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JEE Mains 2025 (24 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
    Consider a parallel plate capacitor of area A (of each plate) and separation 'd' between the plates. If E is the electric field and ε0\varepsilon_{0} is the permittivity of free space between the plates, then potential energy stored in the capacitor is :-

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  2. 2
    What is the relative decrease in focal length of a lens for an increase in optical power by 0.1 D from 2.5 D ? ['D' stands for dioptre]

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  3. 3
    An air bubble of radius 0.1 cm lies at a depth of 20 cm below the free surface of a liquid of density 1000 kg/m31000\text{ }kg/m^{3}. If the pressure inside the bubble is 2100 N/m22100\text{ }N/m^{2} greater than the atmospheric pressure, then the surface tension of the liquid in SI unit is (use g=10 m/s2g = 10\text{ }m/s^{2} )

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  4. 4
    For an experimental expression y=32.3×112527.4y =\frac{32.3 \times 1125}{27.4}, where all the digits are significant. Then to report the value of yy we should write :-

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  5. 5
    During the transition of electron from state A to state C of a Bohr atom, the wavelength of emitted radiation is 2000Å2000\text{Å} and it becomes 6000Å6000\text{Å} when the electron jumps from state BB to state CC. Then the wavelength of the radiation emitted during the transition of electrons from state A to state B is :-

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  6. 6
    Consider the following statements : (A) The junction area of solar cell is made very narrow compared to a photo diode. (B) Solar cells are not connected with any external bias. (C) LED is made of lightly doped p-n junction. (D) Increase of forward current results in continuous increase of LED light intensity. (E) LEDs have to be connected in forward bias for emission of light.

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  7. 7
    The amount of work done to break a big water drop of radius ' R ' into 27 small drops of equal radius is 10 J . The work done required to break the same big drop into 64 small drops of equal radius will be :-

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  8. 8
    An object of mass ' mm ' is projected from origin in a vertical xy plane at an angle 45∘45^{\circ} with the x -axis with an initial velocity v0v_{0}. The magnitude and direction of the angular momentum of the object with respect to origin, when it reaches at the maximum height, will be [ g is acceleration due to gravity]

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  9. 9
    The Young's double slit interference experiment is performed using light consisting of 480 nm and 600 nm wavelengths to form interference patterns. The least number of the bright fringes of 480 nm light that are required for the first coincidence with the bright fringes formed by 600 nm light is :-

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  10. 10
    A car of mass ' mm ' moves on a banked road having radius ' r ' and banking angle θ\theta. To avoid slipping from banked road, the maximum permissible speed of the car is v0v_{0}. The coefficient of friction μ\mu between the wheels of the car and the banked road is :-

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  11. 11
    A uniform solid cylinder of mass ' m ' and radius ' r ' rolls along an inclined rough plane of inclination 45∘45^{\circ}. If it starts to roll from rest from the top of the plane then the linear acceleration of the cylinder axis will be :-

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  12. 12
    A thin plano convex lens made of glass of refractive index 1.5 is immersed in a liquid of refractive index 1.2. When the plane side of the lens is silver coated for complete reflection, the lens immersed in the liquid behaves like a concave mirror of focal length 0.2 m . The radius of curvature of the curved surface of the lens is :-

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  13. 13
    A particle is executing simple harmonic motion with time period 2 s and amplitude 1 cm . If D and d are the total distance and displacement covered by the particle in 12.5 s , then Dd\frac{D}{d} is :-

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  14. 14
    A satellite is launched into a circular orbit of radius ' R ' around the earth. A second statellite is launched into an orbit of radius 1.03 R . The time period of revolution of the second satellite is larger than the first one approximately by :-

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  15. 15
    A plano-convex lens having radius of curvature of first surface 2 cm exhibits focal length of f1f_{1} in air. Another plano-convex lens with first surface radius of curvature 3 cm has focal length of f2f_{2} when it is immersed in a liquid of refractive index 1.2. If both the lenses are made of same glass of refractive index 1.5 , the ratio of f1f_{1} and f2f_{2} will be :-

