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JEE Mains 2026 (5 Apr, Shift 2), Paper 1 (B.E./B.Tech) question paper

75 questions · 300 marks · 180 minutes · +4 per correct answer, -1 per wrong one

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Sit it as a timed mock

Physics

25 questions

  1. 1
    Match List-I with List-II.
    List-IList-II
    (A) Meter (L)(I) hcG\sqrt{\frac{hc}{G}}
    (B) Second (S)(II) Ghc5\sqrt{\frac{Gh}{c^{5}}}
    (C) Kilogram (M)(III) K2L2c3Gh\sqrt{\frac{K^{2}L^{2}c^{3}}{Gh}}
    (D) Kelvin (K)(IV) Ghc3\sqrt{\frac{Gh}{c^{3}}}
    where hh (Planck's constant), GG (gravitational constant) and cc (speed of light in vacuum) are the fundamental units. Choose the correct answer from the options given below:

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  2. 2
    In an experiment to determine the resistance of a given wire using Ohm's law, the voltmeter and ammeter readings are noted as 10 V and 5 A , respectively. The least counts of voltmeter and ammeter are 500 mV and 200 mA , respectively. The estimated error in the resistance measurement is ____\_\_\_\_ Ω\Omega

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  3. 3
    A mass of 1 kg kept on an inclined plane with 30∘30^{\circ} inclination with respect to horizontal plane and it is at rest initially. Then the whole assembly is moved up with constant velocity of 4 m/s4\text{ }m/s. The work done by the frictional force in time 2 s is ____\_\_\_\_ J. (Take g=10 m/s2g = 10\text{ }m/s^{2} )

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  4. 4
    The velocity (v)(v) versus time (t)(t) plot of a particle is shown in the figure, for a time interval of 40s. The total distance travelled by the particle and the average velocity during this period are, respectively ____\_\_\_\_

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  5. 5
    A wheel initially at rest is subjected to a uniform angular acceleration about its axis. In the first 2 s it rotates through an angle θ1\theta_{1} and in the next 2 s it rotates through an angle θ2\theta_{2}. The ratio θ2θ1\frac{\theta_{2}}{\theta_{1}} is ____\_\_\_\_ .

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  6. 6
    An object of uniform density rolls up the curved path with the initial velocity v0v_{0} as shown in the figure. If the maximum height attained by an object is 7v0210 g\frac{7v_{0}^{2}}{10\text{ }g} (g=g = acceleration due to gravity), the object is a ____\_\_\_\_ .

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  7. 7
    A body of mass mm is taken from the surface of earth to a height equal to twice the radius of earth (ReR_{e}). The increase in potential energy will be ____\_\_\_\_ . (g is acceleration due to gravity at the surface of earth)

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  8. 8
    Eight mercury drops, each of radius rr, coalesce to form a bigger drop. The surface energy released in this process is ____\_\_\_\_ . (S is the surface tension of mercury).

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  9. 9
    An ideal gas at pressure PP and temperature TT is expanding such that PT3=PT^{3}= constant. The coefficient of volume expansion of the gas is ____\_\_\_\_ .

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  10. 10
    Match List-I with List-II.
    List-IList-II
    (A) sin⁡2ωt\sin^{2}\omega t(I) Periodic with time period T=πωT = \frac{\pi}{\omega} but not simple harmonic motion (SHM)
    (B) sin⁡3(2ωt)\sin^{3}(2\omega t)(II) Periodic with time period T=2πωT = \frac{2\pi}{\omega} but not SHM
    (C) sin⁡(ωt)+cos⁡(πωt)\sin(\omega t) + \cos(\pi\omega t)(III) Periodic with time period T=πωT = \frac{\pi}{\omega} and SHM
    (D) cos⁡ωt+cos⁡2ωt\cos\omega t + \cos 2\omega t(IV) Non-periodic
    Choose the correct answer from the options given below.

