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JEE Mains 2026 (6 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
    The density ρ\rho of a uniform cylinder is determined by measuring its mass m , length ll and diameter d . The measured values of m,lm,l and d are 97.42±0.02 g,8.35±0.05 mm97.42 \pm 0.02\text{ }g,8.35 \pm 0.05\text{ }mm and 20.20±0.02 mm20.20 \pm 0.02\text{ }mm, respectively. Calculated percentage fractional error in ρ\rho is ____\_\_\_\_.

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  2. 2
    The potential energy of a particle changes with distance xx from a fixed origin as V=Axx+BV=\frac{A\sqrt{x}}{x+B}, where A and B are constant with appropriate dimensions. The dimension of AB are ____\_\_\_\_ .

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  3. 3
    The rain drop of mass 1 g , starts with zero velocity from a height of 1 km . It hits the ground with a speed of 5 m/s5\text{ }m/s. The work done by the unknown resistive force is ____\_\_\_\_ J . (take g=10 m/s2g = 10\text{ }m/s^{2} )

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  4. 4
    Two blocks ( P and Q ) with respectively masses 2 kg and 1.5 kg are joined by a massless thread. These blocks are mounted on a frictionless pulley which is fixed on the edge of a cube ( S ), as shown in the figure below. Block P is positioned on the top surface which has no friction and block Q is in contact with side-surface, having coefficient friction μ\mu. The cube ( S ) moves towards the right with acceleration of g2\frac{g}{2}, where gg is gravitational acceleration. During this movement the block P and Q remain stationary. The value of μ\mu is ____\_\_\_\_. (take g=10 m/s2g = 10\text{ }m/s^{2} )

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  5. 5
    A lift of mass 1600 kg is supported by thick iron wire. If the maximum stress which the wire can withstand is 4×108 N/m24 \times10^{8}\text{ }N/m^{2} and its radius is 4 mm , then maximum acceleration the lift can take is ____\_\_\_\_ m/s2m/s^{2}. (take g=10 m/s2g = 10\text{ }m/s^{2} and π=3.14\pi= 3.14 )

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  6. 6
    A solid sphere of radius 4 cm and mass 5 kg is rotating (rotation axis is passing through the center of the sphere) with an angular velocity of 1200 rpm . It is brought to rest in 10 s by applying a constant torque. The torque applied and the number of rotations it made before it comes to rest are ____\_\_\_\_ and ____\_\_\_\_ respectively.

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  7. 7
    A smooth inclined plane ends in a vertical circular loop, as shown in the figure. A small body is released from height hh as shown. If the body exerts a force of three times its weight on the plane at the highest point of circle, then the height h=αRh=\alpha R. The value of α\alpha is

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  8. 8
    The position of center of mass of three masses 2 kg,3 kg2\text{ }kg,3\text{ }kg and 15 kg placed with respect to mid point (p) of normal bisector, as shown in the figure is ____\_\_\_\_.

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  9. 9
    The two wires A and B of equal cross-section but of different materials are joined together. The ratio of Young's modulus of wire A and wire B is 20/11. When the joined wire is kept under certain tension the elongations in the wires A and B are equal. If the length of wire A is 2.2 m , then the length of wire BB is ____\_\_\_\_ m.

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  10. 10
    Two closed vessels of same volume are joined through a narrow tube and both vessels are filled with air of pressure 90 kPa and temperature 400 K . Keeping the temperature of one vessel constant at 400 K the second vessel temperature is raised to 500 K . The final pressure in the vessels is ____\_\_\_\_ kPa .

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  11. 11
    In interference experiment the path difference between two interfering waves at a point A on the screen is λ/3\lambda/3, where λ\lambda is the wavelength of these waves, and at another point B the path difference is λ/6\lambda/6. The ratio of intensities at points A and B is ____\_\_\_\_.

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  12. 12
    A particle is executing simple harmonic motion. Its amplitude is A and time period is 5 sec . the time required by it to move from x=Ax=A to x=A2x=\frac{A}{\sqrt{2}} is ____\_\_\_\_ sec.

