Complex Numbers PYQs Set 9 (PDF)

Set 9 of 14 · Questions 241-270 of 417

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Complex Numbers · Set 9 (Q241-270)
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241. $If (z - \frac{2}{z}) = 2, then the greatest value of \| z \| is$ (1) $3 - 1$ (2) $3$ (3) $3 + 1$ (4) $3 + 2$ 242. $If the amplitude of (z - 1 - 2 i) is \frac{π}{3}, then the locus of z is$ (1) $y = 3 x + (2 - 3)$ (2) $y = 3 x - 3$ (3) $x = 3 y + (2 - 3)$ (4) $y = 3 x + 2$ 243. $Let A_{r} = (x + \frac{1}{x})^{3} \cdot (x^{2} + \frac{1}{x ^{2}})^{3} \cdot (x^{3} + \frac{1}{x ^{3}})^{3} ..... (x^{r} + \frac{1}{x ^{r}})^{3} . If x^{2} + x + 1 = 0, then \frac{1}{A _{3}} + \frac{1}{A _{6}} + \frac{1}{A _{9}} + \frac{1}{A _{12}} + \dots \infty =$ (1) $\frac{1}{6}$ (2) $\frac{2}{5}$ (3) $1$ (4) $\frac{1}{7}$ 244. $(\frac{1 - i}{1 + i})^{2022} + (\frac{1 + i}{1 - i})^{2021} =$ (1) $- i$ (2) $i$ (3) $i + 1$ (4) $i-1$ 245. $If C is a point on the straight line joining the points A (- 2 + i) and B (3 - 4 i) in the Argand plane and \frac{AC}{CB} = \frac{1}{2}, then the argument of C is$ (1) $tan^{- 1} 3$ (2) $tan^{- 1} 2 - π$ (3) $tan^{- 1} 2$ (4) $π - tan^{- 1} 3$ 246. $If ∣ z^{2} - 1∣ = ∣ z ∣^{2} + 1, then z lies on:$ (1) $the real axis$ (2) $the imaginary axis$ (3) $a circle$ (4) $an ellipse$ 247. $If z = x + i y, x, y \in R and the imaginary part of \frac{z ˉ - 1}{z ˉ - i} is 1, then the locus of z is$ (1) $x + y + 1 = 0$ (2) $x + y + 1 = 0, (x, y) \neq = (0, - 1)$ (3) $x^{2} + y^{2} - x + 3 y + 2 = 0$ (4) $x^{2} + y^{2} - x + 3 y + 2 = 0, (x, y) \neq = (0, - 1)$ 248. $If 1, ω, ω^{2} denote the cube roots of unity, then, the value of (1 - ω + ω^{2})^{5} + (1 + ω - ω^{2})^{5} is$ (1) $32 ω^{2}$ (2) $32 ω$ (3) $-32$ (4) $32$ 249. $Let S be the set of all complex numbers z satisfying ∣ z^{2} + z + 1∣ = 1 . Then which of the following statements is/are TRUE? (JEE Advanced 2020)$ (1) $z + \frac{1}{2} \leq \frac{1}{2} for all z \in S$ (2) $\|z\|\leq2 for all z\inS$ (3) $z + \frac{1}{2} \geq \frac{1}{2} for all z \in S$ (4) $The set S has exactly four elements$ 250. $If 1, α, α^{2}, \dots ., α^{n - 1} are the n^{th} roots of unity. Then \sum_{i = 1}^{n - 1} \frac{1}{2 - a ^{i}} is equal to$ (1) $(n - 2) 2^{n}$ (2) $\frac{( n - 2 ) 2 ^{n - 1} + 1}{2 ^{n} - 1}$ (3) $\frac{( n - 2 ) 2 ^{n - 1} - 1}{2 ^{n} - 1}$ (4) $1 (n - 2) 2^{n}$ 251. $The modulus of the complex number \frac{( 1 + i ) ^{2} ( 1 + 3 i )}{( 2 - 6 i ) ( 2 - 2 i )} is$ (1) $\frac{1}{2}$ (2) $\frac{4}{2}$ (3) $\frac{2}{4}$ (4) $\frac{2}{2}$ 252. $Let (- 2 - \frac{1}{3} i)^{3} = \frac{x + iy}{27} (i = - 1), where x and y are real numbers then y - x equals:$ (1) $91$ (2) $-85$ (3) $85$ (4) $-91$ 253. $The complex number z = \frac{i - 1}{cos \frac{π}{3} + i sin \frac{π}{3}} is equal to$ (1) $2 (cos \frac{5 π}{12} + i sin \frac{5 π}{12})$ (2) $cos \frac{π}{12} - i sin \frac{π}{12}$ (3) $2 (cos \frac{π}{12} + i sin \frac{π}{12})$ (4) $2 i (cos \frac{5 π}{12} - i sin \frac{5 π}{12})$ 254. $If (\frac{1 + i}{1 - i})^{x} = 1, then$ (1) $x = 4n, where n is any positive integer$ (2) $x = 2n, where n is