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  11. 11
    A metal rod of length LL rotates about one end at the origin with a uniform angular velocity ω\omega. The magnetic field radially falls off as B(r)=Boe−λrB(r) =B_{o}e^{-\lambda r}; λ\lambda being a positive constant. The emf induced (neglecting the centripetal force on electrons in the rod) is:

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  12. 12
    Under steady state condition the potential difference across the capacitor in the circuit is ____\_\_\_\_ V.

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  13. 13
    A particle of charge qq and mass mm is projected from origin with an initial velocity v→=(v02x^+v02y^)\overrightarrow{v}=\left( \frac{v_{0}}{\sqrt{2}}\widehat{x}+\frac{v_{0}}{\sqrt{2}}\widehat{y} \right). There exists a uniform magnetic field B→=B0z^\overrightarrow{B}=B_{0}\widehat{z} and a space varying electric field E→=E0e−λxx^\overrightarrow{E}=E_{0}e^{-\lambda x}\widehat{x} within the region 0≤x≤L0 \leq x \leq L. After travelling a distance such that x -coordinate has changed from x=0x= 0 to x=Lx=L, the change in the kinetic energy is ____\_\_\_\_ .

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  14. 14
    Given below are two statements : one is labelled as Assertion (A) : and the other is labelled as Reason (R). Assertion (A) : The electromagnetic wave exerts pressure on the surface on which they are allowed to fall. Reason (R) : There is no mass associated with the electromagnetic waves. In the light of the above statements, choose the correct answer from the options given below:

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  15. 15
    A thin convex lens and a thin concave lens are kept in contact and are co-axial. Which of the following statements is correct for this combination of two lenses?

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  16. 16
    An object AB is placed 15 cm on the left of a convex lens P of focal length 10 cm . Another convex lens Q is now placed 15 cm right of lens P . If the focal length of lens Q is 15 cm , the final image is ____\_\_\_\_

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  17. 17
    The maximum intensity in a Young's double slit experiment is I0I_{0}. Distance between the slits (d) is 5λ5\lambda, where λ\lambda is the wavelength of light used. The intensity of the fringe, exactly opposite to one of the slits on the screen, placed at D=10 dD = 10\text{ }d is ____\_\_\_\_

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  18. 18
    An electron is travelling with a velocity vv in free space and when it enters a medium, its velocity is reduced by 20%20\%. The de Broglie wavelength of electron in the medium is αλ0\alpha\lambda_{0}, where λ0\lambda_{0} is its de Broglie wavelength in free space. The value of α\alpha is ____\_\_\_\_

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  19. 19
    Assuming the experimental mass of  612C\ _{6}^{12}C as 12 u , the mass defect of  612C\ _{6}^{12}C atom is ____\_\_\_\_ MeV/c2MeV/c^{2}. (Mass of proton =1.00727u= 1.00727u. mass of neutron =1.00866u,1u=931.5MeV/c2= 1.00866u,1u = 931.5MeV/c^{2} and c is the speed of the light in vacuum.)

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  20. 20
    In a semiconductor p−np - n diode, the doping concentrations on p -side and n -side are 101510^{15} atoms/ cm3cm^{3} and 101810^{18} atoms /cm3/cm^{3}, respectively. Which one of the following statement is true?

