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JEE Mains 2026 (21 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
    Consider two identical metallic spheres of radius RR each having charge QQ and mass mm. Their centers have an initial separation of 4R4R. Both the spheres are given an initial speed of uu towards each other. The minimum value of uu, so that they can just touch each other is : (Take k=14πϵ0k =\frac{1}{4\pi\epsilon_{0}} and assume kQ2>Gm2kQ^{2}> Gm^{2} where G is the Gravitational constant)

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  2. 2
    The charge stored by the capacitor C in the given circuit in the steady state is ____\_\_\_\_ μC\mu C.

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  3. 3
    The total length of potentiometer wire AB is 50 cm in the arrangement as shown in figure. If P is the point where the galvanometer shows zero reading then the length AP is ____\_\_\_\_ cm .

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  4. 4
    A capacitor C is first charged fully with potential difference of V0V_{0} and disconnected from the battery. The charged capacitor is connected across an inductor having inductance L . In t s 25%25\% of the initial energy in the capacitor is transferred to the inductor. The value of t is ____\_\_\_\_ s.

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  5. 5
    The r.m.s speed of oxygen molecules at 47∘C47^{\circ}C is equal to that of the hydrogen molecules kept at ____\_\_\_\_  ∘C\ ^{\circ}C. (Mass of oxygen molecule/mass of hydrogen molecule=32/2)

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  6. 6
    Two cars A and B each of mass 103 kg10^{3}\text{ }kg are moving on parallel tracks separated by a distance of 10 m , in same direction with speeds 72 km/h72\text{ }km/h and 36 km/h36\text{ }km/h. The magnitude of angular momentum of car A with respect to car B is ____\_\_\_\_ J.s.

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  7. 7
    The pulley shown in figure is made using a thin rim and two rods of length equal to diameter of the rim. The rim and each rod have a mass of M . Two blocks of mass of M and m are attached to two ends of a light string passing over the pulley, which is hinged to rotate freely in vertical plane about its centre. The magnitudes of the acceleration experienced by the blocks is ____\_\_\_\_ (assume no slipping of string on pulley.)

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  8. 8
    The kinetic energy of a simple harmonic oscillator is oscillating with angular frequency of 176rad/s176rad/s. The frequency of this simple harmonic oscillator is___ Hz. [take\left\lbrack take \right.  π=227]\left. \ \pi =\frac{22}{7} \right\rbrack

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  9. 9
    Given below are two statements: Statement I: In a Young's double slit experiment, the angular separation of fringes will increase as the screen is moved away from the plane of the slits Statement II : In a Young's double slit experiment, the angular separation of fringes will increase when monochromatic source is replaced by another monochromatic source of higher wavelength In the light of the above statements, choose the correct answer from the options given below:

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  10. 10
    A battery with EMF E and internal resistance rr is connected across a resistance RR. The power consumption in R will be maximum when :

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  11. 11
    Keeping the significant figures in view, the sum of the physical quantities 52.01 m,153.2 m52.01\text{ }m,153.2\text{ }m and 0.123 m is :

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  12. 12
    A spherical body of radius rr and density σ\sigma falls freely through a viscous liquid having density ρ\rho and viscosity η\eta and attains a terminal velocity v0v_{0}. Estimated maximum error in the quantity η\eta is : (Ignore errors associated with σ,ρ\sigma,\rho and g , gravitational acceleration)

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  13. 13
    Surface tension of two liquids (having same densities), T1T_{1} and T2T_{2}, are measured using capillary rise method utilizing two tubes with inner radii of r1r_{1} and r2r_{2} where r1>r2r_{1}>r_{2}. The measured liquid heights in these tubes are h1h_{1} and h2h_{2} respectively. [Ignore the weight of the liquid about the lowest point of miniscus]. The heights h1h_{1} and h2h_{2} and surface tensions T1T_{1} and T2T_{2} satisfy the relation :

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  14. 14
    A river of width 200 m is flowing from west to east with a speed of 18 km/h18\text{ }km/h. A boat, moving with speed of 36 km/h36\text{ }km/h in still water, is made to travel one-round trip (bank to bank of the river). Minimum time taken by the boat for this journey and also the displacement along the river bank are ____\_\_\_\_ and ____\_\_\_\_ respectively.

