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JEE Mains 2025 (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
    Two capacitors C1C_{1} and C2C_{2} are connected in parallel to a battery. Charge-time graph is shown below for the two capacitors. The energy stored with them are U1U_{1} and U2U_{2}, respectively. Which of the given statements is true?

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  2. 2
    In the experiment for measurement of viscosity ' η\eta ' of given liquid with a ball having radius RR, consider following statements. (A) Graph between terminal velocity V and R will be a parabola (B) The terminal velocities of different diameter balls are constant for a given liquid. (C) Measurement of terminal velocity is dependent on the temperature. (D) This experiment can be utilized to assess the density of a given liquid. (E) If balls are dropped with some initial speed, the value of η\eta will change. Choose the correct answer from the options given below:

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  3. 3
    Consider following statements: (A) Surface tension arises due to extra energy of the molecules at the interior as compared to the molecules at the surface, of a liquid. (B) As the temperature of liquid rises, the coefficient of viscosity increases. (C) As the temperature of gas increases, the coefficient of viscosity increases. (D) The onset of turbulence is determined by Reynold's number. (E) In a steady flow two stream lines never intersect. Choose the correct answer from the options given below :

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  4. 4
    Three infinitely long wires with linear charge density λ\lambda are placed along the xx-axis, yy-axis and zz axis respectively. Which of the following denotes an equipotential surface?

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  5. 5
    A hemispherical vessel is completely - filled with a liquid of refractive index μ\mu. A small coin is kept at the lowest point ( O ) of the vessel as shown in figure. The minimum value of the refractive index of the liquid so that a person can see the coin from point EE (at the level of the vessel) is ____\_\_\_\_

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  6. 6
    Consider a long thin conducting wire carrying a uniform current I . A particle having mass " M " and charge " q " is released at a distance " a " from the wire with a speed v0v_{0} along the direction of current in the wire. The particle gets attracted to the wire due to magnetic force. The particle turns round when it is at distance x from the wire. The value of x is [ μ0\mu_{0} is vacuum permeability]

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  7. 7
    A Carnot engine (E) is working between two temperatures 473 K and 273 K . In a new system two engines - engine E1E_{1} works between 473 K to 373 K and engine E2E_{2} works between 373 K to 273 K . If η12\eta_{12}, η1\eta_{1} and η2\eta_{2} are the efficiencies of the engines E,E1E,E_{1} and E2E_{2}, respectively, then

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  8. 8
    Given below are two statements: one is labelled as Assertion AA and the other is labelled as Reason RR Assertion A: A sound wave has higher speed in solids than gases. Reason R: Gases have higher value of Bulk modulus than solids. In the light of the above statements, choose the correct answer from the options given below

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  9. 9
    For a particular ideal gas which of the following graphs represents the variation of mean squared velocity of the gas molecules with temperature?

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  10. 10
    A bead of mass ' mm ' slides without friction on the wall of a vertical circular hoop of radius ' RR ' as shown in figure. The bead moves under the combined action of gravity and a massless spring (k) attached to the bottom of the hoop. The equilibrium length of the spring is ' RR '. If the bead is released from top of the hoop with (negligible) zero initial speed, velocity of bead, when the length of spring becomes ' RR ', would be (spring constant is ' k ', g is acceleration due to gravity)

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  11. 11
    Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason RR Assertion A: In a central force field, the work done is independent of the path chosen Reason R: Every force encountered in mechanics does not have an associated potential energy. In the light of the above statements, choose the most appropriate answer from the options given below

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  12. 12
    Choose the correct nuclear process from the below options [p\left\lbrack p \right. : proton, n : neutron, e−e^{-}: electron, e+e^{+}: positron, vv : neutrino, vˉ\bar{v} : antineutrino]

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  13. 13
    Which of the following circuits has the same output as that of the given circuit?

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  14. 14
    Find the equivalent resistance between two ends of the following circuit.

