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JEE Mains 2026 (6 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
    The percentage error in the calculated volume of a sphere, if there is 2%2\% error in its diameter measurement, is ____\_\_\_\_ .

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  2. 2
    Match List-I with List-II.
    List-IList-II
    (A) Boltzmann constant(I) [M−1L3T−2][M^{-1}L^{3}T^{-2}]
    (B) Stefan's constant(II) [ML2T−1][ML^{2}T^{-1}]
    (C) Planck's constant(III) [ML2T−2K−1][ML^{2}T^{-2}K^{-1}]
    (D) Gravitational constant(IV) [ML0T−3K−4][ML^{0}T^{-3}K^{-4}]
    Choose the correct answer from the options given below.

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  3. 3
    A solid sphere (A) of mass, 5 m and spherical shell (B) of mass mm, both having same radius, are placed on a rough surface. When a force of same magnitude is applied tangentially at the highest points of A and B , they start rolling without slipping with an acceleration of aAa_{A} and aBa_{B}, respectively. The ratio of aAa_{A} and aBa_{B} is ____\_\_\_\_ .

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  4. 4
    A body of mass 1 kg moves along a straight line with a velocity v=2x2v= 2x^{2}. The work done by the body during displacement from x=0x = 0 to 5 m is ____\_\_\_\_ J.

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  5. 5
    A cylinder with adiabatic walls is closed at both ends and is divided into two compartments by a frictionless adiabatic piston. Ideal gas is filled in both (left and right) the compartments at same P,VP,V, T. Heating is started from left side until pressure changes to 27P8\frac{27P}{8}. If initial volume of each compartment was 9 litres then the final volume in right-hand side compartment is ____\_\_\_\_ litres. (for this ideal gas CP/Cv=1.5C_{P}/C_{v}= 1.5 )

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  6. 6
    For an electromagnetic wave propagating through vacuum, k→,E→\overrightarrow{k},\overrightarrow{E} and ω\omega represent propagation vector, electric field and angular frequency, respectively. The magnetic field associated with this wave is represented by :

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  7. 7
    Two identical bodies A and B of equal masses have initial velocities v→1=4i^m/s{\overrightarrow{v}}_{1}= 4\widehat{i}m/s and v→2=4j^m/s{\overrightarrow{v}}_{2}= 4\widehat{j}m/s respectively. The body A has acceleration a→1=6i^+6j^m/s2{\overrightarrow{a}}_{1}= 6\widehat{i}+ 6\widehat{j}m/s^{2} while the acceleration of the other body BB is zero. The centre of mass of the two bodies moves in ____\_\_\_\_ path.

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  8. 8
    Figure represents the extension ( Δl\Delta l ) of a wire of length 1 meter, suspended from the ceiling of the room at one end with load W connected to the other end. If the cross- sectional area of the wire is 10−5 m210^{- 5}{\text{ }m}^{2} then the Young's modulus of the wire is ____\_\_\_\_ N/m2N/m^{2}.

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  9. 9
    A cylindrical vessel of 40 cm radius is completely filled with water and its capacity is 528dm3(dm528dm^{3}(dm : decimeter). The vessel is placed on a solid block of exactly same height as vessel. If a small hole is made at 70 cm below the top of water level, then horizontal range of water falling on the ground in the beginning is ____\_\_\_\_ cm .

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  10. 10
    If 2 mole of an ideal monoatomic gas at temperature T , is mixed with 6 mole of another ideal monoatomic gas at temperature 2T then the temperature of mixture is :

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  11. 11
    A spring stretches by 2 mm when it is loaded with a mass of 200 g . From equilibrium position the mass is further pulled down by 2 mm and released. The frequency associated with the system and maximum energy in the spring are ____\_\_\_\_ Hz and ____\_\_\_\_ J , respectively. (Take g=10 m/s2g = 10\text{ }m/s^{2} )

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  12. 12
    The electric potential as a function of x,yx,y is given by V=5(x2−y2)VV = 5\left( x^{2}-y^{2} \right)V. The electric field at a point (2,3)m(2,3)m is ____\_\_\_\_ V/m.