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  16. 16
    An alternating current is given by I=IAsin⁡ωt+IBcos⁡ωtI =I_{A}\sin\omega t +I_{B}\cos\omega t. The r.m.s. current will be :-

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  17. 17
    An electron of mass ' mm ' with an initial velocity v→=v0i^(v0>0)\overrightarrow{v}=v_{0}\widehat{i}\left( v_{0}> 0 \right) enters an electric field E→=−E0k^\overrightarrow{E}= -E_{0}\widehat{k}. If the initial de Broglie wavelength is λ0\lambda_{0}, the value after time tt would be :-

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  18. 18
    A parallel plate capacitor was made with two rectangular plates, each with a length of l=3 cml = 3\text{ }cm and breadth of b=1 cmb = 1\text{ }cm. The distance between the plates is 3μ m3\mu\text{ }m. Out of the following, which are the ways to increase the capacitance by a factor of 10 ? (A) l=30 cm, b=1 cm, d=1μ ml = 30\text{ }cm,\text{ }b = 1\text{ }cm,\text{ }d = 1\mu\text{ }m (B) l=3 cm, b=1 cm, d=30μ ml = 3\text{ }cm,\text{ }b = 1\text{ }cm,\text{ }d = 30\mu\text{ }m (C) l=6 cm, b=5 cm, d=3μ ml = 6\text{ }cm,\text{ }b = 5\text{ }cm,\text{ }d = 3\mu\text{ }m (D) l=1 cm, b=1 cm, d=10μ ml = 1\text{ }cm,\text{ }b = 1\text{ }cm,\text{ }d = 10\mu\text{ }m (E) l=5 cm, b=2 cm, d=1μ ml = 5\text{ }cm,\text{ }b = 2\text{ }cm,\text{ }d = 1\mu\text{ }m Choose the correct answer from the options given below :

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  19. 19
    A force F=α+βx2F = \alpha + \beta x^{2} acts on an object in the x -direction. The work done by the force is 5 J when the object is displaced by 1 m . If the constant, α=1 N\alpha = 1\text{ }N then β\beta will be

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  20. 20
    An ideal gas goes from an initial state to final state. During the process, the pressure of gas increases linearly with temperature. (A) The work done by gas during the process is zero. (B) The heat added to gas is different from change in its internal energy. (C) The volume of the gas is increased. (D) The internal energy of the gas is increased. (E) The process is isochoric (constant volume process) Choose the correct answer from the options given below :-

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  21. 21
    A square loop of sides a=1 ma = 1\text{ }m is held normally in front of a point charge q=1Cq= 1C. The flux of the electric field through the shaded region is 5p×1ε0Nm2C\frac{5}{p}\times\frac{1}{\varepsilon_{0}}\frac{Nm^{2}}{C}, where the value of p is _____\_\_\_\_\_ .
  22. 22
    The least count of a screw guage is 0.01 mm . If the pitch is increased by 75%75\% and number of divisions on the circular scale is reduced by 50%50\%, the new least count will be _____\_\_\_\_\_ ×10−3 mm\times10^{- 3}\text{ }mm.
  23. 23
    A wire of resistance 9Ω9\Omega is bent to form an equilateral triangle. Then the equivalent resistance across any two vertices will be ____\_\_\_\_ ohm.
  24. 24
    A current of 5 A exists in a square loop of side 12 m\frac{1}{\sqrt{2}}\text{ }m. Then the magnitude of the magnetic field BB at the centre of the square loop will be p×10−6 Tp \times10^{- 6}\text{ }T. where, value of p is _____\_\_\_\_\_ [Take μ0\left\lbrack Take\ \mu_{0} \right. =4π×10−7 T mA−1= 4\pi\times10^{- 7}\text{ }T{\text{ }mA}^{- 1} ].
  25. 25
    The temperature of 1 mole of an ideal monoatomic gas is increased by 50∘C50^{\circ}C at constant pressure. The total heat added and change in internal energy are E1E_{1} and E2E_{2}, respectively. If E1E2=x9\frac{E_{1}}{E_{2}}=\frac{x}{9} then the value of xx is____

