Practice Questions

Q89.Let α|x| = |y|exy−β, α, β ∈N be the solution of the differential equation x dy −y dx + xy(x dy + y dx) = 0, y(1) = 2. Then α + β is equal to ________ β + γ is equal

Q89.Let 𝑌= 𝑌( 𝑋) be a curve lying in the first quadrant such that the area enclosed by the line 𝑌- 𝑦= 𝑌' (𝑥) (𝑋- 𝑥) and the co-ordinate axes, where ( 𝑥, 𝑦) is any point on the curve, is always -𝑦2 + 1, 𝑌'𝑥≠0. If 𝑌( 1 ) = 1, then 12 𝑌( 2 ) equals ________. 2𝑌' (𝑥)

Q89.Let →a = 9^i −13^j + 25^k,→b = 3^i + 7^j −13^k and →c = 17^i −2^j + ^k be three given vectors. If →r is a vector such |593→r+67→a|2 is equal to___________ that →r × →a = (→b + →c) × →a and →r ⋅(→b −→c) = 0 , then (593)2

Q89.Let the set of all positive values of λ , for which the point of local minimum of the function (1 + x (λ2 −x2)) satisfies x2+x+2 < 0, be (α, β). Then α2 + β2 is equal to _________ x2+5x+6

Q89.If the solution curve, of the differential equation 𝑑𝑦 𝑥+ 𝑦- 2 𝑑𝑥= 𝑥- 𝑦 passing through the point ( 2, 1 ) is tan-1𝑦- 1 - 1 𝑦- 1 2 = 1, then 5𝛽+ 𝛼 is equal to 𝑥- 1 𝛽log𝑒𝛼+ 𝑥- 1 log𝑒𝑥- x - 2 y z - 7 x + 3 y + 2 z + 2

Q89.Let y = y(x) be the solution of the differential equation dx + (1+x2)2 − 1 Then the area enclosed by the curve f(x) = y(x)e (1+x2) and the line y −x = 4 is__________

Q90.A line with direction ratio 2, 1, 2 meets the lines x = y + 2 = z and x + 2 = 2y = 2z respectively at the point P and Q. if the length of the perpendicular from the point (1, 2, 12) to the line PQ is l, then l2 is JEE Main 2024 (29 Jan Shift 1) JEE Main Previous Year Paper

Q90.The lines = = and = = intersect at the point P. If the distance of P from the line 2 -2 16 4 3 1 x + 1 y - 1 = = z - 1 is 𝑙, then 14𝑙2 is equal to _____. 2 3 1 JEE Main 2024 (27 Jan Shift 2) JEE Main Previous Year Paper

Q90.Let the point (−1, α, β) lie on the line of the shortest distance between the lines x+2−3 = y−24 = z−52 and y+6 x+2 −1 = 2 = z−10 . Then (α −β)2 is equal to___________ JEE Main 2024 (05 Apr Shift 2) JEE Main Previous Year Paper

Q90.A line passes through 𝐴4, −6, −2 and 𝐵16, −2, 4. The point 𝑃𝑎, 𝑏, 𝑐 where 𝑎, 𝑏, 𝑐 are non-negative integers, on the line 𝐴𝐵 lies at a distance of 21 units, from the point 𝐴. The distance between the points 𝑃𝑎, 𝑏, 𝑐 and 𝑄4, −12, 3 is equal to ______. JEE Main 2024 (31 Jan Shift 2) JEE Main Previous Year Paper

Q90.Let the line of the shortest distance between the lines 𝐿1: →𝑟= ^𝑖+ 2 ^𝑗+ 3 ^𝑘+ 𝜆 ^𝑖− ^𝑗+ ^𝑘 and 𝐿2: →𝑟= 4 ^𝑖+ 5 ^𝑗+ 6 ^𝑘+ 𝜇 ^𝑖+ ^𝑗− ^𝑘 intersect 𝐿1 and 𝐿2 at 𝑃 and 𝑄 respectively. If 𝛼, 𝛽, 𝛾 is the midpoint of the line segment 𝑃𝑄, then 2𝛼+ 𝛽+ 𝛾 is equal to___________ JEE Main 2024 (01 Feb Shift 1) JEE Main Previous Year Paper

Q4. A car is moving on a circular path of radius 600 m such that the magnitudes of the tangential acceleration and centripetal acceleration are equal. The time taken by the car to complete first quarter of revolution, if it is 2 ) s . The value of t is ______. moving with an initial speed of 54 km h−1 is t(1–e– π

