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JEE Mains 2026 (21 Jan, 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
    Potential energy (VV) versus distance (xx) is given by the graph, Rank various regions as per the magnitudes of the force (F) acting on a particle from high to low.

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  2. 2
    A gas based geyser heats water flowing at the rate of 5.0 litres per minute from 27∘C27^{\circ}C to 87∘C87^{\circ}C. The rate of consumption of the gas is ____\_\_\_\_ g/s. (Take heat of combustion of gas =5.0×104 J/g= 5.0 \times10^{4}\text{ }J/g ) specific heat capacity of water =4200 J/kg⋅ ∘C= 4200\text{ }J/kg \cdot\ ^{\circ}C

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  3. 3
    A conducting circular loop of area 1.0 m21.0{\text{ }m}^{2} is placed perpendicular to a magnetic field which varies as B=sin⁡(100t)B = \sin(100t) Tesla. If the resistance of the loop is 100Ω100\Omega, then the average thermal energy dissipated in the loop in one period is ____\_\_\_\_ J.

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  4. 4
    Water flows through a horizontal tube as shown in the figure. The difference in height between the water columns in vertical tubes is 5 cm and the areas of cross-section at AA and BB are 6 cm26{\text{ }cm}^{2} and 3 cm23{\text{ }cm}^{2} respectively. The rate of flow will be ____\_\_\_\_ cm3/scm^{3}/s. (take g=10 m/s2g = 10\text{ }m/s^{2})

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  5. 5
    In an experiment the values of two spring constants were measured as k1=(10±0.2)N/mk_{1}= (10 \pm 0.2)N/m and k2=(20±0.3)N/mk_{2}= (20 \pm 0.3)N/m. If these springs are connected in parallel, then the percentage error in equivalent spring constant is :

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  6. 6
    A 4 kg mass moves under the influence of a force F→=(4t3i^−3tj^)N\overrightarrow{F}=\left( 4t^{3}\widehat{i}- 3t\widehat{j} \right)N where tt is the time in second. If mass starts from origin at t=0t = 0, the velocity and position after t=2 st = 2\text{ }s will be :

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  7. 7
    Consider a modified Bernoulli equation.
    (P+ABt2)+ρg( h+Bt)+12ρ V2= constant \left( P +\frac{A}{Bt^{2}} \right)+ \rho g(\text{ }h + Bt) +\frac{1}{2}\rho{\text{ }V}^{2}=\text{~constant~}
    If t has the dimension of time then the dimensions of A and B are ____\_\_\_\_ , ____\_\_\_\_ respectively.

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  8. 8
    A current carrying solenoid is placed vertically and a particle of mass m with charge Q is released from rest. The particle moves along the axis of solenoid. If g is acceleration due to gravity then the acceleration (a) of the charged particle will satisfy :

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  9. 9
    A parallel plate capacitor has capacitance C, when there is vacuum within the parallel plates. A sheet having thickness (13)rd \left( \frac{1}{3} \right)^{\text{rd~}} of the separation between the plates and relative permittivity K is introduced between the plates. The new capacitance of the system is :

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  10. 10
    The electric field a plane electromagnetic wave is given by : Ey=69sin⁡[0.6×103x−1.8×1011t]V/mE_{y}= 69\sin\left\lbrack 0.6 \times10^{3}x - 1.8 \times10^{11}t \right\rbrack V/m. The expression for magnetic field associated with this electromagnetic wave is ____\_\_\_\_ T.

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  11. 11
    In a double slit experiment the distance between the slits is 0.1 cm and the screen is placed at 50 cm from the slits plane. When one slit is covered with a transparent sheet having thickness tt and refractive index n(=1.5)n( = 1.5), the central fringe shifts by 0.2 cm. The value of t is ____\_\_\_\_ cm.

