3+→0.8VFe2+→−0.5VFe0FeO_{4}^{2 -}\overset{+ 2.0V}{\rightarrow}Fe^{3 +}\overset{0.8V}{\rightarrow}Fe^{2 +}\overset{- 0.5V}{\rightarrow}Fe^{0} In the above diagram, the standard electrode potentials are given in volts (over the arrow). The value of EFeO42−/Fe2+⊖E_{FeO_{4}^{2 -}/Fe^{2 +}}^{\ominus} is

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  6. 31
    Match the LIST-I with LIST-II
    LIST-I (Name reaction)LIST-II (Product obtainable)
    (A)Swarts reaction(I)Ethyl benzene
    (B)Sandmeyer's reaction(II)Ethyl iodide
    (C)Wurtz-Fittig reaction(III)Cyanobenzene
    (D)Finkelstein reaction(IV)Ethyl fluoride
    Choose the correct answer from the options given below:

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  7. 32
    Given below are two statements: Statement I: Fructose does not contain an aldehydic group but still reduces Tollen's reagent Statement II : In the presence of base, fructose undergoes rearrangement to give glucose. In the light of the above statements, choose the correct answer from the options given below

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  8. 33
    2.8×10−3 mol2.8 \times10^{- 3}\text{ }mol of CO2CO_{2} is left after removing 102110^{21} molecules from its ' xx ' mg sample. The mass of CO2CO_{2} taken initially is Given : NA=6.02×1023 mol−1N_{A}= 6.02 \times10^{23}{\text{ }mol}^{- 1}

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  9. 34
    Ice at −5∘C-5^{\circ}C is heated to become vapor with temperature of 110∘C110^{\circ}C at atmospheric pressure. The entropy change associated with this process can be obtained from :

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  10. 35
    The d-electronic configuration of an octahedral Co (II) complex having magnetic moment of 3.95 BM is :

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  11. 36
    The complex that shows Facial - Meridional isomerism is

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  12. 37
    The major product of the following reaction is : CH3CH2CH=OCH_{3}CH_{2}CH = O →excess HCHOalkalireflux\underset{reflux}{\overset{\begin{array}{r} excess\ HCHO \\ alkali \end{array}}{\rightarrow}}?

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  13. 38
    The correct stability order of the following species/molecules is : p q r

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  14. 39
    Propane molecule on chlorination under photochemical condition gives two di-chloro products, xx and yy. Amongst xx and yy, xx is an optically active molecule. How many tri-chloro products (consider only structural isomers) will be obtained from xx when it is further treated with chlorine under the photochemical condition?

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  15. 40
    What amount of bromine will be required to convert 2 g of phenol into 2, 4, 6-tribromophenol ? (Given molar mass in gmol−1gmol^{- 1} of C,H,O,BrC,H,O,Br are 12,1,16,8012,1,16,80 respectively)

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  16. 41
    The correct set of ions (aqueous solution) with same colour from the following is :

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  17. 42
    Given below are two statements : Statement I : In Lassaigne's test, the covalent organic molecules are transformed into ionic compounds. Statement II : The sodium fusion extract of an organic compound having N and S gives prussian blue colour with FeSO4FeSO_{4} and Na4[Fe(CN)6]Na_{4}\left\lbrack Fe(CN)_{6} \right\rbrack In the light of the above statements, choose the correct answer from the options given below

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  18. 43
    Which of the following happens when NH4OHNH_{4}OH is added gradually to the solution containing 1MA2+1MA^{2 +} and 1MB3+1MB^{3 +} ions ? Given : Ksp [A(OH)2]=9×10−10K_{\text{sp~}}\left\lbrack A(OH)_{2} \right\rbrack= 9 \times10^{- 10} and
    Ksp[ B(OH)3]=27×10−18 at 298 KK_{sp}\left\lbrack \text{ }B(OH)_{3} \right\rbrack= 27 \times10^{- 18}\text{~at~}298\text{ }K

