Practice Questions

Q89.The point of intersection C of the plane 8x + y + 2z = 0 and the line joining the points A(−3, −6, 1) and B(2, 4, −3) divides the line segment AB internally in the ratio k : 1. If a, b, c (|a|, |b|, |c| are coprime) are the direction ratios of the perpendicular from the point C on the line 1−x 1 = y+42 = z+23 , then |a + b + c| is equal to _____ .

Q89.Let 𝑦= 𝑦𝑥 be a solution of the differential equation 𝜋 𝜋 𝜋 𝜋 𝜋 is equal to (𝑥 cos𝑥)𝑑𝑦+ (𝑥𝑦 sin𝑥+ 𝑦 cos𝑥- 1)𝑑𝑥= 0, 0 < 𝑥< 2 . If 3𝑦 3 = √3, then 6𝑦" 6 + 2𝑦'𝜋6 _______ .

Q89.Let λ1, λ2 be the values of λ for which the points ( 25 , 1, λ) and (−2, 0, 1) are at equal distance from the plane 2x + 3y −6z + 7 If λ1 > λ2 then the distance of the point (λ1 −λ2, λ2, λ1) from the line x−51 = y−12 = z+72 is ______

Q89.Let P1 be the plane 3x −y −7z = 11 and P2 be the plane passing through the points (2, −1, 0), (2, 0, −1), and (5, 1, 1). If the foot of the perpendicular drawn from the point (7, 4, −1) on the line of intersection of the JEE Main 2023 (08 Apr Shift 2) JEE Main Previous Year Paper planes P1 and P2 is (α, β, γ), then α + β + γ is equal to

Q89.Let 𝑓: -2, 2 →ℝ be defined by 𝑓𝑥= 𝑥𝑥 , -2 < 𝑥< 0 where 𝑥 denotes the greatest integer function. If 𝑥- 1𝑥 , 0 ≤𝑥< 2 𝑚 and 𝑛 respectively are the number of points in – 2, 2 at which 𝑦= 𝑓𝑥 is not continuous and not differentiable, then 𝑚+ 𝑛 is equal to ________.

Q89.Let the tangent at any point P on a curve passing through the points 1, 1 and 10, 100, intersect positive x-axis and y-axis at the points A and B respectively. If 𝑃 𝐴: 𝑃 𝐵= 1: 𝑘 and y = yx is the solution of the 𝑑𝑦 𝑘 differential equation 𝑒 𝑑𝑥= 𝑘𝑥+ 2, 𝑦0 = 𝑘, then 4y1 - 5loge3 is equal to _______________

Q89.Let the equation of the plane passing through the line x −2y −z −5 = 0 = x + y + 3z −5 and parallel to the line x + y + 2z −7 = 0 = 2x + 3y + z −2 be ax + by + cz = 65. Then the distance of the point (a, b, c) from the plane 2x + 2y −z + 16 = 0 is _____ .

Q89.Let N be the sum of the numbers appeared when two fair dice are rolled and let the probability that N −2, √3 N, N + 2 are in geometric progression be 48k . Then the value of k is (1) 2 (2) 4 (3) 16 (4) 8 Q90. 25% of the population are smokers. A smoker has 27 times more chances to develop lung cancer then a non- smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is k .Then the 10 JEE Main 2023 (25 Jan Shift 2) JEE Main Previous Year Paper value of k is _____ . JEE Main 2023 (25 Jan Shift 2) JEE Main Previous Year Paper

Q89.Let a line L pass through the point P(2, 3, 1) and be parallel to the line x + 3y −2z −2 = 0 = x −y + 2z. If the distance of L from the point (5, 3, 8) is α, then 3α2 is equal to ________

Q90.Let the plane P contain the line 2x + y −z −3 = 0 = 5x −3y + 4z + 9 and be parallel to the line x+2 2 = 3−y−4 = z−75 . Then the distance of the point A(8, −1, −19) from the plane P measured parallel to the line −3 x = y−54 = −122−z is equal to _________. JEE Main 2023 (15 Apr Shift 1) JEE Main Previous Year Paper

Q90.Let the foot of perpendicular from the point 𝐴4, 3, 1 on the plane 𝑃: 𝑥- 𝑦+ 2𝑧+ 3 = 0 be 𝑁. If 𝐵( 5, 𝛼, 𝛽) , 𝛼, 𝛽∈ℤ is a point on plane 𝑃 such that the area of the triangle 𝐴𝐵𝑁 is 3√2, then 𝛼2 + 𝛽2 + 𝛼𝛽 is equal to _____________ JEE Main 2023 (10 Apr Shift 2) JEE Main Previous Year Paper

Q90.If 𝜆1 < 𝜆2 are two values of 𝜆 such that the angle between the planes 𝑃1: →𝑟(3 ^i - 5 ^𝑗+ ^𝑘) = 7 and , then the square of the length of perpendicular from the point 𝑃2: →𝑟· (𝜆 ^𝑖+ ^𝑗- 3 ^𝑘) = 9 is sin-12√65 38𝜆1, 10𝜆2, 2 to the plane 𝑃1 is JEE Main 2023 (30 Jan Shift 1) JEE Main Previous Year Paper

