Complex Numbers PYQs Set 2 (PDF)

Set 2 of 14 · Questions 31-60 of 417

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Complex Numbers · Set 2 (Q31-60)
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31. $If 1, ω, ω^{2} are the cube roots of unity, n \in N and n > 2 then the least value of n such that 1 + ω is a root of x^{n} - x = 0 is$ (1) $3$ (2) $5$ (3) $7$ (4) $4$ 32. $If 2 x = - 1 + 3 i, then the value of (1 - x^{2} + x)^{6} - (1 - x + x^{2})^{6} is$ (1) $32$ (2) $-64$ (3) $64$ (4) $0$ 33. $Let s, t, r be non-zero complex numbers and L be the set of solutions z = x + iy (x, y Î R, i = - 1) of the equation sz + t z + r = 0, where z = x - iy. Then, which of the following statement(s) is (are) TRUE?$ (1) $If L has exactly one element, then \|s\| ¹ \|t\|$ (2) $If \|s\| = \|t\|, then L has infinitely many elements$ (3) $The number of elements in L Ç {z : \|z - 1 + i\| = 5} is at most 2$ (4) $If L has more than one element, then L has infinitely many elements$ 34. $If P (x, y) denotes z = x + i y in argand's plane and \frac{z - 1}{z + 2 i} = 1, then the locus of P is a/an$ (1) $hyperbola$ (2) $circle$ (3) $ellipse$ (4) $straight line$ 35. $Let z = x + i y be a complex number, A = {z : ∣ z ∣ \leq 2} and B = {z : (1 - i) z + (1 + i) z ˉ \geq 4} . Then which one of the following options belongs to A \cap B ?$ (1) $3 + \frac{1}{2} i$ (2) $\frac{1}{2} + \frac{i}{2}$ (3) $2 + \frac{i}{2}$ (4) $2 + 2 i$ 36. $If 1, ω, ω^{2} are the cube roots of unity and (x + y) (xω + (y ω^{2}) (x ω^{2} + yω) = f (x, y), then f (2, 3) =$ (1) $16$ (2) $24$ (3) $35$ (4) $45$ 37. $(\frac{cosθ + isinθ}{sinθ + icosθ})^{8} + (\frac{1 + cosθ - isinθ}{1 + cosθ + isinθ})^{16} =$ (1) $2 cos 8 θ$ (2) $2 cos 16 θ$ (3) $2 sin 8 θ$ (4) $2 sin 16 θ$ 38. ${x \in (0, 2 π) : sin x + i cos 2 x and cos x - i sin 2 x are conjugate to each other} =$ (1) $(\frac{π}{4}, \frac{π}{2}, \frac{3 π}{4}, π, \frac{5 π}{4}, \frac{3 π}{2}, \frac{7 π}{4}, 2 π)$ (2) $(\frac{π}{4}, \frac{3 π}{4}, \frac{5 π}{4}, \frac{7 π}{4})$ (3) $(\frac{π}{2}, π, \frac{3 π}{2}, 2 π)$ (4) $ϕ$ 39. $Let p (x) be a quadratic polynomial with real coefficients. If p (x) = 0 has only purely imaginary roots, then the zeroes of the polynomial p (p (x)) are$ (1) $only real numbers$ (2) $only purely imaginary numbers$ (3) $only rational numbers$ (4) $only complex numbers of the form a + ib with a \neq = 0 and b \neq = 0$ 40. $Consider the following two statements : Statement I: For any two non-zero complex numbers z_{1}, z_{2}, (∣ z_{1} ∣ + ∣ z_{2} ∣) \frac{z _{1}}{∣ z _{1} ∣} + \frac{z _{2}}{∣ z _{2} ∣} \leq 2 (∣ z_{1} ∣ + ∣ z_{2} ∣), and Statement II: If x, y, z are three distinct complex numbers and a, b, c are three positive real numbers such that \frac{a}{∣ y - z ∣} = \frac{b}{∣ z - x ∣} = \frac{c}{∣ x - y ∣}, then \frac{a ^{2}}{y - z} + \frac{b ^{2}}{z - x} + \frac{c ^{2}}{x - y} = 1 . Between the above two statements,$ (1) $Statement I is correct but Statement II is incorrect.$ (2) $both Statement I and Statement II are correct.$ (3) $both Statement I and Statement II are incorrect.$ (4) $Statement I is incorrect but Statement II is correct.