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JEE Mains 2026 (22 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
    A solid sphere of mass 5 kg and radius 10 cm is kept in contact with another solid sphere of mass 10 kg and radius 20 cm . The moment of inertia of this pair of spheres about the tangent passing through the point of contact is ____\_\_\_\_ kg.m2kg.m^{2}.

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  2. 2
    7.9MeVα7.9MeV\alpha-particle scatters from a target material of atomic number 79. From the given data the estimated diameter of nuclei of the target material is (approximately) ____\_\_\_\_ m. [14πϵo=9×109Nm2/C2\left\lbrack \frac{1}{4\pi\epsilon_{o}}= 9 \times10^{9}Nm^{2}/C^{2} \right. and electron charge  =1.6×10−19C]\left. \ = 1.6 \times10^{- 19}C \right\rbrack

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  3. 3
    Six point charges are kept 60∘60^{\circ} apart from each other on the circumference of a circle of radius R as shown in figure. The net electric field at the centre of the circle is ____\_\_\_\_ . (ϵo\epsilon_{o} is permittivity of free space)

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  4. 4
    XPQY is a vertical smooth long loop having a total resistance R where PX is parallel to QY and separation between them is ll. A constant magnetic field B perpendicular to the plane of the loop exists in the entire space. A rod CD of length L ( L>lL > l ) and mass mm is made to slide down from rest under the gravity as shown in figure. The terminal speed acquired by the rod is ____\_\_\_\_ m/sm/s. (g=g = acceleration due to gravity)

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  5. 5
    The escape velocity from a spherical planet A is 10 km/skm/s. The escape velocity from another planet B whose density and radius are 10%10\% of those of planet A, is ____\_\_\_\_ m/sm/s.

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  6. 6
    A meter bridge with two resistances R1R_{1} and R2R_{2} as shown in figure was balanced (null point) at 40 cm from the point P. The null point changed to 50 cm from the point P, when 16Ω16\Omega resistance is connected in parallel to R2R_{2}. The values of resistances R1R_{1} and R2R_{2} are ____\_\_\_\_ .

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  7. 7
    A projectile is thrown upward at an angle 60∘60^{\circ} with the horizontal. The speed of the projectile is 20 m/s20\text{ }m/s when its direction of motion is 45∘45^{\circ} with the horizontal. The initial speed of the projectile is ____\_\_\_\_ m/sm/s.

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  8. 8
    Given below are two statements: Statement I: Pressure of fluid is exerted only on a solid surface in contact as the fluid-pressure does not exist everywhere in a still fluid. Statement II: Excess potential energy of the molecules on the surface of a liquid, when compared to interior, results in surface tension. In the light of the above statements, choose the correct answer from the options given below.

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  9. 9
    Find the correct combination of A, B, C and D inputs which can cause the LED to glow.

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  10. 10
    Net gravitational force at the centre of a square is found to be F1F_{1} when four particles having mass M,2M,3MM,2M,3M and 4M4M are placed at the four corners of the square as shown in figure and it is F2F_{2} when the positions of 3M3M and 4M4M are interchanged. The ratio F1F2\frac{F_{1}}{F_{2}} is α5\frac{\alpha}{\sqrt{5}}. The value of α\alpha is ____\_\_\_\_ .

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  11. 11
    The minimum frequency of photon required to break a particle of mass 15.348 amu into 4α4\alpha particles is ____\_\_\_\_ kHz . [mass of He nucleus = 4.002 amu ,1amu=1.66×10−27 kg, h=6.6×10−34 J.s1amu = 1.66 \times10^{- 27}\text{ }kg,\text{ }h = 6.6 \times10^{- 34}\text{ }J.s and c=3×108 m/sc = 3 \times10^{8}\text{ }m/s ]

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  12. 12
    A cylindrical tube ABAB of length ll, closed at both ends contains an ideal gas of 1 mol having molecular weight MM. The tube is rotated in a horizontal plane with constant angular velocity ω\omega about an axis perpendicular to ABAB and passing through the edge at end A , as shown in the figure. If PAP_{A} and PBP_{B} are the pressures at AA and BB respectively, then (Consider the temperature is same at all points in the tube)