Q90.Let a line 𝐿 pass through the origin and be perpendicular to the lines 𝐿1: →𝑟= ^𝑖- 11 ^𝑗- 7 ^𝑘+ 𝜆 ^𝑖+ 2 ^𝑗+ 3 ^𝑘, 𝜆∈ℝ and 𝐿2: →𝑟= - ^𝑖+ ^𝑘+ 𝜇2 ^𝑖+ 2 ^𝑗+ ^𝑘, 𝜇∈ℝ. If 𝑃 is the point of intersection of 𝐿 and 𝐿1, and ,𝑄𝛼, 𝛽, 𝛾 is the foot of perpendicular from 𝑃 on 𝐿2, then 9𝛼+ 𝛽+ 𝛾 is equal to ________. JEE Main 2023 (11 Apr Shift 1) JEE Main Previous Year Paper

Q90.If 𝑦= 𝑦( 𝑥) is the solution of the differential equation 𝑑𝑦 4𝑥 𝑥+ 25 , 𝑥> 1 such that 𝑑𝑥+ 𝑥2 - 1𝑦= 𝑥2 - 1 2 2 𝑦(2) = 9log𝑒2 + √3 and 𝑦√2 = 𝛼log𝑒√𝛼+ 𝛽+ 𝛽- √𝛾, 𝛼, 𝛽, 𝛾∈ℕ, then 𝛼𝛽𝛾 is equal to JEE Main 2023 (13 Apr Shift 2) JEE Main Previous Year Paper

Q90.Let 𝑦= 𝑝𝑥 be the parabola passing through the points – 1, 0, 0, 1 and 1, 0. If the area of the region 𝑥, 𝑦: 𝑥+ 12 + 𝑦- 12 ≤1, 𝑦≤𝑝𝑥 is 𝐴, then 12𝜋- 4𝐴 is equal to ________ . JEE Main 2023 (10 Apr Shift 1) JEE Main Previous Year Paper

Q14.A series LCR circuit has L = 0. 01 H, R = 10 Ω and C = 1 μF and it is connected to ac voltage of amplitude (Vm) 50 V . At frequency 60% lower than resonant frequency, the amplitude of current will be approximately (1) 466 mA (2) 312 mA (3) 238 mA (4) 196 mA

Q21.A pendulum of length 2 m consists of a wooden bob of mass 50 g . A bullet of mass 75 g is fired towards the stationary bob with a speed v. The bullet emerges out of the bob with a speed v and the bob just completes the 3 vertical circle. The value of v is _____ m s−1 (if g = 10 m s−2 ). JEE Main 2022 (27 Jun Shift 1) JEE Main Previous Year Paper

Q22.A small spherical ball of radius 0. 1 mm and density 104 kg m−3 falls freely under gravity through a distance h before entering a tank of water. If, after entering the water the velocity of ball does not change and it JEE Main 2022 (29 Jun Shift 2) JEE Main Previous Year Paper continue to fall with same constant velocity inside water, then the value of h will be ____ m. (Given g = 10 m s−2 , viscosity of water = 1. 0 × 10−5 N‐s m−2 ).

Q22.A rolling wheel of 12 kg is on an inclined plane at position P and connected to a mass of 3 kg through a string of fixed length and pulley as shown in figure. Consider PR as friction free surface. The velocity of centre of mass of the wheel when it reaches at the bottom Q of the inclined plane PQ will be 1 √xgh m s−1 . The value of x (rounded off to the nearest integer) is _____. 2

Q25.An observer is riding on a bicycle and moving towards a hill at 18 km h-1. He hears a sound from a source at some distance behind him directly as well as after its reflection from the hill. If the original frequency of the sound as emitted by source is 640 Hz and velocity of the sound in air is 320 m s-1, the beat frequency between the two sounds heard by observer will be _____ Hz.

Q26.The potential energy of a particle of mass 4 kg in motion along the x-axis is given by U = 4(1 −cos 4x) J. The time period of the particle for small oscillation (sin θ ≃θ) ( Kπ ) s. The value of K is _____ .

Q27.A ray of light is incident at an angle of incidence 60° on the glass slab of refractive index √3. After refraction, the light ray emerges out from other parallel faces and lateral shift between incident ray and emergent ray is 4√3 cm . The thickness of the glass slab is _____ cm .

Q27.In the given circuit 'a' is an arbitrary constant. The value of m for which the equivalent circuit resistance is minimum, will be √x2 . The value of x is _____ .

Q28.A convex lens of focal length 20 cm is placed in front of convex mirror with principal axis coinciding each other. The distance between the lens and mirror is 10 cm. A point object is placed on principal axis at a distance of 60 cm from the convex lens. The image formed by combination coincides the object itself. The focal length of the convex mirror is _____ cm.

Q29.An AC source is connected to an inductance of 100 mH , a capacitance of 100 μF and a resistance of 120 Ω as shown in figure. The time in which the resistance having a thermal capacity 2 J ∘C−1 will get heated by 16°C is _____ s.