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  21. 21
    A copper wire of length 3 m is stretched by 3 mm by applying an external force. The volume of the wire is 600×10−6 m3600 \times 10^{- 6}{\text{ }m}^{3}. The elastic potential energy stored in the wire in stretched condition would be ____\_\_\_\_ J. (Given Young modulus of copper  =1.1×1011 N/m2)\left. \ = 1.1 \times 10^{11}\text{ }N/m^{2} \right)
  22. 22
    The heat extracted out of x gram of water initially at 50∘C50^{\circ}C to cool it down to 0∘C0^{\circ}C is sufficient to evaporate (1000−x1000 - x) gram of water also initially at 50∘C50^{\circ}C. The value of x (closest integer) is ____\_\_\_\_ . (Take latent heat of water 2256 kJ/kg2256\text{ }kJ/kg, specific heat capacity of water 4200 J/kg K4200\text{ }J/kg\ K)
  23. 23
    A series LCR circuit with R =20Ω, L=1.6H= 20\Omega,\text{ }L = 1.6H and C=40μ FC = 40\mu\text{ }F is connected to a variable frequency a.c. source. The inductive reactance at resonant frequency is ____\_\_\_\_ Ω\Omega.
  24. 24
    When an external resistance of 5Ω5\Omega is connected across terminals of a cell, a current of 0.25 A flows through it. When the 5Ω5\Omega resistor is replaced by a 2Ω2\Omega resistor, a current of 0.5 A flows through it. The internal resistance of the cell is ____\_\_\_\_ Ω\Omega.
  25. 25
    A circular loop of radius 20 cm and resistance 2Ω2\Omega is placed in a time varying magnetic field B→=(2t2+2t+3)T\overrightarrow{B} = \left( 2t^{2} + 2t + 3 \right)T. The plane of the loop is perpendicular to the magnetic field. The induced current in the loop at t=3 st = 3\text{ }s is α50 A\frac{\alpha}{50}\text{ }A. The value of α\alpha is ____\_\_\_\_ (Take π=22/7\pi = 22/7 )

Chemistry

25 questions

  1. 26
    What volume of hydrogen gas at STP would be liberated by action of 50 mL of H2SO4H_{2}SO_{4} of 50%50\% purity (density =1.3 g mL−1= 1.3\text{ }g{\text{ }mL}^{- 1} ) on 20 g of zinc? Given : Molar mass of H, O, S, Zn are 1, 16, 32, 65 gmol−1gmol^{- 1} respectively.

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  2. 27
    Which of the following statement(s) is/are true ? (A) If two orbitals have the same value of ( n+ln + l ), the orbital with lower value of nn will have lower energy. (B) Energies of the orbitals in the same subshell increase with increase in atomic number. (C) The size of 2px2p_{x} orbital is less than the size of 3px3p_{x} orbital. (D) Among 5f,6s, 4d, 5p and 5d orbitals, none of the orbitals have 2 radial nodes. Choose the correct answer from the options given below :

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  3. 28
    The covalent radii of atoms AA and BB are rAr_{A} and rBr_{B}, respectively. The covalent bond length and total length of AB molecule are respectively :

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  4. 29
    Consider the following data for the reaction X2( g)+Y2( g)⇌2XY(g)X_{2}(\text{ }g) +Y_{2}(\text{ }g) \rightleftharpoons 2XY(g) at 600 K. The ΔrG⊖\Delta_{r}G^{\ominus} (in kJ mol−1kJ\,mol^{-1}) for the reaction is:
    Compound
    ΔfH600 KΘ\Delta_{f}H_{600\text{ }K}^{\Theta}
    (kJmol−1)\left( kJmol^{- 1} \right)
    S600 KΘS_{600\text{ }K}^{\Theta}
    (Jmol−1 K−1)\left( Jmol^{- 1}{\text{ }K}^{- 1} \right)
    XY(g)XY(g)
    		42200
    X2( g)X_{2}(\text{ }g)
    		8140
    Y2( g)Y_{2}(\text{ }g)
    		80250

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  5. 30
    The correct order of molar heat capacities measured at 298 K and 1 bar is :

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  6. 31
    The reaction A(g)⇌B(g)+C(g)A(g) \rightleftharpoons B(g) + C(g) was initiated with the amount ' a ' of A(g)A(g). At equilibrium it is found that the amount of A(g)A(g) remaining is (a−x)(a - x) at a total pressure of p. The equilibrium constant KpK_{p} of the reaction can be calculated from the expression :