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  13. 13
    A thin half ring of radius 35 cm is uniformly charged with a total charge of Q coulomb. If the magnitude of the electric field at center of the half ring is 100 V/m100\text{ }V/m, then the value of Q is ____\_\_\_\_ nC. ( ϵ0=8.85×10−12C2/Nm2\epsilon_{0}= 8.85 \times10^{- 12}C^{2}/Nm^{2} and π=3.14\pi= 3.14 )

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  14. 14
    The maximum rated power of the LED is 2 mW and it is used in the circuit with input voltage of 5 V as shown in the figure below. The current through resistance RSR_{S} is 0.5 mA . The minimum value of the resistance of RsR_{s}, to ensure that the LED is not damaged is ____\_\_\_\_ kΩk\Omega.

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  15. 15
    A point light source emits E.M. waves in free space. A detector, placed at a distance of L m , measures the intensity as I0I_{0}. The detector is now shifted to another location on the same spherical surface ensuring the angle between original location and new location as 45∘45^{\circ}. The measured intensity at new location will be ____\_\_\_\_ .

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  16. 16
    A spherical interface lens of radius R separates two media of refractive indices 1 and 1.4 respectively as shown in the figure below. A point source is placed at a distance of 4 R in front of spherical interface. The magnitude of the magnification of point source image is ____\_\_\_\_ .

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  17. 17
    A small cube of side 1 mm is placed at the center of a circular loop of radius 10 cm carrying a current of 2 A . The magnetic energy stored inside the cube is α×10−14 J\alpha \times10^{- 14}\text{ }J. The value of α\alpha is ____\_\_\_\_ . ( μ0=4π×10−7Tm/A,π=3.14\mu_{0}= 4\pi\times10^{- 7}Tm/A,\pi= 3.14 )

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  18. 18
    An inductor of inductance 10mH having resistance of 100Ω100\Omega is connected to battery of E.M.F. 1.0 V through a switch as shown in the figure below. After switch is closed, the ratio of instantaneous voltages across the inductor when the current passing through it is 2 mA and 4 mA is ____\_\_\_\_ .

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  19. 19
    The ratio of momentum of the photons of the 1st 1^{\text{st~}} and 2nd 2^{\text{nd~}} line of Balmer series of Hydrogen atoms is α/β\alpha/\beta. The possible values of α\alpha and β\beta are :-

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  20. 20
    A LCR series circuit driven with Erms=90 VE_{rms}= 90\text{ }V at frequency fd=30 Hzf_{d}= 30\text{ }Hz has resistance R=80ΩR = 80\Omega, an inductance with inductive reactance XL=20.0ΩX_{L}= 20.0\Omega and capacitance with capacitive reactance XC=80.0ΩX_{C}= 80.0\Omega. The power factor of the circuit is ____\_\_\_\_ .

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  21. 21
    Refer to the circuit diagram given below. The heat generated across the 6Ω6\Omega resistance in 100 second is α100 J\frac{\alpha}{100}\text{ }J. The value of α\alpha is ____\_\_\_\_ . (Nearest integer)
  22. 22
    An unpolarized light of intensity I0I_{0} passes through polarizer and then through a certain optically active solution and finally it goes to analyser. If the angle between analyser and polariser is 0∘0^{\circ} and intensity of light emerged from analyzer is 38I0\frac{3}{8}I_{0}, the angle of rotation of the light by the solution with respect to analyser is ____\_\_\_\_ degrees.
  23. 23
    The energy released when 717.13 kg\frac{7}{17.13}\text{ }kg of  37Li\ _{3}^{7}Li is converted into  24He\ _{2}^{4}He by proton bombardment is α×1032eV\alpha \times 10^{32}eV. The value of α\alpha is ____\_\_\_\_ . (Nearest integer) (Mass of  37Li=7.0183u\ _{3}^{7}Li = 7.0183u, mass of  24He=4.004u\ _{2}^{4}He = 4.004u, mass of proton =1.008u= 1.008u and 1u=931MeV/c21u = 931MeV/c^{2} and Avogadro number =6.0×1023= 6.0 \times 10^{23} )
  24. 24
    A three coulomb charge moves from the point ( 0,−2,−50, - 2, - 5 ) to the point ( 5,1,25,1,2 ) in an electric field expressed as E→=2xi^+3y2j^+4k^N/C\overrightarrow{E} = 2x\widehat{i} + 3y^{2}\widehat{j} + 4\widehat{k}N/C. The work done in moving the charge is ____\_\_\_\_ J.
  25. 25
    A certain gas is isothermally compressed to (13)rd \left( \frac{1}{3} \right)^{\text{rd~}} of its initial volume ( V0=3V_{0} = 3 litre) by applying required pressure. If the bulk modulus of the gas is 3×105 N/m23 \times 10^{5}\text{ }N/m^{2}, the magnitude of work done on the gas is ____\_\_\_\_ J.

