Practice Questions

Q61.The equation e4x + 8e3x + 13e2x −8ex + 1 = 0, x ∈R has : (1) four solutions two of which are negative (2) two solutions and both are negative (3) no solution (4) two solutions and only one of them is negative

Q61.Let the complex number 𝑧= 𝑥+ 𝑖𝑦 be such that is purely imaginary. If 𝑥+ 𝑦2 = 0, then 𝑦4 + 𝑦2 - 𝑦 is 2𝑧+ 𝑖 equal to (1) 2 (2) 3 3 2 3 4 (3) (4) 4 3

Q61.Let a ≠b be two non-zero real numbers. Then the number of elements in the set X = {z ∈C : Re(az2 + bz) = a and Re(bz2 + az) = b} is equal to (1) 0 (2) 1 (3) 3 (4) 2

Q61.Let α, β, γ be the three roots of the equation x3 + bx + c = 0 if βγ = 1 = −α then b3 + 2c3 −3α3 −6β3 −8γ 3 is equal to (1) 155 (2) 21 8 (3) 169 (4) 19 8

Q61.The number of integral values of k, for which one root of the equation 2x2 −8x + k = 0 lies in the interval (1, 2) and its other root lies in the interval (2, 3), is : JEE Main 2023 (01 Feb Shift 2) JEE Main Previous Year Paper (1) 2 (2) 0 (3) 1 (4) 3

Q61.Let 𝑆= 𝑧= 𝑥+ 𝑖𝑦: is a real number }. Then which of the following is NOT correct? 4𝑧+ 2𝑖 (1) 𝑦+ 𝑥2 + 𝑦2 ≠- 1 (2) (𝑥, 𝑦) = 0, - 1 4 2 (3) 𝑥= 0 (4) 𝑦∈- ∞, - 1 ∪-1 ∞ 2 2,

Q61.The number of real solutions of the equation 3(x2 + x21 ) −2(x + x1 ) + 5 = 0 , is (1) 4 (2) 0 (3) 3 (4) 2 2π 2π 3 1+sin 9 +i cos 9

Q61.Let λ ≠0 be a real number. Let α, β be the roots of the equation 14x2 −31x + 3λ = 0 and α, γ be the roots of the equation 35x2 −53x + 4λ = 0. Then 3αβ and 4αγ are the roots of the equation : (1) 7x2 + 245x −250 = 0 (2) 7x2 −245x + 250 = 0 (3) 49x2 −245x + 250 = 0 (4) 49x2 + 245x + 250 = 0

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

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 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.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.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.The sum of the first 20 terms of the series 5 + 11 + 19 + 29 + 41 + . . . is (1) 3520 (2) 3450 (3) 3250 (4) 3420

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.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 a, b be two real numbers such that ab < 0 . If the complex number 1+aib+i is of unit modulus and a + ib lies on the circle |z −1| = |2z| , then a possible value of 1+[a]4b , where [t] is greatest integer function, is : (1) 0 (2) −1 (3) 1 (4) 21

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.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.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.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.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.The value of ( 1+sin 2π9 −i cos 2π9 ) is (1) −1 (2) 1 2 (1 −i√3) 2 (1 −i√3) (3) −1 + i) 2 (√3 −i) (4) 12 (√3

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.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