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  7. 32
    One half cell in a voltaic cell is constructed by dipping silver rod in AgNO3AgNO_{3} solution of unknown concentration, other half cell is Zn rod dipped in 1 molar solution of ZnSO4ZnSO_{4}. A voltage of 1.60 V is measured at 298 K for this cell. What is the concentration of Ag+Ag^{+}ions used in terms of log⁡x(x=[Ag+])\log x\left( x =\left\lbrack Ag^{+} \right\rbrack \right)? EZn2+/ZnΘ=−0.76 V,EAg+/AgΘ=+0.80 VE_{Zn^{2 +}/Zn}^{\Theta}= - 0.76\text{ }V,E_{Ag^{+}/Ag}^{\Theta}= + 0.80\text{ }V, 2.303RTF=0.059 V\frac{2.303RT}{F}= 0.059\text{ }V

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  8. 33
    Given below are two statements : Statement-I : The number of pairs among [Al2O3\left\lbrack {Al}_{2}O_{3} \right.,  Cr2O3],[Cl2O7,Mn2O7],[Na2O,V2O3]\left. \ {Cr}_{2}O_{3} \right\rbrack,\left\lbrack {Cl}_{2}O_{7},{Mn}_{2}O_{7} \right\rbrack,\left\lbrack {Na}_{2}O,V_{2}O_{3} \right\rbrack and [CO\lbrack CO,  N2O]\left. \ N_{2}O \right\rbrack that contain oxides of same nature (acidic, basic, neutral or amphoteric) is 4. Statement-II : Among Na2O,Al2O3,CO{Na}_{2}O,{Al}_{2}O_{3},CO and Cl2O7{Cl}_{2}O_{7}, the most basic and acidic oxides are Na2O{Na}_{2}O and Cl2O7{Cl}_{2}O_{7}, respectively. In the light of the above statements, choose the correct answer from the options given below:

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  9. 34
    Given below are two statements : Statement-I : Aluminum upon reaction with NaOH forms [Al(OH)6]3−\left\lbrack Al(OH)_{6} \right\rbrack^{3 -} ion. Statement-II : The geometry of ICl4−,ClO3−{ICl}_{4}^{-},{ClO}_{3}^{-}and IBr2−{IBr}_{2}^{-}is square planar, pyramidal and linear respectively. In the light of the above statements, choose the correct answer from the options given below:

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  10. 35
    Given below are two statements : Statement-I : Presence of large number of unpaired electrons in transition metal atoms results in higher enthalpies of their atomization. Statement-II : dxy=dxz=dyz<dx2−y2=dz2d_{xy} = d_{xz} = d_{yz} < d_{x^{2} - y^{2}} = d_{z^{2}} and dx2−y2=dz2<dxy=dxz=dyzd_{x^{2} - y^{2}} = d_{z^{2}} < d_{xy} = d_{xz} = d_{yz} are the d-orbital splitting in [Fe(H2O)6]3+\left\lbrack Fe\left( H_{2}O \right)_{6} \right\rbrack^{3 +} and [Ni(Cl)4]2−\left\lbrack Ni(Cl)_{4} \right\rbrack^{2 -} complex ions respectively. In the light of the above statements, choose the correct answer from the options given below:

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  11. 36
    Identify the correct statements from the following: (A) [Fe(C2O4)3]3−\left\lbrack Fe\left( C_{2}O_{4} \right)_{3} \right\rbrack^{3 -} is the most stable complex among [Fe(OH)6]3−,[Fe(C2O4)3]3−\left\lbrack Fe(OH)_{6} \right\rbrack^{3 -},\left\lbrack Fe\left( C_{2}O_{4} \right)_{3} \right\rbrack^{3 -} and [Fe(SCN)6]3−\left\lbrack Fe(SCN)_{6} \right\rbrack^{3 -} (B) The stability of [Cu(NH3)4]2+\left\lbrack Cu\left( {NH}_{3} \right)_{4} \right\rbrack^{2 +} is greater than that of [Cu(en)2]2+\left\lbrack Cu(en)_{2} \right\rbrack^{2 +} (C) The hybridization of Fe in K4[Fe(CN)6]K_{4}\left\lbrack Fe(CN)_{6} \right\rbrack is d2sp3d^{2}{sp}^{3} (D) [Fe(NO2)3Cl3]3−\left\lbrack Fe\left( {NO}_{2} \right)_{3}{Cl}_{3} \right\rbrack^{3 -} exhibits linkage isomerism (E) NO2−{NO}_{2}^{-}and SCN−{SCN}^{-}ligands are NOT ambidentate ligands Choose the correct answer from the options given below :