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  13. 13
    A current of 30 A each flows in opposite directions in two conducting wires, placed parallel to each other at a distance of 8 cm . The magnetic field at the midpoint between the two wires is ____\_\_\_\_ μT\mu T. ( μ04π=10−7 N/A2\frac{\mu_{0}}{4\pi}=10^{- 7}\text{ }N/A^{2} )

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  14. 14
    A square loop of side 2 cm is placed in a time varying magnetic field with magnitude as B=0.4sin⁡(300t)B = 0.4\sin(300t) Tesla. The normal to the plane of loop makes an angle of 60∘60^{\circ} with the field. The maximum induced emf produced in the loop is ____\_\_\_\_ mV .

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  15. 15
    A sphere of capacitance 100 pF is charged to a potential of 100 V . Another identical uncharged metal sphere is brought in contact with the charged sphere, then the change in the total energy stored on these spheres, when they touch is α×10−7 J\alpha \times10^{- 7}\text{ }J. The value of α\alpha is ____\_\_\_\_ . (combined capacitance of spheres is 200 pF ).

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  16. 16
    The energy released if two  12H\ _{1}^{2}H (deuterium) atoms are combined to form  24He\ _{2}^{4}He is ____\_\_\_\_ MeV. (Take binding energies per nucleon of  12H\ _{1}^{2}H and  24He\ _{2}^{4}He as 1.1 MeV and 7.2 MeV, respectively).

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  17. 17
    Angle of minimum deviation is equal to the half of the angle of prism in an equilateral prism. The refractive index of the prism is ____\_\_\_\_ .

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  18. 18
    Refer to the logic circuit given below. For two inputs ( A=1, B=1A = 1,\text{ }B = 1 ) and ( A=0, B=1A = 0,\text{ }B = 1 ), output (Y) will be ____\_\_\_\_ .

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  19. 19
    The velocity at which 6 kg mass (shown in figure) strikes the ground when it is released from a height of 6 m above the ground is ____\_\_\_\_ m/sm/s. Assume pulley is massless and string is light and inextensible (Take g=10 m/s2g = 10\text{ }m/s^{2} ).

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  20. 20
    In a Young double slit experiment, the wavelength of the incident light is 6000Å6000\text{Å}, the separation between slits S1S_{1} and S2S_{2} is 5 cm, the distance between slits plane and screen is 50 cm , as shown in the figure below. If the resultant intensity at P is equal to the intensity due to individual slits, the path difference between interfering waves is ____\_\_\_\_ Å.

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  21. 21
    A block takes tt time to slide down a plane inclined at 45∘45^{\circ} to the horizontal. If the surface is made smooth (frictionless), the block takes times t2\frac{t}{2} to slide down the plane. The coefficient of friction between the block and the inclined plane is (α100)\left( \frac{\alpha}{100} \right) . The value of α\alpha is ____\_\_\_\_.
  22. 22
    The de Broglie wavelength for an electron accelerated through the potential difference V1V_{1} volt is λ1\lambda_{1}. When the potential difference is changed to V2V_{2} volt, the associated de Broglie wavelength is increased by 50%50\%. If (V1/V2)=(9/α)\left( V_{1}/V_{2} \right) = (9/\alpha), then the value of α\alpha is ____\_\_\_\_ .
  23. 23
    A moving coil of a galvanometer when shunted with 2Ω2\Omega resistance gives a full-scale deflection for a current of 500 mA . When a resistance of 470Ω470\Omega is connected in series it gives a full-scale deflection for 10 V potential applied on it. The value of resistance of galvanometer coil is ____\_\_\_\_ Ω\Omega.
  24. 24
    Two cells of emfs 1 V and 2 V and internal resistance 2Ω2\Omega and 1Ω1\Omega, respectively connected in parallel, gave current of 1A through an external resistance. If the polarity of one cell is reversed, then value of current through the external resistance will be α5 A\frac{\alpha}{5}\text{ }A. The value of α\alpha is ____\_\_\_\_ .
  25. 25
    A concave mirror of focal length 10 cm forms an image which is double the size of object when the object is placed at two different positions. The distance between the two positions of the object is ____\_\_\_\_ cm.

