Complex Numbers PYQs Set 3 (PDF)

Set 3 of 14 · Questions 61-90 of 417

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Complex Numbers · Set 3 (Q61-90)
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61. $The modulus of the conjugate of z = \frac{- 2 + i}{( 1 - 2 i ) ^{2}} is$ (1) $\frac{1}{5}$ (2) $\frac{1}{5}$ (3) $\frac{1}{25}$ (4) $5$ 62. $\frac{( 1 + i ) x - 2 i}{3 + i} + \frac{( 2 - 3 i ) y + i}{3 - i} = i \Rightarrow x + y =$ (1) $-1$ (2) $1$ (3) $-2$ (4) $2$ 63. $If x + iy = (- 1 + i 3)^{2010}, then x =$ (1) $2^{2010}$ (2) $- 2^{2010}$ (3) $-1$ (4) $1$ 64. $Let C be the circle in the complex plane with centre z_{0} = \frac{1}{2} (1 + 3 i) and radius r = 1. Let z1 = l + i and the complex number z2 be outside circle C such that (z_{1} - z_{0}) (z_{2} - z_{0}) = 1 . If z_{0},z1 and z2 are collinear, then the smaller value of (z_{2})^{2} is equal to$ (1) $\frac{5}{2}$ (2) $\frac{7}{2}$ (3) $\frac{13}{2}$ (4) $\frac{3}{2}$ 65. $If α, β, γ, δ are the roots of the equation x^{4} + x^{3} + x^{2} + x +1 = 0, then α^{2021} + β^{2021} + γ^{2021} + δ^{2021}$ (1) $- 4$ (2) $-1$ (3) $1$ (4) $4$ 66. $Let A = (θ \in (- \frac{π}{2}, π) : \frac{3 + 2 i sin θ}{1 - 2 i sin θ} is purely imaginary) . Then the sum of the elements in A is:$ (1) $\frac{5 π}{6}$ (2) $π$ (3) $\frac{3 π}{4}$ (4) $\frac{2 π}{3}$ 67. $The modulus and amplitude of \frac{1 + 2 i}{1 - ( 1 - i ) ^{2}} are$ (1) $2 and \frac{π}{6}$ (2) $1 and 0$ (3) $1 and \frac{π}{3}$ (4) $1 and \frac{π}{4}$ 68. $If 13 e^{i tan^{- 1} \frac{5}{12}} = a + ib, then the ordered pair (a, b) =$ (1) $(12, 5)$ (2) $(5, 12)$ (3) $(24, 10)$ (4) $(10, 24)$ 69. $If the point (\frac{k - 1}{k}, \frac{k - 2}{k}) lies on the locus of z satisfying the inequality \frac{z + 3 i}{3 z + i} < 1, then the interval in which k lies is$ (1) $(- \infty, 2) \cup (3, \infty)$ (2) $[2, 3]$ (3) $[1, 5]$ (4) $(- \infty, 1) \cup (5, \infty)$ 70. $If z \neq = 0 be a complex number such that (z - \frac{1}{z}) = 2, then the maximum value of \|z\| is:$ (1) $2$ (2) $1$ (3) $2 - 1$ (4) $2 + 1$ 71. $If z = x + iy satisfies \|z\| - 2 = 0 and \|z-i\|-\|z+5i\|=0, then$ (1) $x + 2y - 4 = 0$ (2) $x^{2} + y - 4 = 0$ (3) $x + 2y + 4 = 0$ (4) $x^{2} - y + 3 = 0$ 72. $If α and β are the roots of x^{2} + x + 1 = 0, then α^{16} + β^{16} =$ (1) $2$ (2) $0$ (3) $1$ (4) $-1$ 73. $If ω is a complex cube root of unity, then (1 - ω + ω^{2})^{6} + (1 - ω^{2} + ω)^{6} =$ (1) $0$ (2) $6$ (3) $64$ (4) $128$ 74. $(- i + 3)^{300} + (- i - 3)^{300} =$ (1) $2^{300}$ (2) $2^{301}$ (3) $2^{100}$ (4) $- 2^{300}$ 75. $If 1, α_{1}, α_{2}, α_{3} and α_{4} are the roots of z^{5} - 1 = 0 and ω is a cube roots of unity, then (ω - 1) (ω - α_{1}) (ω - α_{2}) (ω - (α_{3}) (ω - α_{4}) + ω is equal to$ (1) $0$ (2) $-1$ (3) $-2$ (4) $1$ 76. $If α, β are non-real cube roots of 2, then α^{6} + β^{6} equals$ (1) $8$ (2) $4$ (3) $2$ (4) $1$ 77. $If log_{\frac{1}{3}} (\frac{\| z \| ^{2} - \| z \| + 1}{2 + \| z \|}) > - 2, then z lies inside$ (1) $a triangle$ (2) $an ellipse$ (3) $a circle$ (4) $a square$ 78. $Let α and β be the roots of the equation x^{2} + (2 i - 1) = 0 . Then, the value of ∣ α^{8} + β^{8} ∣ is equal to:$ (1) $50$ (2) $250$ (3) $1250$ (4) $1500$ 79. $If a, - a, b are the roots of x^{3} - 5 x^{2} - x + 5 = 0, then b is a root of$ (1) $x^{2} - 5 x + 10 = 0$ (2) $x^{2} + 3 x - 20 = 0$ (3) $x^{2} + 5 x - 30 = 0$ (4) $x^{2} - 3 x - 10 = 0$ 80. $If z is a complex number such that \| z + 4 \| \geq 3, then the smallest value of \| z + 3 \| is$ (1) $3$ (2) $1$ (3) $2$ (4) $0$ 81. $If \frac{z - α}{z + α} (α \in R) is a purely imaginary number and ∣ z ∣ = 2, then a value of α is :$ (1) $2$ (2) $1$ (3) $\frac{1}{2}$ (4) $2$ 82. $If a > 0 and z = x + iy, then log_{cos^{2} θ} \| z - a \| > log_{cos^{2} θ} \| z - ai \|, (θ \in R) implies$ (1) $x > y$ (2) $x < y$ (3) $x + y = cos θ$ (4) $x + y < 0$ 83. $\frac{( 1 + i ) ^{2016}}{( 1 - i ) ^{2014}} is equal to$ (1) $- 2 i$ (2) $2 i$ (3) $2$ (4) $-2$ 84. $Let z be a complex number such that the real part of \frac{z - 2 i}{z + 2 i} is zero. Then, the maximum value of \| z - (6 + 8 i) \| is equal to$ (1) $12$ (2) $10$ (3) $8$ (4) $\infty$ 85. $The solutions of the equation z^{2} (1 - Z^{2}) = 16, z \in C, lie on the curve$ (1) $\| z \| = 1$ (2) $\| z \| = 2 \|z\|$ (3) $\| z \|^{2} = 3 \| z \| + 2$ (4) $\| z \| = 2$ 86. $If α and β are the roots of the equation 2 z^{2} - 3 z - 2 i = 0, where i = - 1, then 16 Re (\frac{α ^{19} + β ^{19} + α ^{11} + β ^{11}}{α ^{15} + β ^{15}}) \cdot lm (\frac{α ^{19} + β ^{19} + α ^{11} + β ^{11}}{α ^{15} + β ^{15}}) is equal to$ (1) $441$ (2) $398$ (3) $312$ (4) $409$ 87. $If the least and the largest real values of α, for which the equation z + α \| z - 1 \| + 2 i = 0 (z \in C and i = - 1) has a solution, are p and q respectively; then 4 (p^{2} + q^{2}) is equal to$ 88. $For a complex number z, let Re (z) denote the real part of z . Let S be the set of all complex numbers z satisfying z^{4} - \| z \|^{4} = 4 i z^{2}, where i = - 1 . Then, the minimum possible value of (z_{1} - z_{2})^{2}, where z_{1}, z_{2} \in S with Re (z_{1}) > 0 and Re (z_{2}) < 0 is [2020 Adv.]$ 89. $If z_{1} = a + ib and z_{2} = c + i d are complex numbers such that ∣ z_{1} ∣ = ∣ z_{2} ∣ = 1 and R (z_{1} z ˉ_{2}) = 0, then the pair of complex numbers w_{1} = a + i c and w_{2} = b + i d, Find R (w_{1} w ˉ_{2})$ 90. $If (\frac{1 + i}{1 - i})^{\frac{m}{2}} = (\frac{1 + i}{i - 1})^{\frac{n}{3}} = 1, (m, n \in N) then the greatest common divisor of the least values of m and n is$