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JEE Mains 2026 (8 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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Physics

25 questions

  1. 1
    A new unit ( α\alpha ) of length is chosen such that it is equal to the speed of light in vacuum. What is the distance between Venus and Earth in terms of α\alpha units if light takes 6 min. 40 s to cover this distance?

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  2. 2
    Consider the equation H=xp∈qErtsH=\frac{x^{p}\in^{q}E^{r}}{t^{s}} Where H=H = magnetic field; E=E = electric field; ϵ=\epsilon= permittivity, x=x = distance, t=t = time. The values of p,q,rp,q,r and ss respectively are :

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  3. 3
    A car moving with a speed of 54 km/h54\text{ }km/h takes a turn of radius 20 m . A simple pendulum is suspended from the ceiling of the car. Determine the angle made by the string of the pendulum with the vertical during the turning. (Take g=10 m/s2g = 10\text{ }m/s^{2} )

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  4. 4
    A gas balloon is going up with a constant velocity of 10 m/s10\text{ }m/s. When this balloon reached a height of 75 m , a stone is dropped from it and balloon keeps moving up with the same velocity. The height of the balloon when the stone hits the ground is ____\_\_\_\_ m . (Take g=10 m/s2g = 10\text{ }m/s^{2})

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  5. 5
    A thin biconvex lens is prepared from the glass ( μ=1.5\mu= 1.5 ) both curved surfaces of which have equal radii of 20 cm each. Left side surface of the lens is silvered from outside to make it reflect. To have the position of image and object at the same place, the object should be placed, from the lens at a distance of ____\_\_\_\_ cm.

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  6. 6
    Two identical bodies projected with the same speed at two different angles cover the same horizontal range R . If the time of flight of these bodies are 5 s and 10 s , respectively, then the value of RR is ____\_\_\_\_ m. (Take g=10 m/s2g = 10\text{ }m/s^{2} )

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  7. 7
    A solid cylinder having radius R and length L is slipping on a rough horizontal plane. At time t=0t = 0 the cylinder has a translational velocity v0=49 m/sv_{0}= 49\text{ }m/s, perpendicular to its axis and a rotational velocity v0/4Rv_{0}/4R about the centre. The time taken by the cylinder to start rolling is ____\_\_\_\_ seconds. (coefficient of kinetic friction μK=0.25\mu_{K}= 0.25 and g=9.8 m/s2g = 9.8\text{ }m/s^{2} )

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  8. 8
    A liquid of density 600 kg/m3600\text{ }kg/m^{3} flowing steadily in a tube of varying cross-section. The cross-section at a point A is 1.0 cm21.0{\text{ }cm}^{2} and that at B is 20 mm220{\text{ }mm}^{2}. Both the points A and B are in same horizontal plane, the speed of the liquid at A is 10 cm/s10\text{ }cm/s. The difference is pressures at A and B points is____\_\_\_\_ Pa.

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  9. 9
    A spherical liquid drop of radius R acquires the terminal velocity v1v_{1} when falls through a gas of viscosity η\eta. Now the drop is broken into 64 identical droplets and each droplet acquires terminal velocity v2v_{2} falling through the same gas. The ratio of terminal velocities v1/v2v_{1}/v_{2} is ____\_\_\_\_ .

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  10. 10
    One mole of diatomic gas having rotational modes only is kept in a cylinder with a piston system. The cross-section area of the cylinder is 0.4 cm20.4{\text{ }cm}^{2}. The gas is heated slowly to raise the temperature by 1.2 ∘C1.2\ ^{\circ}C during which the piston moves by 25 mm . The amount of heat supplied to the gas is ____\_\_\_\_ J. (Atmospheric pressure =100kPa= 100kPa, R=8.3 J/mol.KR = 8.3\text{ }J/mol.K ) (Neglect mass of the piston)

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  11. 11
    Initial pressure and volume of a monoatomic ideal gas are P and V. The change in internal energy of this gas in adiabatic expansion to volume Vfinal =27 VV_{\text{final~}}= 27\text{ }V is ____\_\_\_\_ J.

