Complex Numbers PYQs Set 4 (PDF)

Set 4 of 14 · Questions 91-120 of 417

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Complex Numbers · Set 4 (Q91-120)
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91. $Let z ˉ denote the complex conjugate of a complex number z . If z is a non-zero complex number for which both real and imaginary parts of (z ˉ)^{2} + \frac{1}{z ^{2}} are integer; then which of the following is/are possible value(s) of ∣ z ∣ ? [2022 Adv.]$ (1) $(\frac{43 + 3 205}{2})^{1/4}$ (2) $(\frac{7 + 33}{4})^{1/4}$ (3) $(\frac{9 + 65}{4})^{1/4}$ (4) $(\frac{7 + 13}{6})^{1/4}$ 92. $Let S = (z \in C : z = i (z^{2} + Re (z))) . Then \sum_{z \in S} \| z \|^{2} is equal to$ (1) $\frac{5}{2}$ (2) $4$ (3) $\frac{7}{2}$ (4) $3$ 93. $If α, β are non-zero integers and z = (α + iβ) (2 + 7 i) is a purely imaginary number, then minimum value of \| z \|^{2} is$ (1) $0$ (2) $2809$ (3) $2808$ (4) $1$ 94. $If the value of - 5 - 12 i + 7 + 24 i is a negative real number k, then k =$ (1) $-5$ (2) $-7$ (3) $-6$ (4) $-4$ 95. $Let α, β be real and z be a complex number. If z^{2} + αz + β = 0 has two distinct roots on the line Rez = 1, then it is necessary that:$ (1) $β \in (0, 1)$ (2) $β \in (- 1, 0)$ (3) $\| β \| = 1$ (4) $β \in (1, \infty)$ 96. $Let z be a complex number such that the imaginary part of z is non zero and a = z^{2} + z + 1 is real. Then a cannot take the value$ (1) $-1$ (2) $\frac{1}{3}$ (3) $\frac{1}{2}$ (4) $\frac{3}{4}$ 97. $The real part of z that satisfies i z^{4} + 1 = 0 is$ (1) $sin \frac{π}{4}$ (2) $cos \frac{π}{8}$ (3) $0$ (4) $-1$ 98. $If P, Q and R are points, respectively representing the complex numbers z, z e^{\frac{iπ}{3}} and z (1 + e^{\frac{iπ}{3}}) in argand plane, then the area of the triangle PQR, is$ (1) $3 \| z \|^{2}$ (2) $\frac{3}{2} \| z \|^{2}$ (3) $\frac{3}{4} \| z \|^{2}$ (4) $23 \| z \|^{2}$ 99. $If ω (\neq = 1) is a cube root of unity, and (1 + ω)^{7} = A + Bω . Then (A, B) equals$ (1) $(0, 1)$ (2) $(1, 1)$ (3) $(1, 0)$ (4) $(- 1, 1)$ 100. $The simplified form of i^{n} + i^{n + 1} + i^{n + 2} + i^{n + 3} is$ (1) $0$ (2) $1$ (3) $-1$ (4) $i$ 101. $Among the statements (S1): The set {z \in C - {- i} : \| z \| = 1 and \frac{z - i}{z + i} is purely real} contains exactly two elements, and (S2): The set {z \in C - {- 1} : \| z \| = 1 and \frac{z - 1}{z + 1} is purely imaginary} contains infinitely many elements.$ (1) $both are incorrect$ (2) $both are correct$ (3) $only (S2) is correct$ (4) $only (S1) is correct$ 102. $If ∣ x + i y ∣ = x^{2} + y^{2}, then (1 - 3 i)^{9} + (3 + i)^{9} =$ (1) $2^{9}$ (2) $2^{18}$ (3) $2^{10}$ (4) $2^{\frac{19}{2}}$ 103. $If 1, ω, ω^{2} are the cube roots of unity and 1, α, α^{2}, α^{3} are the fourth roots of unity in usual notation then α + αω - α^{3} ω^{2} =$ (1) $3$ (2) $1$ (3) $0$ (4) $-1$ 104. $The region represented by {z = x + iy \in C : \| z \| - Re (z) \leq 1} is also given by the inequality:$ (1) $y^{2} \leq x + \frac{1}{2}$ (2) $y^{2} \leq 2 (x + \frac{1}{2})$ (3) $y^{2} \geq x + 1$ (4) $y^{2} \geq 2 (x + 1)$ 105. $If (\frac{1 - i}{1 + i})^{96} = a + ib, then (a, b) is$ (1) $(1, 1)$ (2) $(1, 0)$ (3) $(0, 1)$ (4) $(0, - 1)$ 106. $\sum_{k = 0}^{440} i^{k} = x + iy \Rightarrow x^{100} + x^{99} y + x^{242} y^{2} + x^{97} y^{3} =$ (1) $0$ (2) $-4$ (3) $4$ (4) $1$ 107. $The equation ∣ z - i ∣ = ∣ z - 1∣, i = - 1, represents:$ (1) $a circle of radius \frac{1}{2}$ (2) $the line through the origin with slope 1.$ (3) $a circle of radius 1.$ (4) $the line through the origin with slope - 1 .$ 108. $Let a, b, x and y be real numbers such that a - b = 1 and y \neq = 0 . If the complex number z = x + iy satisfies lm (\frac{az + b}{z + 1}) = y, then which of the following is (are) possible value(s) of x ? (JEE Advanced 2017)$ (1) $- 1 - 1 - y^{2}$ (2) $1 + 1 + y^{2}$ (3) $1 - 1 + y^{2}$ (4) $- 1 + 1 - y^{2}$ 109. $The least value of ∣ z ∣ where z is complex number which satisfies the inequality exp (\frac{( ∣ z ∣ + 3 ) ( ∣ z ∣ - 1 )}{∣∣ z ∣ + 1∣} lo g_{e} 2) \geq lo g_{2} 57 + 9 i, i = - 1, is equal to:$ (1) $3$ (2) $5$ (3) $2$ (4) $8$ 110. $Multiplicative inverse of the complex number (sinθ, cosθ) is$ (1) $(sinθ, cosθ)$ (2) $(sinθ, - cosθ)$ (3) $(cosθ, - sinθ)$ (4) $(- cosθ, sinθ)$ 111. $The number of solutions of equation z^{2} + z ˉ = 0, where z \in C are$ (1) $6$ (2) $1$ (3) $4$ (4) $5$ 112. $If the real part of the complex number (1 - cos θ + 2 i sin θ)^{- 1} is \frac{1}{5} for θ \in (0, π), then the value of the integral \int_{0}^{θ} sin x d x is equal to:$ (1) $1$ (2) $2$ (3) $-1$ (4) $0$ 113. $If α and β are roots of the equation x^{2} + x + 1 = 0, then α^{2} + β^{2} is$ (1) $1$ (2) $\frac{- 1 + i 3}{2}$ (3) $\frac{- 1 - i 3}{2}$ (4) $-1$ 114. $If z = (1 - i)^{3} (x + i) is a purely imaginary number for x = x_{1} and if z is a purely real number for x = x_{2}, then x_{1} x_{2} =$ (1) $-1$ (2) $0$ (3) $1$ (4) $2$ 115. $If ω is an imaginary cube root of unity, then the value of (1 - ω + ω^{2}) \cdot (1 - ω^{2} + ω^{4}) . ((1 - ω^{4} + ω^{8}) (2 n factors) is$ (1) $2^{2 n}$ (2) $2^{n}$ (3) $1$ (4) $0$ 116. $The conjugate of the complex number \frac{( 1 + i ) ^{2}}{1 - i} is$ (1) $1 - i$ (2) $1 + i$ (3) $- 1 + i$ (4) $- 1 - i$ 117. $The Cartesian form of complex number z = 4 (cos 300^{\circ} + isin 300^{\circ}), where i = - 1 is$ (1) $2 - 23 i$ (2) $2 + 23 i$ (3) $1 - 3 i$ (4) $1 + 3 i$ 118. $The least positive integer such that \frac{( 2 i ) ^{n}}{( 1 - i ) ^{n - 2}}, i = - 1 is a positive integer, is.$ 119. $For α, β, z \in C and λ > 1, if λ - 1 is the radius of the circle ∣ z - α ∣^{2} + ∣ z - β ∣^{2} = 2 λ, then ∣ α - β ∣ is equal to_____.$ 120. $The sum of 162^{th} power of the roots of the equation x^{3} - 2 x^{2} + 2 x - 1 = 0 is$