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  12. 37
    Match List-I with List-II.
    List-I (Purification technique)List-II (Used to separate)
    (A)Simple distillation(I)Steam volatile compound
    (B)Fractional distillation(II)Two liquids with large difference in boiling points
    (C)Steam distillation(III)Liquid decomposing at its boiling point
    (D)Distillation under reduced pressure(IV)Two liquids with close boiling points
    Choose the correct answer from the options given below :

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  13. 38
    IUPAC name of some alkenes are given below : Find out the correct stability order. (A) 2-Methylbut-2-ene (B) cis-But-2-ene (C) 2,3-Dimethylbut-2-ene (D) Prop-1-ene Choose the correct answer from the options given below:

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  14. 39
    Identify the correct IUPAC name of hydrocarbon (x) containing three primary carbon atoms and with molar mass 72 g mol−172\text{ }g{\text{ }mol}^{- 1}.

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  15. 40
    Complete the following reaction sequence and give the name of major product ' P '.

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  16. 41
    Given below are two statements : Statement-I : The condensation reaction between CH3−CH=O{CH}_{3} - CH = O and semicarbazide, H2N−NH−CO−NH2{H}_{2}N - NH - CO - {NH}_{2}, under optimum pH will produce CH3−CH=N−CO−NH−NH2{CH}_{3} - CH = N - CO - NH - {NH}_{2} Statement-II : The molecule Ph−CH(OH)(OCH3)Ph - CH(OH)(O{CH}_{3}) will generate Ph−CH=OPh - CH = O in the presence of dilute acid. In the light of the above statements, choose the correct answer from the options given below:

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  17. 42
    Given below are two statements : Statement-I : Heating benzamide with bromine in an ethanolic solution of sodium hydroxide will give benzylamine. Statement-II : Nitration of aniline with HNO3/H2SO4{HNO}_{3}/H_{2}{SO}_{4} at 288 K produces mm-nitroaniline in higher amount than oo-nitroaniline ( pH adjusted). In the light of the above statements, choose the correct answer from the options given below:

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  18. 43
    Identify the incorrect statement about tertiary structure of proteins.

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  19. 44
    Given below are two statements : Statement-I : The two structures shown are two anomers of D-(+)-glucose. Statement-II : The open chain forms of D-glucose and D-fructose contain three similar chiral carbons at C3,C4C_{3},C_{4} and C5C_{5}. In the light of the above statements, choose the correct answer from the options given below :

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  20. 45
    A paper dipped in a dil. H2SO4H_{2}SO_{4} solution of ' X ' upon treatment with SO2SO_{2} gas turns into green. The compound ' X ' is :