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  19. 44
    Match the LIST-I with LIST-II
    LIST-I (Classification of molecules based on octet rule)LIST-II (Example)
    (A)Molecules obeying octet rule(I)
    NO,NO2NO,NO_{2}
    (B)Molecules with incomplete octet(II)
    BCl3,AlCl3BCl_{3},AlCl_{3}
    (C)Molecules with incomplete octet with odd electron(III)
    H2SO4,PCl5H_{2}SO_{4},PCl_{5}
    (D)Molecules with expanded octet(IV)
    CCl4,CO2CCl_{4},CO_{2}
    Choose the correct answer from the options given below :

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  20. 45
    Which among the following react with Hinsberg's reagent? (A) Aniline (B) N,N-Dimethylaniline (C) CH3−NH2CH_{3}-NH_{2} (D) N(CH3)3N\left( CH_{3} \right)_{3} (E) Diphenylamine. Choose the correct answer from the options given below :

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  21. 46
    If 1 mM solution of ethylamine produces pH=9pH = 9, then the ionization constant (Kb)\left( K_{b} \right) of ethylamine is 10−x10^{- x}. The value of xx is _____\_\_\_\_\_ (nearest integer). [The degree of ionization of ethylamine can be neglected with respect to unity.]
  22. 47
    During "S" estimation, 160 mg of an organic compound gives 466 mg of barium sulphate. The percentage of Sulphur in the given compound is____\_\_\_\_ %.(Given molar mass in gmol−1gmol^{- 1} of Ba:137, S:32Ba:137,\text{ }S:32, 0:16)0:16)
  23. 48
    Consider the following sequence of reactions to produce major product (A) →(i)Br2,Fe(ii)Sn, HCl                       (iii)NaNO2 HCl 273k(iv)H3PO2   H2O\underset{\begin{array}{r} \left( iii \right)NaNO_{2}\ HCl\ 273k \\ (iv)H_{3}PO_{2\ \ \ }H_{2}O \end{array}}{\overset{\begin{array}{r} \left( i \right)Br_{2,}Fe \\ \left( ii \right)Sn,\ HCl \end{array}}{\rightarrow}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }(A)Major product(A)_{Major\ product} Molar mass of product (A) is ____\_\_\_\_ gmol−1gmol^{- 1}. (Given molar mass in gmol−1gmol^{- 1} of C:12,H:1C:12,H:1, O:16,Br:80, N:14,P:31)O:16,Br:80,\text{ }N:14,P:31)
  24. 49
    For the thermal decomposition of N2O5( g)N_{2}O_{5}(\text{ }g) at constant volume, the following table can be formed, for the reaction mentioned below :
    2 N2O5( g)→2 N2O4( g)+O2( g)2{\text{ }N}_{2}O_{5}(\text{ }g) \rightarrow 2{\text{ }N}_{2}O_{4}(\text{ }g) +O_{2}(\text{ }g)
    S. No.Time/sTotal pressure/atm
    100.6
    2100xx
    x=x = ____\_\_\_\_ ×10−3 atm\times10^{- 3}\text{ }atm [nearest integer] Given : Rate constant for the reaction is 4.606×10−2 s−14.606 \times10^{- 2}{\text{ }s}^{- 1}.
  25. 50
    The standard enthalpy and standard entropy of decomposition of N2O4N_{2}O_{4} to NO2NO_{2} are 55.0 kJ mol−155.0\text{ }kJ{\text{ }mol}^{- 1} and 175.0 J/K/mol175.0\text{ }J/K/mol respectively. The standard free energy change for this reaction at 25∘C25^{\circ}C in Jmol−1Jmol^{- 1} is ______\_\_\_\_\_\_ (Nearest integer)

Mathematics

25 questions

  1. 51
    The value of ∫e2e41x(e((log⁡ex)2+1)−1e((log⁡ex)2+1)−1+e((6−log⁡ex)2+1)−1)dx\int_{e^{2}}^{e^{4}} \frac{1}{x}\left(\frac{e^{\left(\left(\log_{e}x\right)^{2}+1\right)^{-1}}}{e^{\left(\left(\log_{e}x\right)^{2}+1\right)^{-1}}+e^{\left(\left(6-\log_{e}x\right)^{2}+1\right)^{-1}}}\right)dx is

