PYQ Vault

JEE Mains 2026 (23 Jan, 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
    The internal energy of a monoatomic gas is 3 nRT . One mole of helium is kept in a cylinder having internal cross section area of 17 cm217{\text{ }cm}^{2} and fitted with a light movable frictionless piston. The gas is heated slowly by suppling 126 J heat. If the temperature rises by 4 ∘C4\ ^{\circ}C, then the piston will move ____\_\_\_\_ cm. (atmospheric pressure =105 Pa=10^{5}\text{ }Pa )

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  2. 2
    A body of mass 14 kg initially at rest explodes and breaks into three fragments of masses in the ratio 2:2:32:2:3. The two pieces of equal masses fly off perpendicular to each other with a speed of 18 m/s18\text{ }m/s each. The velocity of the heavier fragment is ____\_\_\_\_ m/sm/s.

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  3. 3
    A block is sliding down on an inclined plane of slope θ\theta and at an instant t=0t = 0 this block is given an upward momentum so that it starts moving up on the inclined surface with velocity uu. The distance (S)(S) travelled by the block before its velocity become zero, is ____\_\_\_\_ . (g = gravitational acceleration)

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  4. 4
    The current passing through a conducting loop in the form of equilateral triangle of side 43 cm4\sqrt{3}\text{ }cm is 2A. The magnetic field at its centroid is α×10−5 T\alpha \times 10^{- 5}\text{ }T. The value of α\alpha is ____\_\_\_\_ . (Given : μo=4π×10−7\mu_{o} = 4\pi \times 10^{- 7} SI units)

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  5. 5
    A paratrooper jumps from an aeroplane and opens a parachute after 2 s of free fall and starts deaccelerating with 3 m/s23\text{ }m/s^{2}. At 10 m height from ground, while descending with the help of parachute, the speed of paratrooper is 5 m/s5\text{ }m/s. The initial height of the aeroplane is ____\_\_\_\_ m. (g=10 m/s2)\left( g = 10\text{ }m/s^{2} \right)

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  6. 6
    Two shorts dipoles (A,B),A(A,B),A having charges ±2μC\pm 2\mu C and length 1 cm and BB having charges ±4μC\pm 4\mu C and length 1 cm are placed with their centres 80 cm apart as shown in the figure. The electric field at a point PP, equi-distant from the centres of both dipoles is ____\_\_\_\_ N/C.

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  7. 7
    Two charges 7μC7\mu C and −2μC- 2\mu C are placed at (−9,0,0)cm( - 9,0,0)cm and (9,0,0)cm(9,0,0)cm respectively in an external field E=Ar2r^E =\frac{A}{r^{2}}\widehat{r}, where A=9×105 N/C.m2A = 9 \times10^{5}\text{ }N/C.m^{2}. Considering the potential at infinity is 0 , the electrostatic energy of the configuration is ____\_\_\_\_ J.

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  8. 8
    A bead PP sliding on a frictionless semi-circular string ( ACBACB ) and it is at point SS at t=0t = 0 and at this instant the horizontal component of its velocity is vv. Another bead QQ of the same mass as PP is ejected from point AA at t=0t = 0 along the horizontal string ABAB, with the speed vv, friction between the beads and the respective strings may be neglected in both cases. Let tpt_{p} and tQt_{Q} be the respective times taken by beads PP and QQ to reach the point BB, then the relation between tpt_{p} and tQt_{Q} is

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  9. 9
    A parallel plate capacitor with plate separation 5 mm is charged by a battery. On introducing a mica sheet of 2 mm and maintaining the connections of the plates with the terminals of the battery, it is found that it draws 25%25\% more charge from the battery. The dielectric constant of mica is ____\_\_\_\_ .

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  10. 10
    To compare EMF of two cells using potentiometer the balancing lengths obtained are 200 cm and 150 cm . The least count of scale is 1 cm . The percentage error in the ratio of EMFs is ____\_\_\_\_

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  11. 11
    An air bubble of volume 2.9 cm32.9{\text{ }cm}^{3} rises from the bottom of a swimming pool of 5 m deep. At the bottom of the pool water temperature is 17∘C17^{\circ}C. The volume of the bubble when it reaches the surface, where the water temperature is 27∘C27^{\circ}C, is ____\_\_\_\_ cm3cm^{3}. ( g=10 m/s2g = 10\text{ }m/s^{2}, density of water =103 kg/m3=10^{3}\text{ }kg/m^{3}, and 1 atm pressure is 105 Pa10^{5}\text{ }Pa)

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  12. 12
    Which of the following pair of nuclei are isobars of the element?