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  13. 13
    Rods x and y of equal dimensions but of different materials are joined as shown in figure. Temperatures of end points AA and FF are maintained at 100∘C100^{\circ}C and 40∘C40^{\circ}C respectively. Given the thermal conductivity of rod x is three times of that of rod y , the temperature at junction points BB and EE are (close to) :

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  14. 14
    A thin convex lens of focal length 5 cm and a thin concave lens of focal length 4 cm are combined together (without any gap) and this combination has magnification m1m_{1} when an object is placed 10 cm before the convex lens. Keeping the positions of convex lens and object undisturbed a gap of 1 cm is introduced between the lenses by moving the concave lens away, which lead to a change in magnification of total lens system to m2m_{2}. The value of ∣m1m2∣\left| \frac{m_{1}}{m_{2}} \right| is ____\_\_\_\_ .

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  15. 15
    Consider an equilateral prism (refractive index 2\sqrt{2} ). A ray of light is incident on its one surface at a certain angle ii. If the emergent ray is found to graze along the other surface then the angle of refraction at the incident surface is close to ____\_\_\_\_ .

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  16. 16
    The volume of an ideal gas increases 8 times and temperature becomes (1/4)th (1/4)^{\text{th~}} of initial temperature during a reversible change. If there is no exchange of heat in this process ( ΔQ=0\Delta Q = 0 ) then identify the gas from the following options (Assuming the gases given in the options are ideal gases) :

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  17. 17
    Electric field in a region is given by E→=Axi^+Byj^\overrightarrow{E}= Ax\widehat{i}+ By\widehat{j}, where A=10 V/m2A = 10\text{ }V/m^{2} and B=5 V/m2B = 5\text{ }V/m^{2}. If the electric potential at a point (10,20)(10,20) is 500 V , then the electric potential at origin is ____\_\_\_\_ V.

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  18. 18
    A simple pendulum has a bob with mass mm and charge q . The pendulum string has negligible mass. When a uniform and horizontal electric field E→\overrightarrow{E} is applied, the tension in the string changes. The final tension in the string, when pendulum attains an equilibrium position is ____\_\_\_\_ . (g : acceleration due to gravity)

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  19. 19
    Match the LIST-I with LIST-II
    List-IList-II
    (A) Spring constant(I) ML2T−2K−1ML^{2}T^{-2}K^{-1}
    (B) Thermal conductivity(II) ML0T−2ML^{0}T^{-2}
    (C) Boltzmann constant(III) ML2T−3A−2ML^{2}T^{-3}A^{-2}
    (D) Inductive reactance(IV) MLT−3K−1MLT^{-3}K^{-1}
    Choose the correct answer from the options given below:

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  20. 20
    Three identical coils C1,C2C_{1},C_{2} and C3C_{3} are closely placed such that they share a common axis. C2C_{2} is exactly midway. C1C_{1} carries current II in anti-clockwise direction while C3C_{3} carries current II in clockwise direction. An induced current flows through C2C_{2} will be in clockwise direction when

