Practice Questions

Q90.Let P be the point (10, −2, −1) and Q be the foot of the perpendicular drawn from the point R(1, 7, 6) on the line passing through the points (2, −5, 11) and (−6, 7, −5). Then the length of the line segment PQ is equal to ________ JEE Main 2024 (06 Apr Shift 1) JEE Main Previous Year Paper

Q90.Let the point (−1, α, β) lie on the line of the shortest distance between the lines x+2−3 = y−24 = z−52 and y+6 x+2 −1 = 2 = z−10 . Then (α −β)2 is equal to___________ JEE Main 2024 (05 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.The square of the distance of the image of the point (6, 1, 5) in the line x−13 = 2y = z−24 , from the origin is _________ JEE Main 2024 (09 Apr Shift 2) JEE Main Previous Year Paper

Q90.Let O be the origin, and M and N be the points on the lines x−5 4 = y−41 = z−53 and x+812 = y+25 = z+119 −−→ → respectively such that MN is the shortest distance between the given lines. Then OM ⋅ON is equal to _________. JEE Main 2024 (29 Jan Shift 2) JEE Main Previous Year Paper

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.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 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.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

Q61.Let w = zz + k1z + k2iz + λ(1 + i), k1, k2 ∈R. . Let Re(w) = 0 be the circle C of radius 1 in the first quadrant touching the line y = 1 and the y−axis. If the curve Im(w) = 0 intersects C at A and B, then 30(AB)2 is equal to _______. JEE Main 2023 (13 Apr Shift 1) JEE Main Previous Year Paper

Q61.The number of points, where the curve f(x) = e8x −e6x −3e4x −e2x + 1, x ∈R cuts x-axis, is equal to............ ¯¯¯¯

Q61.Let m and n be the numbers of real roots of the quadratic equations x2 −12x + [x] + 31 = 0 and x2 −5 x + 2 −4 = 0 respectively, where [x] denotes the greatest integer ≤x. Then m2 + mn + n2 is equal to

Q61.Let α1, α2, … , α7α1, α2, … , α7 be the roots of the equation x7 + 3x5 −13x3 −15x = 0 and |α1| ≥|α2| ≥… ≥|α7|. Then, α1α2 −α3α4 + α5α6 is equal to _______ ¯

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.Let S = {α : log2(92α−4 + 13) −log2( 25 ⋅32α−4 + 1) = 2}. Then the maximum value of β for which the equation x2 −2(∑α∈s α) 2x + ∑a∈s (α + 1)2β = 0 has real roots, is _____ .

Q62.The number of ways of selecting two numbers a and b, a ∈{2, 4, 6, … … , 100} and b ∈{1, 3, 5, … … , 99} such that 2 is the remainder when a + b is divided by 23 is (1) 186 (2) 54 (3) 108 (4) 268 JEE Main 2023 (30 Jan Shift 2) JEE Main Previous Year Paper

Q62.The number of seven digit positive integers formed using the digits 1, 2, 3 and 4 only and sum of the digits equal to 12 is _______.

Q62.Let α = 8 −14i, A = {z ∈C : z2−(¯z)2−112iαz−α¯z = 1} and B = {z ∈C : |z + 3i| = 4} Then, ∑z∈A∩B(Re z −Imz) is equal to ________

Q62.For α, β, z ∈C and λ > 1 , if √λ −1 is the radius of the circle |z −α|2 + |z −β|2 = 2λ, then |α −β| is equal to _____.

Q63.If all the six digit numbers x1x2x3x4x5x6 with 0 < x1 < x2 < x3 < x4 < x5 < x6 are arranged in the increasing order, then the sum of the digits in the 72th number is _______.

Q63.Let x and y be distinct integers where 1 ≤x ≤25 and 1 ≤y ≤25. Then, the number of ways of choosing x and y, such that x + y is divisible by 5 , is _____ .

Q63.The number of seven digits odd numbers, that can be formed using all the seven digits 1, 2, 2, 2, 3, 3, 5 is

Q63.Number of integral solutions to the equation x + y + z = 21 , where x ≥1, y ≥3, z ≥4 , is equal to _____ .

Q64.Five digit numbers are formed using the digits 1, 2, 3, 5, 7 with repetitions and are written in descending order with serial numbers. For example, the number 77777 has serial number 1 . Then the serial number of 35337 is

Q64.Let the digits a, b, c be in A.P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed? = p1 p2 p3 . . . pm , where