Practice Questions

Q61.The number of real roots of the equation √𝑥2 - 4𝑥+ 3 + √𝑥2 - 9 = √4𝑥2 - 14𝑥+ 6, is: (1) 0 (2) 1 (3) 3 (4) 2

Q61.The number of points, where the curve f(x) = e8x −e6x −3e4x −e2x + 1, x ∈R cuts x-axis, is equal to............ ¯¯¯¯

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 𝑥2 - 4 𝑥2 - 4 𝑆= 𝑥: 𝑥∈ℝ and √3 + √2 + √3 - √2 = 10. Then 𝑛𝑆 is equal to (1) 2 (2) 4 (3) 6 (4) 0 𝑧- 2

Q61.Let w = zz + k1z + k2iz + λ(1 + i), k1, k2 ∈R. . Let Re(w) = 0 be the circle C of radius 1 in the first quadrant touching the line y = 1 and the y−axis. If the curve Im(w) = 0 intersects C at A and B, then 30(AB)2 is equal to _______. JEE Main 2023 (13 Apr Shift 1) JEE Main Previous Year Paper

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

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.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.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.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.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 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.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.The number of seven digit positive integers formed using the digits 1, 2, 3 and 4 only and sum of the digits equal to 12 is _______.

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 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 α = 8 −14i, A = {z ∈C : z2−(¯z)2−112iαz−α¯z = 1} and B = {z ∈C : |z + 3i| = 4} Then, ∑z∈A∩B(Re z −Imz) is equal to ________

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.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 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 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.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