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  2. 52
    Let I(x)=∫dx(x−11)1113(x+15)1513I(x) = \int\frac{dx}{(x - 11)^{\frac{11}{13}}(x + 15)^{\frac{15}{13}}}. If I(37)−I(24)=14(1b113−1c113),b,c∈NI(37) - I(24) =\frac{1}{4}\left( \frac{1}{b^{\frac{1}{13}}}-\frac{1}{c^{\frac{1}{13}}} \right),b,c \in N, then 3(b+c)3(b + c) is equal to

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  3. 53
    If the function
    f(x)={2x{sin⁡(k1+1)x+sin⁡(k2−1)x},x<04x=02xlog⁡e(2+k1x2+k2x)x>0f(x) =\left\{ \begin{matrix} \frac{2}{x}\left\{ \sin\left( k_{1}+ 1 \right)x + \sin\left( k_{2}- 1 \right)x \right\} & ,x < 0 \\ 4 & x = 0 \\ \frac{2}{x}\log_{e}\left( \frac{2 +k_{1}x}{2 +k_{2}x} \right) & x > 0 \end{matrix} \right.
    is continuous at x=0x = 0, then k1 2+k2 2k_{1}\ ^{2}+k_{2}\ ^{2} is equal to

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  4. 54
    If the line 3x−2y+12=03x - 2y + 12 = 0 intersects the parabola 4y=3x24y = 3x^{2} at the points AA and BB, then at the vertex of the parabola, the line segment ABAB subtends an angle equal to

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  5. 55
    Let a curve y=f(x)y = f(x) pass through the points (0,5)(0,5) and (log⁡e2,k)\left( \log_{e}2,k \right). If the curve satisfies the differential equation 2(3+y)e2xdx−(7+e2x)dy=02(3 + y)e^{2x}dx -\left( 7 +e^{2x} \right)dy = 0, then kk is equal to

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  6. 56
    Let f(x)=log⁡exf(x) =\log_{e}x and g(x)=x4−2x3+3x2−2x+22x2−2x+1g(x) =\frac{x^{4}- 2x^{3}+ 3x^{2}- 2x + 2}{2x^{2}- 2x + 1} Then the domain of fog is

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  7. 57
    Let the arc AC of a circle subtend a right angle at the centre O . If the point B on the arcACarcAC, divides the arcACarcAC such that  length of arcAB length of arcBC=15\frac{\text{~length of~}arcAB}{\text{~length of~}arcBC}=\frac{1}{5}, and OC→=αOA→+βOB→\overrightarrow{OC}= \alpha\overrightarrow{OA}+ \beta\overrightarrow{OB}, then α+2(3−1)β\alpha +\sqrt{2}(\sqrt{3}- 1)\beta is equal to

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  8. 58
    If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to

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  9. 59
    Let P be the foot of the perpendicular from the point Q(10,−3,−1)Q(10, - 3, - 1) on the line x−37=y−2−1=z+1−2\frac{x - 3}{7}=\frac{y - 2}{- 1}=\frac{z + 1}{- 2}. Then the area of the right angled triangle PQRPQR, where RR is the point (3,−2,1)(3, - 2,1), is

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  10. 60
    Let ∣zˉ−i2z‾+i∣=13,z∈C\left| \frac{\bar{z}- i}{2\overline{z}+ i} \right|=\frac{1}{3},z \in\mathbb{C}, be the equation of a circle with center at C . If the area of the triangle, whose vertices are at the points (0,0),C(0,0),C and (α,0)(\alpha,0) is 11 square units, then α2\alpha^{2} equals

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  11. 61
    Let R={(1,2),(2,3),(3,3)}\mathbf{R = \{(1,2),(2,3),(3,3)\}} be a relation defined on the set {1,2,3,4}\mathbf{\{ 1,2,3,4\}}. Then the minimum number of elements, needed to be added in R so the R becomes an equivalence relation, is :

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  12. 62
    The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is

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  13. 63
    Let the area of a △PQR\bigtriangleup PQR with vertices P(5,4),Q(−2,4)P(5,4),Q( - 2,4) and R(a,b)R(a,b) be 35 square units. If its orthocenter and centroid are O(2,145)O\left( 2,\frac{14}{5} \right) and C(c,d)C(c,d) respectively, then c+2 dc + 2\text{ }d is equal to

