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JEE Mains 2026 (28 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
    10 kg of ice at −10∘C-10^{\circ}C is added to 100 kg of water to lower its temperature from 25∘C25^{\circ}C. Consider no heat exchange to surroundings. The decrement to the temperature of water is ____\_\_\_\_  ∘C\ ^{\circ}C. (specific heat of ice =2100 J/Kg. ∘C= 2100\text{ }J/Kg.\ ^{\circ}C, specific heat of water =4200 J/Kg. ∘C= 4200\text{ }J/Kg.\ ^{\circ}C, latent heat of fusion of ice =3.36×105 J/Kg= 3.36 \times10^{5}\text{ }J/Kg )

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  2. 2
    The electric current in the circuit is given as i=io(t/T)i=i_{o}(t/T). The r.m.s current for the period t=0t= 0 to t=Tt=T is ____\_\_\_\_

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  3. 3
    In the potentiometer, when the cell in the secondary circuit is shunted with 4Ω4\Omega resistance, the balance is obtained at the length 120 cm of wire. Now when the same cell in shunted with 12Ω12\Omega resistance, the balance is shifted to a length of 180 cm . The internal resistance of cell is ____\_\_\_\_ Ω\Omega.

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  4. 4
    Two circular discs of radius each 10 cm are joined at their centres by a rod of length 30 cm and mass 600 gm as shown in figure. If the mass of each disc is 600 gm and applied torque between two discs is 43×10543 \times10^{5} dyne cm , the angular acceleration of the discs about the given axis ABAB is ____\_\_\_\_ rad/s2rad/s^{2}.

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  5. 5
    Water drops fall from a tap on the floor, 5 m below, at regular intervals of time, the first drop strikes the floor when the sixth drop begins to fall. The height at which the fourth drop will be from ground, at the instant when the first drop strikes the ground is ____\_\_\_\_ m. ( g=10 m/s2g = 10\text{ }m/s^{2} )

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  6. 6
    An atom  38X\ _{3}^{8}X is bombarded by shower of fundamental particles and in 10 s this atom absorbed 10 electrons, 10 protons and 9 neutrons. The percentage growth in the surface area of the nucleus is recorded by :

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  7. 7
    The electric field of an electromagnetic wave travelling through a medium is given by E→(x,t)=25sin⁡(2.0×1015t−107x)n^\overrightarrow{E}(x,t) = 25\sin\left( 2.0 \times10^{15}t-10^{7}x \right)\widehat{n} then the refractive index of the medium is ____\_\_\_\_.(All given measurement are in SI units)

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  8. 8
    Which of the following best represents the temperature versus heat supplied graph for water, in the range of −20∘C-20^{\circ}C to 120∘C120^{\circ}C ?

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  9. 9
    Three long straight wires carrying current are arranged mutually parallel as shown in the figure. The force experienced by 15 cm length of wire QQ is ____\_\_\_\_ . ( μ0=4π×10−7 T.m/A\mu_{0}= 4\pi\times10^{- 7}\text{ }T.m/A )

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  10. 10
    Assuming in forward bias condition there is a voltage drop of 0.7 V across a silicon diode, the current through diode D1D_{1} in the circuit is ____\_\_\_\_ mA . (Assume all diodes in the given circuit are identical)

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  11. 11
    The magnitudes of power of a biconvex lens (refractive index 1.5) and that of a Plano-concave lens (refractive index =1.7= 1.7 ) are same. If the curvature of plano-concave lens exactly matches with the curvature of back surface of the biconvex lens, then ratio of radius of curvature of front and back surface of the biconvex lens is ____\_\_\_\_.