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  12. 12
    A light wave described by E=60[sin⁡(3×1015t)+sin⁡(12×1015t)]E = 60\left[\sin\left(3 \times 10^{15}t\right) + \sin\left(12 \times 10^{15}t\right)\right] (in SI units) falls on a metal surface of work function 2.8 eV . The maximum kinetic energy of ejected photoelectron is (approximately) ____\_\_\_\_ eV. (h=6.6×10−34 J sh = 6.6 \times 10^{-34}\ J\,s and e=1.6×10−19 Ce = 1.6 \times 10^{-19}\ C)

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  13. 13
    If an alpha particle with energy 7.7 MeV is bombarded on a thin gold foil, the closest distance from nucleus it can reach is ____\_\_\_\_ m . (Atomic number of gold =79= 79 and 14πϵ0=9×109\frac{1}{4\pi\epsilon_{0}}= 9 \times10^{9} in SI units)

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  14. 14
    A uniform rod of mass m and length ll suspended by means of two identical inextensible light strings as shown in figure. Tension in one string immediately after the other string is cut, is ____\_\_\_\_ . (g acceleration due to gravity)

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  15. 15
    An aluminium and steel rods having same lengths and cross-sections are joined to make total length of 120 cm at 30∘C30^{\circ}C. The coefficient of linear expansion of aluminium and steel are 24×10−6/ ∘C24 \times 10^{- 6}/\ ^{\circ}C and 1.2×10−5/ ∘C1.2 \times 10^{- 5}/\ ^{\circ}C, respectively. The length of this composite rod when its temperature is raised to 100∘C100^{\circ}C, is ____\_\_\_\_ cm.

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  16. 16
    A 1 m long metal rod AB completes the circuit as shown in figure. The area of circuit is perpendicular to the magnetic field of 0.10 T. If the resistance of the total circuit is 2Ω2\Omega then the force needed to move the rod towards right with constant speed (v) of 1.5 m/s1.5\text{ }m/s is ____\_\_\_\_ N.

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  17. 17
    The given circuit works as :

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  18. 18
    Two strings (A, B) having linear densities μA=2×10−4 kg/m\mu_{A}= 2 \times10^{- 4}\text{ }kg/m and μB=4×10−4 kg/m\mu_{B}= 4 \times10^{- 4}\text{ }kg/m and lengths LA=2.5 mL_{A}= 2.5\text{ }m and LB=1.5 mL_{B}= 1.5\text{ }m respectively are joined. Free ends of A and B are tied to two rigid supports C and D, respectively creating a tension of 500 N in the wire. Two identical pulses, sent from C and D ends, take time t1t_{1} and t2t_{2}, respectively, to reach the joint. The ratio t1/t2t_{1}/t_{2} is :

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  19. 19
    Initially a satellite of 100 kg is in a circular orbit of radius 1.5RE1.5R_{E}. This satellite can be moved to a circular orbit of radius 3RE3R_{E} by supplying α×106 J\alpha \times10^{6}\text{ }J of energy. The value of α\alpha is ____\_\_\_\_ . (Take Radius of Earth RE=6×106 mR_{E}= 6 \times10^{6}\text{ }m and g=10m/s2g = 10m/s^{2} )

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  20. 20
    A point charge of 10−8C10^{- 8}C is placed at origin. The work done in moving a point charge 2μC2\mu C from point A(4,4,2)mA(4,4,2)m to B(2,2,1)mB(2,2,1)m is ____\_\_\_\_ J. ( 14πϵ0=9×109\frac{1}{4\pi\epsilon_{0}}= 9 \times10^{9} in SI units)

