Practice Questions

Q90.Let 𝑄 and 𝑅 be the feet of perpendiculars from the point 𝑃𝑎, 𝑎, 𝑎 on the lines 𝑥= 𝑦, 𝑧= 1 and 𝑥= −𝑦, 𝑧= −1 respectively. If ∠𝑄𝑃𝑅 is a right angle, then 12𝑎2 is equal to ________ JEE Main 2024 (31 Jan Shift 1) JEE Main Previous Year Paper

Q90.A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tosses required and let a = P(X = 3), b = P(X ≥3) and c = P(X ≥6 ∣X > 3). Then b+ca is equal to JEE Main 2024 (27 Jan Shift 1) JEE Main Previous Year Paper

Q90.Let a line passing through the point ( - 1, 2, 3 ) intersect the lines 𝐿1: 𝑥- 1 = 𝑦- 2 = 𝑧+ 1 at 𝑀( 𝛼, 𝛽, 𝛾) and 3 2 -2 𝑥+ 2 𝑦- 2 𝑧- 1 ( 𝛼+ 𝛽+ 𝛾) 2 equals ________________. = = at 𝑁( 𝑎, 𝑏, 𝑐) . Then the value of 𝐿2: -3 -2 4 ( 𝑎+ 𝑏+ 𝑐) 2 JEE Main 2024 (30 Jan Shift 2) JEE Main Previous Year Paper

Q90.From a lot of 12 items containing 3 defectives, a sample of 5 items is drawn at random. Let the random variable X denote the number of defective items in the sample. Let items in the sample be drawn one by one without replacement. If variance of X is mn , where gcd(m, n) = 1, then n −m is equal to _________ JEE Main 2024 (06 Apr Shift 2) JEE Main Previous Year Paper

Q90.If d1 is the shortest distance between the lines x + 1 = 2 y = −12 z, x = y + 2 = 6 z −6 and d2 is the shortest distance between the lines x−1 2 = y+8−7 = z−45 , x−12 = y−21 = z−6−3 , then the value of 32√3d2 d1 is : JEE Main 2024 (30 Jan Shift 1) JEE Main Previous Year Paper

Q90.Let a, b and c denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked 1, 2, 3, 4. If the probability that ax2 + bx + c = 0 has all real roots is mn , gcd(m, n) = 1, then m + n is equal to ________ JEE Main 2024 (09 Apr Shift 1) JEE Main Previous Year Paper

Q90.Let →a = ^i −3^j + 7^k, b = 2^i −^j + ^k and→cbe a vector such that (→a+ 2b) ×→c= 3(→c×→a) . If →a ⋅→c = 130 , then →b ⋅→c is equal to _______ JEE Main 2024 (05 Apr Shift 1) JEE Main Previous Year Paper

Q90.Let P(α, β, γ) be the image of the point Q(1, 6, 4) in the line x1 = y−12 = z−23 . Then 2α + to_______ JEE Main 2024 (08 Apr Shift 2) JEE Main Previous Year Paper

Q90.Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables X and Y respectively denote the number of blue and yellow balls. If ¯X and ¯Y are the means of X and Y respectively, then 7¯X + 4¯Y is equal to________ JEE Main 2024 (08 Apr Shift 1) JEE Main Previous Year Paper

Q90.In a tournament, a team plays 10 matches with probabilities of winning and losing each match as 1 and 2 3 3 respectively. Let x be the number of matches that the team wins, and y be the number of matches that team loses. If the probability P(|x −y| ≤ 2) is p , then 39p equals ______ JEE Main 2024 (04 Apr Shift 2) JEE Main Previous Year Paper

Q90.Let →𝑎= ^𝑖+ ^𝑗+ ^𝑘, →𝑏= − ^𝑖−8 ^𝑗+ 2 ^𝑘 and →𝑐= 4 ^𝑖+ 𝑐2 ^𝑗+ 𝑐3 ^𝑘 be three vectors such that →𝑏× →𝑎= →𝑐× →𝑎. If the angle between the vector →𝑐 and the vector 3 ^𝑖+ 4 ^𝑗+ ^𝑘 is 𝜃, then the greatest integer less than or equal to tan2𝜃 is: JEE Main 2024 (01 Feb Shift 2) JEE Main Previous Year Paper