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  13. 13
    For the given logic gate circuit, which of the following is the correct truth table?

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  14. 14
    One mole of an ideal diatomic gas expands from volume V to 2 V isothermally at a temperature 27∘C27^{\circ}C and does W joule of work. If the gas undergoes same magnitude of expansion adiabatically from 27∘C27^{\circ}C doing the same amount of work W, then its final temperature will be (close to) ____\_\_\_\_  ∘C\ ^{\circ}C.

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  15. 15
    A prism of angle 75∘75^{\circ} and refractive index 3\sqrt{3} is coated with thin film of refractive index 1.5 only at the back exit surface. To have total internal reflection at the back exit surface the incident angle must be ____\_\_\_\_ . (sin⁡15∘=0.25\sin15^{\circ}= 0.25 and sin⁡25∘=0.43\sin25^{\circ}= 0.43 )

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  16. 16
    A circular loop of radius 7 cm is placed in uniform magnetic field of 0.2 T directed perpendicular to plane of loop. The loop is converted into a square loop in 0.5 s . The EMF induced in the loop is ____\_\_\_\_ mV.

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  17. 17
    Suppose a long solenoid of 100 cm length, radius 2 cm having 500 turns per unit length, carries a current I=10sin⁡(ωt)AI = 10\sin(\omega t)A, where ω=1000rad./s\omega = 1000rad./s. A circular conducting loop (B) of radius 1 cm coaxially slided through the solenoid at a speed =1 cm/s= 1\text{ }cm/s. The r.m.s. current through the loop when the coil B is inserted 10 cm inside the solenoid is α/2μ A\alpha/\sqrt{2}\mu\text{ }A. The value of α\alpha is ____\_\_\_\_ . [Resistance of the loop =10Ω= 10\Omega ]

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  18. 18
    The ratio of speeds of electromagnetic waves in vacuum and a medium, having dielectric constant k=3k = 3 and permeability of μ=2μ0\mu = 2\mu_{0}, is ( μ0=\mu_{0}= permeability of vacuum)

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  19. 19
    When an unpolarized light falls at a particular angle on a glass plate (placed in air), it is observed that the reflected beam is linearly polarized. The angle of refracted beam with respect to the normal is ____\_\_\_\_ . (tan⁡−1(1.52)=57.7∘\left( \tan^{- 1}(1.52) =57.7^{\circ} \right., refractive indices of air and glass are 1.00 and 1.52 , respectively)

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  20. 20
    A small metallic sphere of diameter 2 mm and density 10.5 g/cm310.5\text{ }g/cm^{3} is dropped in glycerine having viscosity 10 Poise and density 1.5 g/cm31.5\text{ }g/cm^{3} respectively. The terminal velocity attained by the sphere is ____\_\_\_\_ cm/scm/s. (π=227\pi =\frac{22}{7} and g=10 m/s2g = 10\text{ }m/s^{2})

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  21. 21
    The average energy released per fission for the nucleus of  92235U\ _{92}^{235}U is 190 MeV. When all the atoms of 47 g pure  92235U\ _{92}^{235}U undergo fission process, the energy released is α×1023MeV\alpha \times 10^{23}MeV. The value of α\alpha is ____\_\_\_\_ . (Avogadro Number =6×1023= 6 \times 10^{23} per mole)
  22. 22
    The size of the images of an object, formed by a thin lens are equal when the object is placed at two different positions 8 cm and 24 cm from the lens. The focal length of the lens is ____\_\_\_\_ cm .
  23. 23
    A ball of radius r and density ρ\rho dropped through a viscous liquid of density σ\sigma and viscosity η\eta attains its terminal velocity at time t , given by t=Aρarbηcσdt = A\rho^{a}r^{b}\eta^{c}\sigma^{d}, where A is a constant and a,b,ca, b, c and dd are integers. The value of b+ca+d\frac{b + c}{a + d} is ____\_\_\_\_ .
  24. 24
    Suppose there is a uniform circular disc of mass M kg and radius r m shown in figure. The shaded regions are cut out from the disc. The moment of inertia of the remainder about the axis A of the disc is given by X256Mr2\frac{X}{256}{Mr}^{2}. The value of x is ____\_\_\_\_ .
  25. 25
    The velocity of sound in air is doubled when the temperature is raised from 0∘C0^{\circ}C to α∘C\alpha^{\circ}C. The value of α\alpha is ____\_\_\_\_ .

