Practice Questions

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

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.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.A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is p. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colours is q . If p : q = m : n, where m and n are co-prime, then m + n is equal to JEE Main 2023 (30 Jan Shift 2) JEE Main Previous Year Paper

Q90.Let the image of the point 𝑃( 1, 2, 3 ) in the plane 2𝑥– 𝑦+ 𝑧= 9 be 𝑄. If the coordinates of the point 𝑅 are ( 6, 10, 7 ) , then the square of the area of the triangle 𝑃𝑄𝑅 is _______ . JEE Main 2023 (06 Apr Shift 1) 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.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.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 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 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 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

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 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.The shortest distance between the lines = = and = = is equal to ______ 3 2 2 3 2 0 JEE Main 2023 (24 Jan Shift 1) JEE Main Previous Year Paper