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  12. 12
    In the following pp - VV diagram the equation of state along the curved path is given by (V−2)2=4ap(V- 2)^{2}= 4ap where aa is a constant. The total work done in the closed path is

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  13. 13
    For the two cells having same EMF E and internal resistance rr, the current passing through the external resistor 6Ω6\Omega is same when both the cells are connected either in parallel or in series. The value of internal resistance rr is ____\_\_\_\_ Ω\Omega

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  14. 14
    Two wires AA and BB made of different materials of length 6.0 cm and 5.4 cm , respectively and area of cross sections 3.0×10−5 m23.0 \times10^{- 5}{\text{ }m}^{2} and 4.5×10−5 m24.5 \times10^{- 5}{\text{ }m}^{2}, respectively are stretched by the same magnitude under a given load. The ratio of the Young's modulus of AA to that of BB is x:3x:3. The value of xx is ____\_\_\_\_.

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  15. 15
    A block of mass 5 kg is moving on an inclined plane which makes an angle of 30∘30^{\circ} with the horizontal. Friction coefficient between the block and inclined plane surface is 32\frac{\sqrt{3}}{2}. The force to be applied on the block so that the block will move down without acceleration is ____\_\_\_\_ N.

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  16. 16
    Given below are two statements : Statement-I : A plane wave after passing through prism remains as plane wave but passing through small pin hole may become spherical wave. Statement-II : The curvature of a spherical wave emerging from a slit will increase for increasing slit width. In the light of the above statements, choose the correct answer from the options given below.

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  17. 17
    When both jaws of Vernier calipers touch each other, zero mark of the vernier scale is right to zero mark of main scale, 4th 4^{\text{th~}} mark on Vernier scale coincides with certain mark on the main scale. While measuring the length of a cylinder, observer observes 15 divisions on main scale and 5th5^{\text{th}} division of Vernier scale coincides with a main scale division. Measured length of cylinder is ____\_\_\_\_ mm. (Least count of Vernier caliper =0.1 mm= 0.1\text{ }mm )

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  18. 18
    The magnetic field at the center of a current carrying circular loop of radius RR is 16μ T16\mu\text{ }T. The magnetic field at a distance x=3Rx = \sqrt{3}R on its axis from the centre is ____\_\_\_\_ μT\mu T.

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  19. 19
    Two-point charges of 1nC and 2nC are placed at the two corners of equilateral triangle of side 3 cm . The work done in bringing a charge of 3nC from infinity to the third corner of the triangle is ____\_\_\_\_ μJ\mu J. 14πϵ0=9×109 N.m2/C2\frac{1}{4\pi\epsilon_{0}}= 9 \times10^{9}\text{ }N.m^{2}/C^{2}

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  20. 20
    A particle of mass mm falls from rest through a resistive medium having resistive force, F=−kvF = - kv, where v is the velocity of the particle and k is a constant. Which of the following graphs represents velocity ( v ) versus time( t ) ?

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  21. 21
    The displacement of a particle, executing simple harmonic motion with time period TT, is expressed as x(t)=Asinωtx(t) = Asin\omega t, where AA is the amplitude. The maximum value of potential energy of this oscillator is found at t=T/2βt = T/2\beta. The value of β\beta is ____\_\_\_\_ .
  22. 22
    The ratio of de Broglie wavelength of a deuteron with kinetic energy EE to that of an alpha particle with kinetic energy 2E2E, is n:1n:1. The value of nn is ____\_\_\_\_ . (Assume mass of proton == mass of neutron) :
  23. 23
    A solid sphere of radius 10 cm is rotating about an axis which is at a distance 15 cm from its center. The radius of gyration about this axis is n cm\sqrt{n}\text{ }cm. The value of nn is
  24. 24
    A convex lens of refractive index 1.5 and focal length f=18 cmf = 18\text{ }cm is immersed in water. The difference in focal lengths of the given lens when it is in water and in air is α×f\alpha \times f. The value of α\alpha is ____\_\_\_\_ . (refractive index of water =4/3= 4/3 )
  25. 25
    The equivalent resistance between the points AA and BB in the following circuit is x5Ω\frac{x}{5}\Omega. The value of xx is ____\_\_\_\_ .