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  21. 21
    A collimated beam of light of diameter 2 mm is propagating along x -axis. The beam is required to be expanded in a collimated beam of diameter 14 mm using a system of two convex lenses. If first lens has focal length 40 mm , then the focal length of second lens is ____\_\_\_\_ mm .
  22. 22
    The heat generated in 1 minute between points A and B in the given circuit, when a battery of 9 V with internal resistance of 1Ω1\Omega is connected across these points is ____\_\_\_\_ J.
  23. 23
    Two identical thin rods of mass M kg and length L m are connected as shown in figure. Moment of inertia of the combined rod system about an axis passing through point P and perpendicular to the plane of the rods is X12ML2\frac{X}{12}ML^{2}. The value of XX is ____\_\_\_\_ .
  24. 24
    10 mole of oxygen is heated at constant volume from 30∘C30^{\circ}C to 40∘C40^{\circ}C. The change in the internal energy of the gas is ____\_\_\_\_ cal. (The molecular specific heat of oxygen at constant pressure, Cp=7cal./mol∘CC_{p} = 7cal./{mol}^{\circ}C and R=2cal./mol∘CR = 2cal./{mol}^{\circ}C.)
  25. 25
    In a microscope the objective is having focal length f0=2 cmf_{0} = 2\text{ }cm and eye-piece is having focal length fe=4 cmf_{e} = 4\text{ }cm. The tube length is 32 cm. The magnification produced by this microscope for normal adjustment is ____\_\_\_\_ .

Chemistry

25 questions

  1. 26
    Consider the following reactions.
    PbCl2+K2CrO4→ A+2KClPbCl_{2}+K_{2}CrO_{4}\rightarrow\text{ }A + 2KCl
    (Hot solution)
    A+NaOH⇌B+Na2CrO4PbSO4+4CH3COONH4→(NH4)2SO4+X\begin{gathered}A + NaOH \rightleftharpoons B +Na_{2}CrO_{4} \\ PbSO_{4}+ 4CH_{3}COONH_{4}\rightarrow\left( NH_{4} \right)_{2}SO_{4}+ X\end{gathered}
    In the above reactions, A,BA,B and X are respectively.

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  2. 27
    Which of the following represents the correct trend for the mentioned property? (A) F>P>S>B−F > P > S > B - First Ionization Energy (B) Cl>F>S>P−Cl > F > S > P - Electron Affinity (C) K>Al>Mg>B−K > Al > Mg > B - Metallic character (D) K2O>Na2O>MgO>Al2O3K_{2}O > {Na}_{2}O > MgO > {Al}_{2}O_{3} - Basic character Choose the correct answer from the option given below.

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  3. 28
    Identify A in the following reaction.

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  4. 29
    A hydrocarbon ' P ' (C4H8)\left( C_{4}H_{8} \right) on reaction with HCl gives an optically active compound ' Q ' (C4H9Cl)\left( C_{4}H_{9}Cl \right) which on reaction with one mole of ammonia gives compound ' R ' (C4H11 N)\left( C_{4}H_{11}\text{ }N \right). ' R ' on diazotization followed by hydrolysis gives ' S '. Identify P,Q,RP,Q,R and S .

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  5. 30
    Given below are two statements : Statement I: The number of pairs among [ SiO2,CO2{SiO}_{2},{CO}_{2} ], [SnO,SnO2],[PbO,PbO2]\left\lbrack SnO,{SnO}_{2} \right\rbrack,\left\lbrack PbO,{PbO}_{2} \right\rbrack and [GeO,GeO2]\left\lbrack GeO,{GeO}_{2} \right\rbrack, which contain oxides that are both amphoteric is 2 . Statement II : BF3{BF}_{3} is an electron deficient molecule can act as a lewis acid, forms adduct with NH3{NH}_{3} and has a trigonal planar geometry. In the light of the above statement, choose the correct answer from the option given below.

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  6. 31
    80 mL of a hydrocarbon on mixing with 264 mL of oxygen in a closed U-tube undergoes complete combustion. The residual gases after cooling to 273 K occupy 224 mL. When the system is treated with KOH solution, the volume decreases to 64 mL. The formula of the hydrocarbon is :

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  7. 32
    14.0 g of calcium metal is allowed to react with excess HCl at 1.0 atm pressure and 273 K . Which of the following statements is incorrect? [Given : Molar mass in gmol−1gmol^{- 1} of Ca−40,Cl−35.5,H−1Ca - 40,Cl - 35.5,H - 1 ]

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  8. 33
    In Carius method, 0.75 g of an organic compound gave 1.2 g of barium sulphate, find percentage of sulphur (molar mass 32 g mol−132\text{ }g{\text{ }mol}^{- 1}). Molar mass of barium sulphate is 233 g mol−1233\text{ }g{\text{ }mol}^{- 1}.