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  21. 21
    Two loudspeakers (L1L_{1} and L2L_{2}) are placed with a separation of 10 m , as shown in figure. Both speakers are fed with an audio input signal of same frequency with constant volume. A voice recorder, initially at point AA, at equidistance to both loud speakers, is moved by 25 m along the line ABAB while monitoring the audio signal. The measured signal was found to undergo 10 cycles of minima and maxima during the movement. The frequency of the input signal is ____\_\_\_\_ Hz (Speed of sound in air is 324 m/s324\text{ }m/s and 5=2.23\sqrt{5} = 2.23 )
  22. 22
    The electric field of a plane electromagnetic wave, travelling in an unknown non-magnetic medium is given by, Ey=20sin⁡(3×106x−4.5×1014t)V/mE_{y} = 20\sin\left( 3 \times 10^{6}x - 4.5 \times 10^{14}t \right)V/m(where x,tx,t and other values have S.I. units). The dielectric constant of the medium is ____\_\_\_\_ (speed of light in free space is 3×108 m/s3 \times 10^{8}\text{ }m/s)
  23. 23
    A parallel beam of light travelling in air (refractive index 1.0) is incident on a convex spherical glass surface of radius of curvature 50 cm. Refractive index of glass is 1.5. The rays converge to a point at a distance x cm from the centre of the curvature of the spherical surface. The value of x is ____\_\_\_\_ cm.
  24. 24
    A circular disc has radius R1R_{1} and thickness T1T_{1}. Another circular disc made of the same material has radius R2R_{2} and thickness T2T_{2}. If the moment of inertia of both discs are same and R1R2=2\frac{R_{1}}{R_{2}} = 2 then T1T2=1α\frac{T_{1}}{T_{2}} = \frac{1}{\alpha}. The value of α\alpha is ____\_\_\_\_ .
  25. 25
    Inductance of a coil with 10410^{4} turns is 10 mH and it is connected to a dc source of 10 V with internal resistance of 10Ω10\Omega. The energy density in the inductor when the current reaches (1e)\left( \frac{1}{e} \right) of its maximum value is απ×1e2 J/m3\alpha\pi \times \frac{1}{e^{2}}\text{ }J/m^{3}. The value of α\alpha is ____\_\_\_\_ . (μ0=4π×10−7Tm/A)\left( \mu_{0} = 4\pi \times 10^{- 7}Tm/A \right).

Chemistry

25 questions

  1. 26
    Consider the transition metal ions Mn3+,Cr3+,Fe3+Mn^{3 +},Cr^{3 +},Fe^{3 +} and Co3+Co^{3 +} and all form low spin octahedral complexes. The correct decreasing order of unpaired electrons in their respective d-orbitals of the complexes is

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  2. 27
    The formal charges on the atoms marked as (1) to (4) in the Lewis representation of HNO3HNO_{3} molecule respectively are

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  3. 28
    Given below are two statements : Statement I: Phenol on treatment with CHCl3/{CHCl}_{3}/ aq. KOH under refluxing condition, followed by acidification produces pp-hydroxy benzladehyde as the major product and oo-hydroxybenzaldehyde as the minor product. Statement II : The mixture of pp-hydroxybenzaldehyde and oo-hydroxybenzaldehyde can be easily separated through steam distillation. In the light of the above statements, choose the correct answer from the options given below :

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  4. 29
    The energy required by electrons, present in the first Bohr orbit of hydrogen atom to be excited to second Bohr orbit is ____\_\_\_\_ Jmol−1Jmol^{- 1}. Given : RH=2.18×10−11R_{H}= 2.18 \times10^{- 11} ergs.

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  5. 30
    A 'p'-block element (E) and hydrogen form a binary cation (EHx)+\left( EH_{x} \right)^{+}, while EH3EH_{3} on treatment with K2HgI4K_{2}HgI_{4} in alkaline medium gives a precipitate of basic mercury(II) amido-iodide. Given below are first ionisation enthalpy values ( kJmol−1kJmol^{- 1} ) for first element each from group 13, 14, 15 and 16. Identify the correct first ionisation enthalpy value for element E .

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  6. 31
    In the reaction, 2Al(s)+6HCl(aq)→2Al3+(aq)+6Cl−(aq)+3H2(g)2Al(s) + 6HCl(aq) \rightarrow 2Al^{3 +}(aq) + 6Cl^{-}(aq) + 3H_{2}(g), which of the following statements is correct?

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  7. 32
    Consider a solution of CO2( g)CO_{2}(\text{ }g) dissolved in water in a closed container. Which one of the following plots correctly represents variation of log (partial pressure of CO2 in vapour phase above water) [y-axis] with log (mole fraction of CO2CO_{2} in water) [ x -axis] at 25∘C25^{\circ}C ?