Chemistry

25 questions

  1. 26
    Given above is the concentration vs time plot for a dissociation reaction : A→nBA \rightarrow nB. Based on the data of the initial phase of the reaction (initial 10 min ), the value of n is ____\_\_\_\_ .

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  2. 27
    It is noticed that Pb2+Pb^{2 +} is more stable than Pb4+Pb^{4 +} but Sn2+Sn^{2 +} is less stable than Sn4+Sn^{4 +}. Observe the following reactions
    PbO2+Pb→2PbO; ΔrG∘(1)PbO_{2}+ Pb \rightarrow 2PbO;\ \Delta_{r}G^{\circ}(1)
    SnO2+Sn→2SnO; ΔrG∘(2)SnO_{2}+ Sn \rightarrow 2SnO;\ \Delta_{r}G^{\circ}(2)
    Identify the correct set from the following.

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  3. 28
    Consider the above electrochemical cell where a metal electrode (M) is undergoing redox reaction by forming M+(M→M++e)M^{+}\left( M \rightarrow M^{+}+ e \right). The cation M+M^{+}is present in two different concentrations c1c_{1} and c2c_{2} as shown above. Which of the following statement is correct for generating a positive cell potential?

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  4. 29
    Both human DNA and RNA are chiral molecules. The chirality in DNA and RNA arises due to the presence of

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  5. 30
    Identify the CORRECT set of details from the following : (A) [Co(NH3)6]3+\left\lbrack Co\left( {NH}_{3} \right)_{6} \right\rbrack^{3 +} : Inner orbital complex ; d2d^{2} sp3{sp}^{3} hybridized (B) [MnCl6]3−\left\lbrack {MnCl}_{6} \right\rbrack^{3 -} : Outer orbital complex; sp3d2{sp}^{3}{d}^{2} hybridized (C) [CoF6]3−\left\lbrack {CoF}_{6} \right\rbrack^{3 -} : Outer orbital complex ; d2sp3d^{2}{sp}^{3} hybridized (D) [FeF6]3−\left\lbrack {FeF}_{6} \right\rbrack^{3 -} : Outer orbital complex: sp3 d2{sp}^{3}{\text{ }d}^{2} hybridized (E) [Ni(CN)4]2−\left\lbrack Ni(CN)_{4} \right\rbrack^{2 -} : Inner orbital complex ; sp3{sp}^{3} hybridized Choose the correct answer from the options given below:

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  6. 31
    Elements X and Y belong to Group 15. The difference between the electronegativity values of ' X ' and phosphorus is higher than that of the difference between phosphorus and ' Y '. ' X ' & ' Y ' are respectively

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  7. 32
    Given below are two statements: Statement I: (CH3)3C⊕\left( {CH}_{3} \right)_{3}\overset{\oplus}{C} is more stable than C⊕H3\overset{\oplus}{C}H_{3} as nine hyperconjugation interactions are possible in (CH3)3C⊕\left( {CH}_{3} \right)_{3}\overset{\oplus}{C}. Statement II: C⊕H3\overset{\oplus}{C}H_{3} is less stable than (CH3)3C⊕\left( {CH}_{3} \right)_{3}\overset{\oplus}{C} as only three hyperconjugation interactions are possible in C⊕H3\overset{\oplus}{C}H_{3}. In the light of the above statements, choose the correct answer from the options given below.

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  8. 33
    Iodoform test can differentiate between (A) Methanol and Ethanol (B) CH3COOH{CH}_{3}COOH and CH3CH2COOH{CH}_{3}{CH}_{2}COOH (C) Cyclohexene and cyclohexanone (D) Diethyl ether and Pentan-3-one (E) Anisole and acetone Choose the correct answer from the options given below:

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  9. 34
    Given below are two statements : Statement I: The second ionisation enthalpy of Na is larger than the corresponding ionisation enthalpy of Mg. Statement II : The ionic radius of O2−O^{2 -} is larger than that of F−F^{-}. In the light of the above statements, choose the correct answer from the options given below.