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  15. 15
    A wire of resistance RR is bent into an equilateral triangle and an identical wire is bent into a square. The ratio of resistance between the two end points of an edge of the triangle to that of the square is

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  16. 16
    Due to presence of an em-wave whose electric component is given by E=100sin⁡(ωt−kx)NC−1E = 100\sin(\omega t - kx)NC^{- 1}, a cylinder of length 200 cm holds certain amount of em-energy inside it. If another cylinder of same length but half diameter than previous one holds same amount of em-energy, the magnitude of the electric field of the corresponding em-wave should be modified as

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  17. 17
    A particle of mass ' mm ' and charge ' qq ' is fastened to one end 'A' of a massless string having equilibrium length l\mathcal{l}, whose other end is fixed at point ' O '. The whole system is placed on a frictionless horizontal plane and is initially at rest. If uniform electric field is switched on along the direction as shown in figure, then the speed of the particle when it crosses the xx-axis is

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  18. 18
    A proton of mass ' mp′m_{p}^{'} ' has same energy EE as that of a photon of wavelength ' λ\lambda '. If the proton is moving at non-relativistic speed, then ratio of its de Broglie wavelength to the wavelength of photon is.

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  19. 19
    The centre of mass of a thin rectangular plate (fig xx ) with sides of length aa and bb, whose mass per unit area (σ)(\sigma) varies as σ=σ0xab\sigma =\frac{\sigma_{0}x}{ab} (where σ0\sigma_{0} is a constant), would be ______\_\_\_\_\_\_

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  20. 20
    A thin prism P1P_{1} with angle 4∘4^{\circ} made of glass having refractive index 1.54 , is combined with another thin prism P2P_{2} made of glass having refractive index 1.72 to get dispersion without deviation. The angle of the prism P2P_{2} in degrees is

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  21. 21
    A tiny metallic rectangular sheet has length and breadth of 5 mm and 2.5 mm , respectively. Using a specially designed screw gauge which has pitch of 0.75 mm and 15 divisions in the circular scale, you are asked to find the area of the sheet. In this measurement, the maximum fractional error will be x100\frac{x}{100} where x is ____\_\_\_\_
  22. 22
    The moment of inertia of a solid disc rotating along its diameter is 2.5 times higher than the moment of inertia of a ring rotating in similar way. The moment of inertia of a solid sphere which has same radius as the disc and rotating in similar way, is n times higher than the moment of inertia of the given ring. Here, n=n =  _____\ \_\_\_\_\_ Consider all the bodies have equal masses.
  23. 23
    In a measurement, it is asked to find modulus of elasticity per unit torque applied on the system. The measured quantity has dimension of [Ma Lb Tc]\left\lbrack M^{a}{\text{ }L}^{b}{\text{ }T}^{c} \right\rbrack. If b=−3b= -3, the value of cc is _____\_\_\_\_\_
  24. 24
    Two iron solid discs of negligible thickness have radii R1R_{1} and R2R_{2} and moment of inertia I1I_{1} and I2I_{2}, respectively. For R2=2R1R_{2}= 2R_{1}, the ratio of I1I_{1} and I2I_{2} would be 1/x1/x, where x=x = ____\_\_\_\_
  25. 25
    A double slit interference experiment performed with a light of wavelength 600 nm forms an interference fringe pattern on a screen with 10th 10^{\text{th~}} bright fringe having its centre at a distance of 10 mm from the central maximum. Distance of the centre of the same 10th 10^{\text{th~}} bright fringe from the central maximum when the source of light is replaced by another source of wavelength 660 nm would be _____\_\_\_\_\_ mm .

Chemistry

25 questions

  1. 26
    The incorrect decreasing order of atomic radii is :

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  2. 27
    Given below are two statements : Statement I : In the oxalic acid vs KMnO4KMnO_{4} (in the presence of dil H2SO4H_{2}SO_{4} ) titration the solution needs to be heated initially to 60∘C60^{\circ}C, but no heating is required in Ferrous ammonium sulphate (FAS) vs KMnO4KMnO_{4} titration (in the presence of dil H2SO4H_{2}SO_{4} ) Statement II : In oxalic acid vs KMnO4KMnO_{4} titration, the initial formation of MnSO4MnSO_{4} takes place at high temperature, which then acts as catalyst for further reaction. In the case of FAS vs KMnO4KMnO_{4}, heating oxidizes Fe2+Fe^{2 +} into Fe3+Fe^{3 +} by oxygen of air and error may be introduced in the experiment. In the light of the above statements, choose the correct answer from the options given below :