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  8. 33
    A first row transition metal (M) does not liberate H2H_{2} gas from dilute HCl. 1 mol of aqueous solution of MSO4MSO_{4} is treated with excess of aqueous KCN and then H2 S( g)H_{2}\text{ }S(\text{ }g) is passed through the solution. The amount of MS (metal sulphide) formed from the above reaction is ____\_\_\_\_ mol.

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  9. 34
    The correct order of reactivity of CH3BrCH_{3}Br in methanol with the following nucleophiles is F−,I−,C2H5O−F^{-},I^{-},C_{2}H_{5}O^{-}and C6H5O−C_{6}H_{5}O^{-}

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  10. 35
    Match the LIST-I with LIST-II
    List-I Thermodynamic ProcessList-II Magnitude in kJ
    (A)Work done in reversible, isothermal expansion of 2 mol of ideal gas from 2 dm3{dm}^{3} to 20dm320{dm}^{3} at 300 K .(I)4
    (B)Work done in irreversible isothermal expansion of 1 mol ideal gas from 1 m31{\text{ }m}^{3} to 3 m33{\text{ }m}^{3} at 300 K against A constant pressure of 3 kPa .(II)11.5
    (C)Change in internal energy for adiabatic expansion of a 1 mol ideal gas with change of temperature =320 K= 320\text{ }K and C‾V=32R{\overline{C}}_{V} = \frac{3}{2}R.(III)6
    (D)Change in enthalpy at constant pressure of 1 mole ideal gas with change of temperature == 337 K and C‾P=52R{\overline{C}}_{P} = \frac{5}{2}R.(IV)7
    Choose the correct answer from the option given below :

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  11. 36
    Given below are two statements : Statement I : Benzene is nitrated to give nitrobenzene, which on further treatment with CH3COCl/AlCl3{CH}_{3}COCl/{AlCl}_{3} will give m-nitroacetophenone (shown below). Statement II : NO2{NO}_{2} group is a mm-directing, and deactivating group. In the light of the above statements, choose the most appropriate answer from the options given below.

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  12. 37
    A → products (First order reaction). Three sets of experiment were performed for a reaction under similar experimental conditions. Run 1⇒100 mL1 \Rightarrow 100\text{ }mL of 10 M solution of reactant A Run 2⇒200 mL2 \Rightarrow 200\text{ }mL of 10 M solution of reactant A Run 3⇒100 mL3 \Rightarrow 100\text{ }mL of 10 M solution of reactant A+A + 100 mL of H2OH_{2}O added. The correct variation of rate of reaction is

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  13. 38
    Match the LIST-I with LIST-II
    List-I ReagentsList-II Name of Reaction involving carbonyl compound
    (A)NH2−NH2, KOH{NH}_{2} - {NH}_{2},\ KOH(I)Tollen's Test
    (B)Ag(NH3)2OHAg\left( {NH}_{3} \right)_{2}OH(II)Clemmensen Reduction
    (C)Aq. CuSO4{CuSO}_{4}, sodium potassium tartrate, KOH(III)Wolff-Kishner Reduction
    (D)Zn−Hg, HClZn - Hg,\ HCl(IV)Fehling's Test
    Choose the correct answer from the options given below

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  14. 39
    Given below are two statements : Statement I: The halogen that makes longest bond with hydrogen in HX, has the smallest covalent radius in its group. Statement II : A group 15 element's hydride EH3{EH}_{3} has the lowest boiling point among corresponding hydrides of other group 15 elements. The maximum covalency of that element E is 4 . 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: Sucrose is dextrorotatory. However sucrose upon hydrolysis gives a solution having mixture of products. This solution shows laevorotation. Statement II : Hydrolysis of sucrose gives glucose and fructose. Since the laevorotation of glucose is more than the dextrorotation of fructose the resulting solution becomes laevorotatory. In the light of the above statements, choose the correct answer from the options given below.