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  12. 12
    The frequency of oscillation of a mass m suspended by spring is v1v_{1}. If the length of the spring is cut to half, the same mass oscillates with frequency v2v_{2}. The value of v2v1\frac{v_{2}}{v_{1}} is ____\_\_\_\_ .

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  13. 13
    A monochromatic source of light operating at 15 kW emits 2.5×10222.5 \times10^{22} photons/s. The region of an electromagnetic spectrum to which the emitted electromagnetic radiation belongs to ____\_\_\_\_ . (Take h=6.6×10−34 Jh = 6.6 \times10^{- 34}\text{ }J.s and c=3×108 m/sc = 3 \times10^{8}\text{ }m/s ).

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  14. 14
    A current carrying circular loop of radius 2 cm with unit normal n^=k^+i^2\widehat{n}=\frac{\widehat{k}+\widehat{i}}{\sqrt{2}} is placed in a magnetic field, B→=B0(3i^+2k^)\overrightarrow{B}=B_{0}(3\widehat{i}+ 2\widehat{k}). If B0=4×10−3 TB_{0}= 4 \times10^{- 3}\text{ }T and current =1002 A= 100\sqrt{2}\text{ }A, the torque experienced by the loop is ____\_\_\_\_ Wb.A. (π=3.14)(\pi= 3.14)

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  15. 15
    A 30 cm long solenoid has 10 turns per cm and area of 5 cm25{\text{ }cm}^{2}. The current through the solenoid coil varies from 2 A to 4 A in 3.14 s. The e.m.f. induced in the coil is α×10−5 V\alpha \times10^{- 5}\text{ }V. The value α\alpha is ____\_\_\_\_ .

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  16. 16
    Two point charges q1=3μCq_{1}= 3\mu C and q2=−4μCq_{2}= - 4\mu C are placed at points (2i^+3j^+3k^)(2\widehat{i}+ 3\widehat{j}+ 3\widehat{k}) and (i^+j^+k^)(\widehat{i}+\widehat{j}+\widehat{k}) respectively. Force on charge q2q_{2} is ____\_\_\_\_ N. (Take 14πϵ0=9×109\frac{1}{4\pi\epsilon_{0}}= 9 \times10^{9} SI Units)

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  17. 17
    Light ray incident along a vector AO→(AO→=2i^−3j^)\overrightarrow{AO}(\overrightarrow{AO}= 2\widehat{i}- 3\widehat{j}) emerges out along vector OB→(OB→=Ci^−4j^)\overrightarrow{OB}(\overrightarrow{OB}= C\widehat{i}- 4\widehat{j}) as shown in the figure below. The value of C is ____\_\_\_\_ .

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  18. 18
    K1K_{1} and K2K_{2} be the maximum kinetic energies of photoelectrons emitted from a surface of a given material for the light of wavelength λ1\lambda_{1} and λ2\lambda_{2}, respectively. If λ1=2λ2\lambda_{1}= 2\lambda_{2} then the work function of material is given by :

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  19. 19
    Two radioactive substances A and B of mass numbers 200 and 212 respectively, shows spontaneous α\alpha-decay with same Q value of 1 MeV . The ratio of energies of α\alpha-rays produced by A and BB is ____\_\_\_\_ .

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  20. 20
    The output Y for the given inputs A and B to the circuit is :