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  3. 28
    Match List-I with List-II.
    List-I (Redox reaction)List-II (Type of redox reaction)
    (A)CH4(g)+2O2(g)→ΔCO2(g)+2H2O(l)CH_{4}(g) + 2O_{2}(g) \overset{\Delta}{\rightarrow} CO_{2}(g) + 2H_{2}O(l)(I)Disproportionation reaction
    (B)2NaH(s)→Δ2Na(s)+H2(g)2NaH(s) \overset{\Delta}{\rightarrow} 2Na(s) + H_{2}(g)(II)Combination reaction
    (C)V2O5(s)+5Ca(s)→Δ2V(s)+5CaO(s)V_{2}O_{5}(s) + 5Ca(s) \overset{\Delta}{\rightarrow} 2V(s) + 5CaO(s)(III)Decomposition reaction
    (D)2H2O2(aq)→Δ2H2O(l)+O2(g)2H_{2}O_{2}(aq) \overset{\Delta}{\rightarrow} 2H_{2}O(l) + O_{2}(g)(IV)Displacement reaction
    Choose the correct answer from the options given below :

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  4. 29
    Given below are two statements : Statement I : Et2N−CH2−CH2−ClEt_{2}N-CH_{2}-CH_{2}-Cl will undergo alkaline hydrolysis at a faster rate than Et2CH−CH2−CH2−ClEt_{2}CH-CH_{2}-CH_{2}-Cl. Statement II : In Et2N−CH2−CH2−ClEt_{2}N-CH_{2}-CH_{2}-Cl, intramolecular substitution takes place first by involving lone pair of electrons on nitrogen. In the light of the above statements, choose the most appropriate answer from the options given below :

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  5. 30
    A weak acid HA has degree of dissociation xx. Which option gives the correct expression of (pH−pKa)(pH - pK_{a}) ?

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  6. 31
    Consider ' nn ' is the number of lone pair of electrons present in the equatorial position of the most stable structure of ClF3ClF_{3}. The ions from the following with ' nn ' number of unpaired electrons are : (A) V3+V^{3 +} (B) Ti3+Ti^{3 +} (C) Cu2+Cu^{2 +} (D) Ni2+Ni^{2 +} (E) Ti2+Ti^{2 +} Choose the correct answer from the options given below :

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  7. 32
    For a given reaction R→P,t1/2R \rightarrow P,t_{1/2} is related to [A]0\lbrack A\rbrack_{0} as given in table :
    [A]0/molL−1\lbrack A\rbrack_{0}/molL^{- 1}t1/2/mint_{1/2}/min
    0.100200
    0.025100
    Given : log2=0.30log2 = 0.30 Which of the following is true? (A) The order of the reaction is 12\frac{1}{2}. (B) If [A]0\lbrack A\rbrack_{0} is 1 M , then t1/2t_{1/2} is 20010 min⁡200\sqrt{10}\text{ }\min (C) The order of the reaction changes to 1 if the concentration of reactant changes from 0.100 M to 0.500 M . (D) t1/2t_{1/2} is 800 min for [A]0=1.6M\lbrack A\rbrack_{0}= 1.6M Choose the correct answer from the options given below :

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  8. 33
    A molecule ("P") on treatment with acid undergoes rearrangement and gives ("Q") ("Q") on ozonolysis followed by reflux under alkaline condition gives ("R"). The structure of ("R") is given below : The structure of ("P") is

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  9. 34
    Ice and water are placed in a closed container at a pressure of 1 atm and temperature 273.15 K . If pressure of the system is increased 2 times, keeping temperature constant, then identify correct observation from following :

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  10. 35
    The molecules having square pyramidal geometry are

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  11. 36
    The metal ion whose electronic configuration is not affected by the nature of the ligand and which gives a violet colour in non-luminous flame under hot condition in borax bead test is