Chemistry

25 questions

  1. 26
    Which of the following contain the same number of atoms? (Given : Molar mass in gmol−1gmol^{- 1} of H,He,OH,He,O and S are 1,4,161,4,16 and 32 respectively) (A) 2 g of O2O_{2} gas (B) 4 g of SO2{SO}_{2} gas (C) 1400 mL of O2O_{2} at STP (D) 0.05 L of He at STP (E) 0.0625 mol0.0625\text{ }mol of H2H_{2} gas Choose the correct answer from the options given below :

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  2. 27
    The Bohr radius of a hydrogen like species is 70.53 pm . The species and the stationary state(n) are respectively (Given: Hydrogen atom Bohr radius is 52.9 pm )

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  3. 28
    Given below are two statements: Statement I: The number of compounds among SO2,SO3,SF4,SF6{SO}_{2},{SO}_{3},{SF}_{4},{SF}_{6} and H2 SH_{2}\text{ }S in which sulphur does not obey the Octet rule is 3 . Statement II : Among [H2O,ClF3,SF4],[NH3\left\lbrack H_{2}O,{ClF}_{3},{SF}_{4} \right\rbrack,\left\lbrack {NH}_{3} \right.,  BrF5,SF4], [BrF5, ClF3, XeF4]\left. \ {BrF}_{5},{SF}_{4} \right\rbrack,\ \left\lbrack {BrF}_{5},\ {ClF}_{3},\ {XeF}_{4} \right\rbrack and  [XeF4\ \left\lbrack {XeF}_{4} \right.,  ClF3,H2O]\left. \ {ClF}_{3},H_{2}O \right\rbrack, the number of sets in which all the molecules have one lone pair of electrons on the central atom is 1 . In the light of the above statements, choose the correct answer from the options given below :

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  4. 29
    Match List-I with List-II. Given V1V_{1} and V2V_{2} are initial and final volumes respectively:
    List-I (Isothermal process)List-II (Expression)
    (A)Reversible expansion(I)
    q=0q = 0
    (B)Free expansion(II)
    q=nRTlnV2 V1q = nRTln\frac{V_{2}}{{\text{ }V}_{1}}
    (C)Irreversible Compression(III)
    w=−pext (V2−V1)w = - p_{\text{ext~}}\left( V_{2} - V_{1} \right)
    (D)Cyclic reversible(IV)
    qrev T=0\frac{q_{\text{rev~}}}{T} = 0
    Choose the correct answer from the options given below :

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  5. 30
    Given below are two statements :
    Chamber 1Semipermeable membraneChamber 2
    18 g glucose in 100 mL aqueous solution30 g glucose in 250 mL aqueous solution
    Statement I: H2OH_{2}O molecules move from the chamber 1 to chamber 2. Statement II: The osmotic pressure of a solution prepared by dissolving 50 mg of potassium sulphate (molar mass =174 g/mol= 174\text{ }g/mol) in 2 L of water (at 27∘C27^{\circ}C) is 0.0107 bar. (Given : R=0.083dm3R = 0.083{dm}^{3} bar K−1 mol−1K^{-1}{\text{ }mol}^{-1} and assume complete dissociation of electrolyte) In the light of the above statements, choose the correct answer from the options given below :

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  6. 31
    Given is a concentrated solution of a weak electrolyte Ax ByA_{x}{\text{ }B}_{y} of concentration ' c ' and dissociation constant ' K '. the degree of dissociation is given by:

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  7. 32
    For a general redox reaction Anode  Red1→Ox1 n1++n1e−\ Red_{1}\rightarrow Ox_{1}\ ^{n_{1}^{+}}+n_{1}e^{-} Cathode Ox2+n2e−→Red2 n2−Ox_{2}+n_{2}e^{-}\rightarrow Red_{2}\ ^{n_{2}-} Which of the following statement is incorrect?

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  8. 33
    In a period, the first ionization enthalpy of the element at extreme left and the negative electron gain enthalpy of the extreme right element, except noble gases, are respectively.