Chemistry

25 questions

  1. 26
    For the given cell
    Fe2+(aq)+Ag+(aq)→Fe3+(aq)+Ag( s)Fe^{2 +}(aq) +Ag^{+}(aq) \rightarrow Fe^{3 +}(aq) + Ag(\text{ }s)
    The standard cell potential of the above reaction is Given :
    Ag++e−→Ag   E0=xVFe2++2e−→Fe   E0=yVFe3++3e−→Fe    E0=zV\begin{matrix} Ag^{+}+e^{-}\rightarrow Ag & \ \ \ E^{0}= xV \\ Fe^{2 +}+ 2e^{-}\rightarrow Fe & \ \ \ E^{0}= yV \\ Fe^{3 +}+ 3e^{-}\rightarrow Fe & \ \ \ \ E^{0}= zV \end{matrix}

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  2. 27
    Following are the four molecules "P", "Q", "R" and "S". Which one among the four molecules will react with H−Br(aq)H - Br(aq) at the fastest rate ?

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  3. 28
    One mole of the octahedral complex compound Co(NH3)5Cl3Co\left( NH_{3} \right)_{5}Cl_{3} gives 3 moles of ions on dissolution in water. One mole of the same complex reacts with excess of AgNO3AgNO_{3} solution to yield two moles of AgCl(s)AgCl_{(s)}. The structure of the complex is :

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  4. 29
    Which one of the carbocations from the following is most stable?

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  5. 30
    Which of the following linear combination of atomic orbitals will lead to formation of molecular orbitals in homonuclear diatomic molecules [internuclear axis in z-direction] ? (A) 2pz2p_{z} and 2px2p_{x} (B) 2 s and 2px2p_{x} (C) 3dxy3d_{xy} and 3dx2−y23d_{x^{2}-y^{2}} (D) 2 s and 2pz2p_{z} (E) 2pz2p_{z} and 3dx2−y23d_{x^{2}-y^{2}}

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  6. 31
    Which of the following ions is the strongest oxidizing agent? (Atomic Number of Ce=58,Eu=63, Tb=65,Lu=71Ce = 58,Eu = 63,\text{ }Tb = 65,Lu = 71)

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  7. 32
    Ksp for Cr(OH)3Cr(OH)_{3} is 1.6×10−301.6 \times10^{- 30}. What is the molar solubility of this salt in water?

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  8. 33
    Let us consider an endothermic reaction which is non-spontaneous at the freezing point of water. However, the reaction is spontaneous at boiling point of water. Choose the correct option.

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  9. 34
    Given below are two statements I and II. Statement I : Dumas method is used for estimation of "Nitrogen" in an organic compound. Statement II : Dumas method involves the formation of ammonium sulphate by heating the organic compound with conc H2SO4H_{2}SO_{4}. In the light of the above statements, choose the correct answer from the options given below

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  10. 35
    Which of the following Statements are NOT true about the periodic table? (A) The properties of elements are function of atomic weights. (B) The properties of elements are function of atomic numbers. (C) Elements having similar outer electronic configuration are arranged in same period. (D) An element's location reflects the quantum numbers of the last filled orbital. (E) The number of elements in a period is same as the number of atomic orbitals available in energy level that is being filled. Choose the correct answer from the options given below:

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  11. 36
    The carbohydrates "Ribose" present in DNA, is (A) A pentose sugar (B) present in pyranose form (C) in "D" configuration (D) a reducing sugar, when free (E) in α\alpha-anomeric form Choose the correct answer from the options given below :

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  12. 37
    Preparation of potassium permanganate from MnO2MnO_{2} involves two step process in which the 1st 1^{\text{st~}} step is a reaction with KOH and KNO3KNO_{3} to produce

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  13. 38
    The large difference between the melting and boiling points of oxygen and sulphur may be explained on the basis of

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  14. 39
    For a reaction, N2O5( g)→2NO2( g)+12O2( g)N_{2}O_{5(\text{ }g)}\rightarrow 2NO_{2(\text{ }g)}+\frac{1}{2}O_{2(\text{ }g)} in a constant volume container, no products were present initially. The final pressure of the system when 50%50\% of reaction gets completed is

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  15. 40
    Which of the following arrangements with respect to their reactivity in nucleophilic addition reaction is correct?