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  15. 15
    Two known resistance of RΩR\Omega and 2RΩ2R\Omega and and one unknown resistance XΩX\Omega are connected in a circuit as shown in the figure. If the equivalent resistance between points A and B in the circuit is XΩX\Omega, then the value of X is ____\_\_\_\_ Ω\Omega.

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  16. 16
    The energy of an electron in an orbit of the Bohr's atom is −0.04E0eV- 0.04E_{0}eV where E0E_{0} is the ground state energy. If L is the angular momentum of the electron in this orbit and h is the Planck's constant, then 2π L h\frac{2\pi\text{ }L}{\text{ }h} is ____\_\_\_\_ :

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  17. 17
    An infinitely long straight wire carrying current I is bent in a planer shape as shown in the diagram. The radius of the circular part is r . The magnetic field at the centre O of the circular loop is :

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  18. 18
    A large drum having radius R is spinning around its axis with angular velocity ω\omega, as shown in figure. The minimum value of ω\omega so that a body of mass M remains stuck to the inner wall of the drum, taking the coefficient of friction between the drum surface and mass M is μ\mu, is :

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  19. 19
    A body of mass 2 kg is moving along x -direction such that its displacement as function of time is given by x(t)=αt2+βt+γmx(t) = \alpha t^{2}+ \beta t + \gamma m, where α=1 m/s2\alpha = 1\text{ }m/s^{2}, β=1 m/s\beta = 1\text{ }m/s and γ=1 m\gamma = 1\text{ }m. The work done on the body during the time interval t=2 st = 2\text{ }s to t=3 st = 3\text{ }s, is ____\_\_\_\_ J.

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  20. 20
    As shown in the diagram, when the incident ray is parallel to base of the prism, the emergent ray grazes along the second surface. If refractive index of the material of prism is 2\sqrt{2}, the angle θ\theta of prism is.

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  21. 21
    An electromagnetic wave of frequency 100 MHz propagates through a medium of conductivity, σ=10mho/m\sigma = 10mho/m. The ratio of maximum conducting current density to maximum displacement current density is ____\_\_\_\_. [Take\left\lbrack Take \right.  14πϵ0=9×109Nm2/C2]\left. \ \frac{1}{4\pi\epsilon_{0}} = 9 \times 10^{9}{Nm}^{2}/C^{2} \right\rbrack
  22. 22
    The terminal velocity of a metallic ball of radius 6 mm in a viscous fluid is 20 cm/s20\text{ }cm/s. The terminal velocity of another ball of same material and having radius 3 mm in the same fluid will be ____\_\_\_\_ cm/scm/s.
  23. 23
    A particle having electric charge 3×10−19C3 \times 10^{- 19}C and mass 6×10−27 kg6 \times 10^{- 27}\text{ }kg is accelerated by applying an electric potential of 1.21 V. Wavelength of the matter wave associated with the particle is α×10−12 m\alpha \times 10^{- 12}\text{ }m. The value of α\alpha is ____\_\_\_\_ . (Take Planck's constant =6.6×10−34 J.s= 6.6 \times 10^{- 34}\text{ }J.s )
  24. 24
    In a Young's double slit experiment set up, the two slits are kept 0.4 mm0.4\text{ }mm apart and screen is placed at 1 m from slits. If a thin transparent sheet of thickness 20μ m20\mu\text{ }m is introduced in front of one of the slits then central bright fringe shifts by 20 mm on the screen. The refractive index of transparent sheet is given by α10\frac{\alpha}{10}, where α\alpha is ____\_\_\_\_ .
  25. 25
    A diatomic gas (γ=1.4)(\gamma = 1.4) does 100 J of work when it is expanded isobarically. Then the heat given to the gas ____\_\_\_\_ J.