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  16. 41
    'A' is a neutral organic compound (M. F : C8H9ONC_{8}H_{9}ON ). On treatment with aqueous Br2/HO(−)Br_{2}/HO^{( - )}, 'A' forms a compound ' B ' which is soluble in dilute acid. ' B ' on treatment with aqueous NaNO2/HCl(0−5∘C)NaNO_{2}/HCl\left( 0 -5^{\circ}C \right) produces a compound ' C ' which on treatment with CuCN/NaCNCuCN/NaCN produces 'D' Hydrolysis of 'D' produces ' E ' which is also obtainable from the hydrolysis of 'A'. 'E' on treatment with acidified KMnO4KMnO_{4} produces 'F'. 'F' contains two different types of hydrogen atoms. The structure of ' A ' is

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  17. 42
    The correct order of the rate of reaction of the following reactants with nucleophile by SN1S_{N}1 mechanism is : (Given : Structure I and II are rigid)

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  18. 43
    Two p-block elements X and Y form fluorides of the type EF3EF_{3}. The fluoride compound XF3XF_{3} is a Lewis acid and YF3YF_{3} is a Lewis base. The hybridization of the central atoms of XF3XF_{3} and YF3YF_{3} respectively are

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  19. 44
    As compared with chlorocyclohexane, which of the following statements correctly apply to chlorobenzene? (A) The magnitude of negative charge is more on chlorine atoms (B) The C−ClC - Cl bond has partial double bond character (C) C−ClC - Cl bond is less polar (D) C−ClC - Cl bond is longer due to repulsion between delocalised electrons of the aromatic ring and lone pairs of electrons of chlorine. (E) The C−ClC - Cl bond is formed using sp2{sp}^{2} hybridised orbital of carbon. Choose the correct answer from the options given below:

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  20. 45
    Given below are two statements: Statement I : The Henry's law constant KHK_{H} is constant with respect to variations in solution's concentration over the range for which the solutions is ideally dilute. Statement II: KHK_{H} does not differ for the same solute in different solvents. In the light of the above statements, choose the correct answer from the options.

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  21. 46
    The cycloalkane (X) on bromination consumes one mole of bromine per mole of (X) and gives the product (Y) in which C:BrC:Br ratio is 3:13:1. The percentage of bromine in the product (Y) is ____\_\_\_\_ %. (Nearest integer) (Given : Molar mass in gmol−1H:1,C:12g{mol}^{- 1}H:1,C:12, O:16,Br:80O:16,Br:80 )
  22. 47
    Consider the following electrochemical cell at 298 K Pt∣HSnO2−(aq)∣Sn(OH)6 2−(aq)∣Bi2O3(s)∣Bi(s)Pt\left| {HSnO}_{2}^{-}(aq) \right|Sn(OH)_{6}\ ^{2 -}(aq)\left| {Bi}_{2}O_{3}(s) \right|Bi(s). If the reaction quotient at a given time is 10610^{6}, then the cell EMF (Ecell E_{\text{cell~}}) is ____\_\_\_\_ ×10−1 V\times 10^{- 1}\text{ }V (Nearest integer). Given the standard half-cell reduction potential as EBi2O3/Bi,OH−0=−0.44 VE_{{Bi}_{2}O_{3}/{Bi}_{,{OH}^{-}}}^{0} = - 0.44\text{ }V and
    ESn(OH)62−/HSnO2−,OH0=−0.90 VE_{Sn(OH)_{6}^{2 -}/{HSnO}_{2}^{-},OH}^{0} = - 0.90\text{ }V
  23. 48
    The temperature at which the rate constants of the given below two gaseous reactions become equal is ____\_\_\_\_ K. (Nearest integer).
    X⟶Y k1=106e−30000 TP⟶Q k2=104e−24000 T\begin{gathered}X \longrightarrow Y\ k_{1} = 10^{6}e^{\frac{- 30000}{\text{ }T}} \\ P \longrightarrow Q\ k_{2} = 10^{4}e^{\frac{- 24000}{\text{ }T}}\end{gathered}
    Given : ln⁡10=2.303\ln10 = 2.303
  24. 49
    Sodium fusion extract of an organic compound (Y) with CHCl3{CHCl}_{3} and chlorine water gives violet color to the CHCl3{CHCl}_{3} layer. 0.15 g of (Y)(Y) gave 0.12 g of the silver halide precipitate in Carius method. Percentage of halogen in the compound (Y)(Y) is ____\_\_\_\_ . (Nearest integer). (Given : molar mass gmol−1C:12,H:1,Cl:35.5g{mol}^{- 1}C:12,H:1,Cl:35.5, Br: 80, I : 127)
  25. 50
    Dissociation of a gas A2A_{2} takes place according to the following chemical reactions. At equilibrium, the total pressure is 1 bar at 300 K.
    A2( g)⇌2 A( g)A_{2}(\text{ }g) \rightleftharpoons 2\text{ }A(\text{ }g)
    The standard Gibbs energy of formation of the involved substances has been provided below:
    SubstanceΔfG∘/kJ mol−1\Delta_{f}G^{\circ}/kJ\ mol^{-1}
    A2A_{2}-100.00
    AA-50.832
    The degree of dissociation of A2(g)A_{2}(g) is given by (x×10−2)1/2\left( x \times 10^{- 2} \right)^{1/2} where x=x = ____\_\_\_\_ . (Nearest integer). [Given : R=8 J mol−1 K−1,log⁡2R = 8\text{ }J{\text{ }mol}^{- 1}{\text{ }K}^{- 1},\log2 = 0.3010, log⁡3=0.48]\log3 = 0.48\rbrack