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  21. 46
    The total number of unpaired electrons present in the d3, d4d^{3},{\text{ }d}^{4} (low spin) d5d^{5} (high spin), d6d^{6} (high spin) and d7d^{7} (low spin) octahedral complex systems is ____\_\_\_\_ .
  22. 47
    RMgI when treated with ice cold water liberated a gas which occupied 1.4dm3/g1.4{dm}^{3}/g at STP. The gas produced is further reacted with iodine in presence of HIO3{HIO}_{3} to give compound (X). Compound (X) in presence of Na and dry ether produced compound (Y)(Y). Molar mass of compound (Y)(Y) is ____\_\_\_\_ gmol−1g{mol}^{- 1}. (Nearest integer)
  23. 48
    20 g hemoglobin in a 1 L aqueous solution (A) at 300 K is separated from pure water by semi permeable membrane. At equilibrium the height of solution in a tube dipped in a solution (A) is found to be 80.0 mm higher than the tube dipped in water. The molar mass of hemoglobin is ____\_\_\_\_ kgmol−1kg{mol}^{- 1}.(Nearest integer) (Given : g=10 ms−2,R=8.3kPadm3 K−1 mol−1g = 10{\text{ }ms}^{- 2},R = 8.3kPa{dm}^{3}{\text{ }K}^{- 1}{\text{ }mol}^{- 1}, density of solution =1000 kg m−3= 1000\text{ }kg{\text{ }m}^{- 3} )
  24. 49
    At 298 K , the molar conductivity of x%(w/w)x\%(w/w) MX solution (aqueous) is 123.5 S cm2 mol−1123.5\text{ }S{\text{ }cm}^{2}{\text{ }mol}^{- 1}. The conductance of same solution is 1.9×10−3 S1.9 \times 10^{- 3}\text{ }S. The value of xx is ____\_\_\_\_ ×10−2\times 10^{- 2}. (Given : cell constant =1.3 cm−1= 1.3{\text{ }cm}^{- 1}; molar mass of MX is 75 g mol−175\text{ }g{\text{ }mol}^{- 1}, density of aqueous solution of MX at 298 K is 1.0 g mL−11.0\text{ }g{\text{ }mL}^{- 1} )
  25. 50
    For a reaction A→PA \rightarrow P at TK, the half life (t1/2)\left( t_{1/2} \right) is plotted as a function of initial concentration [A]0\lbrack A\rbrack_{0} of A as given below : The value of x in the given figure is ____\_\_\_\_ s (Nearest integer)

Mathematics

25 questions

  1. 51
    Let α,β\alpha,\beta be the roots of the equation x2−x+p=0x^{2}- x + p = 0 and γ,δ\gamma,\delta be the roots the equation x2−4x+q=0x^{2}- 4x + q = 0; p,q∈Zp,q \in Z. If α,β,γ,δ\alpha,\beta,\gamma,\delta are in G.P., then ∣p+q∣|p + q| equals:

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  2. 52
    Let z1,z2∈Cz_{1},z_{2}\in C be the distinct solutions of the equation z2+4z−(1+12i)=0z^{2}+ 4z- (1 + 12i) = 0. Then ∣z1∣2+∣z2∣2\left| z_{1} \right|^{2}+\left| z_{2} \right|^{2} is equal to :

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  3. 53
    If f:N→Zf:N \rightarrow Z is defined by f(n)=∣n−1−5−2n23(2k+1)2k+1−3n33k(2k+1)3k(k+2)+1∣,k∈Nf(n) =\left| \begin{matrix} n & - 1 & - 5 \\ - 2n^{2} & 3(2k+ 1) & 2k+ 1 \\ - 3n^{3} & 3k(2k+ 1) & 3k(k+ 2) + 1 \end{matrix} \right|,k\in N, and ∑n=1kf(n)=98\sum_{n= 1}^{k} f(n) = 98, then kk is equal to:

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  4. 54
    Let M be a 3×33 \times 3 matrix such that M(100)=(123),M(010)=(012)M\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}=\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix},M\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}=\begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix} and M(001)=(−111)M\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}=\begin{pmatrix} - 1 \\ 1 \\ 1 \end{pmatrix}. If M(xyz)=(1711)M\begin{pmatrix} x \\ y \\ z \end{pmatrix}=\begin{pmatrix} 1 \\ 7 \\ 11 \end{pmatrix}, then x+y+zx + y + z equals :

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  5. 55
    If the sum of the first 10 terms of the series 11+14×4+21+24×4+31+34×4+41+44×4+…\frac{1}{1 +1^{4}\times 4}+\frac{2}{1 +2^{4}\times 4}+\frac{3}{1 +3^{4}\times 4}+\frac{4}{1 +4^{4}\times 4}+ \ldots is mn,gcd(m,n)=1\frac{m}{n},gcd(m,n) = 1, then m+nm + n is equal to :