Chemistry

25 questions

  1. 26
    Consider the following spectral lines for atomic hydrogen: (A) First line of Paschen series (B) Second line of Balmer series (C) Third line of Paschen series (D) Fourth line of Brackett series The correct arrangement of the above lines in ascending order of energy is:

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  2. 27
    Match List-I with List-II.
    List-I Pair of CompoundsList-II Type of Isomers
    (A)2-Methylpropene and but-1-ene(I)Stereoisomers
    (B)Cis-but-2-ene and trans-but-2-ene(II)Position isomers
    (C)2-Butanol and diethyl ether(III)Chain isomers
    (D)But-1-ene and but-2-ene(IV)Functional group isomers
    Choose the correct answer from the options given below:

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  3. 28
    Consider the above sequence of reactions. The number of bromine atom(s) in the final product (P) will be:

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  4. 29
    Aqueous HCl reacts with MnO2(s)MnO_{2}(s) to form MnCl2(aq)MnCl_{2}(aq), Cl2(g)Cl_{2}(g) and H2O(l)H_{2}O(l). What is the weight (in g) of Cl2Cl_{2} liberated when 8.7 g of MnO2(s)MnO_{2}(s) is reacted with excess aqueous HCl solution ? (Given Molar mass in gmol−1Mn=55,Cl=35.5,O=16,H=1gmol^{- 1}Mn = 55,Cl = 35.5,O = 16,H = 1)

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  5. 30
    By usual analysis, 1.00 g of compound (X)(X) gave 1.79 g of magnesium pyrophosphate. The percentage of phosphorus in compound (X)(X) is: (nearest integer) (Given, molar mass in gmol−1:O=16,Mg=24gmol^{- 1}:O = 16,Mg = 24, P=31P = 31 )

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  6. 31
    Consider the following data: ΔfHΘ(\Delta_{f}H^{\Theta}( methane, g)=−XkJmol−1) = - XkJmol^{- 1} Enthalpy of sublimation of graphite =YkJmol−1= YkJmol^{- 1} Dissociation enthalpy of H2=ZkJmol−1H_{2}= ZkJmol^{- 1} The bond enthalpy of C−HC - H bond is given by:

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  7. 32
    Match List-I with List-II.
    List-I ReagentsList-II Reaction Name (Involving aldehydes)
    (A)H2, Pd−BaSO4H_{2},\ Pd - {BaSO}_{4}(I)Etard Reaction
    (B)SnCl2, HCl{SnCl}_{2},\ HCl(II)Rosenmund Reduction
    (C)CrO2Cl2, CS2{CrO}_{2}{Cl}_{2},\ {CS}_{2}(III)Gattermann-Koch Reaction
    (D)CO, HClCO,\ HCl, anhyd. AlCl3{AlCl}_{3}(IV)Stephen Reaction
    Choose the correct answer from the options given below:

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  8. 33
    Decomposition of A is a first order reaction at T(K)T(K) and is given by A(g)→B(g)+C(g)A(g) \rightarrow B(g) + C(g). In a closed 1 L vessel, 1 bar A(g)A(g) is allowed to decompose at T(K)T(K). After 100 minutes, the total pressure was 1.5 bar. What is the rate constant (in min⁡−1\min^{- 1}) of the reaction ? (log⁡2=0.3\log2 = 0.3)

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  9. 34
    The correct order of reactivity of the following benzyl halides towards reaction with KCN is:

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  10. 35
    Given below are two statements: Statement-I: The correct order in terms of atomic/ionic radii is Al>Mg>Mg2+>Al3+Al > Mg > {Mg}^{2 +} > {Al}^{3 +}. Statement-II: The correct order in terms of the magnitude of electron gain enthalpy is Cl>Br>S>OCl > Br > S > O. In the light of the above statements, choose the correct answer from the options given below:

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  11. 36
    The correct statements are: (A) Activation energy for enzyme catalysed hydrolysis of sucrose is lower than that of acid catalysed hydrolysis. (B) During denaturation, secondary and tertiary structures of a protein are destroyed but primary structure remains intact. (C) Nucleotides are joined together by glycosidic linkage between C1C_{1} and C4C_{4} carbons of the pentose sugar (D) Quaternary structure of proteins represents overall folding of the polypeptide chain. Choose the correct answer from the options given below:

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  12. 37
    The correct order of the rate of the reaction for the following reaction with respect to nucleophiles is:
    CH3Br+NuΘ⟶CH3Nu+BrΘCH_{3}Br +Nu^{\Theta}\longrightarrow CH_{3}Nu +Br^{\Theta}

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  13. 38
    Given below are two statements: Statement I: Crystal Field Stabilization Energy (CFSE) of [Cr(H2O)6]2+\left\lbrack Cr\left( H_{2}O \right)_{6} \right\rbrack^{2 +} is greater than that of [Mn(H2O)6]2+\left\lbrack Mn\left( H_{2}O \right)_{6} \right\rbrack^{2 +}. Statement II: Potassium ferricyanide has a greater spin-only magnetic moment than sodium Ferrocyanide. In the light of the above statements, choose the correct answer from the options given below:

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  14. 39
    The correct increasing order of C−H(A)C - H(A), C−O(B),C=O(C)C - O(B),C = O(C) and C≡N(D)C \equiv N(D) bonds in terms of covalent bond length is:

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  15. 40
    Given below are four compounds: (A) n-propyl chloride (B) iso-propyl chloride (C) sec-butyl chloride (D) neo-pentyl chloride Percentage of carbon in the one which exhibits optical isomerism is:

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  16. 41
    Given below are some of the statements about Mn and Mn2O7Mn_{2}O_{7}. Identify the correct statements (A) Mn forms the oxide Mn2O7{Mn}_{2}O_{7} in which Mn is in its highest oxidation state. (B) Oxygen stabilizes the Mn in higher oxidation states by forming multiple bonds with Mn (C) Mn2O7{Mn}_{2}O_{7} is an ionic oxide. (D) The structure of Mn2O7{Mn}_{2}O_{7} consists of one bridged oxygen. Choose the correct answer from the options given below:

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  17. 42
    For a closed circuit Daniell cell, which of the following plots is the accurate one at a given temperature?

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  18. 43
    Given below are two statements : Statement-I: Compound (X), shown below , dissolves in NaHCO3{NaHCO}_{3} solution and has two chiral carbon atoms. Statement - II : Compound (Y), shown below, has two carbons with sp3{sp}^{3} hybridization, one carbon with sp2{sp}^{2} and one carbon with sp hybridization In the light of the above statements, choose the correct answer from the options given below :

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  19. 44
    Given below are two statements : Statement I : The correct order in terms of bond dissociation enthalpy is Cl2>Br2>F2>I2{Cl}_{2} > {Br}_{2} > F_{2} > I_{2} Statement II : The correct trend in the covalent character of the metal halides is [SnCl4>SnCl2]\left\lbrack {SnCl}_{4} > {SnCl}_{2} \right\rbrack, [PbCl4>PbCl2]\left\lbrack {PbCl}_{4} > {PbCl}_{2} \right\rbrack and [UF4>UF6]\left\lbrack {UF}_{4} > {UF}_{6} \right\rbrack In the light of the above statements, choose the correct answer from the options given below :

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  20. 45
    On heating a mixture of common salt and K2Cr2O7K_{2}Cr_{2}O_{7} in equal amount along with concentrated H2SO4H_{2}SO_{4} in a test tube, a gas is evolved. Formula of the gas evolved and oxidation state of the central metal atom in the gas respectively are :