Mathematics

25 questions

  1. 51
    Let AB→=2i^+4j^−5k^\overrightarrow{AB}= 2\widehat{i}+ 4\widehat{j}- 5\widehat{k} and AD→=i^+2j^+λk^,λ∈R\overrightarrow{AD}=\widehat{i}+ 2\widehat{j}+ \lambda\widehat{k},\lambda \in R. Let the projection of the vector v→=i^+j^+k^\overrightarrow{v}=\widehat{i}+\widehat{j}+\widehat{k} on the diagonal AC→\overrightarrow{AC} of the parallelogram ABCD be of length one unit. If α,β\alpha,\beta, where α>β\alpha > \beta, be the roots of the equation λ2x2−6λx+5=0\lambda^{2}x^{2}- 6\lambda x + 5 = 0, then 2α−β2\alpha - \beta is equal to

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  2. 52
    Let the relation R on the set M={1,2,3,…….16}\mathbf{M = \{ 1,2,3,\ldots\ldots.16\}} be given by R={(x,y):4y=5x−3,x,y∈M}\mathbf{R = \{(x,y):4y = 5x - 3,x,y \in M\}}. Then the minimum number of elements required to be added in R , in order to make the relation symmetric, is equal to

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  3. 53
    Let the line x=−1x = - 1 divide the area of the region {(x,y):1+x2≤y≤3−x}\left\{ (x,y):1 +x^{2}\leq y \leq 3 - x \right\} in the ratio m:n,gcd(m,n)=1m:n,gcd(m,n) = 1. Then m+nm + n is equal to

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  4. 54
    Two distinct numbers a and b are selected at random from 1,2,3,……,501,2,3,\ldots\ldots,50. The probability, that their product ab is divisible by 3 , is

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  5. 55
    Let f(x)=x2025−x2000,x∈[0,1]f(x) =x^{2025}-x^{2000},x \in \lbrack 0,1\rbrack and the minimum value of the function f(x)f(x) in the interval [0,1]\lbrack 0,1\rbrack be (80)80(n)−81(80)^{80}(n)^{- 81}. Then n is equal to

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  6. 56
    Let P(α,β,γ)P(\alpha,\beta,\gamma) be the point on the line x−12=y+1−3=z\frac{x - 1}{2}=\frac{y + 1}{- 3}= z at a distance 4144\sqrt{14} from the point (1,−1,0)(1, - 1,0) and nearer to the origin. Then the shortest distance, between the lines x−α1=y−β2=z−γ3\frac{x - \alpha}{1}=\frac{y - \beta}{2}=\frac{z - \gamma}{3} and x+52=y−101=z−31\frac{x + 5}{2}=\frac{y - 10}{1}=\frac{z - 3}{1} , is equal to

