Practice Questions

Q89.If the shortest distance between the lines x+√6 2 = 3 = z−√64 and x−λ3 = y−2√64 = z+2√65 is 6 , then sum of squares of all possible values(s) of λ is

Q89.Two dice A and B are rolled. Let the numbers obtained on A and B be α and β respectively. If the variance of α −β is pq , where p and q are co-prime, then the sum of the positive divisors of p is equal to (1) 72 (2) 36 (3) 48 (4) 31

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 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.The foot of perpendicular from the origin O to a plane P which meets the co-ordinate axes at the point A , B, C is (2, a, 4), a ∈N . If the volume of the tetrahedron OABC is 144 unit3 , then which of the following points is NOT on P ? (1) (0, 4, 4) (2) (3, 0, 4) (3) (0, 6, 3) (4) (2, 2, 4)

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 𝑓𝑛= ∫0 2 ∑𝑘=𝑛 1 sin𝑘- 1𝑥∑𝑘=𝑛 1 (2𝑘- 1)sin𝑘- 1𝑥cos𝑥𝑑𝑥, 𝑛∈ℕ. Then 𝑓21 - 𝑓20 is equal to

Q89.Let the image of the point ( 53 , 53 , 83 ) in the plane x − 2y + z–2 = 0 be P. If the distance of the point Q(6, −2, α), α > 0, from P is 13, then α is equal to _______. JEE Main 2023 (13 Apr Shift 1) JEE Main Previous Year Paper

Q89.The value of 12 ∫0 𝑥2 - 3𝑥+ 2dx is ______ 𝑥- 2 𝑦+ 1 𝑧- 6 𝑥- 6 1 - 𝑦 𝑧+ 8

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.If the line x = y = z intersects the line x sin A + y sin B + z sin C −18 = 0 = x sin 2A + y sin 2B + z sin 2C −9, where A, B, C are the angles of a triangle ABC , then 80(sin A2 sin B2 sin C2 ) is equal to _________.

Q89.If the lines x−1 1 = y−22 = z+31 and x−a2 = y+23 = z−31 intersects at the point P , then the distance of the point P from the plane z = a is : (1) 16 (2) 28 (3) 10 (4) 22 n ≥2

Q89.Let the line L : x = 1−y−2 = z−3λ , λ ∈R meet the plane P : x + 2 y + 3 z = 4 at the point (α, β, γ). If the angle between the line L and the plane P is , then α + 2β + 6γ is equal to 14 cos−1(√5 )

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 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 ________

Q89.Let the line 𝐿: = = intersect the plane 2𝑥+ 𝑦+ 3𝑧= 16 at the point 𝑃. Let the point 𝑄 be the 2 -1 1 foot of perpendicular from the point 𝑅1, - 1, - 3 on the line 𝐿. If 𝛼 is the area of triangle 𝑃𝑄𝑅. then 𝛼2 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 →𝑣= 𝛼 ^𝑖+ 2 ^𝑗- 3 ^𝑘, →𝑤= 2𝛼 ^𝑖+ ^𝑗- ^𝑘, and →𝑢 be a vector such that →𝑢= 𝛼> 0. If the minimum value of the 2 𝑚 where 𝑚 and 𝑛 are coprime natural numbers, then scalar triple product →𝑢 →𝑣 →𝑤 is -𝛼√3401, and →𝑢. ^𝑖 = 𝑛 𝑚+ 𝑛 is equal to _____ . JEE Main 2023 (01 Feb Shift 1) JEE Main Previous Year Paper Q90.𝐴2, 6, 2, 𝐵-4, 0, 𝜆, 𝐶2, 3, - 1 and 𝐷4, 5, 0, 𝜆≤5 are the vertices of a quadrilateral 𝐴𝐵𝐶𝐷. If its area is 18 square units, then 5 - 6𝜆 is equal to _____ . JEE Main 2023 (01 Feb Shift 1) JEE Main Previous Year Paper

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.If the lines x−1 2 = 2−y−3 = z−3α and x−45 = y−12 = βz intersect, then the magnitude of the minimum value of 8αβ is _____. JEE Main 2023 (06 Apr Shift 2) JEE Main Previous Year Paper

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.For 𝑚, 𝑛> 0, let 𝛼𝑚, 𝑛= ∫0 𝑡𝑚1 + 3𝑡𝑛𝑑𝑡. If ,11𝛼10, 6 + 18𝛼11, 5 = 𝑝146, then 𝑝 is equal to JEE Main 2023 (11 Apr Shift 1) JEE Main Previous Year Paper

Q89.If the equation of the plane passing through the point ( 1, 1, 2 ) and perpendicular to the line 𝑥- 3𝑦+ 2𝑧- 1 = 0 = 4𝑥- 𝑦+ 𝑧 is 𝐴𝑥+ 𝐵𝑦+ 𝐶𝑧= 1, then 140 ( 𝐶- 𝐵+ 𝐴) is equal to

Q89.Fifteen football players of a club-team are given 15 T-shirts with their names written on the backside. If the players pick up the T-shirts randomly, then the probability that at least 3 players pick the correct T-shirt is (1) 5 (2) 2 24 15 (3) 1 (4) 5 6 36

Q90.If the probability that the random variable X takes values x is given by P(X = x) = k(x + 1)3−x , x = 0, 1, 2, 3, … … , where k is a constant, then P(X ≥2) is equal to (1) 7 (2) 7 27 18 (3) 11 (4) 20 18 27 JEE Main 2023 (08 Apr Shift 2) JEE Main Previous Year Paper