Practice Questions

Q89.Let p and p + 2 be prime numbers and let p! (p + 1)! (p + 2)! Δ = (p + 1)! (p + 2)! (p + 3)! (p + 2)! (p + 3)! (p + 4)! Then the sum of the maximum values of α and β , such that pα and (p + 2)β divide Δ , is _______.

Q89.Let f be a differentiable function satisfying f(x) = 2 ∫√30 f( λ2x3 )dλ, √3 passes through the point (α, 6), then α is equal to _______. → → → →

Q89.The largest value of 𝑎, for which the perpendicular distance of the plane containing the lines →𝑟= ^𝑖+ ^𝑗+ 𝜆 ^𝑖+ 𝑎 ^𝑗- ^𝑘and →𝑟= ^𝑖+ ^𝑗+ 𝜇- ^𝑖+ ^𝑗- 𝑎𝑘 from the point 2, 1, 4 is √3, is ______.

Q89.Let →𝑏= ^𝑖+ ^𝑗+ 𝜆 ^𝑘, 𝜆∈ℝ. If →𝑎 is a vector such that →𝑎× →𝑏= 13 ^𝑖- ^𝑗- 4 ^𝑘 and →𝑎· →𝑏+ 21 = 0, then →𝑏- →𝑎· ^𝑘- ^𝑗+ →𝑏+ →𝑎· ^𝑖- ^𝑘 is equal to 1 1

Q89.Let S = {1, 2, 3, 4} . Then the number of elements in the set { f : S × S →S : f is onto and f(a, b) = f(b, a) ≥a∀(a, b) ∈S × S } is

Q89.Let y = y(x) be the solution curve of the differential equation = 0, 0 < x < √π2 sin(2x2) loge(tan x2)dy + (4xy −4√2x sin(x2 −π4 ))dx , which passes through the point (√π6 , 1). Then y(√π3 ) is equal to _______. y−2

Q90.The line of shortest distance between the lines = = and = = makes an angle of 0 1 1 2 2 1 with the plane 𝑃: 𝑎𝑥- 𝑦- 𝑧= 0, 𝑎> 0. If the image of the point 1, 1, - 5 in the plane 𝑃 is 𝛼, 𝛽, 𝛾, sin-1√ 272 then 𝛼+ 𝛽- 𝛾 is equal to _____ . JEE Main 2022 (25 Jul Shift 1) JEE Main Previous Year Paper

Q90.The plane passing through the line 𝐿: 𝑙 𝑥- 𝑦+ 31 - 𝑙 𝑧= 1, 𝑥+ 2𝑦- 𝑧= 2 and perpendicular to the plane 3𝑥+ 2𝑦+ 𝑧= 6 is 3𝑥- 8𝑦+ 7𝑧= 4. If 𝜃 is the acute angle between the line 𝐿 and the 𝑦-axis, then 415 cos2𝜃 is equal to ______. JEE Main 2022 (26 Jul Shift 2) JEE Main Previous Year Paper

Q90.If the probability that a randomly chosen 6 -digit number formed by using digits 1 and 8 only is a multiple of 21 is p, then 96p is equal to _____. JEE Main 2022 (26 Jun Shift 2) JEE Main Previous Year Paper

Q90.Let the line x−3 7 = −1 = z−3−4 intersect the plane containing the lines x−41 = y+1−2 = 1z and 4ax −y + 5z −7a = 0 = 2x −5y −z −3, a ∈R at the point P(α, β, γ). Then the value of α + β + γ equals ______. JEE Main 2022 (27 Jul Shift 1) JEE Main Previous Year Paper

Q90.Let S = {E, E2 … E8} be a sample space of raddom experiment such that P(En) = 36n for every n = 1, 2 … . 8. Then the number of elements in the set {A ⊂S : P(A) ≥45 } is _____. JEE Main 2022 (27 Jun Shift 2) JEE Main Previous Year Paper

Q2. A particle is projected with velocity v0 along x-axis. A damping force is acting on the particle which is proportional to the square of the distance from the origin i.e. ma = −αx2. The distance at which the particle stops: (1) 2v0 13 (2) 3mv20 1 ( 3α ) ( 2α ) 3 (3) 3v20 12 (4) 2v20 12 ( 2α ) ( 3α )

Q3. The normal reaction N for a vehicle of 800 kg mass, negotiating a turn on a 30° banked road at maximum possible speed without skidding is ____ ×103 kg m s−2 . (1) 10. 2 (2) 7. 2 (3) 12. 4 (4) 6. 96