any positive integer$ (3) $x = 4n + 1, where n is any positive integer$ (4) $x = 2n + 1, where n is any positive integer$ 255. $Let the minimum value v_{0} of v = ∣ z ∣^{2} + ∣ z - 3 ∣^{2} + ∣ z - 6 i ∣^{2}, z \in C is attained at z = z_{0} . Then 2 z_{0}^{2} - z ˉ_{0}^{3} + 3^{2} + v_{0}^{2} is equal to$ (1) $1000$ (2) $1024$ (3) $1105$ (4) $1196$ 256. $If x is a cube root of unity other than 1 , then (x + \frac{1}{x})^{2} + (x^{2} + \frac{1}{x ^{2}})^{2} + \dots + (x^{12} + \frac{1}{x ^{12}})^{2} =$ (1) $12$ (2) $64$ (3) $24$ (4) $0$ 257. $The equation whose solutions are the non-zero solutions of the equation z = i z^{2}, is$ (1) $z^{3} + i = 0$ (2) $z^{3} + z + 1 = 0$ (3) $z^{3} - i = 0$ (4) $z^{3} + iz + 1 = 0$ 258. $If z = x + iy, x, y \in R, (x, y) \neq = (0, - 4) and Arg (\frac{2 z - 3}{z + 4 i}) = \frac{π}{4}, then the locus of z is$ (1) $2 x^{2} + 2 y^{2} + 5 x + 5 y - 12 = 0$ (2) $2 x^{2} - 3 xy + y^{2} + 5 x + y - 12 = 0$ (3) $2 x^{2} + 3 xy + y^{2} + 5 x + y + 12 = 0$ (4) $2 x^{2} + 2 y^{2} - 11 x + 7 y - 12 = 0$ 259. $If (a + ib)^{\frac{1}{4}} = 2 + 3 i, then 3 b - 2 a is equal to$ (1) $-22$ (2) $-122$ (3) $-598$ (4) $-698$ 260. $The number of all possible solutions of the equation z^{3} + z ˉ = 0 is$ (1) $4$ (2) $5$ (3) $3$ (4) $6$ 261. $Let a, b, x and y be real numbers such that a - b = 1 and y \neq = 0 . If the complex number z = x + iy satisfies Im (\frac{az + b}{z + 1}) = y, then which of the following is (are) possible value (s) of x ? [JEE (Adv) 2017]$ (1) $- 1 - 1 - y^{2}$ (2) $1 + 1 + y^{2}$ (3) $1 - 1 + y^{2}$ (4) $- 1 + 1 - y^{2}$ 262. $If (2 x - y + 1) + i (x - 2 y - 1) = 2 - 3 i, then the multiplicative inverse of (x - iy) is$ (1) $\frac{15}{41} + \frac{12}{41} i$ (2) $\frac{6}{29} + \frac{15}{29} i$ (3) $\frac{15}{29} + \frac{6}{29} i$ (4) $\frac{12}{41} + \frac{15}{41} i$ 263. $Let r and θ respectively be the modulus and amplitude of the complex number z = 2 - i (2 tan \frac{5 π}{8}), then (r, θ) is equal to$ (1) $(2 sec \frac{3 π}{8}, \frac{3 π}{8})$ (2) $(2 sec \frac{3 π}{8}, \frac{5 π}{8})$ (3) $(2 sec \frac{5 π}{8}, \frac{3 π}{8})$ (4) $(2 sec \frac{11 π}{8}, \frac{11 π}{8})$ 264. $If α, β and γ are roots of x^{3} - 2 x + 1 = 0, then the value of \sum (\frac{1}{α + β - γ}) is$ (1) $\frac{- 1}{2}$ (2) $-1$ (3) $0$ (4) $\frac{1}{2}$ 265. $If x + iy = (1 + i)^{6} - (1 - i)^{6}, then which one of the following is true?$ (1) $x + y = 16$ (2) $x + y = - 16$ (3) $x + y = - 8$ (4) $x + y = 8$ 266. $If z = \frac{1}{2} - 2 i, is such that \| z + 1 \| = αz + β (1 + i), i = - 1 and α, β \in R, then α + β is equal to$ (1) $-4$ (2) $3$ (3) $2$ (4) $-1$ 267. $Let α and β be two roots of the equation x^{2} + 2x + 2 = 0, then α^{15} + β^{15} is equal to:$ (1) $512$ (2) $-256$ (3) $256$ (4) $-512$ 268. $If z, iz and z + iz are the vertices of a triangle whose area is 2 units, then the value of \| z \| is$ 269. $Let S = {z \in C : z^{2} + z ˉ = 0} . Then \sum_{z \in S} (Re (z) + Im (z)) is equal to ________ .$ 270. $Let α, β be the roots of the equation x^{2} - x + 2 = 0 with Im (α) > Im (β) . Then α^{6} + α^{4} + β^{4} - 5 α^{2} is equal to$