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  12. 37
    Both acetaldehyde and acetone (individually) undergo which of the following reactions? (A) Iodoform Reaction (B) Cannizaro Reaction (C) Aldol condensation (D) Tollen's Test (E) Clemmensen Reduction Choose the correct answer from the options given below :

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  13. 38
    In a multielectron atom, which of the following orbitals described by three quantum numbers will have the same energy in the absence of electric and magnetic fields? (A) n=1,l=0,ml=0n = 1,l = 0,m_{l}= 0 (B) n=2,l=0,ml=0n = 2,l = 0,m_{l}= 0 (C) n=2,l=1,ml=1n = 2,l = 1,m_{l}= 1 (D) n=3,l=2,ml=1n = 3,l = 2,m_{l}= 1 (E) n=3,l=2,ml=0n = 3,l = 2,m_{l}= 0 Choose the correct answer from the options given below :

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  14. 39
    The products A and B in the following reactions, respectively are
    A⟵AgNO2CH3−CH2−CH2−Br→AgCN BA\overset{AgNO_{2}}{\longleftarrow}CH_{3}-CH_{2}-CH_{2}- Br\overset{AgCN}{\rightarrow}\text{ }B

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  15. 40
    What is the freezing point depression constant of a solvent, 50 g of which contain 1 g non volatile solute (molar mass 256 g mol−1256\text{ }g{\text{ }mol}^{- 1} ) and the decrease in freezing point is 0.40 K ?

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  16. 41
    Consider the following elements In,Tl,Al,Pb,SnIn,Tl,Al,Pb,Sn and Ge . The most stable oxidation states of elements with highest and lowest first ionisation enthalpies, respectively, are

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  17. 42
    The correct order of stability of following carbocations is :

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  18. 43
    The compounds that produce CO2CO_{2} with aqueous NaHCO3NaHCO_{3} solution are : (A) (B) (C) (D) (E) Choose the correct answer from the options given below :

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  19. 44
    Which of the following oxidation reactions are carried out by both K2Cr2O7K_{2}Cr_{2}O_{7} and KMnO4KMnO_{4} in acidic medium? (A) I−→I2I^{-}\rightarrow I_{2} (B) S2−→SS^{2 -}\rightarrow S (C) Fe2+→Fe3+Fe^{2 +}\rightarrow Fe^{3 +} (D) I−→IO3−I^{-}\rightarrow IO_{3}^{-} (E) S2O3 2−→SO4 2−S_{2}O_{3}\ ^{2 -}\rightarrow SO_{4}\ ^{2 -} Choose the correct answer from the options given below :

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  20. 45
    Given below are two statements : Statement I : D-glucose pentaacetate reacts with 2, 4-dinitrophenylhydrazine. Statement II : Starch, on heating with concentrated sulfuric acid at 100∘C100^{\circ}C and 2−32 - 3 atmosphere pressure produces glucose. In the light of the above statements, choose the correct answer from the options given below

