Complex Numbers PYQs Set 6 (PDF)

Set 6 of 14 · Questions 151-180 of 417

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Complex Numbers · Set 6 (Q151-180)
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151. $For n \in N, If A_{n} = cos (\frac{π}{2 ^{n}}) + isin (\frac{π}{2 ^{n}}), then (A_{1} A_{2} A_{3} A_{4})^{4} =$ (1) $\frac{- 1 - i}{2}$ (2) $1$ (3) $0$ (4) $\frac{1 - i}{2}$ 152. $All the points in the set S = (\frac{α + i}{α - i} : α \in R) (i = - 1) lie on a :$ (1) $straight line whose slope is 1.$ (2) $circle whose radius is 1.$ (3) $circle whose radius is 2 .$ (4) $straight line whose slope is - 1 .$ 153. $For two non - zero complex number z1 and z2, if Re (z_{1} z_{2}) = 0 and Re (z_{1} + z_{2}) = 0, then which of the following are possible? (A) Imz1>0 and Imz2>0 (B) Im (z_{1}) < 0 and Imz2>0 (C)Imz1>0 and Im (Z_{2}) < 0 (D) Im (Z_{1}) < 0 and Im (z_{2}) < 0 Choose the correct answer from the options given below:$ (1) $B and D$ (2) $B and C$ (3) $A and B$ (4) $A and C$ 154. $If the complex cube roots of (- i) are α, β, γ, then α^{2} + β^{2} + γ^{2} =$ (1) $1$ (2) $-1$ (3) $- i$ (4) $0$ 155. $If z is a complex number, then the number of common roots of the equation z^{1985} + z^{100} + 1 = 0 and z^{3} + 2 z^{2} + 2 z + 1 = 0, is equal to:$ (1) $1$ (2) $2$ (3) $0$ (4) $3$ 156. $If z = x - iy and z^{\frac{1}{3}} = a + ib, then \frac{( \frac{x}{a} + \frac{y}{b} )}{a ^{2} + b ^{2}} =$ (1) $-2$ (2) $-1$ (3) $1$ (4) $2$ 157. $Let w_{1} be the point obtained by the rotation of z_{1} = 5 + 4i about the origin through a right angle in the anticlockwise direction, and W_{2} be the point obtained by the rotation of z_{2} = 3 + 5i about the origin through a right angle in the clockwise direction. Then the principal argument w_{1} - w_{2} is equal to$ (1) $π - tan^{- 1} \frac{8}{9}$ (2) $- π + tan^{- 1} \frac{33}{5}$ (3) $- π + tan^{- 1} \frac{8}{9}$ (4) $π - tan^{- 1} \frac{33}{5}$ 158. $If the cube roots of unity are 1, ω, ω^{2}, then the roots of the equation (x - 1)^{3} + 8 = 0, are:$ (1) $- 1, 1 + 2 ω, 1 + 2 ω^{2} .$ (2) $- 1, 1 - 2 ω, 1 - 2 ω^{2} .$ (3) $- 1, - 1, - 1 .$ (4) $- 1, - 1 + 2 ω, - 1 - 2 ω^{2} .$ 159. $If the equation, x^{2} + bx + 45 = 0 (b \in R) has conjugate complex roots and they satisfy \| z + 1 \| = 210 then:$ (1) $b^{2} + b = 12$ (2) $b^{2} - b = 30$ (3) $b^{2} - b = 36$ (4) $b^{2} + b = 30$ 160. $If z_{1}, z_{2} are complex numbers such that Re (z_{1}) = \| z_{1} - 1 \|, Re (z_{2}) = \| z_{2} - 1 \| and arg (z_{1} - z_{2}) = \frac{π}{6} then Im (z_{1} + z_{2}) is equal to$ (1) $\frac{3}{2}$ (2) $\frac{1}{3}$ (3) $\frac{2}{3}$ (4) $23$ 161. $Let complex numbers α and \frac{1}{α} lies on circles (x - x_{0})^{2} + (y - y_{0})^{2} = r^{2} and (x - x_{0})^{2} + (y - y_{0})^{2} = 4 r^{2}, respectively. If z_{0} = x_{0} + i y_{0} satisfies the equation 2 (z_{0})^{2} = r^{2} + 2, then \| α \| =$ (1) $\frac{1}{2}$ (2) $\frac{1}{2}$ (3) $\frac{1}{7}$ (4) $\frac{1}{3}$ 162. $If 1, ω, ω^{2}, \dots, ω^{8} are the roots of the equation x^{9} - 1 = 0, then \sum_{r = 1}^{8} (ω^{r})^{99} =$ (1) $0$ (2) $8$ (3) $1$ (4) $ω$ 163. $If z = x + i y, x y \neq = 0, satisfies the equation z^{2} + i z ˉ = 0, then ∣ z^{2} ∣ is equal to :$ (1) $9$ (2) $1$ (3) $4$ (4) $\frac{1}{4}$ 164. $For real numbers a and b, if 4 a + i (3 a - b) = b - 6 i and z = a + \frac{b}{4} i, then \frac{\| z \|}{a} =$ (1) $22$ (2) $62$ (3) $2$ (4) $2$ 165. $Let S be the set of all complex numbers z satisfying ∣ z^{2} + z + 1∣ = 1 . Then, which of the following statements is/are true? [2020 Adv.]