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  10. 35
    Identify (P)

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  11. 36
    Observe the following reactions at T(K)T(K) (I) A → products. (II) 5Br−(aq)+BrO3−(aq)+6H+(aq)→3Br2(aq)+3H2O(l)5{Br}^{-}(aq) + {BrO}_{3}^{-}(aq) + 6H^{+}(aq) \rightarrow 3{Br}_{2}(aq) + 3H_{2}O(l) Both the reactions are started at 10.00 am . The rates of these reactions at 10.10 am are same. The value of −Δ[Br−]Δt-\frac{\Delta\left\lbrack Br^{-} \right\rbrack}{\Delta t} at 10.10 am is 2×10−4 mol L−1Min−12 \times10^{- 4}\text{ }mol{\text{ }L}^{- 1}Min^{- 1}. The concentration of A at 10.10 am is 10−2 mol L−110^{- 2}\text{ }mol{\text{ }L}^{- 1}. What is the first order rate constant (in min⁡−1\min^{- 1} ) of reaction I?

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  12. 37
    Which of the following statements are TRUE about Haloform reaction? (A) Sodium hypochlorite reacts with KI to give KOI. (B) KOI is a reducing agent. (C) α,β\alpha,\beta-unsaturated methylketone will give iodoform reaction. (D) Isopropyl alcohol will not give iodoform test. (E) Methanoic acid will give positive iodoform test. Choose the correct answer from the options given below:

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  13. 38
    Which statements are NOT TRUE about XeO2 F2XeO_{2}{\text{ }F}_{2} ? (A) It has a see-saw shape. (B) Xe has 5 electron pairs in its valence shell in XeO2 F2{XeO}_{2}{\text{ }F}_{2}. (C) The O−Xe−OO - Xe - O bond angle is close to 180∘180^{\circ}. (D) The F−Xe−FF - Xe - F bond angle is close to 180∘180^{\circ} (E) Xe has 16 valence electrons in XeO2 F2{XeO}_{2}{\text{ }F}_{2}. Choose the correct answer from the options given below.

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  14. 39
    Identify the INCORRECT statements from the following: (A) Notation  1224Mg\ _{12}^{24}Mg represents 24 protons and 12 neutrons. (B) Wavelength of a radiation of frequency 4.5×1015 s−14.5 \times 10^{15}{\text{ }s}^{- 1} is 6.7×10−8 m6.7 \times 10^{- 8}\text{ }m. (C) One radiation has wavelength =λ1(900 nm)= \lambda_{1}(900\text{ }nm) and energy =E1= E_{1}. Other radiation has wavelength =λ2(300 nm)= \lambda_{2}(300\text{ }nm) and energy =E2⋅E1:E2=3:1= E_{2} \cdot E_{1}:E_{2} = 3:1. (D) Number of photons of light of wavelength 2000 pm that provides 1 J of energy is 1.006×10161.006 \times 10^{16}. Choose the correct answer from the options given below:

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  15. 40
    In Carius method 0.2425 g of an organic compounds gave 0.5253 g silver chloride. The percentage of chlorine in the organic compound is

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  16. 41
    A mixed ether ( P ), when heated with excess of hot concentrated hydrogen iodide produces two different alkyl iodides which when treated with aq. NaOH give compounds (Q) and (R). Both (Q) and (R) give yellow precipitate with NaOI. Identify the mixed ether (P):

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  17. 42
    The oxidation state of chromium in the final product formed in the reaction between KI and acidified K2Cr2O7K_{2}Cr_{2}O_{7} solution is:

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  18. 43
    The work functions of two metals (MAM_{A} and MBM_{B}) are in the 1:21:2 ratio. When these metals are exposed to photons of energy 6 eV , the kinetic energy of liberated electrons of MA:MBM_{A}:M_{B} is in the ratio of 2.642 : 1. The work functions (in eV ) of MAM_{A} and MBM_{B} are respectively.