Chemistry

25 questions

  1. 26
    20.0dm320.0dm^{3} of an ideal gas ' X ' at 600 K and 0.5 MPa undergoes isothermal reversible expansion until pressure of the gas is 0.2 MPa . Which of the following option is correct? (Given: log⁡2=0.3010\log2 = 0.3010 and log⁡5=0.6989\log5 = 0.6989 )

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  2. 27
    CORRECT order of stability for the following is CH2=CH−,CH3−CH2−,CH≡C−CH_{2}=CH^{-},CH_{3}-CH_{2}^{-},CH \equiv C^{-}

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  3. 28
    At T(K),2T(K),2 moles of liquid A and 3 moles of liquid BB are mixed. The vapour pressure of ideal solution formed is 320 mm Hg . At this stage, one mole of A and one mole of BB are added to the solution. The vapor pressure is now measured as 328.6 mm Hg . The vapour pressure (in mm Hg ) of A and B are respectively:

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  4. 29
    Consider the following reaction sequence Compound (x) → (i) CO2,NaOH,120∘C, high pressure (ii) H3O+\underset{(ii)\ H_{3}O^{+}}{\overset{\begin{matrix} \text{~(i)~}CO_{2},NaOH,120^{\circ}C, \\ \text{~high pressure~} \end{matrix}}{\rightarrow}} Compound (y) (Major Product) [76.6%C,6.38%H,vapour density 47]\lbrack 76.6\% C,6.38\% H\text{,vapour density~}47\rbrack Compound (y) develops characteristic colour with neutral FeCl3FeCl_{3} solution. Identify the INCORRECT statement from the following for the above sequence.

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  5. 30
    Method used for separation of mixture of products ( B and C ) obtained in the following reaction is:

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  6. 31
    Consider a weak base ' B ' of pKb=5.699pK_{b}= 5.699. ' x ' mL of 0.02 M HCl and ' y ' mL of 0.02 M weak base ' B ' are mixed to make 100 mL of a buffer of pH 9 at 25∘C25^{\circ}C. The values of ' xx ' and ' yy ' respectively are: [Given: log⁡2=0.3010,log⁡3=0.4771,log⁡5=0.699\log2 = 0.3010,\log3 = 0.4771,\log5 = 0.699 ]

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  7. 32
    An organic compound undergoes first order decomposition. The time taken for decomposition to (18)th \left( \frac{1}{8} \right)^{\text{th~}} and (110)th \left( \frac{1}{10} \right)^{\text{th~}} of its initial concentration are t1/8t_{1/8} and t1/10t_{1/10} respectively. What is the value of t1/8t1/10×10\frac{t_{1/8}}{t_{1/10}}\times 10 ?
    (log⁡2=0.3)(\log2 = 0.3)

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  8. 33
    Given below are two statements for the following reaction sequence. Statement I: Compound ' ZZ ' will give yellow precipitate with NaOI . Statement II: Compound 'Q' has two different types of ' H ' atoms (aromatic : aliphatic) in the ratio 1:3. In the light of the above statements, choose the correct answer from the option given below:

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  9. 34
    Given below are the four isomeric compounds (P, Q, R, S) (P) (Q) (R) (S) Identify correct statements from below. (A) Q,RQ,R and SS will give precipitate with 2,4 - DNP. (B) P and Q will give positive Bayer's test. (C) Q and R will give sooty flame. (D) R and S will give yellow precipitate with I2/NaOHI_{2}/NaOH. (E) Q alone will deposit silver with Tollen's reagent Choose the correct option.

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  10. 35
    Given below are two statements: Statement I: The number of species among BF4−,SiF4,XeF4{BF}_{4}^{-},{SiF}_{4},{XeF}_{4} and SF4{SF}_{4}, that have unequal E-F bond lengths is two. Here, E is the central atom. Statement II: Among O2−,O22−,F2O_{2}^{-},O_{2}^{2 -},F_{2} and O2+,O2−O_{2}^{+},O_{2}^{-}has the highest bond order. In the light of the above statements, choose the correct answer from the options given below

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  11. 36
    Regarding the hydrides of group 15 elements EH3EH_{3} (E = N, P, As, Sb), select the correct statement from the following: (A) The stability of hydrides decreases down the group. (B) The basicity of hydrides decreases down the group. (C) The reducing character increases down the group. (D) The boiling point increases down the group. Choose the correct answer from the options given below:

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  12. 37
    Given below are two statements: Statement I: Griess-Ilosvay test is used for the detection of nitrite ion, which involves the use of sulphanilic acid and α\alpha-naphthylamine reagent. Statement II: In the above test, sulphanilic acid is diazotized by the acidified nitrite ion, which on further coupling with α\alpha-naphthylamine forms an azo-dye. In the light of the above statements, choose the correct answer from the options given below

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  13. 38
    Consider the following reactions giving major product. Identify the correct reaction.