$ 41. $The sum of all possible values of θ \in [- π, 2 π], for which \frac{1 + icos θ}{1 - 2 icos θ} is purely imaginary, is equal$ (1) $3 π$ (2) $2 π$ (3) $5 π$ (4) $4 π$ 42. $Let z be a complex number such that the imaginary part of z is nonzero and a = z^{2} + z + 1 is real. Then a cannot take the value (IIT-JEE, 2012)$ (1) $-1$ (2) $\frac{1}{3}$ (3) $\frac{1}{2}$ (4) $\frac{3}{4}$ 43. $Let z and w be two distinct non-zero complex numbers if \| z \|^{2} w - \| w \|^{2} z = z - w, then$ (1) $w = z^{2}$ (2) $z w = 2$ (3) $z w = 1$ (4) $w = z$ 44. $If the set {Re (\frac{z - z ˉ + z z ˉ}{2 - 3 z + 5 z ˉ}) : z \in C, Re z = 3} is equal to the interval (α, β], then 24 (β - α) is equal to$ (1) $36$ (2) $27$ (3) $30$ (4) $42$ 45. $The value of (\frac{1 + sin \frac{2 π}{9} + i cos \frac{2 π}{9}}{1 + sin \frac{2 π}{9} - i cos \frac{2 π}{9}}) is$ (1) $\frac{- 1}{2} (1 - i 3)$ (2) $\frac{1}{2} (1 - i 3)$ (3) $- \frac{1}{2} (3 - i)$ (4) $\frac{1}{2} (3 + i)$ 46. $If the equation a ∣ z ∣^{2} + α z ˉ + α ˉ z + d = 0 represents a circle where a, d are real constants then which of the following condition is correct?$ (1) $\| α \|^{2} - ad \neq = 0$ (2) $\| α \|^{2} - ad > 0 and a \in R -0$ (3) $\| α \|^{2} - ad \geq 0 and a \in R$ (4) $α = 0, a, d \in R^{+}$ 47. $Let n denote the number of solutions of the equation z^{2} + 3 z ˉ = 0, where z is a complex number. Then the value of \sum_{k = 0}^{\infty} \frac{1}{n ^{k}} is equal to$ (1) $1$ (2) $\frac{4}{3}$ (3) $\frac{3}{2}$ (4) $2$ 48. $If α + iβ and γ + iδ are the roots of x^{2} - (3 - 2 i) x - (2 i - 2) = 0, i = - 1, then αγ + βδ is equal to:$ (1) $-2$ (2) $6$ (3) $-6$ (4) $2$ 49. $The complex number \frac{1 + 2 i}{1 - i} lies in$ (1) $fourth quadrant$ (2) $first quadrant$ (3) $second quadrant$ (4) $third quadrant$ 50. $If z_{1} and z_{2} are two non-zero complex numbers such that ∣ z_{1} + z_{2} ∣ = ∣ z_{1} ∣ + ∣ z_{2} ∣, then ar g z_{1} - ar g z_{2} is equal to:$ (1) $- \frac{π}{2}$ (2) $0$ (3) $- π$ (4) $\frac{π}{2} .$ 51. $If x^{2} + x + 1 = 0, then the value of (x + \frac{1}{x})^{4} + (x^{2} + \frac{1}{x ^{2}})^{4} + (x^{3} + \frac{1}{x ^{3}})^{4} + \dots + (x^{25} + \frac{1}{x ^{25}})^{4} is:$ (1) $162$ (2) $175$ (3) $145$ (4) $128$ 52. $Imaginary part of (3 - i)^{2016} + (- 3 - i)^{2019} is$ (1) $2^{2016}$ (2) $- 2^{2016}$ (3) $- 2^{2019}$ (4) $2^{2019}$ 53. $The least positive integer n, for which \frac{( 1 + i ) ^{n}}{( 1 - i ) ^{n - 2}} is positive is$ (1) $4$ (2) $3$ (3) $2$ (4) $1$ 54. $For any two complex numbers z_{1} and z_{2} and any real numbers a and b, ∣ a z_{1} - b z_{2} ∣^{2} + ∣ b z_{1} + a z_{2} ∣^{2} = (a^{2} + b^{2}) (∣ z_{1} ∣^{A} + ∣ z_{2} ∣^{B}) . Find A + B .$ 55. $Let α = \frac{- 1 + i 3}{2} and β = \frac{- 1 - i 3}{2}, i = - 1 . If (7 - 7 α + 9 β)^{20} + (9 + 7 α - 7 β)^{20} + (- 7 + 9 α + 7 β)^{20} + (14 + 7 α + 7 β)^{20} = m^{10}, then m is____.$ 56. $Let z and w be two complex numbers such that \| z \| \leq 1, \| w \| \leq 1 and \| z + iw \| = \| z - iw \| = 2 . Then sum ofall possible z is equal to$ 57. $If a = cos α + isin α, b = cos β + isin β, c = cos γ + isin γ and \frac{b}{c} + \frac{c}{a} + \frac{a}{b} = 1, then cos (β - γ) + cos (γ - α) + cos (α - β) is equal to$ 58. $Let z = 1 + i and z_{1} = \frac{1 + i z ˉ}{z ˉ ( 1 - z ) + \frac{1}{z}} . Then \frac{12}{π} ar g (z_{1}) is equal to _________$ 59. $For any integer k, let α_{k} = cos \frac{kπ}{7} + isin \frac{kπ}{7}, where i = - 1 . Value of the expression \frac{\sum _{k = 1}^{12} ( α _{k + 1} - α _{k} )}{\sum _{k = 1}^{3} ( α _{4 k - 1} - α _{4 k - 2} )} is____. (JEE Advanced 2015)$ 60. $Let S = (z \in C - (1, 2 i) : \frac{z ^{2} + 8 iz - 15}{z ^{2} - 3 iz - 2} \in R) . α - \frac{13}{11} i \in S, α \in R - (0), then 242 α 2 is equal to$