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  21. 46
    Given below is the plot of the molar conductivity vs  concentration \sqrt{\text{~concentration~}} for KCl in aqueous solution. If, for the higher concentration of KCl solution, the resistance of the conductivity cell is 100Ω100\Omega, then the resistance of the same cell with the dilute solution is ' xx ' Ω\Omega. The value of xx is ______\_\_\_\_\_\_ (Nearest integer)
  22. 47
    Quantitative analysis of an organic compound (X) shows following % composition. C : 14.5% Cl : 64.46% H: 1.8%, the rest being oxygen. The empirical formula mass of the compound (X) is _____\_\_\_\_\_ ×10−1\times10^{- 1} (Given molar mass in gmol−1gmol^{- 1} of C:12,H:1C:12,H:1, O:16,Cl:35.5)O:16,Cl:35.5)
  23. 48
    The molarity of a 70%70\% (mass // mass) aqueous solution of a monobasic acid (X) is ____\_\_\_\_ M(Nearest integer) [Given : Density of aqueous solution of (X)(X) is 1.25 g mL−11.25\text{ }g{\text{ }mL}^{- 1} Molar mass of the acid is 70 g mol−170\text{ }g{\text{ }mol}^{- 1} ]
  24. 49
    Consider the following sequence of reactions : →(i) Mg , dry ether(ii) CO2,H3O+ (iii)NH3,Δ\underset{\begin{array}{r} {\left( ii \right)\ CO}_{2},H_{3}O^{+ \ } \\ (iii)NH_{3},\Delta \end{array}}{\overset{\left( i \right)\ Mg\ ,\ dry\ ether}{\rightarrow}} A→Br2, NaOH\overset{Br_{2},\ NaOH}{\rightarrow}B 11.25 mg of chlorobenzene will produce_____\_\_\_\_\_ ×10−1mg\times10^{- 1}mg of product B . (Consider the reactions result in complete conversion.) [Given molar mass of C,H,O,NC,H,O,N and Cl as 12,1 , 16,14 and 35.5 g mol−135.5\text{ }g{\text{ }mol}^{- 1} respectively]
  25. 50
    The formation enthalpies, ΔHfΘ\Delta H_{f}^{\Theta} for H(g)H_{(g)} and O(g)O_{(g)} are 220.0 and 250.0 kJ mol−1250.0\text{ }kJ{\text{ }mol}^{- 1}, respectively, at 298.15 K , and ΔHf⊖\Delta H_{f}^{\ominus} for H2O(g)H_{2}O_{(g)} is −242.0 kJ mol−1- 242.0\text{ }kJ{\text{ }mol}^{- 1} at the same temperature. The average bond enthalpy of the O−HO - H bond in water at 298.15 K is ____\_\_\_\_ kJmol−1kJmol^{- 1} (nearest integer).

Mathematics

25 questions

  1. 51
    The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0,1 , 2,3,4,5,6,72,3,4,5,6,7, such that the sum of their first and last digits should not be more than 8 , is

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  2. 52
    Let ABCDABCD be a trapezium whose vertices lie on the parabola y2=4xy^{2}= 4x. Let the sides ADAD and BCBC of the trapezium be parallel to yy-axis. If the diagonal AC is of length 254\frac{25}{4} and it passes through the point (1,0)(1,0), then the area of ABCDABCD is :

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  3. 53
    Two number k1k_{1} and k2k_{2} are randomly chosen from the set of natural numbers. Then, the probability that the value of ik1+ik2,(i=−1)i^{k_{1}}+i^{k_{2}},(i =\sqrt{- 1}) is non-zero, equals

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  4. 54
    If f(x)=2x2x+2,x∈Rf(x) =\frac{2^{x}}{2^{x}+\sqrt{2}},x \in R, then ∑k=181f(k82)\sum_{k = 1}^{81} f\left( \frac{k}{82} \right) is equal to :

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  5. 55
    Let f:R→Rf:R \rightarrow R be a function defined by f(x)=(2+3a)x2+(a+2a−1)x+b,a≠1f(x) = (2 + 3a)x^{2}+\left( \frac{a + 2}{a - 1} \right)x + b,a \neq 1. If f(x+y)=f(x)+f(y)+1−27xyf(x + y) = f(x) + f(y) + 1 -\frac{2}{7}xy, then the value of 28∑i=15∣f(i)∣28\sum_{i = 1}^{5} |f(i)| is:

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  6. 56
    Let A(x,y,z)A(x,y,z) be a point in xy-plane, which is equidistant from three points (0,3,2),(2,0,3)(0,3,2),(2,0,3) and (0,0,1)(0,0,1). Let B=(1,4,−1)B = (1,4, - 1) and C=(2,0,−2)C = (2,0, - 2). Then among the statements (S1): △ABC\bigtriangleup ABC is an isosceles right angled triangle and (S2) : the area of △ABC\bigtriangleup ABC is 922\frac{9\sqrt{2}}{2}.