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  19. 44
    Given below are two statements: Statement I: Benzene-1,2-diamine can be synthesized from 1,2-di-n-propylbenzene using simpler reagents in the order (i) Acidic KMnO4{KMnO}_{4}, (ii) Ammonia, (iii) Bromine and alkali Statement II : 4-Methylaniline can be converted into 3,5-dibromotoluene using reagents in the order (i) Bromine- H2OH_{2}O (ii) NaNO2/HCl(0−5 ∘C){NaNO}_{2}/HCl\left( 0 - 5\ ^{\circ}C \right) (iii) aq. H3PO2H_{3}{PO}_{2}. In the light of the above statements, choose the correct answer from the options given below

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  20. 45
    A student has been given a compound " x " of molecular formula C6H7NC_{6}H_{7}N. ' x ' is sparingly soluble in water. However, on addition of dilute mineral acid , ' x ' becomes soluble in water. ' x ' when treated with CHCl3CHCl_{3} and KOH (alc.) ' y ' is produced. ' y ' has a specific unpleasant smell. On treatment with benzenesulphonyl chloride, ' x ' gives a compound ' z ' which is soluble in alkali. The number of different "H" atoms present in ' zz ' is:-

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  21. 46
    X2(g)+Y2(g)⇌2Z(g)X_{2}(g) + Y_{2}(g) \rightleftharpoons 2Z(g) X2(g)X_{2}(g) and Y2(g)Y_{2}(g) are added to a 1 L flask and it is found that the system attains the above equilibrium at T(K)T(K) with the number of moles of X2( g),Y2( g)X_{2}(\text{ }g),Y_{2}(\text{ }g) and Z(g)Z(g) being 3, 3 and 9 mol respectively (equilibrium moles). Under this conditions of equilibrium, 10 mol of Z(g)Z(g) is added to the flask and the temperature is maintained at T(K)T(K). Then the number of moles of Z(g)Z(g) in the flask when the new equilibrium is established is ____\_\_\_\_ . (Nearest integer).
  22. 47
    Two liquids A and B form an ideal solution. At 320 K, the vapour pressure of the solution, containing 3 mol of A and 1 mol of B is 500 mm Hg. At the same temperature, if 1 mol of A is further added to this solution, vapour pressure of the solution increases by 20 mm Hg. Vapour pressure (in mm Hg ) of B in the pure state is ____\_\_\_\_ . (Nearest integer)
  23. 48
    200 cc of x×10−3Mx \times 10^{- 3}M potassium dichromate is required to oxidise 750 cc of 0.6 M Mohr's salt solution in acidic medium. Here x=x =
  24. 49
    Total number of unpaired electrons present in the central metal atoms/ions of [Ni(CO)4],[NiCl4]2−,[PtCl2(NH3)2],[Ni(CN)4]2−\left\lbrack Ni(CO)_{4} \right\rbrack,\left\lbrack {NiCl}_{4} \right\rbrack^{2 -},\left\lbrack {PtCl}_{2}\left( {NH}_{3} \right)_{2} \right\rbrack,\left\lbrack Ni(CN)_{4} \right\rbrack^{2 -} and [Pt(CN)4]2−\left\lbrack Pt(CN)_{4} \right\rbrack^{2 -} is ____\_\_\_\_ .
  25. 50
    In compound (Q), the percentage of oxygen is ____\_\_\_\_ %. (Nearest integer)

Mathematics

25 questions

  1. 51
    If f(x)={a∣x∣+x2−2(sin⁡∣x∣)(cos⁡∣x∣)x,x≠0b,x=0f(x) =\left\{ \begin{matrix} \frac{a|x| +x^{2}- 2(\sin|x|)(\cos|x|)}{x} & ,x \neq 0 \\ b & ,x = 0 \end{matrix} \right. is continuous at x=0x = 0, then a+ba + b is equal to :

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  2. 52
    Let π2<θ<π\frac{\pi}{2}< \theta < \pi and cot⁡θ=−122\cot\theta = -\frac{1}{2\sqrt{2}}. Then the value of sin⁡(15θ2)(cos⁡8θ+sin⁡8θ)+cos⁡(15θ2)(cos⁡8θ−sin⁡8θ)\sin\left( \frac{15\theta}{2} \right)(\cos8\theta + \sin8\theta) + \cos\left( \frac{15\theta}{2} \right)(\cos8\theta - \sin8\theta) is equal to :