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  21. 46
    The first and second ionization constants of H2XH_{2}X are 2.5×10−82.5 \times 10^{- 8} and 1.0×10−131.0 \times 10^{- 13} respectively. The concentration of X2−X^{2 -} in 0.1MH2X0.1MH_{2}X solution is ____\_\_\_\_ ×10−15M\times 10^{- 15}M. (Nearest Integer)
  22. 47
    The osmotic pressure of a living cell in 12 atm at 300 K . The strength of sodium chloride solution that is isotonic with the living cell at this temperature is ____\_\_\_\_ gL−1gL^{- 1}. (Nearest integer) Given : R=0.08 L atm K−1 mol−1R = 0.08\text{ }L\text{ }atm{\text{ }K}^{- 1}{\text{ }mol}^{- 1} Assume complete dissociation of NaCl (Given : Molar mass of Na and Cl are 23 and 35.5 gmol−1g{mol}^{- 1} respectively.)
  23. 48
    A substance ' X ' ( 1.5 g ) dissolved in 150 g of a solvent ' Y ' (molar mass =300 g mol−1= 300\text{ }g{\text{ }mol}^{- 1} ) led to an elevation of the boiling point by 0.5 K . The relative lowering in the vapour pressure of the solvent ' Y ' is ____\_\_\_\_ ×10−2\times 10^{- 2}. (Nearest integer) [Given : KbK_{b} of the solvent =5.0 K kg mol−1= 5.0\text{ }K\text{ }kg{\text{ }mol}^{- 1} ] Assume the solution to be dilute and no association or dissociation of X takes place in solution.
  24. 49
    Identify the metal ions among Co2+,Ni2+,Fe2+,V3+{Co}^{2 +},{Ni}^{2 +},{Fe}^{2 +},V^{3 +} and Ti2+{Ti}^{2 +} having a spin-only magnetic moment value more than 3.0 BM. The sum of unpaired electrons present in the high spin octahedral complexes formed by those metal ions is____\_\_\_\_ .
  25. 50
    MX is a sparingly soluble salt that follows the given solubility equilibrium at 298 K
    MX(s)⇌M+(aq)+X−(aq);Ksp=10−10MX(s) \rightleftharpoons M^{+}(aq) + X^{-}(aq);K_{sp} = 10^{- 10}
    If the standard reduction potential for M+(aq)→+e− M(s)M^{+}(aq)\overset{+ e^{-}\ }{\rightarrow}M(s) is (EM+/M⊖)=0.79 V\left( E_{M^{+}/M}^{\ominus} \right) = 0.79\text{ }V, then the value of the standard reduction potential for the metal/metal insoluble salt electrode EX−/MX(s)/M⊖E_{X^{-}/MX(s)/M}^{\ominus} is ____\_\_\_\_ mV. (nearest integer) [Given: 2.303RTF=0.059 V\frac{2.303RT}{F} = 0.059\text{ }V ]

Mathematics

25 questions

  1. 51
    The positive integer n , for which the solutions of the equation x(x+2)+(x+2)(x+4)+….+(x+2n−2)(x+2n)=8n3x(x + 2) + (x + 2)(x + 4) + \ldots. +\left( x + 2n - 2 \right)\left( x + 2n \right)=\frac{8n}{3}are two consecutive even integers, is :-

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  2. 52
    Let f:R→Rf:R \rightarrow R be a twice differentiable function such that f′′(x)>0f^{''}(x) > 0 for all x∈Rx \in R and f′(a−1)=0f^{'}(a - 1) = 0, where a is real number. Let g(x)=f(tan⁡2x−2tan⁡x+a)g(x) = f\left( \tan^{2}x - 2\tan x + a \right), 0<x<π20 < x <\frac{\pi}{2}. Consider the following two statements : (I) g is increasing in (0,π4)\left( 0,\frac{\pi}{4} \right) (II) g is decreasing in (π4,π2)\left( \frac{\pi}{4},\frac{\pi}{2} \right) Then,

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  3. 53
    Let f(x)=x3+x2f′(1)+2xf′′(2)+f′′′(3)f(x) =x^{3}+x^{2}f^{'}(1) + 2xf^{''}(2) +f^{'''}(3), x∈Rx \in R. Then the value of f′(5)f^{'}(5) is :

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  4. 54
    If the line αx+4y=7\alpha x + 4y =\sqrt{7}, where α∈R\alpha \in R, touches the ellipse 3x2+4y2=13x^{2}+ 4y^{2}= 1 at the point PP in the first quadrant, then one of the focal distances of P is :

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  5. 55
    Let y2=12xy^{2}= 12x be the parabola with its vertex at O . Let P be a point on the parabola and A be a point on the x -axis such that ∠OPA=90∘\angle OPA =90^{\circ}. Then the locus of the centroid of such triangles OPA is :

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  6. 56
    Let one end of a focal chord of the parabola y2=16xy^{2}= 16x be (16, 16). If P(α,β)P(\alpha,\beta) divides this focal chord internally in the ratio 5:25:2, then the minimum value of α+β\alpha + \beta is equal to :