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  16. 41
    Aman has been asked to synthesise the molecule (x) He thought of preparing the molecule using an aldol condensation reaction. He found a few cyclic alkenes in his laboratory. He thought of performing ozonolysis reaction on alkene to produce a dicarbonyl compound followed by aldol reaction to prepare " xx ". Predict the suitable alkene that can lead to the formation of "x".

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  17. 42
    Consider the given plots of vapour pressure (VP) vs temperature ( T/KT/K ) Which amongst the following options is correct graphical representation showing ΔTf\Delta T_{f}, depression in the freezing point of solvent in a solution?

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  18. 43
    Which of the following statement is true with respect to H2O,NH3H_{2}O,NH_{3} and CH4CH_{4} ? (A) The central atoms of all the molecules are sp3sp^{3} hybridized. (B) The H−O−H,H−N−HH - O - H,H - N - H and H−C−HH - C - H angles in the above molecules are 104.5∘,107.5∘104.5^{\circ},107.5^{\circ} and 109.5∘109.5^{\circ} respectively. (C) The increasing order of dipole moment is CH4<NH3<H2OCH_{4}<NH_{3}<H_{2}O. (D) Both H2OH_{2}O and NH3NH_{3} are Lewis acids and CH4CH_{4} is a Lewis base (E) A solution of NH3NH_{3} in H2OH_{2}O is basic. In this solution NH3NH_{3} and H2OH_{2}O act as Lowry-Bronsted acid and base respectively. Choose the correct answer from the options given below :

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  19. 44
    Given below are two statements : Statement-I : The conversion proceeds well in the less polar medium. CH3−CH2−CH2−CH2−Cl⟶HO−CH3−CH2−CH2−CH2−OH+Cl−CH_{3}-CH_{2}-CH_{2}-CH_{2}-Cl \overset{HO^{-}}{\longrightarrow} CH_{3}-CH_{2}-CH_{2}-CH_{2}-OH + Cl^{-} Statement-II : The conversion proceeds well in the more polar medium. CH3−CH2−CH2−CH2−Cl⟶R3NCH3−CH2−CH2−CH2−N+R3 Cl−CH_{3}-CH_{2}-CH_{2}-CH_{2}-Cl \overset{R_{3}N}{\longrightarrow} CH_{3}-CH_{2}-CH_{2}-CH_{2}-\overset{+}{N}R_{3}\ Cl^{-} In the light of the above statements, choose the correct answer from the options given below.

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  20. 45
    The product (A) formed in the following reaction sequence is :