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  21. 21
    A parallel plate capacitor is having separation between plates 0.885 mm . It has a capacitance of 1μ F1\mu\text{ }F when the space between the plates is filled with an insulating material of resistivity 1×1013Ω m1 \times 10^{13}\Omega\text{ }m and resistance 17.7×1014Ω17.7 \times 10^{14}\Omega. Relative permittivity of the insulating material is α×107\alpha \times 10^{7}. The value of α\alpha is ____\_\_\_\_ . (Take permittivity of free space =8.85×10−12 F/m= 8.85 \times 10^{- 12}\text{ }F/m)
  22. 22
    A distant star is to be observed by a telescope whose objective lens has diameter aa, at an angular resolution of 3.0×10−73.0 \times 10^{- 7} radian. If the wavelength of light from the star reaching the telescope is 500 nm , the minimum diameter of the objective lens of the telescope is ____\_\_\_\_ cm. (Nearest integer)
  23. 23
    A 5 mg particle carrying a charge of 5π×10−6C5\pi \times 10^{- 6}C is moving with velocity of (3i^+2k^)×10−2 m/s(3\widehat{i} + 2\widehat{k}) \times 10^{- 2}\text{ }m/s in a region having magnetic field B→=0.1k^Wb/m2\overrightarrow{B} = 0.1\widehat{k}Wb/m^{2}. It moves a distance of α\alpha meter along k^\widehat{k} when it completes 5 revolutions. The value of α\alpha is ____\_\_\_\_ .
  24. 24
    The stored charge in the capacitor in steady state of the following circuit is ____\_\_\_\_ μC\mu C.
  25. 25
    Two masses of 3.4 kg and 2.5 kg are accelerated from an initial speed of 5 m/s5\text{ }m/s and 12 m/s12\text{ }m/s, respectively. The distances traversed by the masses in the 5th 5^{\text{th~}} second are 104 m and 129 m , respectively. The ratio of their momenta after 10 s is x8\frac{x}{8}. The value of xx is ____\_\_\_\_ .

Chemistry

25 questions

  1. 26
    Match List-I with List-II.
    List-I Mass of substanceList-II Number of atoms
    (A)1.8mg1.8mg water(I)
    2×10−4×NA2 \times 10^{- 4} \times N_{A}
    (B)9.8mg9.8mg sulphuric acid(II)
    1.5×10−4×NA1.5 \times 10^{- 4} \times N_{A}
    (C)1.8 mg carbon(III)
    3×10−4×NA3 \times 10^{- 4} \times N_{A}
    (D)5.85 mg salt (NaCl)(NaCl)(IV)
    7×10−4×NA7 \times 10^{- 4} \times N_{A}
    Choose the correct answer from the options given below:

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  2. 27
    Given below are two statements : Given : Molar mass of C,H,O,ClC,H,O,Cl are 12,1,1612,1,16 and 35.5 g mol−135.5\text{ }g{\text{ }mol}^{- 1}, respectively. Statement I : In 30%(w/w)30\%(w/w) solution of methanol in CCl4{CCl}_{4} (at T K ), the mole fraction of CCl4{CCl}_{4} is equal to 0.33 . Statement II : Mixture of methanol and CCl4{CCl}_{4} shows positive deviation from Raoult's law. In the light of the above statements, choose the correct answer from the option given below :

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  3. 28
    Bromine trifluoride autoionizes to form BrF2⊕BrF_{2}^{\oplus} and BrF4⊖BrF_{4}^{\ominus}. The shapes of the cation and anion are respectively ____\_\_\_\_ , and ____\_\_\_\_ .

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  4. 29
    Which of the following are not correct? (A) For water, magnitude of KbK_{b} is more than the magnitude of KfK_{f}. (B) The elevation in boiling point of water when a non-volative solute is added to it is larger in magnitude than its depression in freezing point. (C) Osmotic pressure measurement is preferred over any other colligative property to determine molar mass of proteins and polymers. (D) The dimerized form of benzoic acid in benzene is Choose the correct answer from the options given below :

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  5. 30
    Consider the following reactions in which all the reactants and products are present in gaseous state 2xy⇌x2+y2 K1=2.5×1052xy \rightleftharpoons x_{2}+y_{2}{\text{ }K}_{1}= 2.5 \times10^{5}
    xy+12z2⇌xyz K2=5×10−3xy +\frac{1}{2}z_{2}\rightleftharpoons xyz\ K_{2}= 5 \times10^{- 3}
    The value of K3K_{3} for the equilibrium 12x2+12y2+12z2⇌xyz\frac{1}{2}x_{2}+\frac{1}{2}y_{2}+\frac{1}{2}z_{2}\rightleftharpoons xyz is :

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  6. 31
    Given at 298 K:EFe2+/Fe⊖=X298\text{ }K:E_{Fe^{2 +}/Fe}^{\ominus}= X Volt
    EFe3+/Fe⊖=Y Volt E_{Fe^{3 +}/Fe}^{\ominus}= Y\text{~Volt~}
    The EFe3+/Fe2+⊖E_{Fe^{3 +}/Fe^{2 +}}^{\ominus} in Volt at 298 K is given by :