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  9. 34
    Given below are two statements : Statement I: F2O<H2O<Cl2OF_{2}O < H_{2}O < {Cl}_{2}O is the correct trend in terms of bond angle. Statement II : SiF4,SnF4{SiF}_{4},{SnF}_{4} and PbF4{PbF}_{4} are ionic in nature. In the light of the above statements, choose the correct answer from the options given below :

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  10. 35
    The correct order of first (ΔiH1)\left( \Delta_{i}H_{1} \right) and second (ΔiH2)\left( \Delta_{i}H_{2} \right) ionisation enthalpy values of Cr and Mn are : (A) ΔiH1:Cr>Mn\Delta_{i}H_{1}:Cr > Mn (B) ΔiH2:Cr>Mn\Delta_{i}H_{2}:Cr > Mn (C) ΔiH1:Mn>Cr\Delta_{i}H_{1}:Mn > Cr (D) ΔiH2:Mn>Cr\Delta_{i}H_{2}:Mn > Cr Choose the correct answer from the options given below :

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  11. 36
    Which of the following sequence of hybridization, geometry and magnetic nature are correct for the given coordination compounds ? (A) [NiCl4]2−−sp3\left\lbrack {NiCl}_{4} \right\rbrack^{2 -} - {sp}^{3}, tetrahedral, paramagnetic (B) [Ni(NH3)6]2+−sp3 d2\left\lbrack Ni\left( {NH}_{3} \right)_{6} \right\rbrack^{2 +} - {sp}^{3}{\text{ }d}^{2}, octahedral, paramagnetic (C) [Ni(CO)4]−sp3\left\lbrack Ni(CO)_{4} \right\rbrack - {sp}^{3}, tetrahedral, paramagnetic (D) [Ni(CN)4]2−−dsp2\left\lbrack Ni(CN)_{4} \right\rbrack^{2 -} - {dsp}^{2}, square planar diamagnetic Choose the correct answer from the options given below :

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  12. 37
    Given below are two statements : Statement I: A mixture of C12H22O11C_{12}H_{22}O_{11} (sugar) and NaCl can be separated by dissolving sugar in alcohol, due to differential solubility Statement II : Rose essence from rose petals is separated by steam distillation due to its high volatility and insolubility in H2OH_{2}O. In the light of the above statements, choose the correct answer from the options given below :

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  13. 38
    Shown below is the structure of methyl acetate with three different α,β\alpha,\beta and γ\gamma carbon-oxygen bonds. The correct order of bond length of these bonds is :

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  14. 39
    ' x ' is the product which is obtained by the hydrolysis of prop-1-yne in the presence of mercuric sulphate under dilute acidic medium at 333 K . ' y ' is the product which is obtained by the reaction of ethane nitrile with methyl magnesium bromide in dry ether followed by hydrolysis. IUPAC name of product obtained from ' x ' and ' y ' in the presence of barium hydroxide followed by heating is:

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  15. 40
    An optically active alkyl bromide C4H9BrC_{4}H_{9}Br, reacts with ethanolic KOH to form major compound [A] which reacts with bromine to give compound [B]. Compound [B] reacts with ethanolic KOH and sodamide to give compound [C]. One molecule of water adds to compound [C] on warming with mercuric sulphate and dilute sulphuric acid at 333 K to form compound [D]. The functional group in compound D will be confirmed by:

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  16. 41
    Consider the following reaction. Statement I: In the above reaction, product formed will be a mixture of benzyl alcohol and iodobenzamide. Statement II : In the above reaction, the −O−CH2−- O - {CH}_{2} - bond is cleaved to give the product. 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 organic reaction sequence. Choose the final product (X) from the following (consider the major product in all intermediate reactions)

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  18. 43
    The number of compounds from the following which can undergo reaction with Br2/KOH/Br_{2}/KOH/ (alcoholic) to give respective products and these respective products can also be obtained separately by Gabriel phthalimide reaction is:

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  19. 44
    Consider the following reactions. Total number of electrons in the π\pi bonds and lone pair of electrons in the product ( X ) is:

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  20. 45
    Treatment of a gas ' X ' with a freshly prepared ferrous sulphate solution gives a compound ' Y ' as a brown ring. The compounds X and Y are.

