Complex Numbers PYQs Set 8 (PDF)

Set 8 of 14 · Questions 211-240 of 417

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Complex Numbers · Set 8 (Q211-240)
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211. $The locus of z such that \frac{z - i}{z + i} = 2, where z = x + i y, is$ (1) $3 x^{2} + 3 y^{2} + 10 y + 3 = 0$ (2) $3 x^{2} - 3 y^{2} - 10 y - 3 = 0$ (3) $3 x^{2} + 3 y^{2} + 10 y - 3 = 0$ (4) $x^{2} + y^{2} - 5 y + 3 = 0$ 212. $One of the values of (3 - i)^{\frac{2}{5}} is$ (1) $2^{\frac{2}{5}} (1 - 3 i)$ (2) $2^{\frac{- 3}{5}} (3 + i)$ (3) $2^{\frac{2}{5}} (3 - i)$ (4) $2^{\frac{- 3}{5}} (1 + 3 i)$ 213. $If (\frac{3 + i}{3 - i})^{m} = 1, 2022 < m < 2029, then m =$ (1) $2022$ (2) $2024$ (3) $2028$ (4) $2026$ 214. $If Z_{1} = 4 i^{40} - 5 i^{35} + 6 i^{17} + 2, Z_{2} = - 1 + i, where i = - 1, then ∣ Z_{1} + Z_{2} ∣ =$ (1) $5$ (2) $13$ (3) $12$ (4) $15$ 215. $Let the complex number z = x + iy be such that \frac{2 z - 3 i}{2 z + i} is purely imaginary. If x + y2 = 0, then y^{4} + y2 - y is equal to$ (1) $\frac{2}{3}$ (2) $\frac{3}{2}$ (3) $\frac{3}{4}$ (4) $\frac{4}{3}$ 216. $(sin θ - icos θ)^{3} is equal to$ (1) $i^{3} (cos 3 θ + isin 3 θ)$ (2) $cos 3 θ + isin 3 θ$ (3) $sin 3 θ - icos 3 θ$ (4) $(- i)^{3} (cos 3 θ + isin 3 θ)$ 217. $If (x - iy) (3 + 5 i) is the conjugate of - 6 - 24 i (where x, y \in R and i = - 1), then the values of x and y are respectively.$ (1) $5, - 3$ (2) $3, - 3$ (3) $- 3, 3$ (4) $5, 3$ 218. $If z = x + i y is a complex number such that z z ˉ^{3} + z ˉ z^{3} = 350 and x, y are integers, then ∣ z ∣ =$ (1) $41$ (2) $5$ (3) $25$ (4) $13$ 219. $If ω_{0}, ω_{1}, \dots ., ω_{n - 1} are the n th roots of unity, then (1 + 2 ω_{0}) (1 + 2 ω_{1}) (1 + 2 ω_{2}) \dots (1 + 2 ω_{n - 1}) =$ (1) $1 + (- 1)^{n} 2^{n}$ (2) $1 + 2^{n}$ (3) $(- 1)^{n} + 2^{n}$ (4) $1 + (- 1)^{n - 1} 2^{n}$ 220. $Let ω be a complex cube root of unity with ω \neq = 1 and P = (p_{ij}) be a n \times n matrix with p_{ij} = ω^{i + j} . Then P^{2} \neq = 0, when n =$ (1) $57$ (2) $55$ (3) $58$ (4) $56$ 221. $If y^{2} + z^{2} = 3 yz, z^{2} + x^{2} = 8 zx, x^{2} + y^{2} = 4 xy, then the value of \frac{y ^{2}}{xz} + \frac{xz}{y ^{2}} is$ (1) $2$ (2) $3$ (3) $4$ (4) $5$ 222. $If a and b are real numbers such that (2 + α)^{4} = a + bα where α = \frac{- 1 + i 3}{2} then a + b is equato$ (1) $33$ (2) $9$ (3) $24$ (4) $57$ 223. $If z \in C, then the minimum value of \| z \| + \| 2 z - 3 \| + \| z - 1 \| is$ (1) $2$ (2) $1$ (3) $3$ (4) $0$ 224. $If (\frac{1 + i}{1 - i})^{x} = 1 then$ (1) $x = 4 n + 1; n \in N$ (2) $x = 2 n + 1; n \in N$ (3) $x = 2 n; n \in N$ (4) $x = 4 n; n \in N$ 225. $If 1, ω, ω^{2} are the cube roots of unity then the value of (x + y)^{2} + (xω + y ω^{2})^{2} + (x ω^{2} + yω)^{2} is$ (1) $2 x^{2} \cdot 3 y^{2}$ (2) $4 xy$ (3) $6 xy$ (4) $2 x^{2} \cdot 2 y^{2}$ 226. $Let a = \frac{- 1 + i 3}{2} is a = (1 + α) \sum α^{2 k} and b = \sum α^{3 k} then a and b are the roots of thequadratic equation:$ (1) $x^{2} - 102 x + 101 = 0$ (2) $x^{2} - 101 x + 100 = 0$ (3) $x^{2} + 101 x + 100 = 0$ (4) $x^{2} + 102 x + 100 = 0$ 227. $A value of θ for which \frac{2 + 3 i sin θ}{1 - 2 i sin θ} is purely imaginary, is:$ (1) $\frac{π}{6}$ (2) $s i n^{- 1} (\frac{3}{4})$ (3) $s i n^{- 1} (\frac{1}{3})$ (4) $\frac{π}{3}$ 228. $The value of \sum_{K = 1}^{6} (sin \frac{2 Kπ}{7} - icos \frac{2 Kπ}{7}) is$ (1) $- i$ (2) $-1$ (3) $i$ (4) $0$ 229. $If a + bi = \frac{i}{1 - i}, then (a, b) =$ (1) $(\frac{- 1}{2}, \frac{- 1}{2})$ (2) $(\frac{1}{2}, \frac{1}{2})$ (3) $(\frac{1}{2}, \frac{- 1}{2})$ (4) $(\frac{- 1}{2}, \frac{1}{2})$ 230. $If 2 x = - 1 + 3 i, then the value of (1 - x^{2} + x)^{6} - (1 - x + x^{2})^{6} =$ (1) $32$ (2) $-64$ (3) $64$ (4) $0$ 231. $If 1, z_{1}, z_{2}, \dots, z_{n - 1} are the n th roots of unity, then (1 - z_{1}) (1 - z_{2}) \dots (1 - z_{n - 1}) is equal to$ (1) $0$ (2) $n - 1$ (3) $n$ (4) $1$ 232. $Let z be a complex number such that \| z + 2 \| = 1 and Im (\frac{z + 1}{z + 2}) = \frac{1}{5} . Then the value of \| Re (z + 2) \| is$ (1) $\frac{2 6}{5}$ (2) $\frac{24}{5}$ (3) $\frac{1 + 6}{5}$ (4) $\frac{6}{5}$ 233. $If ∣ z_{1} ∣ = 1, ∣ z_{2} ∣ = 2, ∣ z_{3} ∣ = 3 and ∣9 z_{1} z_{2} + 4 z_{1} z_{3} + z_{2} z_{3} ∣ = 12, then the value of ∣ z_{1} + z_{2} + z_{3} ∣ is$ (1) $3$ (2) $4$ (3) $8$ (4) $2$ 234. $Let u = \frac{2 z + i}{z - ki}, z = x + iy and k > 0 . If the curve represented by Re (u) + lm (u) = 1 intersects the y -axis at the point P and Q where PQ = 5, then the value of K is:$ (1) $1/2$ (2) $3/2$ (3) $2$ (4) $4$ 235. $Let a \neq = b be two non-zero real numbers. Then the number of elements in the set X = {z \in C: R e (a z^{2} + bz) = a and R e (b z^{2} + az) = b} is equal to$ (1) $0$ (2) $1$ (3) $3$ (4) $2$ 236. $If z = x - iy and z^{1/3} = p + iq, then (\frac{x}{p} + \frac{y}{q}) / (p^{2} + q^{2}) is equal to:$ (1) $1$ (2) $-1$ (3) $2$ (4) $-2$ 237. $Let i = - 1 . If \frac{( - 1 + i 3 ) ^{21}}{( 1 - i ) ^{24}} + \frac{( 1 + i 3 ) ^{21}}{( 1 + i ) ^{24}} = k, and n = [\| k \|] be the greatest integral part of \| k \| . Then \sum (j + 5)^{2} - \sum (j + 5) is equal to$ 238. $If z is a complex number, then the minimum value of \| z \| + \| z - 1 \| is$ 239. $Let z be a complex number with non-zero imaginary part. If \frac{2 + 3 z + 4 z ^{2}}{2 - 3 z + 4 z ^{2}} is a real number, then find the value of \| z \|^{2} . [2022 Adv.]$ 240. $Let z = (1 + i) (1 + 2 i) (1 + 3 i) \dots (1 + ni), where i = - 1 . If \| z \|^{2} = 44200, then n is equal to____.$