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  3. 53
    Let a→=i^−2j^+3k^,b→=2i^+j^−k^,c→=λi^+j^+k^\overrightarrow{a}=\widehat{i}- 2\widehat{j}+ 3\widehat{k},\overrightarrow{b}= 2\widehat{i}+\widehat{j}-\widehat{k},\overrightarrow{c}= \lambda\widehat{i}+\widehat{j}+\widehat{k} and v→=a→×b→\overrightarrow{v}=\overrightarrow{a}\times\overrightarrow{b}. If v→⋅c→=11\overrightarrow{v}\cdot\overrightarrow{c}= 11 and the length of the projection of b→\overrightarrow{b} on c→\overrightarrow{c} is pp, then 9p29p^{2} is equal to :

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  4. 54
    Let A(1,2)A(1,2) and C(−3,−6)C( - 3, - 6) be two diagonally opposite vertices of a rhombus, whose sides AD an BC are parallel to the line 7x−y=147x - y = 14. If B(α,β)B(\alpha,\beta) and D(γ,δ)D(\gamma,\delta) are the other two vertices, then ∣α+β+γ+δ∣|\alpha + \beta + \gamma + \delta| is equal to :

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  5. 55
    Consider two sets A={x∈Z:∣∣x−3∣−3∣≤1}A=\{x\in Z:||x-3|-3|\le 1\} and B={x∈R−{1,2}:(x−2)(x−4)x−1log⁡e(∣x−2∣)=0}B=\left\{x\in R-\{1,2\}:\frac{(x-2)(x-4)}{x-1}\log_{e}(|x-2|)=0\right\}. Then the number of onto functions f:A→Bf:A\rightarrow B is equal to:

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  6. 56
    The system of linear equations x+y+z=6x + y + z = 6, 2x+5y+az=362x + 5y + az = 36, x+2y+3z=bx + 2y + 3z = b has

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  7. 57
    The sum of all the real solutions of the equation log⁡(x+3)(6x2+28x+30)=5−2log⁡(6x+10)(x2+6x+9)\log_{(x + 3)}\left( 6x^{2}+ 28x + 30 \right)= 5 - 2\log_{(6x + 10)}\left( x^{2}+ 6x + 9 \right) is equal to :

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  8. 58
    If the points of intersection of the ellipses x2+2y2−6x−12y+23=0x^{2}+ 2y^{2}- 6x - 12y + 23 = 0 and 4x2+2y2−20x−12y+35=04x^{2}+ 2y^{2}- 20x - 12y + 35 = 0 lie on a circle of radius rr and centre (a,b)(a,b), then the value of ab+18r2ab + 18r^{2} is

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  9. 59
    Let PQ be a chord of the hyperbola x24−y2 b2=1\frac{x^{2}}{4}-\frac{y^{2}}{{\text{ }b}^{2}}= 1, perpendicular to the x -axis such that OPQ is an equilateral triangle, O being the centre of the hyperbola. If the eccentricity of the hyperbola is 3\sqrt{3}, then the area of the triangle OPQ is :

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  10. 60
    Let a→,b→,c→\overrightarrow{a},\overrightarrow{b},\overrightarrow{c} be three vectors such that a→×b→=2(a→×c→)\overrightarrow{a}\times\overrightarrow{b}= 2(\overrightarrow{a}\times\overrightarrow{c}).If ∣a→∣=1,∣ b→∣=4,∣c→∣=2|\overrightarrow{a}| = 1,|\overrightarrow{\text{ }b}| = 4,|\overrightarrow{c}| = 2, and the angle between b→\overrightarrow{b} and c→\overrightarrow{c} is 60∘60^{\circ}, then ∣a→.c→∣|\overrightarrow{a}.\overrightarrow{c}| is :

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  11. 61
    Bag A contains 9 white and 8 black balls, while bag B contains 6 white and 4 black balls. One ball is randomly picked up from the bag B and mixed up with the balls in the bag A . Then a ball is randomly drawn from the bag A . If the probability, that the ball drawn is white, is pq,gcd(p,q)=1\frac{p}{q},gcd(p,q) = 1, then p+qp + q is equal to