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  7. 57
    If a random variable x has the probability distribution
    xx01234567
    p(x)p(x)02k2kkk3k3k2k22k^{2}2k2kk2+kk^{2}+k7k27k^{2}
    then P(3<x≤6)P(3 < x \leq 6) is equal to

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  8. 58
    The number of distinct real solutions of the equation x∣x+4∣+3∣x+2∣+10=0x|x + 4| + 3|x + 2| + 10 = 0 is

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  9. 59
    Let f:[1,∞)→Rf:\lbrack 1,\infty) \rightarrow R be a differentiable function, If 6∫1xf(t)dt=3xf(x)+x3−46\int_{1}^{x} f(t)dt = 3xf(x) +x^{3}- 4 for all x≥1x \geq 1, then the value of f(2)−f(3)f(2) - f(3) is

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  10. 60
    If the line αx+2y=1\alpha x + 2y = 1, where α∈R\alpha \in R, does not meet the hyperbola x2−9y2=9x^{2}- 9y^{2}= 9, then a possible value of α\alpha is :

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  11. 61
    If the image of the point P(1,2,a)P(1,2,a) in the line x−63=y−72=7−z2\frac{x - 6}{3}=\frac{y - 7}{2}=\frac{7 - z}{2} is Q(5, b,c)Q(5,\text{ }b,c), then a2+b2+c2a^{2}+b^{2}+c^{2} is equal to

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  12. 62
    Let the set of all values of r , for which the circles (x+1)2+(y+4)2=r2(x + 1)^{2}+ (y + 4)^{2}=r^{2} and x2+y2−4x−2y−4=0x^{2}+y^{2}- 4x - 2y - 4 = 0 intersect at two distinct points be the interval ( α\alpha, β\beta ). Then αβ\alpha\beta is equal to

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  13. 63
    If A=[2335]A =\begin{bmatrix} 2 & 3 \\ 3 & 5 \end{bmatrix}, then the determinant of the matrix ( A2025−3 A2024+A2023A^{2025}- 3{\text{ }A}^{2024}+A^{2023} ) is

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  14. 64
    If the domain of the function f(x)=sin⁡−1(5−x3+2x)+1log⁡e(10−x)f(x) =\sin^{- 1}\left( \frac{5 - x}{3 + 2x} \right)+\frac{1}{\log_{e}(10 - x)} is (−∞,α]∪[β,γ)−{δ}\mathbf{( - \infty,\alpha\rbrack \cup \lbrack\beta,\gamma) - \{\delta\}}, then 6(α+β+γ+δ)6(\alpha + \beta + \gamma + \delta) is equal to

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  15. 65
    The value of ∫−π2π2(1[x]+4)dx\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( \frac{1}{\lbrack x\rbrack + 4} \right)dx, where [•] denotes the greatest integer function, is

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  16. 66
    The coefficient of x48x^{48} in (1+x)+2(1+x)2+3(1+x)3+….+100(1+x)100(1 + x) + 2(1 + x)^{2}+ 3(1 + x)^{3}+ \ldots. + 100(1 + x)^{100} is equal to :

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  17. 67
    If the chord joining the points P1(x1,y1)P_{1}\left( x_{1},y_{1} \right) and P2(x2P_{2}\left( x_{2} \right., y2y_{2} ) on the parabola y2=12xy^{2}= 12x subtends a right angle at the vertex of the parabola, then x1x2−y1y2x_{1}x_{2}-y_{1}y_{2} is equal to

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  18. 68
    The number of solutions of tan⁡−14x+tan⁡−16x=π6\tan^{- 1}4x +\tan^{- 1}6x =\frac{\pi}{6}, where −126<x<126-\frac{1}{2\sqrt{6}}< x <\frac{1}{2\sqrt{6}} is equal to