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  14. 39
    In the given pentapeptide, find out an essential amino acid (Y) and the sequence present in the pentapeptide: Choose the correct answer from the options given below:

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  15. 40
    Ph−CH=CH2Ph - CH =CH_{2} →(PhCOO)2HBr Product\underset{HBr\ }{\overset{(PhCOO)_{2}}{\rightarrow}}Product Consider the above reaction (A) The reaction proceeds through a more stable radical intermediate. (B) The role of peroxide is to generate H˙\dot{H} (Hydrogen radical). (C) During this reaction, benzene is formed as a by-product. (D) 1-Bromo-2-phenylethane is formed as the minor product. (E) The same reaction in absence of peroxide proceeds via carbocation intermediate. Identify the correct statements. Choose the correct answer from the options given below:

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  16. 41
    The wave numbers of three spectral lines of H atom are considered. Identify the set of spectral lines belonging to Balmer series. ( R=R = Rydberg constant)

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  17. 42
    Figure 1 (top): electron probability density for the 2s orbital. Figure 2 (bottom): wave function for the 2s orbital. Which point in Figure 2 accurately represents the same nodal surface as Figure 1?

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  18. 43
    Given below are two statements : Statement I: The number of pairs, following, in which both the ions are coloured in aqueous solution is 3. [  Sc3+,Ti3+],[Mn2+,Cr2+],[Cu2+,Zn2+]\left. \ {Sc}^{3 +},{Ti}^{3 +} \right\rbrack,\left\lbrack {Mn}^{2 +},{Cr}^{2 +} \right\rbrack,\left\lbrack {Cu}^{2 +},{Zn}^{2 +} \right\rbrack and [Ni2+,Ti4+]\lbrack{Ni}^{2 +},{Ti}^{4 +}\rbrack Statement II : Th4+{Th}^{4 +} is the strongest reducing agent among Th4+,Ce4+,Gd3+{Th}^{4 +},{Ce}^{4 +},{Gd}^{3 +} and Eu2+{Eu}^{2 +}. In the light of the above statements, choose the correct answer from the options given below:

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  19. 44
    In period 4 of the periodic table, the elements with highest and lowest atomic radii are respectively.

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  20. 45
    The correct statement among the following is :