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  12. 62
    The area of the region enclosed between the circles x2+y2=4x^{2}+y^{2}= 4 and x2+(y−2)2=4x^{2}+ (y - 2)^{2}= 4 is :

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  13. 63
    The least value of (cos⁡2θ−6sin⁡θcos⁡θ+3sin⁡2θ+2)\left( \cos^{2}\theta - 6\sin\theta \cos\theta + 3\sin^{2}\theta + 2 \right) is

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  14. 64
    Let I(x)=∫3dx(4x+6)(4x2+8x+3)I(x) = \int\frac{3dx}{(4x + 6)\left( \sqrt{4x^{2}+ 8x + 3} \right)} and I(0)=34+20I(0) =\frac{\sqrt{3}}{4}+ 20. If I(12)=a2 b+cI\left( \frac{1}{2} \right)=\frac{a\sqrt{2}}{\text{ }b}+ c, where a,b,c∈N,gcd(a,b)=1a,b,c \in N,gcd(a,b) = 1, then a+b+ca + b + c is equal to :

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  15. 65
    The number of ways, in which 16 oranges can be distributed to four children such that each child gets at least one orange, is

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  16. 66
    Let A={0,1,2,….9)\mathbf{A = \{ 0,1,2,\ldots.9)}. Let R be a relation on A defined by (x,y)∈R(x,y) \in R if and only if ∣x−y∣|x - y| is a multiple of 3. Given below are two statements: Statement I: n(R)=36n(R) = 36 Statement II: RR is an equivalence relation. In the light of the above statements, choose the correct answer from the options given below

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  17. 67
    If z=32+i2,i=−1z =\frac{\sqrt{3}}{2}+\frac{i}{2},i =\sqrt{- 1}, then (z201−i)8\left( z^{201}- i \right)^{8} is equal to

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  18. 68
    If the mean and the variance of the data
    Class4-88-1212-1616-20
    Frequency3λ\lambda47
    are μ\mu and 19 respectively, then the value of λ+μ\lambda + \mu is

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  19. 69
    An equilateral triangle OAB is inscribed in the parabola y2=4xy^{2}= 4x with the vertex O at the vertex of the parabola. Then the minimum distance of the circle having AB as a diameter from the origin is

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  20. 70
    Let ∑k=1nak=αn2+βn\sum_{k = 1}^{n} a_{k}= \alpha n^{2}+ \beta n. If a10=59a_{10}= 59 and a6=7a1a_{6}= 7a_{1} then α+β\alpha + \beta is equal to

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  21. 71
    If the solution curve y=f(x)y = f(x) of the differential equation (x2−4)y′−2xy+2x(4−x2)2=0, x>2\left( x^{2} - 4 \right)y' - 2xy + 2x\left( 4 - x^{2} \right)^{2} = 0,\ x > 2, passes through the point ( 3,15 ), then the local maximum value of f is ____\_\_\_\_
  22. 72
    If the image of the point P(a,2,a)P(a,2,a) in the line x2=y+a1=z1\frac{x}{2} = \frac{y + a}{1} = \frac{z}{1} is Q and the image of Q in the line x−2b2=y−a1=z+2b−5\frac{x - 2b}{2} = \frac{y - a}{1} = \frac{z + 2b}{- 5} is PP, then a+ba + b is equal to ____\_\_\_\_
  23. 73
    The number of elements in the set S={x:x∈[0,100]S = \left\{ x:x \in \lbrack 0,100\rbrack \right. and  ∫0xt2sin⁡(x−t)dt=x2}\left. \ \int_{0}^{x} t^{2}\sin(x - t)dt = x^{2} \right\} is.....
  24. 74
    Let A=[02−3−2013−10]A = \begin{bmatrix} 0 & 2 & - 3 \\ - 2 & 0 & 1 \\ 3 & - 1 & 0 \end{bmatrix} and B be a matrix such that B(I−A)=I+AB(I - A) = I + A. Then the sum of the diagonal elements of BTBB^{T}B is equal to
  25. 75
    Let S denote the set of 4-digit numbers abcd such that a>b>c>da > b > c > d and P denote the set of 5-digit numbers having product of its digits equal to 20. Then n(S)+n(P)n(S) + n(P) is equal to........... .

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