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  6. 56
    Let A1, A2, A3,……..,A39A_{1},{\text{ }A}_{2},{\text{ }A}_{3},\ldots\ldots..,A_{39} be 39 arithmetic means between the numbers 59 and 159. Then the mean of A25, A28, A31A_{25},{\text{ }A}_{28},{\text{ }A}_{31} and A36A_{36} is equal to:

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  7. 57
    The coefficient of x2x^{2} in the expansion of (2x2+1x)10,x≠0\left( 2x^{2}+\frac{1}{x} \right)^{10},x \neq 0, is :

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  8. 58
    The probabilities that players A and B of a team are selected for the captaincy for a tournament are 0.6 and 0.4 , respectively. If A is selected the captain, the probability that the team wins the tournament is 0.8 and if B is selected the captain, the probability that the team wins the tournament is 0.7 Then the probability, that the team wins the tournament, is:

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  9. 59
    A box contains 5 blue, 6 yellow and 4 red balls. The number of ways, of drawing 8 balls containing at least two balls of each colour, is:

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  10. 60
    A variable XX takes values 0,0,2,6,12,20,…,n(n−1)0,0,2,6,12,20,\ldots,n(n - 1) with frequencies  nC0, nC1, nC2, nC3, nC4\ ^{n}C_{0},\ ^{n}C_{1},\ ^{n}C_{2},\ ^{n}C_{3},\ ^{n}C_{4},  nC5,….. nCn\ ^{n}C_{5},\ldots..\ ^{n}C_{n}, respectively. If the mean of this data is 60 , then its median is:

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  11. 61
    Let the point P be the vertex of the parabola y=x2−6x+12y=x^{2}- 6x+ 12. If a line passing through the point PP intersects the circle x2+y2−2x−4y+3=0x^{2}+y^{2}- 2x- 4y+ 3 = 0 at the points R and S , then the maximum value of (PR+PS)2(PR + PS)^{2} is:

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  12. 62
    Let the directrix of the parabola P:y2=8xP:y^{2}= 8x, cut x -axis at the point A . Let B(α,β),α>1B(\alpha,\beta),\alpha> 1, be a point on P such that the slope of AB is 3/53/5. If BC is a focal chord of P , then six times the area of △ABC\bigtriangleup ABC is:

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  13. 63
    Let the eccentricity e of a hyperbola satisfy the equation 6e2−11e+3=06e^{2}- 11e + 3 = 0. If the foci of the hyperbola are (3,5)(3,−4)(3,5)(3, - 4), then the length of its latus rectum is:

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  14. 64
    Let a triangle PQR be such that P and Q lie on the line x+38=y−42=z+12\frac{x+ 3}{8}=\frac{y- 4}{2}=\frac{z+ 1}{2} and are at a distance of 6 units from R(1,2,3)R(1,2,3). If (α,β,γ)(\alpha,\beta,\gamma) is the centroid of ΔPQR\Delta PQR, then α+β+γ\alpha+\beta+\gamma is equal to:

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  15. 65
    If the distance of the point (a,2,5)(a,2,5) from the image of the point (1,2,7)(1,2,7) in the line x1=y−11=z−22\frac{x}{1}=\frac{y- 1}{1}=\frac{z- 2}{2} is 4, then the sum of all possible values of a is equal to :

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  16. 66
    Let O be the origin, OP→=a→\overrightarrow{OP}=\overrightarrow{a} and OQ→=b→\overrightarrow{OQ}=\overrightarrow{b}. If RR is the point on OP→\overrightarrow{OP} such that OP→=5OR→\overrightarrow{OP}= 5\overrightarrow{OR}, and M is the point such that OQ→=5RM→\overrightarrow{OQ}= 5\overrightarrow{RM}, then PM→\overrightarrow{PM} is equal to :

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  17. 67
    Let f(x)=lim⁡y→0(1−cos⁡(xy))tan⁡(xy)y3f(x) =\lim_{y\rightarrow 0} \frac{(1 - \cos(xy))\tan(xy)}{y^{3}}. Then the number of solutions of the equation f(x)=sin⁡xf(x) = \sin x, x∈Rx \in R is :