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  7. 32
    Given below are two statements : R=8.314 J K−1 mol−1R = 8.314\text{ }J{\text{ }K}^{- 1}{\text{ }mol}^{- 1} and 1cal=4.2 J1cal = 4.2\text{ }J Statement I : When Ea=12.6kcal/molEa = 12.6kcal/mol, the room temperature rate constant is doubled by a 10∘C10^{\circ}C increase in temperature ( 298 K to 308 K ) Statement II : For a first order reaction A→BA \rightarrow B, the plot of t1/2t_{1/2} against [A]0\lbrack A\rbrack_{0} is as shown in the figure. Here [A]0\lbrack A\rbrack_{0} is the initial concentration of A and t1/2t_{1/2} is half-life of reaction. In the light of the above statements, choose the correct answer from the option given below :

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  8. 33
    Match List-I with List-II.
    List-I Electronic configuration of neutral atom (where n=2n = 2)List-II 1st1^{st} Ionization Energy (kJ mol−1kJ\,mol^{-1})
    (A)ns2{ns}^{2}(I)2080
    (B)ns2np1ns^{2}np^{1}(II)899
    (C)ns2np3ns^{2}np^{3}(III)800
    (D)ns2np6ns^{2}np^{6}(IV)1402
    Choose the correct answer from the options given below :

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  9. 34
    Find the correct statements related to group 15 hydrides. (A) Reducing nature increases from NH3{NH}_{3} to BiH3{BiH}_{3} (B) Tendency to donate lone pair of electrons decreases from NH3{NH}_{3} to BiH3{BiH}_{3} (C) The stability of hydrides decreases from NH3{NH}_{3} to BiH3{BiH}_{3} (D) HEH bond angle decreases from NH3{NH}_{3} to SbH3{SbH}_{3} ( E=E = Elements of group 15) Choose the correct answer from the options given below :

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  10. 35
    Given below are two statements : Statement I : The number of pairs among [Ti4+,V2+],[V2+,Mn2+],[Mn2+,Fe3+]\left\lbrack {Ti}^{4 +},V^{2 +} \right\rbrack,\left\lbrack V^{2 +},{Mn}^{2 +} \right\rbrack,\left\lbrack {Mn}^{2 +},{Fe}^{3 +} \right\rbrack and [V2+,Cr2+]\left\lbrack V^{2 +},{Cr}^{2 +} \right\rbrack in which both ions are coloured is 3 . Statement II : The number pairs among [ La3+{La}^{3 +},  Yb2+],[Lu3+,Ce4+]\left. \ {Yb}^{2 +} \right\rbrack,\left\lbrack {Lu}^{3 +},{Ce}^{4 +} \right\rbrack and [Ac3+,Lr3+]\left\lbrack {Ac}^{3 +},{Lr}^{3 +} \right\rbrack ions in which both are diamagnetic is 3 . In the light of the above statements, choose the correct answer from the option given below :

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  11. 36
    Given below are two statements for catalytic properties of transition metals : Statement I : First row transition metals which act as catalyst utilise their 3d electrons only for formation of bonds between reactant molecules and atoms on the surface of catalyst. Statement II : There is increase in the concentration of reactants on the surface of catalyst which strengthens the bonds in reacting molecules. In the light of the above statements, choose the correct answer from the option given below :

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  12. 37
    Given below are two statements : Statement I: Vapors of the liquid with higher boiling point condense before vapours of the liquid with lower boiling points in fractional distillation. Statement II : The vapors rising up in the fractionating column become richer in high boiling component of the mixture. In the light of the above statements, choose the correct answer from the option given below :

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  13. 38
    The major product of which of the following reaction is not obtained by rearrangement reaction?

