Practice Questions

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

Q62.Let 𝑤1 be the point obtained by the rotation of 𝑧1 = 5 + 4𝑖 about the origin through a right angle in the anticlockwise direction, and 𝑤2 be the point obtained by the rotation of 𝑧2 = 3 + 5𝑖 about the origin through a right angle in the clockwise direction. Then the principal argument 𝑤1 - 𝑤2 is equal to (1) 𝜋- tan-18 (2) -𝜋+ tan-133 9 5 (3) -𝜋+ tan-18 (4) 𝜋- tan-133 9 5

Q62.Let z1 = 2 + 3i and z2 = 3 + 4i . The set S = {z ∈C : |z −z1|2 −|z −z2|2 = |z1 −z2|2} represents a (1) straight line with sum of its intercepts on the (2) hyperbola with the length of the transverse axis 7 coordinate axes equals 14 (3) straight line with the sum of its intercepts on the (4) hyperbola with eccentricity 2 coordinate axes equals −18

Q62.Let the first term a and the common ratio 𝑟 of a geometric progression be positive integers. If the sum of squares of its first three terms is 33033, then the sum of these three terms is equal to (1) 241 (2) 231 (3) 210 (4) 220 1 13 1 13

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.Let z be a complex number such that z−2iz+i = 2, z ≠−i. Then z lies on the circle of radius 2 and centre (1) (2, 0) (2) (0, 2) (3) (0, 0) (4) (0, −2)

Q62.For all 𝑧∈𝐶 on the curve 𝐶1: | 𝑧| = 4, let the locus of the point z + 1 be the curve 𝐶2. Then z (1) the curves C1 and C2intersect at 4 points (2) the curves 𝐶1 lies inside 𝐶2 (3) the curves 𝐶1 and 𝐶2 intersect at 2 points (4) the curves 𝐶2 lies inside 𝐶1

Q62.Eight persons are to be transported from city A to city B in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is (1) 1120 (2) 3360 (3) 1680 (4) 560 1

Q62.If the center and radius of the circle = 2 are respectively 𝛼, 𝛽 and 𝛾, then 3𝛼+ 𝛽+ 𝛾 is equal to 𝑧- 3 (1) 11 (2) 9 (3) 10 (4) 12

Q62.For a ∈C, let A = {z ∈C :Re (a + z) >Im (a + z)} and B = {z ∈C :Re (a + z) <Im (a + z)} . Then among the two statements: (S1) : If Re (a), Im (a) > 0, then the set A contains all the real numbers (S2) : If Re (a), Im (a) < 0, then the set B contains all the real numbers, (1) Only (S2) is true (2) only (S1) is true (3) Both are true (4) Both are false z2+8iz−15 : α −1311 i ∈S, α ∈R −{0}, then 242α2 is equal to

Q62.If 𝑎𝑛= 4𝑛2 - 16𝑛+ 15, then 𝑎1 + 𝑎2 + … . + 𝑎25 is equal to: (1) 51 (2) 49 144 138 50 52 (3) (4) 141 147 1 15

Q62.If for z = α + iβ, |z + 2| = z + 4(1 + i), then α + β and αβ are the roots of the equation (1) x2 + 3x −4 = 0 (2) x2 + 7x + 12 = 0 (3) x2 + x −12 = 0 (4) x2 + 2x −3 = 0

Q62.Let C be the circle in the complex plane with centre z0 = 12 (1 + 3i) and radius r = 1. Let z1 = 1 + i and the complex number z2 be outside circle C such that |z1 −z0||z2 −z0| = 1 . If z0, z1 and z2 are collinear, then the smaller value of |z2|2 is equal to (1) 5 (2) 7 2 2 (3) 13 (4) 3 2 2

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 _____.

Q62.Let A = {θ ∈(0, 2π) : 1+2i1−i sinsinθθ is purely imaginary} Then the sum of the elements is in A is (1) 4π (2) 3π (3) π (4) 2π

Q63.The number of integers, greater than 7000 that can be formed, using the digits 3, 5, 6, 7, 8 without repetition is (1) 120 (2) 168 (3) 220 (4) 48 13+23+33......upto n terms

Q63.The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is : (1) 89 (2) 84 (3) 86 (4) 79

Q63.Let S = {z ∈C −{i, 2i} z2−3iz−2 ∈R}. JEE Main 2023 (11 Apr Shift 2) JEE Main Previous Year Paper

Q63.The number of numbers, strictly between 5000 and 10000 can be formed using the digits 1, 3, 5, 7, 9 without repetition, is (1) 6 (2) 12 (3) 120 (4) 72

Q63.If the coefficient of 𝑥15 in the expansion of 𝑎𝑥3 + 1 is equal to the coefficient of 𝑥-15 in the expansion of 𝑏𝑥 3 1 15 1 𝑎𝑥 3 - , where 𝑎 and 𝑏 are positive real numbers, then for each such ordered pair 𝑎, 𝑏: 𝑏𝑥3 (1) 𝑎= 𝑏 (2) 𝑎𝑏= 1 (3) 𝑎= 3𝑏 (4) 𝑎𝑏= 3

Q63.Let s1, s2, s3. . . . , s10 respectively be the sum of 12 terms of 10 A. Ps whose first terms are 1, 2, 3, . . . . , 10 and the common differences are 1, 3, 5, . . . , 19 respectively. Then ∑10i=1 si is equal to (1) 7220 (2) 7360 (3) 7260 (4) 7380

Q63.Let 𝑎1, 𝑎2, 𝑎3, . . . . , 𝑎𝑛 be n positive consecutive terms of an arithmetic progression. If 𝑑> 0 is its common difference, then lim 𝑑 1 + 1 + … + 1 is 𝑛→∞√ 𝑛 √𝑎1 + √𝑎2 √𝑎2 + √𝑎3 √𝑎𝑛- 1 + √𝑎𝑛 (1) 1 (2) √𝑑 √𝑑 (3) 1 (4) 2 𝑛

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.If the sum and product of four positive consecutive terms of a G.P., are 126 and 1296, respectively, then the sum of common ratios of all such GPs is 9 (1) 7 (2) 2 (3) 3 (4) 14