Q6. A body of mass 2 kg is initially at rest. It starts moving unidirectionally under the influence of a source of constant power P . Its displacement in 4 s is 31 α2√P m. The value of α will be ______. JEE Main 2023 (30 Jan Shift 2) JEE Main Previous Year Paper

Q7. Two discs of same mass and different radii are made of different materials such that their thicknesses are 1 cm and 0. 5 cm respectively. The densities of materials are in the ratio 3 : 5 . The moment of inertia of these discs respectively about their diameters will be in the ratio of x . The value of x is ______. 6

Q8. A uniform disc of mass 0. 5 kg and radius r is projected with velocity 18 m s−1 at t = 0 s on a rough horizontal surface. It starts off with a purely sliding motion at t = 0 s. After 2 s it acquires a purely rolling motion (see figure). The total kinetic energy of the disc after 2 s will be ______ J. (given, coefficient of friction is 0. 3 and g = 10 m s−2 ).

Q11.A body cools from 80oC to 60o in 5 minutes. The temperature for the surrounding is 20oC. The time it takes to cool from 60oC to 40oC is (1) 450 s (2) 420 s (3) 500 s (4) 253 s

Q12.Two long straight wires 𝑃 and 𝑄 carrying equal current 10 A each were kept parallel to each other at 5 cm distance. Magnitude of magnetic force experienced by 10 cm length of wire 𝑃 is 𝐹1. If distance between wires is halved and currents on them are doubled, force 𝐹2 on 10 cm length of wire 𝑃 will be: (1) 8𝐹1 (2) 10 𝐹1 𝐹1 𝐹1 (3) (4) 8 10

Q14.A wire of density 8 × 103 kg m−3 is stretched between two clamps 0. 5 m apart. The extension developed in the wire is 3. 2 × 10−4 m. If Y = 8 × 1010 N m−2 , the fundamental frequency of vibration in the wire will be _____ Hz

Q16.Two parallel plate capacitors C1 and C2 each having capacitance of 10 μF are individually charged by a 100 V D.C. source. Capacitor C1 is kept connected to the source and a dielectric slab is inserted between it plates. Capacitor C2 is disconnected from the source and then a dielectric slab is inserted in it. Afterwards the capacitor C1 is also disconnected from the source and the two capacitors are finally connected in parallel combination. The common potential of the combination will be _____ V . (Assuming Dielectric constant = 10 )

Q16.For a charged spherical ball, electrostatic potential inside the ball varies with r as V = 2ar2 + b. Here, a and b are constant and r is the distance from the center. The volume charge density inside the ball is −λaε. The value of λ is ______. ε = permittivity of medium.

Q16.A thin infinite sheet charge and an infinite line charge of respective charge densities +σ and +λ are placed parallel at 5 m distance from each other. Points P and Q are at π3 m and π4 m perpendicular distances from line charge towards sheet charge, respectively. Ep and Eq are the magnitudes of resultant electric field intensities at point P and Q respectively. If EP = a4 for 2|σ| = |λ|, then the value of a is _____. EQ

Q16.As shown in figure, a cuboid lies in a region with electric field E = 2x2ˆi −4yˆj + 6ˆk N C−1 . The magnitude of charge within the cuboid is nε0 C. The value of n is ______ (if dimension of cuboid is 1 × 2 × 3 m3 )

Q16.Experimentally it is found that 12. 8 eV energy is required to separate a hydrogen atom into a proton and an electron. So the orbital radius of the electron in a hydrogen atom is 9 × 10−10 m. x The value of the x is: _____. ( 1 eV = 1. 6 × 10−19 J, 1 = 9 × 109 Nm2C2 and electronic charge = 1. 6 × 10−19 C) 4πε0

Q17.In the given circuit. C1 = 2μF, C2 = 0. 2μF, C3 = 2μF, C4 = 4μF, C5 = 2μF, C6 = 2μF. The charge stored on capacitor C4 is ______ μC. JEE Main 2023 (11 Apr Shift 2) JEE Main Previous Year Paper

Q19.Given below are two statements: Statement I : The diamagnetic property depends on temperature. Statement II : The induced magnetic dipole moment in a diamagnetic sample is always opposite to the magnetising field. In the light of given statements, choose the correct answer from the options given below (1) Both Statement I and Statement II are False (2) Statement I is incorrect but Statement II is true (3) Statement I is correct but Statement II is false (4) Both Statement I and Statement II are true