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  21. 46
    500 mL of 1.2 M KI solution is mixed with 500 mL of 0.2MKMnO40.2M{KMnO}_{4} solution in basic medium. The liberated iodine was titrated with standard 0.1MNa2 S2O30.1M{Na}_{2}{\text{ }S}_{2}O_{3} solution in the presence of starch indicator till the blue color disappeared. The volume (in L ) of Na2 S2O3{Na}_{2}{\text{ }S}_{2}O_{3} consumed is ____\_\_\_\_ .(Nearest integer)
  22. 47
    Consider the following redox reaction taking place in acidic medium
    BH4−(aq)+ClO3−(aq)→H2BO3−(aq)+Cl−(aq){BH}_{4}^{-}(aq) + {ClO}_{3}^{-}(aq) \rightarrow H_{2}{BO}_{3}^{-}(aq) + {Cl}^{-}(aq)
    If the Nernst equation for the above balanced reaction is Ecell =Ecell 0−RTnFln⁡QE_{\text{cell~}} = E_{\text{cell~}}^{0} - \frac{RT}{nF}\ln Q, Then the value of nn is ____\_\_\_\_ (Nearest integer)
  23. 48
    X is the number of geometrical isomers exhibited by [Pt(NH3)(H2O)BrCl]\left\lbrack Pt\left( {NH}_{3} \right)\left( H_{2}O \right)BrCl \right\rbrack. Y is the number of optically inactive isomer(s) exhibited by [CrCl2(ox)2]3−\left\lbrack {CrCl}_{2}(ox)_{2} \right\rbrack^{3 -} Z is the number of geometrical isomers exhibited by [Co(NH3)3(NO2)3]\left\lbrack Co\left( {NH}_{3} \right)_{3}\left( {NO}_{2} \right)_{3} \right\rbrack The value of X+Y+ZX + Y + Z is ____\_\_\_\_ .
  24. 49
    0.53 g of an organic compound (x)(x) when heated with excess of nitric acid (concentrated) and then with silver nitrate gave 0.75 g of silver bromide precipitate. 1.0 g of (x)(x) gave 1.32 g of CO2{CO}_{2} gas on combustion. The percentage of hydrogen in the compound ( x ) is ____\_\_\_\_ %. [Nearest Integer] [Given : Molar mass in gmol−1H:1,C:12,Br:g{mol}^{- 1}H:1,C:12,Br: 80, Ag:108,O:16Ag:108,O:16; Compound (x):CxHyBrz(x):C_{x}H_{y}{Br}_{z} ]
  25. 50
    Consider the dissociation equilibrium of the following weak acid HA⇌H+(aq)+A−(aq)HA \rightleftharpoons H^{+}(aq) + A^{-}(aq) If the pKa of the acid is 4 , then the pH of 10 mM HA solution is ____\_\_\_\_ . (Nearest integer) [Given : The degree of dissociation can be neglected with respect to unity]

Mathematics

25 questions

  1. 51
    If g(x)=3x2+2x−3,f(0)=−3g(x) = 3x^{2}+ 2x- 3,f(0) = - 3 and 4g(f(x))=3x2−32x+724g(f(x)) = 3x^{2}- 32x + 72, then f( g(2))f(\text{ }g(2)) is equal to:

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  2. 52
    The value of ∑k=1∞(−1)k+1(k(k+1)k!)\sum_{k= 1}^{\infty} ( - 1)^{k+ 1}\left( \frac{k(k+ 1)}{k!} \right) is :

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  3. 53
    Let y=xy=x be the equation of a chord of the circle C1C_{1} (in the closed half-plane x≥0x\geq 0 ) of diameter 10 passing through the origin. Let C2C_{2} be another circle described on the given chord as its diameter. If the equation of the chord of the circle C2C_{2}, which passes through the point (2,3)(2,3) and is farthest from the center of C2C_{2}, is x+ay+b=0x + ay + b = 0, then a−ba - b is equal to :

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  4. 54
    If tan⁡(A−B)tan⁡A+sin⁡2Csin⁡2A=1, A,B,C∈(0,π2)\frac{\tan(A - B)}{\tan A}+\frac{\sin^{2}C}{\sin^{2}A}= 1,\ A,B,C \in\left( 0,\frac{\pi}{2} \right), then

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  5. 55
    Let zz be a complex number such that ∣z−6∣=5|z- 6| = 5 and ∣z+2−6i∣=5|z+ 2 - 6i| = 5. Then the value of z3+3z2−15z+141z^{3}+ 3z^{2}- 15z+ 141 is equal to

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  6. 56
    Let ABC be an equilateral triangle with orthocenter at the origin and the side BC on the line x+22y=4x+ 2\sqrt{2}y= 4. If the co-ordinates of the vertex A are (α,β)(\alpha,\beta), then the greatest integer less than or equal to ∣α+2β∣|\alpha+\sqrt{2}\beta| is

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  7. 57
    Let S={1,2,3,4,5,6,7,8,9}S\mathbf{= \{ 1,2,3,4,5,6,7,8,9\}}. Let xx be the number of 9 -digit numbers formed using the digits of the set S such that only one digit is repeated, and it is repeated exactly twice. Let yy be the number of 9-digit numbers formed using the digits of the set S such that only two digits are repeated and each of these is repeated exactly twice. Then,