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  9. 34
    Elements P and Q form two types of non-volatile, non-ionizable compounds PQ and PQ2PQ_{2}. When 1 g of PQ is dissolved in 50 g of solvent ' A '. ΔTb\Delta T_{b} was 1.176 K while when 1 g of PQ2PQ_{2} is dissolved in 50 g of solvent ' A ', ΔTb\Delta T_{b} was 0.689 K. (KbK_{b} of ' A ' =5 Kkgmol−1= 5\text{ }Kkgmol^{- 1}). The molar masses of elements P and Q (in gmol−1gmol^{- 1}) respectively, are :

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  10. 35
    An organic compound (P)(P) on treatment with aqueous ammonia under hot condition forms compound (Q)(Q) which on heating with Br2Br_{2} and KOH forms compound (R) having molecular formula C6H7NC_{6}H_{7}N. Names of P,QP,Q and R respectively are.

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  11. 36
    An organic compound "P" of molecular formula C6H12O3C_{6}H_{12}O_{3} gives positive lodoform test but negative Tollen's test. When "P" is treated with dilute acid, it produces "Q". "Q" gives positive Tollen's test and also iodoform test. The structure of " P " is :

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  12. 37
    From the following, the least stable structure is :

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  13. 38
    MnO42−MnO_{4}^{2 -}, in acidic medium, disproportionates to :

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  14. 39
    Given below are two statements: Statement I: The number of species among SF4{SF}_{4}, NH4+, [NiCl4]2−, XeF4, [PtCl4]2−, SeF4{NH}_{4}^{+},\ \left\lbrack {NiCl}_{4} \right\rbrack^{2 -},\ {XeF}_{4},\ \left\lbrack {PtCl}_{4} \right\rbrack^{2 -},\ {SeF}_{4} and [Ni(CN)4]2−\left\lbrack Ni(CN)_{4} \right\rbrack^{2 -}, that have tetrahedral geometry is 3 . Statement II: In the set [NO2,BeH2,BF3,AlCl3]\left\lbrack {NO}_{2},{BeH}_{2},{BF}_{3},{AlCl}_{3} \right\rbrack, all the molecules have incomplete octet around central atom. 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: Among [Cu(NH3)4]2+,[Ni(en)3]2+\left\lbrack Cu\left( {NH}_{3} \right)_{4} \right\rbrack^{2 +},\left\lbrack Ni(en)_{3} \right\rbrack^{2 +}, [Ni(NH3)6]2+\left\lbrack Ni\left( {NH}_{3} \right)_{6} \right\rbrack^{2 +} and [Mn(H2O)6]2+\left\lbrack Mn\left( H_{2}O \right)_{6} \right\rbrack^{2 +}, [Mn(H2O)6]2+\left\lbrack Mn\left( H_{2}O \right)_{6} \right\rbrack^{2 +} has the maximum number of unpaired electrons. Statement II: The number of pairs among {[NiCl4]2−,[Ni(CO)4]},{[NiCl4]2−,[Ni(CN)4]2−}\left\{ \left\lbrack {NiCl}_{4} \right\rbrack^{2 -},\left\lbrack Ni(CO)_{4} \right\rbrack \right\},\left\{ \left\lbrack {NiCl}_{4} \right\rbrack^{2 -},\left\lbrack Ni(CN)_{4} \right\rbrack^{2 -} \right\} and {[Ni(CO)4], [Ni(CN)4]2−}\left\{ \left\lbrack Ni(CO)_{4} \right\rbrack,\ \left\lbrack Ni(CN)_{4} \right\rbrack^{2 -} \right\} that contain only diamagnetic species is two. In the light of the above statements, choose the correct answer from the options given below:

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  16. 41
    Identify correct statement from the following : (A) Propanal and propanone are functional isomers. (B) Ethoxyethane and methoxypropane are metamers. (C) But-2-ene shows optical isomerism. (D) But-1-ene and but-2-ene are functional isomers. (E) Pentane and 2, 2-dimethyl propane are chain isomers. Choose the correct answer from the options given below :

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  17. 42
    Identify the correct statements. (A) Arginine and Tryptophan are essential amino acids. (B) Histidine does not contain heterocyclic ring in its structure. (C) Proline is a six membered cyclic ring amino acid. (D) Glycine does not have chiral centre. (E) Cysteine has characteristic feature of side chain as MeS−CH2−CH2−MeS - {CH}_{2} - {CH}_{2} -. Choose the correct answer from the options given below.