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  7. 57
    Let the line L pass through the point (−3,5,2)( - 3,5,2) and make equal angles with the positive coordinate axes. If the distance of L from the point ( −2,r,1- 2,r,1 ) is 143\sqrt{\frac{14}{3}}, then the sum of all possible values of rr is :

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  8. 58
    Let the line L1L_{1} be parallel to the vector −3i^+2j^+4k^- 3\widehat{i}+ 2\widehat{j}+ 4\widehat{k} and pass through the point (2,6,7)(2,6,7) and the line L2L_{2} be parallel to the vector 2i^+j^+3k^2\widehat{i}+\widehat{j}+ 3\widehat{k} and pass through the point (4,3,5)(4,3,5). If the line L3L_{3} is parallel to the vector −3i^+5j^+16k^- 3\widehat{i}+ 5\widehat{j}+ 16\widehat{k} and intersects the lines L1L_{1} and L2L_{2} at the points C and D , respectively, then ∣CD→∣2|\overrightarrow{CD}|^{2} is equal to :

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  9. 59
    Let α\alpha and β\beta be the roots of equation x2+2ax+(3a+10)=0x^{2}+ 2ax + (3a + 10) = 0 such that α<1<β\alpha < 1 < \beta. Then the set of all possible values of aa is :

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  10. 60
    A random variable XX takes values 0,1,2,30,1,2,3 with probabilities 2a+130,8a−130,4a+130\frac{2a + 1}{30},\frac{8a - 1}{30},\frac{4a + 1}{30}, b respectively, where a,b∈Ra,b \in R. Let μ\mu and σ\sigma respectively be the mean and standard deviation of XX such that σ2+μ2=2\sigma^{2}+\mu^{2}= 2. Then ab\frac{a}{b} is equal to :

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  11. 61
    If the area of the region {(x,y):1−2x≤y≤4−x2\left\{ (x,y):1 - 2x \leq y \leq 4 -x^{2} \right., x≥0,y≥0}x \geq 0,y \geq 0\} is αβ,α,β,∈N,gcd(α,β)=1\frac{\alpha}{\beta},\alpha,\beta, \in N,gcd(\alpha,\beta) = 1, then the value of (α+β)(\alpha + \beta) is :

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  12. 62
    Let a1,a22,a322,…,a1029a_{1},\frac{a_{2}}{2},\frac{a_{3}}{2^{2}},\ldots,\frac{a_{10}}{2^{9}} be a G.P. of common ratio 12\frac{1}{\sqrt{2}}. If a1+a2+…+a10=62a_{1}+a_{2}+ \ldots +a_{10}= 62, then a1a_{1} is equal to :

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  13. 63
    Let A={x:∣x2−10∣≤6}A =\left\{ x:\left| x^{2}- 10 \right|\leq 6 \right\} and B={x:∣x−2∣>1}\mathbf{B = \{ x:|x - 2| > 1\}}. Then

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  14. 64
    For the matrices A=[3−41−1]A =\begin{bmatrix} 3 & - 4 \\ 1 & - 1 \end{bmatrix} and B=[−2949−1318]B =\begin{bmatrix} - 29 & 49 \\ - 13 & 18 \end{bmatrix}, if (A15+B)[xy]=[00]\left( A^{15}+ B \right)\begin{bmatrix} x \\ y \end{bmatrix}=\begin{bmatrix} 0 \\ 0 \end{bmatrix}, then among the following which one is true?

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  15. 65
    For a triangle ABCABC, let p→=BC→,q→=CA→\overrightarrow{p}=\overrightarrow{BC},\overrightarrow{q}=\overrightarrow{CA} and r→=BA→\overrightarrow{r}=\overrightarrow{BA}. If ∣p→∣=23,∣q→∣=2|\overrightarrow{p}| = 2\sqrt{3},|\overrightarrow{q}| = 2 and cos⁡θ=13\cos\theta =\frac{1}{\sqrt{3}}, where θ\theta is the angle between p→\overrightarrow{p} and q→\overrightarrow{q}, then ∣p→×(q→−3r→)∣2+3∣r→∣2|\overrightarrow{p}\times (\overrightarrow{q}- 3\overrightarrow{r})|^{2}+ 3|\overrightarrow{r}|^{2} is equal to:

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  16. 66
    Let y=y(x)y = y(x) be the solution of the differential equation secxdydx−2y=2+3sin⁡x,x∈(−π2,π2)secx\frac{dy}{dx}- 2y = 2 + 3\sin x,x \in\left( -\frac{\pi}{2},\frac{\pi}{2} \right), y(0)=−74y(0) = -\frac{7}{4}. Then y(π6)y\left( \frac{\pi}{6} \right) is equal to:

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  17. 67
    Let A={2,3,5,7,9}\mathbf{A = \{ 2,3,5,7,9\}}. Let R be the relation on A defined by x Ry if and only if 2x≤3y2x \leq 3y. Let l\mathcal{l} be the number of elements in R , and m be the minimum number of elements required to be added in R to make it a symmetric relation. Then l+m\mathcal{l +}m is equal to:

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  18. 68
    If the system of equations 3x+y+4z=33x + y + 4z = 3, 2x+αy−z=−32x + \alpha y - z = - 3, x+2y+z=4x + 2y + z = 4 has no solution, then the value of α\alpha is equal to:

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  19. 69
    Let zz be the complex number satisfying ∣z−5∣≤3|z - 5| \leq 3 and having maximum positive principal argument. Then 34∣5z−125iz+16∣234\left| \frac{5z - 12}{5iz + 16} \right|^{2} is equal to:

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  20. 70
    The largest n∈Nn \in N, for which 7n7^{n} divides 101!, is :

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  21. 71
    Let [ • ] denote the greatest integer function and f(x)=lim⁡n→∞1n3∑k=1n[k23x]f(x) = \lim_{n \rightarrow \infty} \frac{1}{n^{3}}\sum_{k = 1}^{n} \left\lbrack \frac{k^{2}}{3^{x}} \right\rbrack. Then 12∑j=1∞f(j)12\sum_{j = 1}^{\infty} f(j) is equal to ____\_\_\_\_ .
  22. 72
    If ∫014cot⁡−1(1−2x+4x2)dx=atan⁡−1(2)−bloge(5)\int_{0}^{1} 4\cot^{- 1}\left( 1 - 2x + 4x^{2} \right)dx = a\tan^{- 1}(2) - {blog}_{e}(5), where a,b∈Na,b \in N, then (2a+b)(2a + b) is equal to ____\_\_\_\_ .
  23. 73
    Let the maximum value of (sin⁡−1x)2+(cos⁡−1x)2\left( \sin^{- 1}x \right)^{2} + \left( \cos^{- 1}x \right)^{2} for x∈[−32,12]x \in \left\lbrack - \frac{\sqrt{3}}{2},\frac{1}{\sqrt{2}} \right\rbrack be mnπ2\frac{m}{n}\pi^{2}, where gcd(m,n)=1gcd(m,n) = 1. Then m+nm + n is equal to ____\_\_\_\_ .
  24. 74
    If (1 15C0+1 15C1)(1 15C1+1 15C2)…(1 15C12+1 15C13)=α13 14C0 14C1… 14C12\left( \frac{1}{\ ^{15}C_{0}} + \frac{1}{\ ^{15}C_{1}} \right)\left( \frac{1}{\ ^{15}C_{1}} + \frac{1}{\ ^{15}C_{2}} \right)\ldots\left( \frac{1}{\ ^{15}C_{12}} + \frac{1}{\ ^{15}C_{13}} \right) = \frac{\alpha^{13}}{\ ^{14}C_{0}\ ^{14}C_{1}\ldots\ ^{14}C_{12}}, then 30α30\alpha is equal to ____\_\_\_\_ .
  25. 75
    If P is a point on the circle x2+y2=4,Qx^{2} + y^{2} = 4,Q is a point on the straight line 5x+y+2=05x + y + 2 = 0 and x−y+1=0x - y + 1 = 0 is the perpendicular bisector of PQ , then 13 times the sum of abscissa of all such point P is ____\_\_\_\_ .

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