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  19. 69
    Let the solution curve of the differential equation xdy−ydx=x2+y2dx,x>0,y(1)=0xdy - ydx =\sqrt{x^{2}+y^{2}}dx,x > 0,y(1) = 0, be y=y(x)y = y(x). Then y(3)y(3) is equal to

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  20. 70
    If the sum of the first four terms of an A.P. is 6 and the sum of its first six terms is 4 , then the sum of its first twelve terms is

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  21. 71
    Let α=−1+i32\alpha = \frac{- 1 + i\sqrt{3}}{2} and β=−1−i32,i=−1\beta = \frac{- 1 - i\sqrt{3}}{2},i = \sqrt{- 1}. If (7−7α+9β)20+(9+7α−7β)20+(−7+9α+7β)20+(14+7α+7β)20=m10(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + ( - 7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}, then m is ____\_\_\_\_ .
  22. 72
    Let A be a 3×33 \times 3 matrix such that A+AT=OA + A^{T} = O. If A[1−10]=[332],A2[1−10]=[−319−24]A\begin{bmatrix} 1 \\ - 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 3 \\ 3 \\ 2 \end{bmatrix},A^{2}\begin{bmatrix} 1 \\ - 1 \\ 0 \end{bmatrix} = \begin{bmatrix} - 3 \\ 19 \\ - 24 \end{bmatrix} and det(adj(2adj(A+I)))=(2)α⋅(3)β⋅(11)γ,α,β,γdet(adj(2adj(A + I))) = (2)^{\alpha} \cdot (3)^{\beta} \cdot (11)^{\gamma},\alpha,\beta,\gamma are non-negative integers, then α+β+γ\alpha + \beta + \gamma is equal to ____\_\_\_\_
  23. 73
    If ∫ (sin⁡x)−112(cos⁡x)−52dx=\int_{}^{}\ (\sin x)^{\frac{- 11}{2}}(\cos x)^{\frac{- 5}{2}}dx = −p1q1(cot⁡x)92−p2q2(cot⁡x)52−p3q3(cot⁡x)12+p4q4(cot⁡x)−32+C- \frac{p_{1}}{q_{1}}(\cot x)^{\frac{9}{2}} - \frac{p_{2}}{q_{2}}(\cot x)^{\frac{5}{2}} - \frac{p_{3}}{q_{3}}(\cot x)^{\frac{1}{2}} + \frac{p_{4}}{q_{4}}(\cot x)^{\frac{- 3}{2}} + C, where pip_{i} and qiq_{i} are positive integers with gcd⁡(pi,qi)=1\gcd\left( p_{i},q_{i} \right) = 1 for i=1,2,3,4i = 1,2,3,4 and C is the constant of integration, then 15p1p2p3p4q1q2q3q4\frac{15p_{1}p_{2}p_{3}p_{4}}{q_{1}q_{2}q_{3}q_{4}} is equal to ____\_\_\_\_ .
  24. 74
    If cos⁡248∘−sin⁡212∘sin⁡224∘−sin⁡26∘=α+β52\frac{\cos^{2}48^{\circ} - \sin^{2}12^{\circ}}{\sin^{2}24^{\circ} - \sin^{2}6^{\circ}} = \frac{\alpha + \beta\sqrt{5}}{2}, where α,β∈N\alpha,\beta \in \mathbb{N}, then α+β\alpha + \beta is equal to ____\_\_\_\_ .
  25. 75
    Let ABC be a triangle. Consider four points p1,p2p_{1},p_{2}, p3,p4p_{3},p_{4} on the side AB , five points p5,p6,p7,p8,p9p_{5},p_{6},p_{7},p_{8},p_{9} on the side BC and four points p10,p11,p12,p13p_{10},p_{11},p_{12},p_{13} on the side AC . None of these points is a vertex of the triangle ABC . Then the total number of pentagons, that can be formed by taking all the vertices from the points p1,p2,….p13p_{1},p_{2},\ldots.p_{13}, is ____\_\_\_\_ .

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