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  18. 43
    Which of the following graphs between pressure 'P' versus volume 'V' represent the maximum work done?

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  19. 44
    For the reaction, N2O4⇌2NO2N_{2}O_{4}\rightleftharpoons 2NO_{2}, graph is plotted as shown below. Identify correct statements. (A) Standard free energy change for the reaction is −5.40 kJ mol−1- 5.40\text{ }kJ{\text{ }mol}^{- 1}. (B) As AG⊖{AG}^{\ominus} in graph is positive, N2O4N_{2}O_{4} will not dissociate into NO2{NO}_{2} at all. (C) Reverse reaction will go to completion. (D) When 1 mole of N2O4N_{2}O_{4} changes into equilibrium mixture, value of ΔG⊖=−0.84 kJ mol−1\Delta G^{\ominus} = - 0.84\text{ }kJ{\text{ }mol}^{- 1} (E) When 2 mole of NO2{NO}_{2}, changes into equilibrium mixture, ΔG⊖\Delta G^{\ominus} for equilibrium mixture is −6.24 kJ mol−1- 6.24\text{ }kJ{\text{ }mol}^{- 1}. Choose the correct answer from the options given below :

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  20. 45
    Given below are two statements: Statement I: When an electric discharge is passed through gaseous hydrogen, the hydrogen molecules dissociate and the energetically excited hydrogen atoms produce electromagnetic radiation of discrete frequencies. Statement II: The frequency of second line of Balmer series obtained from He+{He}^{+}is equal to that of first line of Lyman series obtained from hydrogen atom. In the light of the above statements, choose the correct answer from the options given below:

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  21. 46
    Pre-exponential factors of two different reactions of same order are identical. Let activation energy of first reaction exceeds the activation energy of second reaction by 20 kJ mol−120\text{ }kJ{\text{ }mol}^{- 1}. If k1k_{1} and k2k_{2} are the rate constants of first and second reaction respectively at 300 K , then In k2k1\frac{k_{2}}{k_{1}} will be (nearest integer) [R=8.3 J K−1 mol−1]\left\lbrack R = 8.3\text{ }J{\text{ }K}^{- 1}{\text{ }mol}^{- 1} \right\rbrack
  22. 47
    The pH and conductance of a weak acid (HX) was found to be 5 and 4×10−5 S4 \times 10^{- 5}\text{ }S, respectively. The conductance was measured under standard condition using a cell where the electrode plates having a surface area of 1 cm21{\text{ }cm}^{2} were at a distance of 15 cm apart. The value of the limiting molar conductivity is ____\_\_\_\_ Sm2 mol−1Sm^{2}{\text{ }mol}^{- 1}. (nearest integer) (Given: degree of dissociation of the weak acid (a) << 1)
  23. 48
    Use the following data:
    SubstanceΔfH⊖(500 K)/kJ mol−1\Delta_{f}H^{\ominus}(500\text{ }K)/kJ\,mol^{-1}S⊖(500 K)/J K−1mol−1S^{\ominus}(500\text{ }K)/J\,K^{-1}mol^{-1}
    AB(g)AB(g)32222
    A2(g)A_{2}(g)6146
    B2(g)B_{2}(g)x280
    One mole each of A2(g)A_{2}(g) and B2(g)B_{2}(g) are taken in a 1L closed flask and allowed to establish the equilibrium at 500 K .
    A2( g)+B2( g)⇌2AB(g)A_{2}(\text{ }g) + B_{2}(\text{ }g) \rightleftharpoons 2AB(g)
    The value of xx (in kJ mol−1kJ\,mol^{-1}) is ____\_\_\_\_ (Nearest integer) (Given: log⁡K=2.2, R=8.3 JK−1mol−1\log K = 2.2,\ R = 8.3\ JK^{- 1}mol^{- 1} )
  24. 49
    Consider the following reaction sequence The percentage of nitrogen in product ' T ' formed is ____\_\_\_\_ %. (Nearest integer) (Given molar mass in gmol−1H:1,C:12, N:14,O:16g{mol}^{- 1}H:1,C:12,\text{ }N:14,O:16 )
  25. 50
    Consider the following reactions:
    NaCl+K2Cr2O7+H2SO4→ A+KHSO4+NaHSO4+H2OA+NaOH→B+NaCl+H2OB+H2SO4+H2O2→C+Na2SO4+H2O\begin{gathered}NaCl + K_{2}{Cr}_{2}O_{7} + H_{2}{SO}_{4} \rightarrow \text{ }A + {KHSO}_{4} + {NaHSO}_{4} + H_{2}O \\ A + NaOH \rightarrow B + NaCl + H_{2}O \\ B + H_{2}{SO}_{4} + H_{2}O_{2} \rightarrow C + {Na}_{2}{SO}_{4} + H_{2}O\end{gathered}
    In the product ' C ', ' X ' is the number of O22−O_{2}^{2 -} units, ' Y ' is the total number oxygen atoms present and ' ZZ ' is the oxidation state of Cr . The value of X+Y+ZX + Y + Z is ____.