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  14. 39
    The total number of aromatic compounds/species from the following is:

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  15. 40
    n-Butane on monochlorination under photochemical conditions gives an optically active compound "P". "P" on further chlorination gives dichloro compounds. The number of dichloro compounds obtained (ignore stereoisomers) is :

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  16. 41
    Given below are two statements : Statement-I : Due to increase in van der Waals forces, the order of boiling points is CH3CH2CH2I>CH3CH2I>CH3I{CH}_{3}{CH}_{2}{CH}_{2}I > {CH}_{3}{CH}_{2}I > {CH}_{3}I. Statement-II : As p-dichlorobenzene (1,4-dichlorobenzene) is more symmetric, its melting point is higher than that of o-dichlorobenzene (1,2-dichlorobenzene); however, its boiling point is lower than that of o-dichlorobenzene. 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 reaction. The major product (P)(P) formed is :

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  18. 43
    Which statements are True? (A) In Hoffmann bromamide degradation, 4 moles of NaOH and 2 moles of Br2{Br}_{2} are consumed per mole of an amide (B) Hoffmann bromamide reaction is not given by alkyl amides. (C) Primary amines can be synthesized by Hoffmann bromamide degradation. (D) Secondary amide on reaction with Br2{Br}_{2} and NaOH will give secondary amine. (E) The by-products of Hoffmann degradation are Na2CO3,NaBr{Na}_{2}{CO}_{3},NaBr and H2OH_{2}O. Choose the correct answer from the options given below :

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  19. 44
    The incorrect statement from the following with respect to carbohydrates is :

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  20. 45
    Which of the following amino acid will give violet coloured complex with neutral ferric chloride solution?

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  21. 46
    Number of paramagnetic complexes among the following is ____\_\_\_\_ . [MnBr4]2−\left\lbrack {MnBr}_{4} \right\rbrack^{2 -}, [NiCl4]2−\left\lbrack {NiCl}_{4} \right\rbrack^{2 -}, [Ni(CN)4]2−\left\lbrack Ni(CN)_{4} \right\rbrack^{2 -}, [Ni(CO)4]\left\lbrack Ni(CO)_{4} \right\rbrack, [CoF6]3−\left\lbrack {CoF}_{6} \right\rbrack^{3 -}, [Fe(CN)6]4−, [Mn(CN)6]3−, [Ti(CN)6]3−\lbrack Fe(CN)_{6}\rbrack^{4 -},\ \left\lbrack Mn(CN)_{6} \right\rbrack^{3 -},\ \left\lbrack Ti(CN)_{6} \right\rbrack^{3 -}, [Cu(H2O)6]2+,[Co(C2O4)3]3−\left\lbrack Cu\left( H_{2}O \right)_{6} \right\rbrack^{2 +},\left\lbrack Co\left( C_{2}O_{4} \right)_{3} \right\rbrack^{3 -}
  22. 47
    ' x ' is the product which is obtained from benzene by reacting it with carbon monoxide and hydrogen chloride in the presence of cuprous chloride. ' y ' is the major product obtained from the benzene by reacting it with ethanoyl chloride in the presence of anhydrous AlCl3{AlCl}_{3}. Product (major) obtained by heating x and y in the presence of alkali is z . Total number of π\pi (pi) electrons in z is ____\_\_\_\_ .
  23. 48
    Consider two radiations of wavelengths : 1. λ1=2000Å\lambda_{1} = 2000\text{Å} 2. λ2=6000Å\lambda_{2} = 6000\text{Å} The ratio of the energies of these two radiations (E1E2)\left( \frac{E_{1}}{E_{2}} \right) is ____\_\_\_\_ . (Nearest integer)
  24. 49
    Consider the reaction :
    2H2 S( g)+3O2( g)→2H2O(l)+2SO2( g)2H_{2}\text{ }S(\text{ }g) + 3O_{2}(\text{ }g) \rightarrow 2H_{2}O(l) + 2{SO}_{2}(\text{ }g)
    The magnitude of enthalpy change for the reaction in kJmol−1kJ{mol}^{- 1} is ____\_\_\_\_ . (Nearest integer) Given : ΔfH⊖(H2 S)=−20.1 kJ mol−1\Delta_{f}H^{\ominus}\left( H_{2}\text{ }S \right) = - 20.1\text{ }kJ{\text{ }mol}^{- 1}
    ΔfHΘ(H2O)=−286.0 kJ mol−1\Delta_{f}H^{\Theta}\left( H_{2}O \right) = - 286.0\text{ }kJ{\text{ }mol}^{- 1}
    ΔfHΘ(SO2)=−297.0 kJ mol−1\Delta_{f}H^{\Theta}\left( {SO}_{2} \right) = - 297.0\text{ }kJ{\text{ }mol}^{- 1}
  25. 50
    Solid carbon, CaO and CaCO3{CaCO}_{3} are mixed and allowed to attain equilibrium at T K .
    CaCO3(s)⇌CaO(s)+CO2(g),Kp1=0.08 atm{CaCO}_{3}(s) \rightleftharpoons CaO(s) + {CO}_{2}(g),\quad K_{p_{1}} = 0.08\text{ atm}
    C(s)+CO2(g)⇌2CO(g),Kp2=2 atmC(s) + {CO}_{2}(g) \rightleftharpoons 2CO(g),\quad K_{p_{2}} = 2\text{ atm}
    The partial pressure of CO is ____\_\_\_\_ ×10−1 atm\times 10^{- 1}\text{ }atm.