$ (1) $\|z+1/2\|\leq1/2,\forall z\inS$ (2) $∣ z ∣ \leq 2, \forall z \in S$ (3) $\|z+1/2\|\geq1/2,\forall z\inS$ (4) $The set S has exactly four elements$ 166. $If z_{1} = (2, - 1) and z_{2} = (6, 3), then amp (\frac{z _{1} - z _{2}}{z _{1} + z _{2}}) =$ (1) $- \frac{3 π}{4} - tan^{- 1} (\frac{1}{4})$ (2) $\frac{π}{4} tan^{- 1} (\frac{1}{4})$ (3) $\frac{3 π}{4} + tan^{- 1} (\frac{1}{4})$ (4) $\frac{π}{4} + tan^{- 1} (\frac{1}{4})$ 167. $The value of \frac{1 + i 3}{( 1 + \frac{1}{i + 1} ) ^{2}} is$ (1) $20$ (2) $9$ (3) $\frac{5}{4}$ (4) $\frac{4}{5}$ 168. $If z is a complex number with ∣ z ∣ \geq 5 . Then the least value of z + \frac{2}{z} is$ (1) $\frac{24}{5}$ (2) $\frac{26}{5}$ (3) $\frac{23}{5}$ (4) $\frac{29}{5}$ 169. $If i = - 1 then \sum_{n = 0}^{\infty} (\frac{i}{3})^{n} =$ (1) $\frac{9 - 3 i}{10}$ (2) $9 - 3 i$ (3) $9 + 3 i$ (4) $\frac{9 + 3 i}{10}$ 170. $If the roots of the equation x^{3} + a x^{2} + bx + c = 0 are in A.P., then 2 a^{3} - 9 ab =$ (1) $9 c$ (2) $18 c$ (3) $27 c$ (4) $- 27 c$ 171. $(- 3 + 4 i) (8 + 6 i) =$ (1) $\pm (1 + 2 i)$ (2) $\pm (3 + i)$ (3) $\pm (1 + 7 i)$ (4) $\pm (7 - i)$ 172. $If P is a complex number whose modulus is one, then the equation (\frac{1 + iz}{1 - iz})^{4} = P has$ (1) $real and equal roots$ (2) $real and distinct roots$ (3) $two real and two complex roots$ (4) $all complex roots$ 173. $If a > 0 and z = \frac{( 1 + i ) ^{2}}{a - i}, has magnitude \frac{2}{5}, then z is equal to:$ (1) $- \frac{1}{5} - \frac{3}{5} i$ (2) $- \frac{3}{5} - \frac{1}{5} i$ (3) $\frac{1}{5} - \frac{3}{5} i$ (4) $- \frac{1}{5} + \frac{3}{5} i$ 174. $If z = \frac{3}{2} + \frac{i}{2} (i = - 1), then (1 + iz + z^{5} + i z^{8})^{9} is equal to :$ (1) $0$ (2) $1$ (3) $- 1 +2i9$ (4) $- 1$ 175. $If z = e^{iθ} and \frac{3 cos 3 θ + 2 cos 2 θ + 5 cos 5 θ}{3 sin 3 θ + 2 sin 2 θ + 5 sin 5 θ} = \frac{i \sum _{r = 0}^{10} a _{r} z ^{r}}{\sum _{r = 0}^{10} b _{r} z ^{r}} then \frac{( \sum _{r = 0}^{10} a _{r} + \sum _{r = 0}^{10} b _{r} )}{10} =$ (1) $0$ (2) $1$ (3) $2$ (4) $3$ 176. $If α is a complex number satisfying the equation α^{2} + α + 1 = 0 then α^{31} is equal to$ (1) $1$ (2) $0$ (3) $α$ (4) $α^{2}$ 177. $The imaginary part of (3 + 2 - 54)^{1/2} - (3 - 2 - 54)^{1/2} can be:$ (1) $6$ (2) $- 6$ (3) $- 26$ (4) $2$ 178. $The equation of a circle is Re(z^{2}) + 2 (I m (z))^{2} + 2Re(z) = 0, where z = x+iy A line which passes through the center of the given circle and the vertex of the parabola, x^{2} - 6x - y + 13 = 0, has y-intercept equal to ___$ 179. $If 1, ω, ω^{2}, ω^{3} \dots \dots, ω^{n - 1} are the n, n^{th} roots of unity, then (1 - ω) (1 - ω^{2}) \dots .. (1 - ω^{n - 1}) equal M . Find M / n .$ 180. $Let A = z \in C : \| z-2-i \| =3, B = z \in C : Re (z- iz) =2 and S = A \cap B . Then \sum \| z \|^{2} is equal to ____.$