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  8. 58
    Let S={x3+ax2+bx+c:a,b,c∈NS=\left\{ x^{3}+ax^{2}+bx+c:a,b,c\in N \right. and a,b,c≤20}a,b,c\leq 20\} be a set of polynomials. Then the number of polynomials in S , which are divisible by x2+2x^{2}+ 2, is

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  9. 59
    A bag contains 10 balls out of which k are red and (10−k)(10 - k) are black, where 0≤k≤100 \leq k \leq 10. If three balls are drawn at random without replacement and all of them are found to be black, then the probability that the bag contains 1 red and 9 black balls is :

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  10. 60
    If α,β\alpha,\beta, where α<β\alpha<\beta, are the roots of the equation λx2−(λ+3)x+3=0\lambda x^{2}- (\lambda+ 3)x + 3 = 0 such that 1α−1β=13\frac{1}{\alpha}-\frac{1}{\beta}=\frac{1}{3}, then the sum of all possible values of λ\lambda is :

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  11. 61
    If ∫ (1−5cos⁡2xsin⁡5xcos⁡2x)dx=f(x)+C\int_{}^{}\ \left( \frac{1 - 5\cos^{2}x}{\sin^{5}x\cos^{2}x} \right)dx=f(x) +C where CC is the constant of integration, then f(π6)−f(π4)f\left( \frac{\pi}{6} \right)- f\left( \frac{\pi}{4} \right) is equal to

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  12. 62
    Let ff be a polynomial function such that f(x2+1)=x4+5x2+2f\left( x^{2}+ 1 \right)=x^{4}+ 5x^{2}+ 2, for all x∈Rx\in\mathbb{R}. Then ∫03f(x)dx\int_{0}^{3} f(x)dx is equal to

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  13. 63
    The area of the region R={(x,y):xy≤8,1≤y≤x2,x≥0}R=\left\{ (x,y):xy\leq 8,1 \leq y\leq x^{2},x\geq 0 \right\} is

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  14. 64
    The value of lim⁡x→0log⁡e(sec(ex)⋅sec(e2x)⋅…⋅sec(e10x))e2−e2cos⁡x\underset{x\rightarrow 0}{\lim} \frac{\log_{e}\left( sec(ex) \cdot sec\left( e^{2}x \right)\cdot \ldots \cdot sec\left( e^{10}x \right) \right)}{e^{2}-e^{2\cos x}} is equal to

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  15. 65
    The mean and variance of 10 observations are 9 and 34.2 , respectively. If 8 of these observations are 2,3,5,10,11,13,15,212,3,5,10,11,13,15,21, then the mean deviation about the median of all the 10 observations is

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  16. 66
    Let A,BA,B and C be three 2×22 \times 2 matrices with real entries such that B=(I+A)−1B = (I + A)^{- 1} and A+C=IA + C = I If BC=[1−5−12]BC =\begin{bmatrix} 1 & - 5 \\ - 1 & 2 \end{bmatrix} and CB[x1x2]=[12−6],CB\begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix}=\begin{bmatrix} 12 \\ - 6 \end{bmatrix}, then x1+x2x_{1}+x_{2} is

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  17. 67
    The common difference of the A.P.: a1,a2,….,ama_{1},a_{2},\ldots.,a_{m} is 13 more than the common difference of the A.P.: b1,b2,…,bnb_{1},b_{2},\ldots,b_{n}. If b31=−277,b43=−385b_{31}= - 277,b_{43}= - 385 and a18=a_{18}= 327 , then a1a_{1} is equal to

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  18. 68
    If the distances of the point (1,2,a)(1,2,a) from the line x−11=y2=z−11\frac{x- 1}{1}=\frac{y}{2}=\frac{z- 1}{1} along the lines L1:x−13=y−24=z−abL_{1}:\frac{x - 1}{3}=\frac{y - 2}{4}=\frac{z - a}{b} and L2:x−11=y−24=z−acL_{2}:\frac{x - 1}{1}=\frac{y - 2}{4}=\frac{z - a}{c} are equal, then a+b+ca + b + c is equal to

