Practice Questions

Q80.If the mirror image of the point (2, 4, 7) in the plane 3x −y + 4z = 2 is (a, b, c), the 2a + b + 2c is equal to (1) 54 (2) −6 (3) 50 (4) −42 ¯

Q80.If the lines →r= (ˆi −ˆj + ˆk) λ(3ˆj −ˆk) and →r (αˆi −ˆj) μ(2ˆi −3ˆk) are co-planar, the the distance of the plane containing these two lines from the point (α, 0, 0) is (1) 2 (2) 2 9 11 (3) 4 (4) 2 11

Q80.If a point A(x, y) lies in the region bounded by the y-axis, straight lines 2y + x = 6 and 5x −6y = 30, then the probability that y < 1 is (1) 16 (2) 56 (3) 2 (4) 6 3 7

Q80.Let the plane ax + by + cz = d pass through (2, 3, −5) and is perpendicular to the planes 2x + y −5z = 10 and 3x + 5y −7z = 12 If a, b, c, d are integers d > 0 and gcd(|a|, |b|, |c|, d) = 1 then the value of a + 7b + c + 20d is equal to JEE Main 2022 (28 Jun Shift 2) JEE Main Previous Year Paper (1) 18 (2) 20 (3) 24 (4) 22 ¯

Q80.Let S be the sample space of all five digit numbers. If p is the probability that a randomly selected number from S , is a multiple of 7 but not divisible by 5 , then 9p is equal to (1) 1. 0146 (2) 1. 2085 (3) 1. 0285 (4) 1. 1521 ¯

Q81.Let S ={ z ∈C : |z −3| ≤1 and z(4 + 3i) + z(4 −3i) ≤24}. If α + iβ is the point in S which is closest to 4i , then 25(α + β) is equal to ______.

Q81.If p and q are real number such that p + q = 3, p4 + q4 = 369 , then the value of −2 ( p1 + 1q ) is equal to is equal to _____.

Q81.Let S = {z ∈C : z2 + z = 0}. Then ∑z∈S(Re (z)+ Im (z)) is equal to _______.

Q81.The total number of four digit numbers such that each of the first three digits is divisible by the last digit, is equal to ______.

Q81.The number of real solutions of the equation e4x + 4e3x −58e2x + 4ex + 1 = 0 is _____.

Q81.If for some p, q, r ∈R, all have positive sign, one of the roots of the equation q2+r2 (p2 + q2)x2 −2q(p + r)x + q2 + r2 = 0 is also a root of the equation x2 + 2x −8 = 0 , then p2 is equal to-

Q81.Let α, β be the roots of the equation x2 −4λx + 5 = 0 and α, γ be the roots of the equation + + 7 + 3λ√3 = 0. If β + γ = 3√2, then (α + 2β + γ)2 is equal to x2 −(3√2 2√3)x is 939,

Q81.Sum of squares of modulus of all the complex numbers z satisfying z = iz2 + z2 −z is equal to

Q81.The total number of three-digit numbers, with one digit repeated exactly two times, is ______. JEE Main 2022 (25 Jun Shift 2) JEE Main Previous Year Paper 3 10

Q81.In an examination, there are 5 multiple choice questions with 3 choices, out of which exactly one is correct. There are 3 marks for each correct answer, −2 marks for each wrong answer and 0 mark if the question is not attempted. Then, the number of ways a student appearing in the examination gets 5 marks is _____ JEE Main 2022 (24 Jun Shift 1) JEE Main Previous Year Paper

Q81.Let z = a + ib, b ≠0 be complex numbers satisfying z2 = ¯z ⋅21−|z| . Then the least value of n ∈N , such that zn = (z + 1)n , is equal to _____ .

Q81.Let S = {4, 6, 9} and T = {9, 10, 11, … , 1000}. If A = {a1 + a2 + … + ak : k ∈N, a1, a2, a3, … , ak ∈S} then the sum of all the elements in the set T −A is equal to _______.

Q81.For a natural number 𝑛, let 𝛼𝑛= 19𝑛- 12𝑛. Then, the value of is ______ 57𝛼8

Q81.The number of ways, 16 identical cubes, of which 11 are blue and rest are red, can be placed in a row so that between any two red cubes there should be at least 2 blue cubes, is ______.

Q81.Let 𝛼, 𝛽𝛼> 𝛽 be the roots of the quadratic equation 𝑥2 - 𝑥- 4 = 0. If 𝑃𝑛= 𝛼𝑛- 𝛽𝑛, 𝑛∈ℕ, then 𝑃15𝑃16 - 𝑃14𝑃16 - 𝑃152 + 𝑃14𝑃15 is equal to _____. 𝑃13𝑃14

Q81.Let S = {z ∈C : |z −2| ≤1, z(1 + i) + z(1 −i) ≤2} . Let |z −4 i| attains minimum and maximum values, + = α + β√5 , where α and β are integers, then the value respectively, at z1 ∈S and z2 ∈S . If 5(|z1|2 |z2|2) of α + β is equal to ______.

Q81.The sum of all real values of 𝑥 for which 3𝑥2 - 9𝑥+ 17 = 5𝑥2 - 7𝑥+ 19 is equal to 𝑥2 + 3𝑥+ 10 3𝑥2 + 5𝑥+ 12

Q81.Let f(x) be a quadratic polynomial with leading coefficient 1 such that f(0) = p, p ≠0 , and f(1) = 31 . If the equations f(x) = 0 and fofofof(x) = 0 have a common real root, then f(−3) is equal to ______. JEE Main 2022 (25 Jul Shift 2) JEE Main Previous Year Paper + = k + 6√3 + 8√6 ,

Q81.Numbers are to be formed between 1000 and 3000, which are divisible by 4, using the digits 1, 2, 3, 4, 5 and 6 without repetition of digits. Then the total number of such numbers is _______.

Q81.The sum of the cubes of all the roots of the equation x4 −3x3 −2x2 + 3x + 1 = 0 is _____.