Mathematics

25 questions

  1. 51
    Consider the relation R on the set {−2,−1,0,1,2}\mathbf{\{ - 2, - 1,0,1,2\}} defined by (a,b)∈R(a,b) \in R if and only if 1+ab>01 + ab > 0. Then among the statements : (I) The number of elements in R is 17 (II) R is an equivalence relation

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  2. 52
    The number of values of z∈Cz\in\mathbb{C}, satisfying the equations ∣z−(4+8i)∣=10|z- (4 + 8i)| =\sqrt{10} and ∣z−(3+5i)∣+∣z−(5+11i)∣=45|z- (3 + 5i)| + |z- (5 + 11i)| = 4\sqrt{5}, is :

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  3. 53
    If the system of linear equations : x+y+z=6x+y+z= 6, x+2y+5z=10x+ 2y+ 5z= 10, 2x+3y+λz=μ2x + 3y +\lambda z =\mu. has infinitely many solutions, then the value of λ+μ\lambda+\mu equals.

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  4. 54
    Let A=[α12230045]A=\begin{bmatrix} \alpha & 1 & 2 \\ 2 & 3 & 0 \\ 0 & 4 & 5 \end{bmatrix} and B=[1000−5α004α−2α]+adj(A)B =\begin{bmatrix} 1 & 0 & 0 \\ 0 & - 5\alpha & 0 \\ 0 & 4\alpha & - 2\alpha \end{bmatrix}+ adj(A). If det(B)=66det(B) = 66, then det(adj(A))det(adj(A)) equals :

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  5. 55
    Let α=3+4+8+9+13+14+…\alpha= 3 + 4 + 8 + 9 + 13 + 14 + \ldots.. upto 40 terms. If (tan⁡β)α1020(\tan\beta)^{\frac{\alpha}{1020}} is a root of the equation x2+x−2=0,β∈(0,π2)x^{2}+x- 2 = 0,\beta\in\left( 0,\frac{\pi}{2} \right), then sin⁡2β+3cos⁡2β\sin^{2}\beta+ 3\cos^{2}\beta is equal to :

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  6. 56
    A candidate has to go to the examination center to appear in an examination. The candidate uses only one means of transportation for the entire distance out of bus, scooter and car. The probabilities of the candidate going by bus, scooter and car, respectively, are 25,15\frac{2}{5},\frac{1}{5} and 25\frac{2}{5}. The probabilities that the candidate reaches late at the examination center are 15,13\frac{1}{5},\frac{1}{3} and 14\frac{1}{4} if the candidate uses bus, scooter and car, respectively. Given that the candidate reached late at the examination centre, the probability that the candidate travelled by bus is :

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  7. 57
    A set of four observations has mean 1 and variance 13. Another set of six observations has mean 2 and variance 1. Then, the variance of all these 10 observations is equal to :

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  8. 58
    If 26(233( 13C2)+255( 12C4)+277( 12C6)+...+21313( 12C12))=313−α26\left( \frac{2^{3}}{3}\left( \ ^{13}C_{2} \right)+\frac{2^{5}}{5}\left( \ ^{12}C_{4} \right)+\frac{2^{7}}{7}\left( \ ^{12}C_{6} \right)+ ... +\frac{2^{13}}{13}\left( \ ^{12}C_{12} \right) \right)=3^{13}-\alpha, then α\alpha is equal to :