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  19. 69
    For three-unit vectors a→,b→,c→\overrightarrow{a},\overrightarrow{b},\overrightarrow{c} satisfying ∣a→−b→∣2+∣b→−c→∣2+∣c→−a→∣2=9 and |\overrightarrow{a}-\overrightarrow{b}|^{2}+ |\overrightarrow{b}-\overrightarrow{c}|^{2}+ |\overrightarrow{c}-\overrightarrow{a}|^{2}= 9\text{~and~} ∣2a→+kb→+kc→∣=3|2\overrightarrow{a}+k\overrightarrow{b}+k\overrightarrow{c}| = 3 the positive value of kk is :

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  20. 70
    Let y=y(x)y=y(x) be the solution of the differential equation xdydx−sin⁡2y=x3(2−x3)cos⁡2y,x≠0.x\frac{dy}{dx}- \sin{2y}=x^{3}\left( 2 -x^{3} \right)\cos^{2}y,x\neq 0. If y(2)=0y(2) =0, then tan⁡(y(1))\tan(y(1)) is equal to

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  21. 71
    In a G.P., if the product of the first three terms is 27 and the set of all possible values for the sum of its first three terms is R−(a,b)\mathbb{R} - (a,b), then a2+b2a^{2} + b^{2} is equal to ____\_\_\_\_ .
  22. 72
    For some θ∈(0,π2)\theta \in \left( 0,\frac{\pi}{2} \right), let the eccentricity and the length of the latus rectum of the hyperbola x2−y2sec⁡2θ=8x^{2} - y^{2}\sec^{2}\theta = 8 be e1e_{1} and l1\mathcal{l}_{1}, respectively, and let the eccentricity and the length of the latus rectum of the ellipse x2sec⁡2θ+y2=6x^{2}\sec^{2}\theta + y^{2} = 6 be e2e_{2} and l2\mathcal{l}_{2}, respectively. If e12=e22(sec⁡2θ+1)e_{1}^{2} = e_{2}^{2}\left( \sec^{2}\theta + 1 \right), then (l1l2e1e2)tan⁡2θ\left( \frac{\mathcal{l}_{1}\mathcal{l}_{2}}{e_{1}e_{2}} \right)\tan^{2}\theta is equal to ____\_\_\_\_ .
  23. 73
    If k=tan⁡(π4+12cos⁡−1(23))+tan⁡(12sin⁡−1(23))k = \tan\left( \frac{\pi}{4} + \frac{1}{2}\cos^{- 1}\left( \frac{2}{3} \right) \right) + \tan\left( \frac{1}{2}\sin^{- 1}\left( \frac{2}{3} \right) \right) then the number of solutions of the equation sin⁡−1(kx−1)=sin⁡−1x−cos⁡−1x\sin^{- 1}(kx - 1) = \sin^{- 1}x - \cos^{- 1}x is......... .
  24. 74
    The value of ∑r=120(∣π(∫0rx∣sin⁡πx∣dx)∣)\sum_{r = 1}^{20} \left( \left| \sqrt{\pi\left( \int_{0}^{r} x|\sin\pi x|dx \right)} \right| \right) is ....... .
  25. 75
    Let PQRPQR be a triangle such that PQ→=−2i^−j^+2k^\overrightarrow{PQ} = - 2\widehat{i} - \widehat{j} + 2\widehat{k} and PR→=ai^+bj^−4k^,a,b∈Z\overrightarrow{PR} = a\widehat{i} + b\widehat{j} - 4\widehat{k},a,b \in \mathbb{Z}. Let SS be the point on QR, which is equidistant from the lines PQ and PR. If ∣PR→∣=9|\overrightarrow{PR}| = 9 and PS→=i^−7j^+2k^\overrightarrow{PS} = \widehat{i} - 7\widehat{j} + 2\widehat{k}, then the value of 3a−4b3a - 4b is ____\_\_\_\_ .

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