Chemistry

25 questions

  1. 26
    An oxide of iron contains 69.9%69.9\% iron, its empirical formula, is : (Given : Molar mass of Fe and O are 56 and 16 g mol−116\text{ }g{\text{ }mol}^{- 1} respectively.)

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  2. 27
    If shortest wavelength of hydrogen atom in Lyman series is x , then longest wavelength in Balmer series of He+He^{+}is :

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  3. 28
    Match List-I with List-II :
    List-I (Orbital)List-II (Radial nodes and nodal plane)
    (A)2s(I)1 Radial node + two nodal planes
    (B)3s(II)1 Radial node + one nodal plane
    (C)3p(III)2 Radial nodes + No nodal plane
    (D)4d(IV)1 Radial node + No nodal plane.
    Choose the correct answer from the options given below:

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  4. 29
    The pairs among A=[SO3 2−,CO3 2−],B=[O2 2−,F2],C=[CN−,CO]A =\left\lbrack SO_{3}\ ^{2 -},CO_{3}\ ^{2 -} \right\rbrack,B =\left\lbrack O_{2}\ ^{2 -},F_{2} \right\rbrack,C =\left\lbrack CN^{-},CO \right\rbrack, D=[NH3,H3O+]D =\left\lbrack NH_{3},H_{3}O^{+} \right\rbrackand E=[MnO4 2−,CrO4 2−]E =\left\lbrack MnO_{4}\ ^{2 -},CrO_{4}\ ^{2 -} \right\rbrack that do not have similar Lewis dot structure are :

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  5. 30
    Arrange the following isothermal processes in order of the magnitude of the work ( p−Vp-V ) involved between states 1 and 2 . (A) Expansion in single stage wAw_{A} (B) Expansion in multi stages wBw_{B} (C) Compression in single stage wCw_{C} (D) Compression in multi stages wDw_{D}

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  6. 31
    When 0.25 moles of a non-volatile, non-ionizable solute was dissolved in 1 mole of a solvent the vapor pressure of solution was x%x\% of vapor pressure of pure solvent. What is the value of x%x\%?

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  7. 32
    One mole each of He and A(g)A(g) are taken in a 10 L closed flask and heated to 400 K to establish the following equilibrium.
    A(g)⇌B(g)A(g) \rightleftharpoons B(g)
    KCK_{C} for this reaction at 400 K is 4.0. The partial pressures (in atm ) of He and B(g)B(g) are respectively (at equilibrium). (Assume He,A(g)He,A(g) and B(g)B(g) behave as ideal gases) (Given : R=0.082 L atm K−1 mol−1R = 0.082\text{ }L\text{ }atm{\text{ }K}^{- 1}{\text{ }mol}^{- 1} )

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  8. 33
    Consider the following data :
    ElectrolyteΛm∘\Lambda_{m}^{\circ} (S cm2 mol−1S\ cm^{2}\ mol^{-1})
    BaCl2BaCl_{2}x1x_{1}
    H2SO4H_{2}SO_{4}x2x_{2}
    HClHClx3x_{3}
    BaSO4BaSO_{4} is sparingly soluble in water. If the conductivity of the saturated BaSO4BaSO_{4} solution is x S cm−1x\ S\ cm^{-1}, then the solubility product of BaSO4BaSO_{4} can be given as (here Λm=Λm∘\Lambda_{m} = \Lambda_{m}^{\circ}) :

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  9. 34
    Given below are two statements : Statement-I : Aluminum is more electropositive than thallium as the standard electrode potential value of EAl3+/AloE_{{Al}^{3 +}/Al}^{o} is negative and ETl3+/TloE_{{Tl}^{3 +}/Tl}^{o} is positive. Statement-II : The sum of first three ionization enthalpies of boron is very high when compared to that of aluminum. Due to this reason boron forms covalent compounds only and aluminium forms Al3+{Al}^{3 +} ion. In the light of the above statements, choose the correct answer from the options given below.