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  18. 68
    Let (21−a+21+a),f(a),(3a+3−a)\left( 2^{1 - a}+2^{1 + a} \right),f(a),\left( 3^{a}+3^{- a} \right) be a A. P. and α\alpha be the minimum value of f(a)f(a). Then the value of the integral ∫log⁡e(α−1)log⁡e(α)dx(e2x−e−2x)\int_{\log_{e}(\alpha- 1)}^{\log_{e}(\alpha)} \frac{dx}{\left( e^{2x}-e^{- 2x} \right)} is :

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  19. 69
    Let f:[1,∞)→Rf:\lbrack 1,\infty) \rightarrow R be a differentiable function defined as f(x)=∫1xf(t)dt+(1−x)(log⁡ex−1)+ef(x) =\int_{1}^{x} f(t)dt+ (1 -x)\left( \log_{e}x- 1 \right)+e. Then the value of f(f(1))f(f(1)) is :

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  20. 70
    Let f(x)f(x) and g(x)g(x) be twice differentiable functions satisfying f′′(x)=g′′(x)f^{''}(x) =g^{''}(x) for all x∈Rx \in R, f′(1)=2 g′(1)=4f^{'}(1) = 2{\text{ }g}^{'}(1) = 4 and g(2)=3f(2)=9g(2) = 3f(2) = 9. Then f(25)−g(25)f(25) - g(25) is equal to :

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  21. 71
    Let A={1,4,7}A = \{ 1,4,7\} and B={2,3,8}B = \{ 2,3,8\}. Then the number of elements, in the relation R={((a1, b1) R = \left\{ \left( \left( a_{1},{\text{ }b}_{1} \right) \right.\ \right.,  (a2, b2))∈((A×B)×(A×B)):a1+b2\left. \ \left( a_{2},{\text{ }b}_{2} \right) \right) \in ((A \times B) \times (A \times B)):a_{1} + b_{2} divides  a2+b1}\left. \ a_{2} + b_{1} \right\} is ____\_\_\_\_
  22. 72
    From the point (−1,−1)( - 1, - 1), two rays are sent making angles of 45∘45^{\circ} with the line x+y=0x + y = 0. These rays get reflected from the mirror x+2y=1x + 2y = 1. If the equations of the reflected rays are ax+by=9ax + by = 9 and cx+dy=7,a,b,c,d∈Zcx + dy = 7,a,b,c,d \in Z, then the value of ad+bcad + bc is ____\_\_\_\_
  23. 73
    If S={θ∈[−π,π]:cos⁡θcos⁡5θ2=cos⁡7θcos⁡7θ2}S = \left\{ \theta \in \lbrack - \pi,\pi\rbrack:\cos\theta \cos\frac{5\theta}{2} = \cos7\theta\cos\frac{7\theta}{2} \right\}, then n(S)n(S) is equal to .........
  24. 74
    Let f:R→Rf:R \rightarrow R be a function such that f(x)+3f(π2−x)=sin⁡x,x∈Rf(x) + 3f\left( \frac{\pi}{2} - x \right) = \sin x,x \in R. Let maximum value of ff on RR be α\alpha. If the area of the region bounded by the curves g(x)=x2g(x) = x^{2} and h(x)=βx3h(x) = \beta x^{3}, β>0\beta > 0, is α2\alpha^{2}, then 30β330\beta^{3} is equal to ____\_\_\_\_
  25. 75
    Let y=y(x)y = y(x) be the solution of the differential equation (tan⁡x)1/2dy=(sec⁡3x−(tan⁡x)3/2y)dx(\tan x)^{1/2}dy = \left( \sec^{3}x - (\tan x)^{3/2}y \right)dx, 0<x<π2,y(π4)=6250 < x < \frac{\pi}{2},y\left( \frac{\pi}{4} \right) = \frac{6\sqrt{2}}{5}. If y(π3)=45αy\left( \frac{\pi}{3} \right) = \frac{4}{5}\alpha, then α4\alpha^{4} equals ____\_\_\_\_

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