Practice Questions

Q90.Three dice are rolled. If the probability of getting different numbers on the three dice is p q , where p and q are co-prime, then q −p is equal to (1) 2 (2) 1 (3) 3 (4) 4 JEE Main 2023 (06 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.There rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable X denote the number of rotten apples. If μ and σ2 represent mean and variance of X, respectively, then 10(μ2 + σ2) is equal to (1) 20 (2) 250 (3) 25 (4) 30 JEE Main 2023 (29 Jan Shift 1) JEE Main Previous Year Paper

Q90.Let S = {w1, w2, … . } be the sample space associated to a random experiment. Let P(wn) = P(wn−1)2 , . Let A = {2k + 3l; k, l ∈N} and B = {wn; n ∈A} . Then P(B) is equal to (1) 3 (2) 3 32 64 (3) 1 (4) 1 16 32 JEE Main 2023 (29 Jan Shift 2) JEE Main Previous Year Paper

Q90.In a bolt factory, machines A, B and C manufacture respectively 20%, 30% and 50% of the total bolts. Of their output 3, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found the defective then the probability that it is manufactured by the machine C is (1) 5 (2) 9 14 28 (3) 3 (4) 2 7 7 JEE Main 2023 (08 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.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.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

Q90.Two dice are thrown independently. Let A be the event that the number appeared on the 1st die is less than the number appeared on the 2nd die, B be the event that the number appeared on the 1st die is even and that on the second die is odd, and C be the event that the number appeared on the 1st die is odd and that on the 2nd is even. Then JEE Main 2023 (01 Feb Shift 2) JEE Main Previous Year Paper (1) The number of favourable cases of the event (2) A and B are mutually exclusive (A ∪B) ∩C is 6 (3) The number of favourable cases of the events A , (4) B and C are independent B and C are 15, 6 and 6 respectively JEE Main 2023 (01 Feb 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.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 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 M be the maximum value of the product of two positive integers when their sum is 66 . Let the sample space S = {x ∈Z : x(66 −x) ≥59 M} and the event A = {x ∈S : x is a multiple of 3 }. Then P(A) is equal to (1) 15 (2) 1 44 3 (3) 1 (4) 7 5 22 JEE Main 2023 (25 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 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.A coin is biased so that the head is 3 times as likely to occur as tail. This coin is tossed until a head or three tails occur. If X denotes the number of tosses of the coin, then the mean of X is (1) 38 (2) 15 16 16 (3) 21 (4) 81 16 64 JEE Main 2023 (13 Apr Shift 1) 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.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 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.The urns A, B and C contains 4 red, 6 black; 5 red, 5 black and λ red, 4 black balls respectively. One of the urns is selected at random and a ball is drawn. If the ball drawn is red and the probability that it is drawn from urn C is 0. 4, then the square of length of the side of largest equilateral triangle, inscribed in the parabola y2 = λx with one vertex at vertex of parabola is JEE Main 2023 (24 Jan Shift 2) 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

Q61.If z = 2 + 3i, then z5 + (z)5 is equal to: (1) 244 (2) 224 (3) 245 (4) 265

Q61.If 𝑧≠0 be a complex number such that 𝑧- 𝑧= 2, then the maximum value of 𝑧 is (1) √2 (2) 1 (3) √2 - 1 (4) √2 + 1

Q61.Let f(x) be a quadratic polynomial such that f(−2) +f(3) = 0. If one of the roots of f(x) = 0 is −1, then the sum of the roots of f(x) = 0 is equal to (1) 11 (2) 7 3 3 (3) 12 (4) 14 3 3

Q61.Let α be a root of the equation 1 + x2 + x4 = 0. Then the value of α1011 + α2022 −α3033 is equal to: (1) 1 (2) α (3) 1 + α (4) 1 + 2α