Practice Questions

Q79.If the foot of the perpendicular from the point A(−1, 4, 3) on the plane P : 2x + my + nz = 4, is (−2, 72 , 32 ), then the distance of the point A from the plane P , measured parallel to a line with direction ratios 3, −1, −4, is equal to (1) 1 (2) √26 (3) 2√2 (4) √14

Q79.Let Q be the mirror image of the point P(1, 2, 1) with respect to the plane x + 2y + 2z = 16 . Let T be a λ ∈R. Then, which of the plane passing through the point Q and contains the line →r= −ˆk + λ(ˆi + ˆj + 2ˆk), following points lies on T ? (1) (2, 1, 0) (2) (1, 2, 1) (3) (1, 2, 2) (4) (1, 3, 2)

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.A biased die is marked with numbers 2, 4, 8, 16, 32, 32 on its faces and the probability of getting a face with 1 mark 𝑛 is 𝑛. If the die is thrown thrice, then the probability, that the sum of the numbers obtained is 48, is (1) 7 (2) 7 211 212 3 13 (3) (4) 210 212

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

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.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.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.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 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 ______.

Q82.Let A( √a3 , √a), a > 0 , be a fixed point in the xy-plane. The image of A in y-axis be B and the image of B in x-axis be C . If D(3 cos θ, a sin θ), is a point in the fourth quadrant such that the maximum area of ΔACD is 12 square units, then a is equal to _____

Q82.Let b1b2b3b4 be a 4-element permutation with bi ∈{1, 2, 3, … … … , 100} for 1 ≤i ≤4 and bi ≠bj for i ≠j , such that either b1, b2, b3 are consecutive integers or b2, b3, b4 are consecutive integers. Then the number of such permutations b1b2b3b4 is equal to ______.

Q82.The number of natural numbers lying between 1012 and 23421 that can be formed using the digits 2, 3, 4, 5, 6 (repetition of digits is not allowed) and divisible by 55 is _____.

Q82.The number of 5 -digit natural numbers, such that the product of their digits is 36 , is

Q82.Let f(x) = 2x2 −x −1 and S = {n ∈Z : |f(n)| ≤800} . Then, the value of ∑n∈S f(n) is equal to _______.

Q83.Let A = ∑10i=1 ∑10j=1 min{i, j} and B = ∑10i=1 ∑10j=1 max{i, j}. Then A + B is equal to _____.

Q83.A common tangent T to the curves C1 : x24 + y29 = 1 and C2 : x242 − 143y2 = 1 quadrant. If T touches C1 at (x1, y1) and C2 at (x2, y2), then |2x1 + x2| is equal to _______. JEE Main 2022 (27 Jul Shift 2) JEE Main Previous Year Paper Q84. ⎡ α β γ ⎤ Consider a matrix A = α2 β2 γ 2 , where α, β, γ are three distinct natural numbers. ⎣β + γ γ + α α + β⎦ If det(adj(adj(adj(adjA))) = 232 × 316 , then the number of such 3 - tuples (α, β, γ) is _______. (α−β)16(β−γ)16(γ−α)16

Q83.Let a circle C of radius 5 lie below the x-axis. The line L1 = 4x + 3y + 2 passes through the centre P of the circle C and intersects the line L2 : 3x −4y −11 = 0 at Q . The line L2 touches C at the point Q . Then the distance of P from the line 5x −12y + 51 = 0 is

Q83.If two tangents drawn from a point (α, β) lying on the ellipse 25x2 + 4y2 = 1 to the parabola y2 = 4x are such that the slope of one tangent is four times the other, then the value of (10α + 5)2 + (16β2 + 50)2 equals ______

Q83.Let A = {1, 2, 3, 4, 5, 6, 7} . Define B ={ T ⊆A : either 1 ∉T or 2 ∈T } and C ={ T ⊆A : T the sum of all the elements of T is a prime number.} Then the number of elements in the set B ∪C is _______. Q84. 1 a a 1 48 2160 Let A = ⎡0 1 b ⎤ , a, b ∈R. If for some n ∈N, An = ⎡0 1 96 ⎤ then n + a + b is equal to _______. 0 0 1 0 0 1 ⎣ ⎦ ⎣ ⎦

Q83.Different A.P.'s are constructed with the first term 100, the last term 199, And integral common differences. The sum of the common differences of all such, A.P's having at least 3 terms and at most 33 terms is.