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  21. 46
    37.8 g N2O537.8\text{ }g{\text{ }N}_{2}O_{5} was taken in a 1 L reaction vessel and allowed to undergo the following reaction at 500 K
    2 N2O5( g)→2 N2O4( g)+O2( g)2{\text{ }N}_{2}O_{5(\text{ }g)}\rightarrow 2{\text{ }N}_{2}O_{4(\text{ }g)}+O_{2(\text{ }g)}
    The total pressure at equilibrium was found to be 18.65 bar. Then, Kp=Kp = _____\_\_\_\_\_ ×10−2\times10^{- 2} [nearest integer] Assume N2O5N_{2}O_{5} to behave ideally under these conditions Given : R=0.082R = 0.082 bar Lmol−1 K−1Lmol^{- 1}{\text{ }K}^{- 1}
  22. 47
    Standard entropies of X2,Y2X_{2},Y_{2} and XY5XY_{5} are 70,50 and 110 J K−1 mol−1110\text{ }J{\text{ }K}^{- 1}{\text{ }mol}^{- 1} respectively. The temperature in Kelvin at which the reaction
    12X2+52Y2→XY5ΔH⊖=−35 kJ mol−1\frac{1}{2}X_{2}+\frac{5}{2}Y_{2}\rightarrow XY_{5}\quad \Delta H^{\ominus}= - 35\text{ }kJ{\text{ }mol}^{- 1}
    will be at equilibrium is _______\_\_\_\_\_\_\_ (Nearest integer)
  23. 48
    Xg of benzoic acid on reaction with aq. NaHCO3NaHCO_{3} release CO2CO_{2} that occupied 11.2 L volume at STP. X is ____\_\_\_\_ g .
  24. 49
    Among the following cations, the number of cations which will give characteristic precipitate in their identification tests with K4[Fe(CN)6]K_{4}\left\lbrack Fe(CN)_{6} \right\rbrack is :
    Cu2+,Fe3+,Ba2+,Ca2+,NH4+,Mg2+,Zn2+Cu^{2 +},Fe^{3 +},Ba^{2 +},Ca^{2 +},NH_{4}^{+},Mg^{2 +},Zn^{2 +}
  25. 50
    Consider the following reaction occurring in the blast furnace.
    Fe3O4( s)+4CO(g)→3Fe(l)+4CO2( g)Fe_{3}O_{4(\text{ }s)}+ 4CO_{(g)}\rightarrow 3Fe_{(l)}+ 4CO_{2(\text{ }g)}
    'xx' kg of iron is produced when 2.32×103 kg2.32 \times10^{3}\text{ }kg Fe3O4Fe_{3}O_{4} and 2.8×102 kgCO2.8 \times10^{2}\text{ }kgCO are brought together in the furnace. The value of ' xx ' is ____\_\_\_\_ . (nearest integer) {Given : Molar mass of Fe3O4=232 g mol−1Fe_{3}O_{4}= 232\text{ }g{\text{ }mol}^{- 1} Molar mass of CO=28 g mol−1CO = 28\text{ }g{\text{ }mol}^{- 1} Molar mass of  Fe=56 g mol−1}\left. \ Fe = 56\text{ }g{\text{ }mol}^{- 1} \right\}

Mathematics

25 questions

  1. 51
    Let a→=i^+2j^+3k^,b→=3i^+j^−k^\overrightarrow{a}=\widehat{i}+ 2\widehat{j}+ 3\widehat{k},\overrightarrow{b}= 3\widehat{i}+\widehat{j}-\widehat{k} and c→\overrightarrow{c} be three vectors such that c→\overrightarrow{c} is coplanar with a→\overrightarrow{a} and b→\overrightarrow{b}. If the vector c→\overrightarrow{c} is perpendicular to b→\overrightarrow{b} and a→.c→=5\overrightarrow{a}.\overrightarrow{c}= 5, then ∣c→∣|\overrightarrow{c}| is equal to

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  2. 52
    In I(m,n)=∫01xm−1(1−x)n−1dx,m,n>0I(m,n) =\int_{0}^{1} x^{m - 1}(1 - x)^{n - 1}dx,m,n > 0, then I(9,14)+I(10,13)I(9,14) + I(10,13) is

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  3. 53
    Let f:R−{0}→R\mathbf{f:R - \{ 0\} \rightarrow R} be a function such that f(x)−6f(1x)=353x−52f(x) - 6f\left( \frac{1}{x} \right)=\frac{35}{3x}-\frac{5}{2}. If the lim⁡x→0(1αx+f(x))=β\lim_{x \rightarrow 0} \left( \frac{1}{\alpha x}+ f(x) \right)= \beta; α,β∈R\alpha,\beta \in R, then α+2β\alpha + 2\beta is equal to

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  4. 54
    Let Sn=12+16+112+120+…S_{n}=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+ \ldots upto n terms. If the sum of the first six terms of an A.P. with first term -p and common difference p is 2026 S2025\sqrt{2026{\text{ }S}_{2025}}, then the absolute difference between 20th 20^{\text{th~}} and 15th 15^{\text{th~}} terms of the A.P. is

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  5. 55
    Let f(x)=2x+2+1622x+1+2x+4+32f(x) =\frac{2^{x + 2}+ 16}{2^{2x + 1}+2^{x + 4}+ 32}. Then the value 8(f(115)+f(215)+…+f(5915))8\left( f\left( \frac{1}{15} \right)+ f\left( \frac{2}{15} \right)+ \ldots + f\left( \frac{59}{15} \right) \right) is equal to