Q9. Find out the surface charge density at the intersection of point x = 3 m plane and x-axis, in the region of uniform line charge of 8 nC m−1 lying along the z-axis in free space. (1) 0. 424 nC m−2 (2) 47. 88 nC m−2 (3) 0. 07 nC m−2 (4) 4. 0 nC m−2

Q10.Two electrons each are fixed at a distance 2d. A third charge proton placed at the midpoint is displaced slightly by a distance x(x ≪d) perpendicular to the line joining the two fixed charges. Proton will execute simple harmonic motion having angular frequency: ( m = mass of charged particle) (1) πε0md3 12 (2) 2πε0md3 12 ( 2q2 ) ( q2 ) 2 (3) 1 1 ( 2πε0md3q2 ) 2 (4) ( πε0md32q2 )

Q21.A small bob tied at one end of a thin string of length 1 m is describing a vertical circle so that the maximum and minimum tension in the string is in the ratio 5 : 1. The velocity of the bob at the highest position is ______ m s−1 . (Take g = 10 m s−2 ) JEE Main 2021 (25 Feb Shift 1) JEE Main Previous Year Paper

Q23.Consider a badminton racket with length scales as shown in the figure. JEE Main 2021 (26 Aug Shift 1) JEE Main Previous Year Paper If the mass of the linear and circular portions of the badminton racket are same ( M ) and the mass of the threads are negligible, the moment of inertia of the racket about an axis perpendicular to the handle and in the plane of the ring at, r distance from the end A of the handle will be ______ Mr2 . 2

Q23.A rod of mass M and length L is lying on a horizontal frictionless surface. A particle of mass m travelling along the surface hits at one end of the rod with a velocity u in a direction perpendicular to the rod. The collision is completely elastic. After collision, particle comes to rest. The ratio of masses ( Mm ) is x1 . The value of x will be

Q24.Consider an electrical circuit containing a two way switch S. Initially S is open and then T1 is connected to T2. As the current in R = 6 Ω attains a maximum value of steady-state level, T1 is disconnected from T2 and JEE Main 2021 (27 Jul Shift 1) JEE Main Previous Year Paper immediately connected to T3. Potential drop across r = 3 Ω resistor immediately after T1 is connected to T3 is ______V. (Round off to the Nearest Integer)

Q24.A ball of mass 10 kg moving with a velocity 10√3 m s−1 along X -axis, hits another ball of mass 20 kg which is at rest. After the collision, the first ball comes to rest and the second one disintegrates into two equal pieces. One of the pieces starts moving along Y -axis at a speed of 10 m s−1 . The second piece starts moving at a speed of 20 m s−1 at an angle θ (degree) with respect to the X -axis. The configuration of pieces after the collision is shown in the figure. The value of θ to the nearest integer is _________.

Q24.A ball of mass 10 kg moving with a velocity 10√3 m s−1 along the x -axis, hits another ball of mass 20 kg which is at rest. After the collision, first ball comes to rest while the second ball disintegrates into two equal pieces. One piece starts moving along y -axis with a speed of 10 m s−1. The second piece starts moving at an angle of 30° with respect to the x -axis. The velocity of the ball moving at 30° with x -axis is x m s−1. The configuration of pieces after the collision is shown in the figure below. The value of x to the nearest integer is JEE Main 2021 (18 Mar Shift 1) JEE Main Previous Year Paper

Q25.If the maximum value of accelerating potential provided by a radio frequency oscillator is 12 kV . The number of revolution made by a proton in a cyclotron to achieve one sixth of the speed of light is: [mp = 1. 67 × 10−27 kg, e = 1. 6 × 10−19 C, Speed of light = 3 × 108 m s−1]

Q25.A particle of mass 1 mg and charge 𝑞 is lying at the mid-point of two stationary particles kept at a distance 2 m when each is carrying same charge 𝑞. If the free charged particle is displaced from its equilibrium position through distance 𝑥 𝑥< < 1 m. The particle executes SHM. Its angular frequency of oscillation will be _______ × 105 rad s-1 (if 𝑞2 = 10𝐶2)

Q27.A closed organ pipe of length L and an open organ pipe contain gases of densities ρ1 and ρ2 respectively. The compressibility of gases are equal in both the pipes. Both the pipes are vibrating in their first overtone with same frequency. The length of the open pipe is x , where x is _______. 3 L√ρ1ρ2 (Round off to the Nearest Integer)

Q27.Two short magnetic dipoles m1 and m2 each having magnetic moment of 1 A m2 are placed at point O and P respectively. The distance between OP is 1 m . The torque experienced by the magnetic dipole m2 due to the presence of m1 is ________ × 10−7N m