Q61.Let a ≠b be two non-zero real numbers. Then the number of elements in the set X = {z ∈C : Re(az2 + bz) = a and Re(bz2 + az) = b} is equal to (1) 0 (2) 1 (3) 3 (4) 2

Q61.Let λ ≠0 be a real number. Let α, β be the roots of the equation 14x2 −31x + 3λ = 0 and α, γ be the roots of the equation 35x2 −53x + 4λ = 0. Then 3αβ and 4αγ are the roots of the equation : (1) 7x2 + 245x −250 = 0 (2) 7x2 −245x + 250 = 0 (3) 49x2 −245x + 250 = 0 (4) 49x2 + 245x + 250 = 0

Q61.The number of real solutions of the equation 3(x2 + x21 ) −2(x + x1 ) + 5 = 0 , is (1) 4 (2) 0 (3) 3 (4) 2 2π 2π 3 1+sin 9 +i cos 9

Q61.The sum of all the roots of the equation 𝑥2 - 8𝑥+ 15 - 2𝑥+ 7 = 0 is (1) 9 - √3 (2) 9 + √3 (3) 11 - √3 (4) 11 + √3

Q61.Let the complex number 𝑧= 𝑥+ 𝑖𝑦 be such that is purely imaginary. If 𝑥+ 𝑦2 = 0, then 𝑦4 + 𝑦2 - 𝑦 is 2𝑧+ 𝑖 equal to (1) 2 (2) 3 3 2 3 4 (3) (4) 4 3

Q61.Let 𝑆= 𝑧= 𝑥+ 𝑖𝑦: is a real number }. Then which of the following is NOT correct? 4𝑧+ 2𝑖 (1) 𝑦+ 𝑥2 + 𝑦2 ≠- 1 (2) (𝑥, 𝑦) = 0, - 1 4 2 (3) 𝑥= 0 (4) 𝑦∈- ∞, - 1 ∪-1 ∞ 2 2,

Q61.Let a ∈R and let α, β be the roots of the equation x2 + 60 41 x + a = 0. If α4 + β4 = −30, then the product of all possible values of a is _____ .

Q61.The number of integral solution 𝑥 of 7 ≥0 is log𝑥+ 2𝑥- 3 2 (1) 7 (2) 8 (3) 6 (4) 5

Q61.Let 𝑥2 - 4 𝑥2 - 4 𝑆= 𝑥: 𝑥∈ℝ and √3 + √2 + √3 - √2 = 10. Then 𝑛𝑆 is equal to (1) 2 (2) 4 (3) 6 (4) 0 𝑧- 2

Q61.The number of integral values of k, for which one root of the equation 2x2 −8x + k = 0 lies in the interval (1, 2) and its other root lies in the interval (2, 3), is : JEE Main 2023 (01 Feb Shift 2) JEE Main Previous Year Paper (1) 2 (2) 0 (3) 1 (4) 3

Q61.The equation e4x + 8e3x + 13e2x −8ex + 1 = 0, x ∈R has : (1) four solutions two of which are negative (2) two solutions and both are negative (3) no solution (4) two solutions and only one of them is negative

Q61.Let α, β, γ be the three roots of the equation x3 + bx + c = 0 if βγ = 1 = −α then b3 + 2c3 −3α3 −6β3 −8γ 3 is equal to (1) 155 (2) 21 8 (3) 169 (4) 19 8

Q62.The complex number z = πi−1 π is equal to: cos 3 +i sin 3 (1) √2i(cos 5π12 −i sin 5π12 ) (2) cos 12π −i sin 12π (3) √2(cos 12π + i sin 12π ) (4) √2(cos 5π12 + i sin 5π12 )

Q62.For two non-zero complex number z1 and z2 , if Re (z1z2) = 0 and Re (z1 + z2) = 0, then which of the following are possible? (A) Im (z1) > 0 and Im (z2) > 0 (B) Im (z1) < 0 and Im (z2) > 0 (C) Im (z1) > 0 and Im (z2) < 0 (D) Im (z1) < 0 and Im (z2) < 0 Choose the correct answer from the options given below: (1) B and D (2) B and C (3) A and B (4) A and C