Mathematics

25 questions

  1. 51
    If the domain of the function f(x)=cos⁡−1(2x−511−3x)+sin⁡−1(2x2−3x+1)f(x) =\cos^{- 1}\left( \frac{2x - 5}{11 - 3x} \right)+\sin^{- 1}\left( 2x^{2}- 3x + 1 \right) is the interval [α,β]\lbrack\alpha,\beta\rbrack, then α+2β\alpha + 2\beta is equal to :

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  2. 52
    The area of the region, inside the ellipse x2+4y2=4x^{2}+ 4y^{2}= 4 and outside the region bounded by the curves y=∣x∣−1y = |x| - 1 and y=1−∣x∣y = 1 - |x|, is :

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  3. 53
    The number of relations, defined on the set {a,b,c,d}\mathbf{\{ a,b,c,d\}}, which are both reflexive and symmetric, is equal to:

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  4. 54
    Let a point A lie between the parallel lines L1L_{1} and L2L_{2} such that its distances from L1L_{1} and L2L_{2} are 6 and 3 units, respectively. Then the area (in sq. units) of the equilateral triangle ABC , where the points B and C lie on the lines L1L_{1} and L2L_{2} respectively, is :

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  5. 55
    Let a→=−i^+2j^+2k^,b→=8i^+7j^−3k^\overrightarrow{a}= -\widehat{i}+ 2\widehat{j}+ 2\widehat{k},\overrightarrow{b}= 8\widehat{i}+ 7\widehat{j}- 3\widehat{k} and c→\overrightarrow{c} be a vector such that a→×c→=b→\overrightarrow{a}\times\overrightarrow{c}=\overrightarrow{b}. If c→⋅(i^+j^+k^)=4\overrightarrow{c}\cdot (\widehat{i}+\widehat{j}+\widehat{k}) = 4, then ∣a→+c→∣2|\overrightarrow{a}+\overrightarrow{c}|^{2} is equal to :

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  6. 56
    Let a1,a2,a3,…a_{1},a_{2},a_{3},\ldots. be a G.P. of increasing positive terms such that a2⋅a3⋅a4=64a_{2}\cdot a_{3}\cdot a_{4}= 64 and a1+a3+a5=8137a_{1}+a_{3}+a_{5}=\frac{813}{7}. Then a3+a5+a7a_{3}+a_{5}+a_{7} is equal to :