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  10. 35
    The correct statements among the following are : (A) Basic vanadium oxide is used in the manufacture of H2SO4H_{2}{SO}_{4}. (B) The spin-only magnetic moment value of the transition metal halide employed in ZieglerNatta polymerization is 2.84 BM . (C) The p-block metal compound employed in Ziegler-Natta polymerization has the metal in +3 oxidation state. (D) The number of electrons present in the outer most 'd' orbital of metal halide employed in Wacker process is 8. Choose the correct answer from the options given below :

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  11. 36
    Match List-I with List-II
    List-I Electronic configuration of tetrahedral metal ionList-II Crystal Field Stabilization Energy(Δt)\left( \Delta_{t} \right)
    (A)
    d2d^{2}
    		(I)-0.6
    (B)
    d4d^{4}
    		(II)-0.8
    (C)
    d6d^{6}
    		(III)-1.2
    (D)
    d8d^{8}
    		(IV)-0.4
    Choose the correct answer from the options given below :

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  12. 37
    Which of the following are true about the energy of the given d-orbitals of a tetrahedral complex ? (A) dxy=dxz>dx2−y2d_{xy} = d_{xz} > d_{x^{2} - y^{2}} (B) dxy=dyz>dz2d_{xy} = d_{yz} > d_{z^{2}} (C) dx2−y2>dz2>dxzd_{x^{2} - y^{2}} > d_{z^{2}} > d_{xz} (D) dx2−y2=dz2<dxzd_{x^{2} - y^{2}} = d_{z^{2}} < d_{xz} Choose the correct answer from the options given below :

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  13. 38
    RfR_{f} value for 2-methylpropene in a solvent system (Ethyl acetate + ether) is 0.42 . 2-methylpropene is treated with dilute H2SO4H_{2}SO_{4} to give major organic product (X). RfR_{f} value for (X) in the same solvent system under identical condition will be :

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  14. 39
    Given below are two statements : Statement-I : 2, 6-diethylcyclohexanone and 6-methyl-2-n-propylcyclohexanone are metamers. Statement-II : 2, 2, 6, 6-tetramethylcyclohexanone exhibits keto-enol tautomerism. In the light of the above statements, choose the correct answer from the options given below.

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  15. 40
    Given below are two statements : Statement-I : Methane can be prepared by decarboxylation of sodium ethanoate, Kolbe's electrolysis of sodium acetate and reaction of CH3MgBr{CH}_{3}MgBr with water. Statement-II : Methane cannot be prepared from unsaturated hydrocarbons and by Wurtz reaction. In the light of the above statements, choose the correct answer from the options given below.

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  16. 41
    Given below are two statements : Statement-I : 3-phenylpropene reacts with HBr and gives secondary alkyl bromide having a chiral carbon atom as the major product. Statement-II : Aryl chlorides and aryl cyanides can be prepared by Sandmeyer reaction as well as Gattermann reaction. In the light of the above statements, choose the correct answer from the options given below.

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  17. 42
    Consider the following sequence of reactions The major product P is :

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  18. 43
    Arrange the following compounds according to increasing order of boiling points. n−C4H9OH(A),n−C4H9NH2( B)n -C_{4}H_{9}OH(A),n -C_{4}H_{9}NH_{2}(\text{ }B), n−C4H10(C)n -C_{4}H_{10}(C) and C2H5NHC2H5(D)C_{2}H_{5}NHC_{2}H_{5}(D).

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  19. 44
    Match List-I with List-II:
    List-I Deficiency DiseaseList-II Vitamin
    (A)Scurvy(I)Pyridoxine
    (B)Convulsions(II)Vitamin A
    (C)Cheilosis(III)Ascorbic Acid
    (D)Xerophthalmia(IV)Riboflavin
    Choose the correct answer from the options given below :

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  20. 45
    Match List-I with List-II
    List-I Amino acidList-II Positive reaction/Test for functional group present in side chain of amino acid
    (A)Glutamine(I)Hinsberg's test
    (B)Lysine(II)Neutral FeCl3{FeCl}_{3} test
    (C)Tyrosine(III)Ceric ammonium nitrate test
    (D)Serine(IV)Hoffmann bromamide degradation
    Choose the correct answer from the options given below :