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  7. 57
    The relation R={(x,y):x,y∈z\mathbf{R = \{(x,y):x,y \in z} and x+yx + y is even }\} is :

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  8. 58
    Let the equation of the circle, which touches xx-axis at the point (a,0),a>0(a,0),a > 0 and cuts off an intercept of length bb on yy-axis be x2+y2−αx+βy+γ=0x^{2}+y^{2}- \alpha x + \beta y + \gamma = 0. If the circle lies below x-axis, then the ordered pair (2a,b2)\left( 2a,b^{2} \right) is equal to :

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  9. 59
    Let ⟨an⟩\left\langle a_{n} \right\rangle be a sequence such that a0=0,a1=12a_{0}= 0,a_{1}=\frac{1}{2} and 2an+2=5an+1−3an,n=0,1,2,3,……2a_{n + 2}= 5a_{n + 1}- 3a_{n},n = 0,1,2,3,\ldots\ldots Then ∑k=1100ak\sum_{k = 1}^{100} a_{k} is equal to :

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  10. 60
    cos⁡(sin⁡−135+sin⁡−1513+sin⁡−13365)\cos\left( \sin^{- 1}\frac{3}{5}+\sin^{- 1}\frac{5}{13}+\sin^{- 1}\frac{33}{65} \right) is equal to :

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  11. 61
    Let TrT_{r} be the rth r^{\text{th~}} term of an A.P. If for some mm, Tm=125, T25=120T_{m}=\frac{1}{25},{\text{ }T}_{25}=\frac{1}{20} and 20∑r=125 Tr=1320\sum_{r = 1}^{25} {\text{ }T}_{r}= 13, then 5 m∑r=m2 m Tr5\text{ }m\sum_{r = m}^{2\text{ }m} {\text{ }T}_{r} is equal to :

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  12. 62
    If the image of the point (4,4,3)(4,4,3) in the line x−12=y−21=z−13\frac{x - 1}{2}=\frac{y - 2}{1}=\frac{z - 1}{3} is (α,β,γ)(\alpha,\beta,\gamma), then α+β+γ\alpha + \beta + \gamma is equal to

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  13. 63
    If ∫−π2π296x2cos⁡2x(1+ex)dx=π(απ2+β),α,β∈Z\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{96x^{2}\cos^{2}x}{\left( 1 +e^{x} \right)}dx = \pi\left( \alpha\pi^{2}+ \beta \right),\alpha,\beta \in Z, then (α+β)2(\alpha + \beta)^{2} equals :

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  14. 64
    The sum of all local minimum values of the function f(x)={1−2x,x<−113(7+2∣x∣),−1≤x≤21118(x−4)(x−5),x>2f(x)=\begin{cases} 1-2x, & x<-1 \\ \frac{1}{3}(7+2|x|), & -1\leq x\leq 2 \\ \frac{11}{18}(x-4)(x-5), & x>2 \end{cases} is

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  15. 65
    The sum, of the squares of all the roots of the equation x2+∣2x−3∣−4=0x^{2}+ |2x - 3| - 4 = 0, is :

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  16. 66
    Let for some function y=f(x),∫0xtf(t)dt=x2f(x)y = f(x),\int_{0}^{x} tf(t)dt =x^{2}f(x), x>0x > 0 and f(2)=3f(2) = 3. Then f(6)f(6) is equal to :

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  17. 67
    Let  nCr−1=28, nCr=56\ ^{n}C_{r - 1}= 28,\ ^{n}C_{r}= 56 and  nCr+1=70\ ^{n}C_{r + 1}= 70. Let A(4cos⁡t,4sin⁡t),B(2sin⁡t,−2cos⁡t)A(4\cos t,4\sin t),B(2\sin t, - 2\cos t) and C(3r−n,r2−n−1)C\left( 3r - n,r^{2}- n - 1 \right) be the vertices of a triangle ABCABC, where tt is a parameter. If (3x−1)2+(3y)2=α(3x - 1)^{2}+ (3y)^{2}= \alpha, is the locus of the centroid of triangle ABC , then α\alpha equals :

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  18. 68
    Let OO be the origin, the point AA be z1=3+22iz_{1}=\sqrt{3}+ 2\sqrt{2}i, the point B(z2)B\left( z_{2} \right) be such that 3∣z2∣=∣z1∣\sqrt{3}\left| z_{2} \right|=\left| z_{1} \right| and arg(z2)=arg(z1)+π6arg\left( z_{2} \right)= arg\left( z_{1} \right)+\frac{\pi}{6}. Then

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  19. 69
    Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If x denote the number of defective oranges, then the variance of xx is :