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  6. 56
    If α\alpha and β\beta are the roots of the equation 2z2−3z−2i=02z^{2}- 3z - 2i = 0, where i=−1i =\sqrt{- 1}, then 16. Re(α19+β19+α11+β11α15+β15)⋅Im(α19+β19+α11+β11α15+β15)Re\left( \frac{\alpha^{19}+\beta^{19}+\alpha^{11}+\beta^{11}}{\alpha^{15}+\beta^{15}} \right)\cdot Im\left( \frac{\alpha^{19}+\beta^{19}+\alpha^{11}+\beta^{11}}{\alpha^{15}+\beta^{15}} \right) is equal to

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  7. 57
    lim⁡x→0csc⁡x(2cos⁡2x+3cos⁡x−cos⁡2x+sin⁡x+4)\lim_{x \rightarrow 0} \csc x\left( \sqrt{2\cos^{2}x + 3\cos x}-\sqrt{\cos^{2}x + \sin x + 4} \right) is

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  8. 58
    Let in a △ABC\bigtriangleup ABC, the length of the side AC be 6 , the vertex BB be (1,2,3)(1,2,3) and the vertices A,CA,C lie on the line x−63=y−72=z−7−2\frac{x - 6}{3}=\frac{y - 7}{2}=\frac{z - 7}{- 2}. Then the area (in sq. units) of △ABC\bigtriangleup ABC is

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  9. 59
    Let y=y(x)y = y(x) be the solution of the differential equation (xy−5x21+x2)dx+(1+x2)dy=0\left( xy - 5x^{2}\sqrt{1 +x^{2}} \right)dx +\left( 1 +x^{2} \right)dy = 0, y(0)=0y(0) = 0. Then y(3)y(\sqrt{3}) is equal to

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  10. 60
    Let the product of the focal distances of the point (3,12)\left( \sqrt{3},\frac{1}{2} \right) on the ellipse x2a2+y2 b2=1,(a>b)\frac{x^{2}}{a^{2}}+\frac{y^{2}}{{\text{ }b}^{2}}= 1,(a > b), be 74\frac{7}{4}. Then the absolute difference of the eccentricities of two such ellipses is

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  11. 61
    A and B alternately throw a pair of dice. A wins if he throws a sum of 5 before B throws a sum of 8 , and BB wins if he throws a sum of 8 before AA throws a sum of 5 . The probability, that A wins if A makes the first throw, is

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  12. 62
    Consider the region R={(x,y):x≤y≤9−113x2,x≥0}R =\left\{ (x,y):x \leq y \leq 9 -\frac{11}{3}x^{2},x \geq 0 \right\}. The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in RR, is :

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  13. 63
    The area of the region {(x,y):x2+4x+2≤y≤∣x+2∣}\left\{ (x,y):x^{2}+ 4x + 2 \leq y \leq |x + 2| \right\} is equal to

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  14. 64
    For a statistical data x1,x2,…,x10x_{1},x_{2},\ldots,x_{10} of 10 values, a student obtained the mean as 5.5 and ∑i=110xi2=371\sum_{i = 1}^{10} x_{i}^{2}= 371. He later found that he had noted two values in the data incorrectly as 4 and 5 , instead of the correct values 6 and 8 , respectively. The variance of the corrected data is

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  15. 65
    Let circle CC be the image of x2+y2−2x+4y−4=0x^{2}+y^{2}- 2x + 4y - 4 = 0 in the line 2x−3y+5=02x - 3y + 5 = 0 and AA be the point on CC such that OA is parallel to xx-axis and A lies on the right hand side of the centre OO of CC. If B(α,β)B(\alpha,\beta), with β<4\beta < 4, lies on CC such that the length of the arc AB is (1/6)th (1/6)^{\text{th~}} of the perimeter of C , then β−3α\beta -\sqrt{3}\alpha is equal to