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  21. 46
    First and second ionization enthalpies of lithium are 520 kJ mol−1520\text{ }kJ{\text{ }mol}^{- 1} and 7297 kJ mol−17297\text{ }kJ{\text{ }mol}^{- 1} respectively. Energy required to convert 3.5 mg lithium (g) into Li2+(g)[Li(g)→Li2+(g)]{Li}^{2 +}(g)\left\lbrack Li(g) \rightarrow {Li}^{2 +}(g) \right\rbrack is ____\_\_\_\_ kJkJ. (nearest integer) [Molar mass of Li=7 g mol−1Li = 7\text{ }g{\text{ }mol}^{- 1} ]
  22. 47
    Consider the following sequence of reactions. The percentage of nitrogen in the yellow product ( X ) formed is ____\_\_\_\_ %. (Nearest Integer) (Given Molar mass in gmol−1H:1,C:12, N:14g{mol}^{- 1}H:1,C:12,\text{ }N:14 )
  23. 48
    4.7 g of phenol is heated with Zn to give product X. If this reaction goes to 60%60\% completion then the number of moles of compound X formed will be ____\_\_\_\_ ×10−2\times 10^{- 2} (Nearest Integer) (Given molar mass in gmol−1:H:1,C:12,O:16g{mol}^{- 1}:H:1,C:12,O:16 )
  24. 49
    Sucrose hydrolyses in acidic medium into glucose and fructose by first order rate law with t1/2=3t_{1/2} = 3 hour. The percentage of sucrose remaining after 6 hours is ____\_\_\_\_ . (Nearest integer) (Given : log⁡2=0.3010\log2 = 0.3010 and log⁡3=0.4771\log3 = 0.4771 )
  25. 50
    Consider the reaction X⇌YX \rightleftharpoons Y at 300 K . If ΔHθ\Delta H^{\theta} and K are 28.40 kJ mol−128.40\text{ }kJ{\text{ }mol}^{- 1} and 1.8×10−71.8 \times 10^{- 7} at the same temperature, then the magnitude of ΔSθ\Delta S^{\theta} for the reaction in JK−1 mol−1JK^{- 1}{\text{ }mol}^{- 1} is ____\_\_\_\_ .(Nearest Integer). (Given : R=8.3JK−1 mol−1,ln⁡10=2.3R = 8.3{JK}^{- 1}{\text{ }mol}^{- 1},\ln10 = 2.3, log⁡3=0.48,log⁡2=0.30\log3 = 0.48,\log2 = 0.30

Mathematics

25 questions

  1. 51
    Let [.] denote the greatest integer function. If the domain of the function f(x)=sin⁡−1(x+[x]3)f(x) =\sin^{- 1}\left( \frac{x+ \lbrack x\rbrack}{3} \right) is [α,β)\lbrack\alpha,\beta), then α2+β2\alpha^{2}+\beta^{2} is equal to:

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  2. 52
    Let one root of the quadratic equation in x : (k2−15k+27)x2+9(k−1)x+18=0\left( k^{2}- 15k+ 27 \right)x^{2}+ 9\left( k- 1 \right)x+ 18 = 0 be twice the other. Then the length of the latus rectum of the parabola y2=6kxy^{2}= 6kx is equal to:

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  3. 53
    Let e1e_{1} and e2e_{2} be two distinct roots of the equation x2−ax+2=0x^{2}-ax+2=0. Let the set of all a∈Ra\in R for which e1e_{1} and e2e_{2} are the eccentricities of two hyperbolas be (α,β)(\alpha,\beta), and the set of all a∈Ra\in R for which e1e_{1} and e2e_{2} are the eccentricities of an ellipse and a hyperbola, respectively, be (γ,∞)(\gamma,\infty). Then α2+β2+γ2\alpha^{2}+\beta^{2}+\gamma^{2} is equal to:

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  4. 54
    Let the set of all values of k∈Rk \in R such that the equation z(z‾+2+i)+k(2+3i)=0,z∈Cz(\overline{z}+ 2 + i) + k(2 + 3i) = 0,z \in C, has at least one solution, be the interval [α,β]\lbrack\alpha,\beta\rbrack. Then 9(α+β)9(\alpha+\beta) is equal to:

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  5. 55
    The value of 13−23+33−….+1531^{3}-2^{3}+3^{3}- \ldots. +15^{3} is:

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  6. 56
    The sum of the first ten terms of an A.P. is 160 and the sum of the first two terms of a G.P. is 8 . If the first term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to common difference of the A.P., then the sum of all possible values of the first term of the G.P. is:

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  7. 57
    The number of 4-letter words, with or without meaning, each consisting of two vowels and two consonants that can be formed from the letters of the word INCONSEQUENTIAL, without repeating any letter, is:

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  8. 58
    If the coefficients of the middle terms in the binomial expansions of (1+αx)26(1 +\alpha x)^{26} and (1−αx)28(1 -\alpha x)^{28}, α≠0\alpha\neq 0, are equal, then the value of α\alpha is:

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  9. 59
    A data consists of 20 observations x1,x2,…..,x20x_{1},x_{2},\ldots..,x_{20}. If ∑i=120(xi+5)2=2500\sum_{i= 1}^{20} \left( x_{i}+ 5 \right)^{2}= 2500 and ∑i=120(xi−5)2=100\sum_{i= 1}^{20} \left( x_{i}- 5 \right)^{2}= 100, then the ratio of mean to standard deviation of this data is :

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  10. 60
    A bag contains (N+1)(N + 1) coins -N fair coins, and one coin with 'Head' on both sides. A coin is selected at random and tossed. If the probability of getting 'Head' is 916\frac{9}{16}, then N is equal to:

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  11. 61
    If the eccentricity e of the hyperbola x2a2−y2 b2=1\frac{x^{2}}{a^{2}}-\frac{y^{2}}{{\text{ }b}^{2}}= 1, passing through (6,43)(6,4\sqrt{3}), satisfies 15(e2+1)=34e15\left( e^{2}+ 1 \right)= 34e, then the length of the latus rectum of the hyperbola x2 b2−y22(a2+1)=1\frac{x^{2}}{{\text{ }b}^{2}}-\frac{y^{2}}{2\left( a^{2}+ 1 \right)}= 1 is:

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  12. 62
    Let chord PQ of length 3133\sqrt{13} of the parabola y2=12xy^{2}= 12x be such that the ordinates of points PP and Q are in the ratio 1:21:2. If the chord PQ subtends an angle α\alpha at the focus of the parabola, then sin⁡α\sin\alpha is equal to :

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  13. 63
    Let 0<α<1,β=13α0 <\alpha< 1,\beta=\frac{1}{3\alpha} and tan⁡−1(1−α)+tan⁡−1(1−β)=π4\tan^{- 1}(1 -\alpha) +\tan^{- 1}(1 -\beta) =\frac{\pi}{4}. Then 6(α+β)6(\alpha+\beta) is equal to:

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  14. 64
    Let S={θ∈(−2π,2π):cos⁡θ+1=3sin⁡θ}S = \{\theta\in ( - 2\pi,2\pi):\cos\theta+ 1 =\sqrt{3}\sin\theta\}. Then ∑θ∈Sθ\sum_{\theta\in S} \theta is equal to:

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  15. 65
    Let the image of the point P(1,6,a)P(1,6,a) in the line L:x1=y−12=z−a+1 b, b>0L:\frac{x}{1}=\frac{y - 1}{2}=\frac{z - a + 1}{\text{ }b},\text{ }b > 0, be (a3,0,a+c)\left( \frac{a}{3},0,a + c \right). If S(α,β,γ),α>0S(\alpha,\beta,\gamma),\alpha> 0, is the point on L such that the distance of S from the foot of perpendicular from the point P on L is 2142\sqrt{14}, then α+β+γ\alpha+\beta+\gamma is equal to :

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  16. 66
    Let a line L be perpendicular to both the lines L1:x+13=y+35=z+57L_{1}:\frac{x + 1}{3}=\frac{y + 3}{5}=\frac{z + 5}{7} and L2:x−21=y−44=z−67L_{2}:\frac{x - 2}{1}=\frac{y - 4}{4}=\frac{z - 6}{7}. If θ\theta is the acute angle between the lines L and L3:x−872=y−471=z2L_{3}:\frac{x -\frac{8}{7}}{2}=\frac{y -\frac{4}{7}}{1}=\frac{z}{2}, then tan⁡θ\tan\theta is equal to:

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  17. 67
    The value of lim⁡x→0(x2sin⁡2xx2−sin⁡2x)\lim_{x\rightarrow 0} \left( \frac{x^{2}\sin^{2}x}{x^{2}-\sin^{2}x} \right) is:

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  18. 68
    The value of the integral ∫−π4π4(32cos⁡4x1+esin⁡x)dx\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \left( \frac{32\cos^{4}x}{1 +e^{\sin x}} \right)dx is :