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  20. 70
    The area (in sq. units) of the region {(x,y):0≤y≤2∣x∣+1,0≤y≤x2+1,∣x∣≤3}\left\{ (x,y):0 \leq y \leq 2|x| + 1,0 \leq y \leq x^{2}+ 1,|x| \leq 3 \right\} is

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  21. 71
    Let M denote the set of all real matrices of order 3×33 \times 3 and let S={−3,−2,−1,1,2}\mathbf{S = \{ - 3, - 2, - 1,1,2\}}. Let S1={A=[aij]∈M:A=ATS_{1}=\left\{ A =\left\lbrack a_{ij} \right\rbrack\in M:A =A^{T} \right. and  aij∈S,∀i,j}\left. \ a_{ij}\in S,\forall i,j \right\} S2={A=[aij]∈M:A=−ATS_{2}=\left\{ A =\left\lbrack a_{ij} \right\rbrack\in M:A = -A^{T} \right. and  aij∈S,∀i,j}\left. \ a_{ij}\in S,\forall i,j \right\} S3={A=[aij]∈M:a11+a22+a33=0S_{3}=\left\{ A =\left\lbrack a_{ij} \right\rbrack\in M:a_{11}+a_{22}+a_{33}= 0 \right. and  aij∈S,∀i,j}\left. \ a_{ij}\in S,\forall i,j \right\} If n(S1∪S2∪S3)=125αn\left( S_{1}\cup S_{2}\cup S_{3} \right)= 125\alpha, then α\alpha equals.
  22. 72
    If α=1+∑r=16(−3)r−1 12C2r−1\alpha= 1 +\sum_{r = 1}^{6} ( - 3)^{r - 1}\ ^{12}C_{2r - 1}, then the distance of the point (12,3)(12,\sqrt{3}) from the line αx−3y+1=0\alpha x-\sqrt{3}y+ 1 = 0 is.....
  23. 73
    Let a→=i^+j^+k^,b→=2i^+2j^+k^\overrightarrow{a}=\widehat{i}+\widehat{j}+\widehat{k},\overrightarrow{b}= 2\widehat{i}+ 2\widehat{j}+\widehat{k} and d→=a→×b→\overrightarrow{d}=\overrightarrow{a}\times\overrightarrow{b}. If c→\overrightarrow{c} is a vector such that a→⋅c→=∣c→∣,∣c→−2a→∣2=8\overrightarrow{a}\cdot\overrightarrow{c}= |\overrightarrow{c}|,|\overrightarrow{c}- 2\overrightarrow{a}|^{2}= 8 and the angle between d→\overrightarrow{d} and c→\overrightarrow{c} is π4\frac{\pi}{4}, then ∣10−3 b→⋅c→∣+∣d→×c→∣2|10 - 3\overrightarrow{\text{ }b}\cdot\overrightarrow{c}| + |\overrightarrow{d}\times\overrightarrow{c}|^{2} is equal to ….\ldots..
  24. 74
    Let
    f(x)={3x,x<0min⁡{1+x+[x],x+2[x]},0≤x≤25,x>2f(x) =\left\{ \begin{matrix} 3x, & x < 0 \\ \min\{ 1 + x + \lbrack x\rbrack,x + 2\lbrack x\rbrack\}, & 0 \leq x \leq 2 \\ 5, & x > 2 \end{matrix} \right.
    where [.] denotes greatest integer function. If α\alpha and β\beta are the number of points, where f is not continuous and is not differentiable, respectively, then α+β\alpha+\beta equals. ………\ldots\ldots\ldots
  25. 75
    Let E1:x29+y24=1E_{1}:\frac{x^{2}}{9}+\frac{y^{2}}{4}=1 be an ellipse. Ellipses are constructed such that their centres and eccentricities are the same as that of E1E_{1}, and the length of the minor axis of EiE_{i} is the length of the major axis of Ei+1 (i≥1)E_{i+1}\ (i\geq 1). If AiA_{i} is the area of the ellipse EiE_{i}, then 5π(∑i=1∞Ai)\frac{5}{\pi}\left(\sum_{i=1}^{\infty} A_{i}\right) is equal to

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