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  21. 46
    An excess of AgNO3{AgNO}_{3} is added to 100 mL of a 0.05 M solution of tetraaquadichloridochromium (III) chloride. The number of moles of AgCl precipitated will be ____\_\_\_\_ ×10−3\times 10^{- 3}. (Nearest integer)
  22. 47
    An alkane (Y) requires 8 moles of oxygen for complete combustion and on chlorination with Cl2/hv{Cl}_{2}/hv, (Y) gives only one monochlorinated product (Z). The total number of primary carbon atoms in (Y) is ____\_\_\_\_ .
  23. 48
    500 mL of 0.2MMnO4−0.2M{MnO}_{4}^{-}solution in basic medium when mixed with 500 mL of 1.5 M KI solution, oxidises iodide ions to liberate molecular iodine. This liberated iodine is then titrated with a standard x M thiosulphate solution in presence of starch till the end point. If 300 mL of thiosulphate was consumed, then the value of x is ____\_\_\_\_ .
  24. 49
    In a closed flask at 600 K , one mole of X2Y4( g)X_{2}Y_{4}(\text{ }g) attains equilibrium as given below:
    X2Y4( g)⇌2XY2( g)X_{2}Y_{4}(\text{ }g) \rightleftharpoons 2{XY}_{2}(\text{ }g)
    At equilibrium, 75%X2Y4( g)75\% X_{2}Y_{4}(\text{ }g) was dissociated and the total pressure is 1 atm . The magnitude of ΔrG⊖\Delta_{r}G^{\ominus} (in kJmol−1kJ{mol}^{- 1} ) at this temperature is ____\_\_\_\_ . (Nearest Integer) (Given: R=8.3 J mol−1 K−1;ln⁡10=2.3R = 8.3\text{ }J{\text{ }mol}^{- 1}{\text{ }K}^{- 1};\ln10 = 2.3, log⁡2=0.3,log⁡3=0.48,log⁡5=0.69,log⁡7=0.84)\log2 = 0.3,\log3 = 0.48,\log5 = 0.69,\log7 = 0.84)
  25. 50
    Decomposition of a hydrocarbon follows the equation k=(5.5×1011 s−1)e−28000 K Tk = \left( 5.5 \times 10^{11}{\text{ }s}^{- 1} \right)e^{\frac{- 28000\text{ }K}{\text{ }T}}. The activation energy of reaction is ____\_\_\_\_ kJmol−1kJ{mol}^{- 1}. (Nearest Integer) Given : R=8.3JK−1 mol−1R = 8.3{JK}^{- 1}{\text{ }mol}^{- 1}

Mathematics

25 questions

  1. 51
    Let f:R→Rf:R \rightarrow R be defined as f(x)=2x2−3x+23x2+x+3f(x) =\frac{2x^{2}- 3x + 2}{3x^{2}+ x + 3}. Then ff is :

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  2. 52
    Consider the quadratic equation (n2−2n+2)x2−3x+(n2−2n+2)2=0,n∈R\left( n^{2}- 2n + 2 \right)x^{2}- 3x +\left( n^{2}- 2n + 2 \right)^{2}= 0,n \in R. Let α\alpha be the minimum value of the product of its roots and β\beta be the maximum value of the sum of its roots. Then the sum of the first six terms of the G.P., whose first term is α\alpha and the common ratio is αβ\frac{\alpha}{\beta}, is :

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  3. 53
    Let S={z∈C:z2+6iz−3=0}S=\left\{ z\in C:z^{2}+\sqrt{6}iz- 3 = 0 \right\}. Then ∑z∈Sz8\sum_{z\in S} z^{8} is equal to :

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  4. 54
    The sum of all possible values of θ∈[0,2π]\theta\in \lbrack 0,2\pi\rbrack, for which the system of equations :
    xcos3θ−8y−12z=0xcos3\theta- 8y- 12z= 0
    xcos2θ+3y+3z=0xcos2\theta+ 3y+ 3z= 0
    x+y+3z=0x+y+ 3z= 0
    has a non-trivial solution, is equal to:

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  5. 55
    Let A=[100310931]A =\begin{bmatrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1 \end{bmatrix} and B=[bij],1≤i,j≤3B =\left\lbrack b_{ij} \right\rbrack,1 \leq i,j \leq 3. If B=A99−IB=A^{99}-I, then the value of b31−b21b32\frac{b_{31}-b_{21}}{b_{32}} is :

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  6. 56
    The sum 1+12(12+22)+13(12+22+32)+...1 +\frac{1}{2}\left( 1^{2}+2^{2} \right)+\frac{1}{3}\left( 1^{2}+2^{2}+3^{2} \right)+ ... up to 10 terms is equal to :

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  7. 57
    A building has ground floor and 10 more floors. Nine persons enter in a lift at the ground floor. The lift goes up to the 10th 10^{\text{th~}} floor. The number of ways, in which any 4 persons exit at a floor and the remaining 5 persons exit at a different floor, if the lift does not stop at the first and the second floors, is equal to :

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  8. 58
    Let the mean and the variance of seven observations 2,4,α,8,β,12,14,α<β2,4,\alpha,8,\beta,12,14,\alpha<\beta, be 8 and 16 respectively. Then the quadratic equation whose roots are 3α+23\alpha+ 2 and 2β+12\beta+ 1 is :

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  9. 59
    A bag contains 6 blue and 6 green balls. Pairs of balls are drawn without replacement until the bag is empty. The probability that each drawn pair consists of one blue and one green ball is :

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  10. 60
    Let C be a circle having centre in the first quadrant and touching the x -axis at a distance 3 units from the origin. If the circle C has an intercept of length 636\sqrt{3} on yy-axis, then the length of the chord of the circle CC on the line x−y=3x-y= 3 is :

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  11. 61
    The eccentricity of an ellipse E with center at the origin OO is 32\frac{\sqrt{3}}{2} and its directrices are x=±463x= \pm\frac{4\sqrt{6}}{3}. Let H:x2a2−y2b2=1H:\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}= 1 be a hyperbola whose eccentricity is equal to the length of semi-major axis of E , and whose length of latus rectum is equal to the length of minor axis of E . Then the distance between the foci of H is :

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  12. 62
    Let x=9x = 9 be a directrix of an ellipse E , whose centre is at the origin and eccentricity is 13\frac{1}{3}. Let P(α,0),α>0P(\alpha,0),\alpha> 0, be a focus of E and AB be a chord passing through P . Then the locus of the mid point of AB is :

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  13. 63
    If sin⁡(tan⁡−1(x2))=cot⁡(sin⁡−11−x2),x∈(0,1)\sin\left( \tan^{- 1}(x\sqrt{2}) \right)= \cot\left( \sin^{- 1}\sqrt{1 -x^{2}} \right),x \in (0,1), then the value of x is :

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  14. 64
    The shortest distance between the lines x−41=y−32=z−2−3\frac{x - 4}{1}=\frac{y - 3}{2}=\frac{z - 2}{- 3} and x+22=y−64=z−5−5\frac{x + 2}{2}=\frac{y - 6}{4}=\frac{z - 5}{- 5} is:

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  15. 65
    Let a→=2i^+3j^+3k^\overrightarrow{a}= 2\widehat{i}+ 3\widehat{j}+ 3\widehat{k} and b→=6i^+3j^+3k^\overrightarrow{b}= 6\widehat{i}+ 3\widehat{j}+ 3\widehat{k}. Then the square of the area of the triangle with adjacent sides determined by the vectors (2a→+3b→)(2\overrightarrow{a}+ 3\overrightarrow{b}) and (a→−b→)(\overrightarrow{a}-\overrightarrow{b}) is :

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  16. 66
    Let lim⁡x→2(tan⁡(x−2))(rx2+(p−2)x−2p)(x−2)2=5\lim_{x \rightarrow 2} \frac{(\tan(x - 2))\left( rx^{2}+ (p - 2)x - 2p \right)}{(x - 2)^{2}}= 5 for some r,p∈Rr,p \in R. If the set of all possible values of q , such that the roots of the equation rx2−px+q=0rx^{2}- px + q = 0 lie in (0,2)(0,2), be the interval (α,β](\alpha,\beta\rbrack, then 4(α+β)4(\alpha+\beta) equals.