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  7. 57
    Let c→\overrightarrow{c} and d→\overrightarrow{d} be vectors such that ∣c→+d→∣=29|\overrightarrow{c}+\overrightarrow{d}| =\sqrt{29} and c→×(2i^+3j^+4k^)=(2i^+3j^+4k^)×d→\overrightarrow{c}\times (2\widehat{i}+ 3\widehat{j}+ 4\widehat{k}) = (2\widehat{i}+ 3\widehat{j}+ 4\widehat{k}) \times\overrightarrow{d}. If λ1,λ2(λ1>λ2)\lambda_{1},\lambda_{2}\left( \lambda_{1}>\lambda_{2} \right) are the possible values of (c→+d→)⋅(−7i^+2j^+3k^)(\overrightarrow{c}+\overrightarrow{d}) \cdot ( - 7\widehat{i}+ 2\widehat{j}+ 3\widehat{k}), then the equation K2x2+(K2−5 K+λ1)xy+(3 K+λ22)y2−8x+12y+λ2=0K^{2}x^{2}+\left( K^{2}- 5\text{ }K +\lambda_{1} \right)xy +\left( 3\text{ }K +\frac{\lambda_{2}}{2} \right)y^{2}- 8x + 12y +\lambda_{2}= 0 represents a circle, for k equal to :

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  8. 58
    Let y=y(x)y = y(x) be the solution curve of the differential equation (1+x2)dy+(y−tan⁡−1x)dx=0\left( 1 +x^{2} \right)dy +\left( y -\tan^{- 1}x \right)dx = 0, y(0)=1y(0) = 1. Then the value of y(1)y(1) is :

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  9. 59
    The number of strictly increasing functions f from the set {1,2,3,4,5,6}\mathbf{\{ 1,2,3,4,5,6\}} to the set (1,2,3,…,9)(1,2,3,\ldots,9) such that f(i)≠if(i) \neq i for 1≤i≤61 \leq i \leq 6, is equal to :

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  10. 60
    Let f:R→(0,∞)f:R \rightarrow (0,\infty) be a twice differentiable function such that f(3)=18,f′(3)=0f(3) = 18,f^{'}(3) = 0 and f′′(3)=4f^{''}(3) = 4. Then lim⁡x→1(log⁡e(f(2+x)f(3))18(x−1)2)\lim_{x \rightarrow 1} \left( \log_{e}\left( \frac{f(2 + x)}{f(3)} \right)^{\frac{18}{(x - 1)^{2}}} \right) is equal to :

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  11. 61
    Let the foci of hyperbola coincide with the foci of the ellipse x236+y216=1\frac{x^{2}}{36}+\frac{y^{2}}{16}= 1. If the eccentricity of the hyperbola is 5 , then the length of its latus rectum is:

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  12. 62
    The value of ∫−π/6π/6(π+4x111−sin⁡(∣x∣+π/6))dx\int_{- \pi/6}^{\pi/6} \left( \frac{\pi + 4x^{11}}{1 - \sin(|x| + \pi/6)} \right)dx is equal to

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  13. 63
    Let the mean and variance of 7 observations 2,4,10,x,12,14,y,x>y2,4,10,x,12,14,y,x > y, be 8 and 16 respectively. Two numbers are chosen from {1,2,3,x−4,y,5}\mathbf{\{ 1,2,3,x - 4,y,5\}} one after another without replacement, then the probability, that the smaller number among the two chosen numbers is less than 4 , is:

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  14. 64
    Let (α,β,γ)(\alpha,\beta,\gamma) be the co-ordinates of the foot of the perpendicular drawn from the point (5,4,2)(5,4,2) on the line r→=(−i^+3j^+k^)+λ(2i^+3j^−k^)\overrightarrow{r}= ( -\widehat{i}+ 3\widehat{j}+\widehat{k}) + \lambda(2\widehat{i}+ 3\widehat{j}-\widehat{k}). Then the length of the projection of the vector αi^+βj^+γk^\alpha\widehat{i}+ \beta\widehat{j}+ \gamma\widehat{k} on the vector 6i^+2j^+3k^6\widehat{i}+ 2\widehat{j}+ 3\widehat{k} is :