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  19. 69
    The area of the region {(x,y):0≤y≤6−x, y2≥4x−3, x≥0}\{(x,y):0 \leq y \leq 6 - x,\ y^{2}\geq 4x - 3,\ x \geq 0\} is :

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  20. 70
    Let e be the base of natural logarithm and let f:{1,2,3,4}→{1,e,e2,e3}f:\{ 1,2,3,4\} \rightarrow\left\{ 1,e,e^{2},e^{3} \right\} and g:{1,e,e2,e3}→{1,12,13,14}g:\left\{ 1,e,e^{2},e^{3} \right\}\rightarrow\left\{ 1,\frac{1}{2},\frac{1}{3},\frac{1}{4} \right\} be two bijective functions such that ff is strictly decreasing and g is strictly increasing. If ϕ(x)=[f−1{ g−1(12)}]x\phi(x) =\left\lbrack f^{- 1}\left\{ {\text{ }g}^{- 1}\left( \frac{1}{2} \right) \right\} \right\rbrack^{x}, then the area of the region R={(x,y):x2≤y≤ϕ(x)R =\left\{ (x,y):x^{2}\leq y \leq\phi(x) \right., 0≤x≤1}0 \leq x \leq 1\} is :

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  21. 71
    Let A=[−11−1101001]A = \begin{bmatrix} - 1 & 1 & - 1 \\ 1 & 0 & 1 \\ 0 & 0 & 1 \end{bmatrix} satisfy A2+α(adj(adj(A)))+β(adj(A)(adj(adj(A))))=[2−22−20−100−1]A^{2} + \alpha(adj(adj(A))) + \beta(adj(A)(adj(adj(A)))) = \begin{bmatrix} 2 & - 2 & 2 \\ - 2 & 0 & - 1 \\ 0 & 0 & - 1 \end{bmatrix} for Some α,β∈R\alpha,\beta \in R. Then (α−β)2(\alpha - \beta)^{2} is equal to ____\_\_\_\_
  22. 72
    Let the centre of the circle x2+y2+2gx+2fy+25=0x^{2} + y^{2} + 2gx + 2fy + 25 = 0 be in the first quadrant and lie on the line 2x−y=42x - y = 4. Let the area of an equilateral triangle inscribed in the circle be 27327\sqrt{3}. Then the square of the length of the chord of the circle on the line x=1x = 1 is____\_\_\_\_ .
  23. 73
    If a→=i^+j^+k^,b→=j^−k^\overrightarrow{a} = \widehat{i} + \widehat{j} + \widehat{k},\overrightarrow{b} = \widehat{j} - \widehat{k} and c→\overrightarrow{c} be three vectors such that a→×c→=b→\overrightarrow{a} \times \overrightarrow{c} = \overrightarrow{b} and a→⋅c→=3\overrightarrow{a} \cdot \overrightarrow{c} = 3, then c→⋅(a→−2b→)\overrightarrow{c} \cdot (\overrightarrow{a} - 2\overrightarrow{b}) is equal to ____\_\_\_\_ .
  24. 74
    For the functions f(θ)=αtan⁡2θ+βcot⁡2θf(\theta) = \alpha\tan^{2}\theta + \beta\cot^{2}\theta, and g(θ)=αsin⁡2θ+βcos⁡2θ,α>β>0g(\theta) = \alpha\sin^{2}\theta + \beta\cos^{2}\theta,\alpha > \beta > 0, let min⁡0<θ<π2f(θ)=max⁡0<θ<πg(θ)\min_{0 < \theta < \frac{\pi}{2}} f(\theta) = \max_{0 < \theta < \pi} g(\theta). If the first term of a G.P. is (α2β)\left( \frac{\alpha}{2\beta} \right), its common ratio is (2βα)\left( \frac{2\beta}{\alpha} \right) and the sum of its first 10 terms is mn,gcd(m,n)=1\frac{m}{n},gcd(m,n) = 1, then m+nm + n is equal to ____\_\_\_\_ .
  25. 75
    Let y=y(x)y = y(x) be the solution of the differential equation (x2−xx2−1)dy+(y(x−x2−1)−x)dx=0\left( x^{2} - x\sqrt{x^{2} - 1} \right)dy + \left( y\left( x - \sqrt{x^{2} - 1} \right) - x \right)dx = 0, x≥1x \geq 1. If y(1)=1y(1) = 1, then the greatest integer less than y(5)y(\sqrt{5}) is ____\_\_\_\_ .

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