Practice Questions

Q79.Let →a be a vector which is perpendicular to the vector 3ˆi + 2 1 ˆj + 2ˆk. If →a× (2ˆi ˆk) the projection of the vector →a on the vector 2ˆi + 2ˆj + ˆk is (1) 1 (2) 1 3 (3) 5 (4) 7 3 3

Q79.Let 𝑃 be the plane passing through the intersection of the planes →𝑟· ^𝑖+ 3 ^𝑗- ^𝑘= 5 and →𝑟· 2 ^𝑖- ^𝑗+ ^𝑘= 3, and the point 2, 1, - 2. Let the position vectors of the points 𝑋 and 𝑌 be ^𝑖- 2 ^𝑗+ 4 ^𝑘 and 5 ^𝑖- ^𝑗+ 2 ^𝑘 respectively. Then the points (1) 𝑋 and 𝑋+ 𝑌 are on the same side of 𝑃 (2) 𝑌 and 𝑌- 𝑋 are on the opposite sides of 𝑃 (3) 𝑋 and 𝑌 are on the opposite sides of 𝑃 (4) 𝑋+ 𝑌 and 𝑋- 𝑌 are on the same side of 𝑃

Q79.If the plane 2x + y −5z = 0 is rotated about its line of intersection with the plane 3x −y + 4z −7 = 0 by an angle of π , then the plane after the rotation passes through the point 2 (1) (2, −2, 0) (2) (−2, 2, 0) (3) (1, 0, 2) (4) (−1, 0, −2) + = +

Q79.The foot of the perpendicular from a point on the circle 𝑥2 + 𝑦2 = 1, 𝑧= 0 to the plane 2𝑥+ 3𝑦+ 𝑧= 6 lies on which one of the following curves? (1) 6𝑥+ 5𝑦- 122 + 43𝑥+ 7𝑦- 82 = 1, 𝑧= 6 - 2𝑥- 3𝑦(2) 5𝑥+ 6𝑦- 122 + 43𝑥+ 5𝑦- 92 = 1, 𝑧= 6 - 2𝑥- 3𝑦 (3) 6𝑥+ 5𝑦- 142 + 93𝑥+ 5𝑦- 72 = 1, 𝑧= 6 - 2𝑥- 3𝑦(4) 5𝑥+ 6𝑦- 142 + 93𝑥+ 7𝑦- 82 = 1, 𝑧= 6 - 2𝑥- 3𝑦

Q79.Let the plane 2x + 3y + z + 20 = 0 be rotated through a right angle about its line of intersection with the plane x −3y + 5z = 8 . If the mirror image of the point (2, −12 , 2) in the rotated plane is B(a, b, c), then (1) a 8 = 5b = −4c (2) a4 = 5b = −2c (3) a 8 = −5b = 4c (4) a4 = 5b = 2c JEE Main 2022 (26 Jun Shift 1) JEE Main Previous Year Paper

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.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 ¯

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

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.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 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.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.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.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 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 _____ .

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 _______.

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

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.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.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 _____

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