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  14. 64
    If π2≤x≤3π4\frac{\pi}{2}\leq x\leq\frac{3\pi}{4}, then cos⁡−1(1213cos⁡x+513sin⁡x)\cos^{- 1}\left( \frac{12}{13}\cos x+\frac{5}{13}\sin x \right) is equal to

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  15. 65
    The value of (sin⁡70∘)(cot⁡10∘cot⁡70∘−1)\left( \sin70^{\circ} \right)\left( \cot10^{\circ}\cot70^{\circ}- 1 \right) is

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  16. 66
    Marks obtained by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12 . If the number of students whose marks are less than 12 is 18 , then the total number of students is

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  17. 67
    Let the position vectors of the vertices A,BA,B and C of a tetrahedron ABCD be i^+2j^+k^,i^+3j^−2k^\widehat{i}+ 2\widehat{j}+\widehat{k},\widehat{i}+ 3\widehat{j}- 2\widehat{k} and 2i^+j^−k^2\widehat{i}+\widehat{j}-\widehat{k} respectively. The altitude from the vertex DD to the opposite face ABCABC meets the median line segment through AA of the triangle ABCABC at the point EE. If the length of ADAD is 1103\frac{\sqrt{110}}{3} and the volume of the tetrahedron is 80562\frac{\sqrt{805}}{6\sqrt{2}}, then the position vector of EE is

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  18. 68
    If A,BA,B and (adj(A−1)+adj(B−1))\left( adj\left( A^{- 1} \right)+ adj\left( B^{- 1} \right) \right) are non-singular matrices of same order, then the inverse of A(adj(A−1)+adj(B−1))−1 BA\left( adj\left( A^{- 1} \right)+ adj\left( B^{- 1} \right) \right)^{- 1}\text{ }B, is equal to

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  19. 69
    If the system of equations
    (λ−1)x+(λ−4)y+λz=5λx+(λ−1)y+(λ−4)z=7 (λ+1)x+(λ+2)y−(λ+2)z=9\begin{matrix} & (\lambda - 1)x + (\lambda - 4)y + \lambda z = 5 \\ & \lambda x + (\lambda - 1)y + (\lambda - 4)z = 7 \\ & \ (\lambda + 1)x + (\lambda + 2)y - (\lambda + 2)z = 9 \end{matrix}
    has infinitely many solutions, then λ2+λ\lambda^{2}+ \lambda is equal to

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  20. 70
    One die has two faces marked 1 , two faces marked 2 , one face marked 3 and one face marked 4 . Another die has one face marked 1, two faces marked 2, two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5 , when both the dice are thrown together, is

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  21. 71
    If the area of the larger portion bounded between the curves x2+y2=25x^{2}+y^{2}= 25 and y=∣x−1∣y = |x - 1| is 14(bπ+c)\frac{1}{4}(b\pi + c), b,c∈Nb,c \in N, then b+cb + c is equal to ___\_\_\_
  22. 72
    The sum of all rational terms in the expansion of (1+21/3+31/2)6\left( 1 +2^{1/3}+3^{1/2} \right)^{6} is equal to ____\_\_\_\_
  23. 73
    Let the circle CC touch the line x−y+1=0x-y+ 1 = 0, have the centre on the positive xx-axis, and cut off a chord of length 413\frac{4}{\sqrt{13}} along the line −3x+2y=1- 3x+ 2y= 1. Let HH be the hyperbola x2α2−y2β2=1\frac{x^{2}}{\alpha^{2}}-\frac{y^{2}}{\beta^{2}}= 1, whose one of the foci is the centre of C and the length of the transverse axis is the diameter of C. Then 2α2+3β22\alpha^{2}+ 3\beta^{2} is equal to ___\_\_\_
  24. 74
    If the set of all values of a, for which the equation 5x3−15x−a=05x^{3}- 15x - a = 0 has three distinct real roots, is the interval (α,β)(\alpha,\beta), then β−2α\beta - 2\alpha is equal to ____\_\_\_\_
  25. 75
    If the equation a(b−c)x2+b(c−a)x+c(a−b)=0a(b-c)x^{2}+b(c-a)x+c(a-b)=0 has equal roots, where a+c=15a+c=15 and b=365b=\frac{36}{5}, then a2+c2a^{2}+c^{2} is equal to

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