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  15. 65
    Let PQ and MN be two straight lines touching the circle x2+y2−4x−6y−3=0x^{2}+y^{2}- 4x - 6y - 3 = 0 at the points A and B respectively. Let O be the centre of the circle and ∠AOB=π/3\angle AOB = \pi/3. Then the locus of the point of intersection of the lines PQ and MN is:

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  16. 66
    If the coefficient of xx in the expansion of (ax2+bx+c)(1−2x)26\left( ax^{2}+ bx + c \right)(1 - 2x)^{26} is -56 and the coefficients of x2x^{2} and x3x^{3} are both zero, then a+b+ca + b + c is equal to

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  17. 67
    If x2+x+1=0x^{2}+ x + 1 = 0, then the value of (x+1x)4+(x2+1x2)4+(x3+1x3)4+…+(x25+1x25)4\left( x +\frac{1}{x} \right)^{4}+\left( x^{2}+\frac{1}{x^{2}} \right)^{4}+\left( x^{3}+\frac{1}{x^{3}} \right)^{4}+ \ldots +\left( x^{25}+\frac{1}{x^{25}} \right)^{4} is :

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  18. 68
    The value of csc⁡10∘−3sec⁡10∘\csc10^{\circ}-\sqrt{3}\sec10^{\circ} is equal to:

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  19. 69
    The sum of all the roots of the equation (x−1)2−5∣x−1∣+6=0(x - 1)^{2}- 5|x - 1| + 6 = 0, is:

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  20. 70
    Let O be the vertex of the parabola x2=4yx^{2}= 4y and Q be any point on it. Let the locus of the point P , which divides the line segment OQ internally in the ratio 2: 3 be the conic C . Then the equation of the chord of C , which is bisected at the point (1,2)(1,2), is:

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  21. 71
    Let f:R→Rf:R \rightarrow R be a twice differentiable function such that the quadratic equation f(x)m2−2f′(x)m+f′(x)=0f(x)m^{2} - 2f'(x)m + f'(x) = 0 in m , has two equal roots for every x∈Rx \in R. If f(0)=1f(0) = 1, f′(0)=2f'(0) = 2 and (α,β)(\alpha,\beta) is the largest interval in which the function f(log⁡ex−x)f\left( \log_{e}x - x \right) is increasing, then α+β\alpha + \beta is equal to
  22. 72
    Let a1=1a_{1} = 1 and for n≥1,an+1n \geq 1,a_{n + 1} =12an+n2−2n−1n2(n+1)2= \frac{1}{2}a_{n} + \frac{n^{2} - 2n - 1}{n^{2}(n + 1)^{2}}. Then ∣∑n=1∞(an−2n2)∣\left| \sum_{n = 1}^{\infty} \left( a_{n} - \frac{2}{n^{2}} \right) \right| is equal to ............. .
  23. 73
    Let S={(m,n):m,n∈{1,2,3,…..,50}}S = \{(m,n):m,n \in \{ 1,2,3,\ldots..,50\}\}. If the number of elements (m,nm,n) in S such that 6m+9n6^{m} + 9^{n} is a multiple of 5 is p and the number of elements (m,n)(m,n) in S such that m+nm + n is a square of a prime number is q , then p+qp + q is equal to ____\_\_\_\_
  24. 74
    For some α,β∈R\alpha,\beta \in R, let A=[α212]A = \begin{bmatrix} \alpha & 2 \\ 1 & 2 \end{bmatrix} and B=[111β]B = \begin{bmatrix} 1 & 1 \\ 1 & \beta \end{bmatrix} be such that A2−4A+2I=B2−3B+I=OA^{2} - 4A + 2I = B^{2} - 3B + I = O. Then (det⁡(adj(A3−B3)))2\left( \det\left( adj\left( A^{3} - B^{3} \right) \right) \right)^{2} is equal to ____\_\_\_\_
  25. 75
    6∫0π∣(sin⁡3x+sin⁡2x+sin⁡x)∣dx6\int_{0}^{\pi} |(\sin3x + \sin2x + \sin x)|dx is equal to

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