Practice Questions

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

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 𝜃 be the angle between the planes 𝑃1 = →𝑟· ^𝑖+ ^𝑗+ 2 ^𝑘= 9 and 𝑃2 = →𝑟· 2 ^𝑖- ^𝑗+ ^𝑘= 15. Let L be the line that meets 𝑃2 at the point 4, - 2, 5 and makes an angle 𝜃 with the normal of 𝑃2. If 𝛼 is the angle between 𝐿 and 𝑃2 then tan2𝜃cot2𝛼 is equal to _____ . JEE Main 2023 (31 Jan Shift 1) JEE Main Previous Year Paper

Q90.Let A be the event that the absolute difference between two randomly chosen real numbers in the sample space [0, 60] is less than or equal to a. If P(A) = 1136 , then a is equal to _____ . JEE Main 2023 (31 Jan Shift 2) JEE Main Previous Year Paper

Q90.Let the probability of getting head for a biased coin be 1 . It is tossed repeatedly until a head appears. Let N 4 be the number of tosses required. If the probability that the equation 64x2 + 5Nx + 1 = 0 has no real root is p , where p and q are co-prime, then q −p is equal to.......... q JEE Main 2023 (11 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.A fair n (n > 1) faces die is rolled repeatedly until a number less than n appears. If the mean of the number of tosses required is n , then n is equal to 9 JEE Main 2023 (12 Apr 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.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