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  16. 66
    For some n≠10n \neq 10, let the coefficients of the 5th ,6th 5^{\text{th~}},6^{\text{th~}} and 7th 7^{\text{th~}} terms in the binomial expansion of (1+x)n+4(1 + x)^{n + 4} be in A.P. Then the largest coefficient in the expansion of (1+x)n+4(1 + x)^{n + 4} is :

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  17. 67
    The product of all the rational roots of the equation (x2−9x+11)2−(x−4)(x−5)=3\left( x^{2}- 9x + 11 \right)^{2}- (x - 4)(x - 5) = 3, is equal to :

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  18. 68
    Let the line passing through the points (−1,2,1)( - 1,2,1) and parallel to the line x−12=y+13=z4\frac{x - 1}{2}=\frac{y + 1}{3}=\frac{z}{4} intersect the line x+23=y−32=z−41\frac{x + 2}{3}=\frac{y - 3}{2}=\frac{z - 4}{1} at the point PP. Then the distance of P from the point Q(4,−5,1)Q(4, - 5,1) is :

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  19. 69
    Let the lines 3x−4y−α=0,8x−11y−33=03x - 4y - \alpha = 0,8x - 11y - 33 = 0, and 2x−3y+λ=02x - 3y + \lambda = 0 be concurrent. If the image of the point (1,2)(1,2) in the line 2x−3y+λ=02x - 3y + \lambda = 0 is (5713,−4013)\left( \frac{57}{13},\frac{- 40}{13} \right), then ∣αλ∣|\alpha\lambda| is equal to :

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  20. 70
    If the system of equations
    2x−y+z=45x+λy+3z=12100x−47y+μz=212\begin{matrix} & 2x - y + z = 4 \\ & 5x + \lambda y + 3z = 12 \\ & 100x - 47y + \mu z = 212 \end{matrix}
    has infinitely many solutions, then μ−2λ\mu - 2\lambda is equal to

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  21. 71
    Let f be a differentiable function such that (x+2)2f(x)−3(x+2)2=10∫0x(t+2)f(t)dt(x + 2)^{2}f(x) - 3(x + 2)^{2}= 10\int_{0}^{x} (t + 2)f(t)dt x≥0x \geq 0. Then f(2)f(2) is equal to _____\_\_\_\_\_ .
  22. 72
    If for some α,β;α≤β,α+β=8\alpha,\beta;\alpha \leq \beta,\alpha + \beta = 8 and sec⁡2(tan⁡−1α)+csc⁡2(cot⁡−1β)=36\sec^{2}\left( \tan^{- 1}\alpha \right)+\csc^{2}\left( \cot^{- 1}\beta \right)= 36, then α2+β\alpha^{2}+ \beta is  ____\ \_\_\_\_ .
  23. 73
    The number of 3-digit numbers, that are divisible by 2 and 3 , but not divisible by 4 and 9 , is
  24. 74
    Let be a 3×33 \times 3 matrix such that XTAX=OX^{T}AX = O for all nonzero 3×13 \times 1 matrices X=[xyz]X =\begin{bmatrix} x \\ y \\ z \end{bmatrix}. If A[111]=[14−5],A[121]=[04−8]A\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}=\begin{bmatrix} 1 \\ 4 \\ - 5 \end{bmatrix},A\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}=\begin{bmatrix} 0 \\ 4 \\ - 8 \end{bmatrix}, and det(adj(2( A+I)))=2α3β5γ,α,β,γ,∈Ndet(adj(2(\text{ }A + I))) =2^{\alpha}3^{\beta}5^{\gamma},\alpha,\beta,\gamma, \in N, then α2+β2+γ2\alpha^{2}+\beta^{2}+\gamma^{2} is
  25. 75
    Let S={p1,p2,…,p10}S=\left\{p_{1},p_{2},\ldots,p_{10}\right\} be the set of first ten prime numbers. Let A=S∪PA=S\cup P, where PP is the set of all possible products of distinct elements of SS. Then the number of all ordered pairs (x,y)(x,y), x∈Sx\in S, y∈Ay\in A, such that xx divides yy, is

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