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  17. 67
    Let A=∣13−121α01−1∣A =\left| \begin{matrix} 1 & 3 & - 1 \\ 2 & 1 & \alpha \\ 0 & 1 & - 1 \end{matrix} \right| be a singular matrix. Let f(x)=∫0x(t2+2t+3)dt,x∈[1,α]f(x) =\int_{0}^{x} \left( t^{2}+ 2t+ 3 \right)dt,x\in \lbrack 1,\alpha\rbrack. If MM and mm are respectively the maximum and the minimum values of ff in [1,α]\lbrack 1,\alpha\rbrack then 3(M−m)3(M-m) is equal to :

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  18. 68
    Let f:R→Rf:R\rightarrow R be such that f(xy)=f(x)f(y)f(xy) =f(x)f(y), for all x,y∈Rx,y\in R and f(0)≠0f(0) \neq 0. Let g:[1,∞)→Rg:\lbrack 1,\infty) \rightarrow R be a differentiable function such that x2g(x)=∫1x(t2f(t)−tg(t))dtx^{2}g(x) =\int_{1}^{x} \left( t^{2}f(t) - tg(t) \right)dtThen g(2)g(2) is equal to :

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  19. 69
    The area of the region {(x,y):x2−8x≤y≤−x}\left\{ (x,y):x^{2}- 8x \leq y \leq - x \right\} is:

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  20. 70
    The value of the integral ∫−11(x3+∣x∣+1x2+2∣x∣+1)dx\int_{- 1}^{1} \left( \frac{x^{3}+ |x| + 1}{x^{2}+ 2|x| + 1} \right)dx is equal to

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  21. 71
    Let R={(x,y)∈N×N:log⁡e(x+y)≤2}R = \left\{ (x,y) \in N \times N:\log_{e}(x + y) \leq 2 \right\}. Then the minimum number of elements, required to be added in R to make it a transitive relation, is ____\_\_\_\_ .
  22. 72
    If (1−x3)10=∑r=010arxr(1−x)30−2r\left( 1 - x^{3} \right)^{10} = \sum_{r = 0}^{10} a_{r}x^{r}(1 - x)^{30 - 2r}, then 9a9a10\frac{9a_{9}}{a_{10}} is equal to ____\_\_\_\_
  23. 73
    Let the line x−y=4x - y = 4 intersect the circle C:(x−4)2+(y+3)2=9C:(x - 4)^{2} + (y + 3)^{2} = 9 at the points Q and R . If P(α,β)P(\alpha,\beta) is a point on C such that PQ=PRPQ = PR, then (6α+8β)2(6\alpha + 8\beta)^{2} is equal to ____\_\_\_\_ .
  24. 74
    Let the image of the point P(0,−5,0)P(0, - 5,0) in the line x−12=y1=z+1−2\frac{x - 1}{2} = \frac{y}{1} = \frac{z + 1}{- 2} be the point RR and the image of the point Q(0,−12,0)Q\left( 0,\frac{- 1}{2},0 \right) in the line x−1−1=y+94=z+11\frac{x - 1}{- 1} = \frac{y + 9}{4} = \frac{z + 1}{1} be the point S. Then the square of the area of the parallelogram PQRS is ____\_\_\_\_ .
  25. 75
    Let f(x)={x3+8x<0x2−4x≥0f(x) = \left\{ \begin{matrix} x^{3} + 8 & x < 0 \\ x^{2} - 4 & x \geq 0 \end{matrix} \right. and g(x)={(x−8)1/3;x<0(x+4)1/2;x≥0g(x) = \left\{ \begin{matrix} (x - 8)^{1/3}; & x < 0 \\ (x + 4)^{1/2}; & x \geq 0 \end{matrix} \right. Then the number of points, where the function gof is discontinuous, is ____\_\_\_\_ .

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