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  9. 59
    A person has three different bags and four different books. The number of ways, in which he can put these books in the bags so that no bag is empty, is

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  10. 60
    If a straight line drawn through the point of intersection of the lines 4x+3y−1=04x+ 3y- 1 = 0 and 3x+4y−1=03x+ 4y- 1 = 0, meet the co-ordinate axes at the points P and Q , then the locus of the mid point of PQ is:

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  11. 61
    Let O be the vertex of the parabola y2=4xy^{2}= 4x and its chords OP and OQ are perpendicular to each other. If the locus of the mid-point of the line segment PQ is a conic C , then the length of its latus rectum is :

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  12. 62
    Let α=3sin⁡−1(611)\alpha= 3\sin^{- 1}\left( \frac{6}{11} \right) and β=3cos⁡−1(49)\beta= 3\cos^{- 1}\left( \frac{4}{9} \right), where inverse trigonometric functions take only the principal values. Given below are two statements: Statement I : cos⁡(α+β)>0\cos(\alpha + \beta) > 0 Statement II : cos⁡(α)<0\cos(\alpha) < 0 In the light of the above statements, choose the correct answer from the options given below:

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  13. 63
    For the function f(x)=e∣sin⁡x∣−∣x∣,x∈Rf(x) =e^{|\sin x|}- |x|,x \in R, consider the following statements: Statement I : f is differentiable for all x∈Rx \in R. Statement II : f is increasing in (−π,−π2)\left( - \pi, - \frac{\pi}{2} \right). In the light of the above statements, choose the correct answer from the options given below:

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  14. 64
    Let a→=4i^−j^+3k^,b→=10i^+2j^−k^\overrightarrow{a}= 4\widehat{i}-\widehat{j}+ 3\widehat{k},\overrightarrow{b}= 10\widehat{i}+ 2\widehat{j}-\widehat{k} and a vector c→\overrightarrow{c} be such that 2(a→×b→)+3(b→×c→)=0→2(\overrightarrow{a}\times\overrightarrow{b}) + 3(\overrightarrow{b}\times\overrightarrow{c}) =\overrightarrow{0}. If a→⋅c→=15\overrightarrow{a}\cdot\overrightarrow{c}= 15, then c→⋅(i^+j^−3k^)\overrightarrow{c}\cdot (\widehat{i}+\widehat{j}- 3\widehat{k}) is equal to :

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  15. 65
    Let the foot of perpendicular from the point (λ,2,3)(\lambda,2,3) on the line x−41=y−92=z−51\frac{x- 4}{1}=\frac{y- 9}{2}=\frac{z- 5}{1} be the point ( 1,μ,21,\mu,2 ). Then the distance between the lines x−12=y−23=z+46\frac{x- 1}{2}=\frac{y- 2}{3}=\frac{z+ 4}{6} and x−λ2=y−μ3=z+56\frac{x-\lambda}{2}=\frac{y-\mu}{3}=\frac{z+ 5}{6} is equal to

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  16. 66
    The value of the integral ∫02x(x2+x+1)(x+1)(x4+x2+1)dx\int_{0}^{2} \frac{\sqrt{x\left( x^{2}+x+ 1 \right)}}{(\sqrt{x+ 1})\left( \sqrt{x^{4}+x^{2}+ 1} \right)}dx is equal to :

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  17. 67
    Let y=y(x)y=y(x) be the solution of the differential equation x1−x2dy+(y1−x2−xcos⁡−1x)dx=0x\sqrt{1 -x^{2}}dy+\left( y\sqrt{1 -x^{2}}-x\cos^{- 1}x \right)dx= 0, x∈(0,1),lim⁡x→1−y(x)=1x\in (0,1),\lim_{x\rightarrow1^{-}} y(x) = 1. Then y(12)y\left( \frac{1}{2} \right) equals :

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  18. 68
    Let f:(1,∞)→Rf:(1,\infty) \rightarrow R be function defined as f(x)=x−1x+1f(x) =\frac{x - 1}{x + 1}. Let fi+1(x)=f(fi(x))f^{i + 1}(x) = f\left( f^{i}(x) \right), i=1,2,…,25i = 1,2,\ldots,25, where f1(x)=f(x)f^{1}(x) = f(x). If g(x)+f26(x)=0g(x) +f^{26}(x) = 0, x∈(1,∞)x\in (1,\infty), then the area of the region bounded by the curves y=g(x),2y=2x−3,y=0y = g(x),2y = 2x - 3,y = 0 and x=4x = 4 is:

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  19. 69
    Let f(x)={13,x≤π/2b(1−sin⁡x)(π−2x)2,x>π/2f(x) =\left\{ \begin{matrix} \frac{1}{3} & , & x\leq\pi/2 \\ \frac{b(1 - \sin x)}{(\pi- 2x)^{2}} & , & x>\pi/2 \end{matrix} \right., If ff is continuous at x=π/2x =\pi/2, then the value of ∫03 b−6∣x2+2x−3∣dx\int_{0}^{3\text{ }b - 6} \left| x^{2}+ 2x - 3 \right|dx is :

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  20. 70
    Let x2f(a2+7a+3)+y2f(3a+15)=1\frac{x^{2}}{f\left( a^{2}+ 7a+ 3 \right)}+\frac{y^{2}}{f(3a+ 15)}= 1 represent an ellipse with major axis along y-axis, where ff is a strictly decreasing positive function on RR. If the set of all possible values of aa is R−[α,β]R - \lbrack\alpha,\beta\rbrack, then α2+β2\alpha^{2}+\beta^{2} is equal to :

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  21. 71
    The sum of squares of all real solution of the equation log⁡(x+1)(2x2+5x+3)=4−log⁡(2x+3)(x2+2x+1)\log_{(x + 1)}\left( 2x^{2} + 5x + 3 \right) = 4 - \log_{(2x + 3)}\left( x^{2} + 2x + 1 \right) is equal to ____\_\_\_\_ .
  22. 72
    If ∫π/6π/4(cot⁡(x−π3)cot⁡(x+π3)+1)dx=αlog⁡e(3−1)\int_{\pi/6}^{\pi/4} \left( \cot\left( x - \frac{\pi}{3} \right)\cot\left( x + \frac{\pi}{3} \right) + 1 \right)dx = \alpha\log_{e}(\sqrt{3} - 1), then 9α29\alpha^{2} is equal to
  23. 73
    Let a line L1L_{1} pass through the origin and be perpendicular to the lines L2:r→=(3+t)i^+(2t−1)j^+(2t+4)k^L_{2}:\overrightarrow{r} = (3 + t)\widehat{i} + (2t - 1)\widehat{j} + (2t + 4)\widehat{k} and L3:r→=(3+2s)i^+(3+2s)j^+(2+s)k^,tL_{3}:\overrightarrow{r} = (3 + 2s)\widehat{i} + (3 + 2s)\widehat{j} + (2 + s)\widehat{k},t, s∈Rs \in R. If (a,b,c),a∈Z(a,b,c),a \in Z, is the point on L3L_{3} at a distance of 17\sqrt{17} from the point of intersection of L1L_{1} and L2L_{2}, then (a+b+c)2(a + b + c)^{2} is equal to ____\_\_\_\_.
  24. 74
    Consider the circle C : x2+y2−6x−8y−11=0x^{2} + y^{2} - 6x - 8y - 11 = 0. Let a variable chord AB of the circle C subtend a right angle at the origin. If the locus of the foot of the perpendicular drawn from the origin on the chord AB is the circle x2+y2−αx−βy−γ=0x^{2} + y^{2} - \alpha x - \beta y - \gamma = 0, then α+β+2γ\alpha + \beta + 2\gamma is equal to ____\_\_\_\_ .
  25. 75
    Let f be a polynomial function such that log⁡2(f(x))=(log⁡2(2+23+29+...∞))\log_{2}(f(x)) = \left( \log_{2}\left( 2 + \frac{2}{3} + \frac{2}{9} + ...\infty \right) \right). log⁡3(1+f(x)f(1/x)),x>0\log_{3}\left( 1 + \frac{f(x)}{f(1/x)} \right),x > 0 and f(6)=37f(6) = 37. Then ∑n=110f(n)\sum_{n = 1}^{10} f(n) is equal to ____\_\_\_\_ .

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