Complex Numbers PYQs Set 13 (PDF)

Set 13 of 14 · Questions 361-390 of 417

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Complex Numbers · Set 13 (Q361-390)
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361. $If z and ω are two complex numbers such that ∣ z ω ∣ = 1 and ar g (z) - ar g (ω) = π /2, then:$ (1) $z ˉ ω = i$ (2) $z ˉ ω = \frac{- 1 + i}{2}$ (3) $z ˉ ω = - i$ (4) $z ˉ ω = \frac{1 - i}{2}$ 362. $Let ω be a complex number such that 2 ω + 1 = z where z = - 3 . If 111 1 - ω^{2} - 1 ω^{2} 1 ω^{2} ω^{7} = 3 k, then k is equal to:$ (1) $-z$ (2) $z$ (3) $-1$ (4) $1$ 363. $If z and ω are two non-zero complex numbers such that ∣ z ω ∣ = 1, and ar g (z) - ar g (ω) = \frac{π}{2}, then z ˉ ω is equal to:$ (1) $1$ (2) $-1$ (3) $i$ (4) $- i$ 364. $One of the values of (3 - i)^{\frac{1}{6}} is$ (1) $2^{\frac{1}{6}} cis \frac{61 π}{36}$ (2) $2^{\frac{1}{6}} cis \frac{37 π}{36}$ (3) $2^{\frac{1}{6}} cis \frac{59 π}{36}$ (4) $2^{\frac{1}{6}} cis \frac{49 π}{36}$ 365. $For all z \in C on the curve C_{1} : ∣ z ∣ = 4, let the locus of the point z + \frac{1}{z} be the curve C_{2} Then$ (1) $The curves C_{1} and C_{2} intersect at 4 points$ (2) $The curves C_{1} lies inside C_{2}$ (3) $The curves C_{1} and C_{2} intersect at 2 points$ (4) $The curves C_{2}, lies inside C_{1} .$ 366. $Assertion : If the arguments of z ˉ_{1} and z_{2} are \frac{π}{5} and \frac{π}{3} respectively, then ar g (z_{1} z_{2}) is \frac{2 π}{15} . Reason : For any complex number z, ar g z ˉ = \frac{π}{2} + ar g z The correct option among the following is$ (1) $(A) is true, (R) is true and (R) is the correct explanation for (A)$ (2) $(A) is true (R) is true but (R) is not the correct explanation for (A)$ (3) $(A) is true but (R) is false$ (4) $(A) is false but (R) is true$ 367. $Let p, q \in R and (1 - 3 i)^{200} = 2^{199} (p + i q), i = - 1 . Then p + q + q^{2} and p - q + q^{2} are roots of the equation.$ (1) $x^{2} + 4 x - 1 = 0$ (2) $x^{2} - 4 x + 1 = 0$ (3) $x^{2} + 4 x + 1 = 0$ (4) $x^{2} - 4 x - 1 = 0$ 368. $Let S = {z \in C : 4 z^{2} + z ˉ = 0} . Then \sum_{z \in S} ∣ z ∣^{2} is equal to:$ (1) $\frac{5}{64}$ (2) $\frac{1}{16}$ (3) $\frac{3}{16}$ (4) $\frac{7}{64}$ 369. $Let the lines (2 - i) z = (2 + i) z ˉ and (2 + i) z + (i - 2) z ˉ - 4 i = 0, (here i^{2} = - 1) be normal to a circle C . If the line i z + z ˉ + 1 + i = 0 is tangent to this circle C, then its radius is:$ (1) $\frac{3}{2}$ (2) $32$ (3) $\frac{3}{2 2}$ (4) $\frac{1}{2 2}$ 370. $If z is a complex number of unit modulus and argument θ, then ar g (\frac{1 + z}{1 + z ˉ}) equals :$ (1) $- θ$ (2) $\frac{π}{2} - θ$ (3) $θ$ (4) $π - θ$ 371. $Let A = {θ \in (0, 2 π) : \frac{1 + 2 i s i n θ}{1 - i s i n θ} is purely imaginary} . Then the sum of the elements in A is$ (1) $4 π$ (2) $3 π$ (3) $7 π$ (4) $2 π$ 372. $Let z_{1}, z_{2} and z_{3} be three complex numbers on the circle ∣ z ∣ = 1 with ar g (z_{1}) = \frac{- π}{4}, ar g (z_{2}) = 0 and ar g (z_{3}) = \frac{π}{4} . If ∣ z_{1} z ˉ_{2} + z_{2} z ˉ_{3} + z_{3} z ˉ_{1} ∣^{2} = α + β 2, α, β \in Z, then the value of α^{2} + β^{2} is:$ (1) $24$ (2) $29$ (3) $41$ (4) $31$ 373. $If for z = α + i β, ∣ z + 2∣ = z + 4 (1 + i), then α + β and α β are the roots of the equation$ (1) $x^{2} + 3x-4 = 0$ (2) $x^{2} + 7x + 12 = 0$ (3) $x^{2} + x - 12 = 0$ (4) $x^{2} + 2x - 3 = 0$ 374. $Let z_{1} and z_{2} be two complex numbers such that z_{1} = i z_{2} and ar g (\frac{z _{1}}{z _{2}}) = π . Then$ (1) $ar g z_{2} = \frac{π}{4}$ (2) $ar g z_{2} = - \frac{3 π}{4}$ (3) $ar g z_{1} = \frac{π}{4}$ (4) $ar g z_{1} = - \frac{3 π}{4}$ 375. $If z_{1} = 2 + 5 i, z_{2} = - 1 + 4 i and z_{3} = i, then \frac{z _{1} - z _{3}}{z _{3} - z _{2}} =$ (1) $2$ (2) $22$ (3) $52$ (4) $42$ 376. $Let z = cos θ + i sin θ . Then the value of \sum_{m = 1}^{15} Im (z^{2 m - 1}) at θ = 2^{\circ} is$ (1) $\frac{1}{sin 2 ^{\circ}}$ (2) $\frac{1}{3 sin 2 ^{\circ}}$ (3) $\frac{1}{2 sin 2 ^{\circ}}$ (4) $\frac{1}{4 sin 2 ^{\circ}}$ 377. $z = cos θ + i sin θ \Rightarrow z^{r} + (z ˉ)^{r} =$ (1) $cosrθ$ (2) $2 cosrθ$ (3) $sinrθ$ (4) $2 sinrθ$ 378. $Let S = {z = x + i y : \frac{2 z - 3 i}{4 z + 2 i} is a real number} . Then which of the following is NOT correct?$ (1) $y + x^{2} + y^{2} \neq = - \frac{1}{4}$ (2) $(x, y) = (0, - \frac{1}{2})$ (3) $x = 0$ (4) $y \in (- \infty, - \frac{1}{2}) \cup (- \frac{1}{2}, \infty)$ 379. $Let a, b \in R and a^{2} + b^{2} \neq = 0 . Suppose S = {z \in C : z = \frac{1}{a + ib t}, t \in R, t \neq = 0}, where i = - 1 . If z = x + i y and z \in S, then (x, y) lies on$ (1) $the circle with radius \frac{1}{2 a} and centre (\frac{1}{2 a}, 0) for a > 0, b \neq = 0$ (2) $the circle with radius - \frac{1}{2 a} and centre (\frac{1}{2 a}, 0) for a < 0, b \neq = 0$ (3) $the x-axis for a \neq = 0, b = 0$ (4) $the y-axis for a = 0, b \neq = 0$ 380. $If z and ω are two complex numbers such that ∣ z ω ∣ = 1 and ar g (z) - ar g (ω) = \frac{3 π}{2}, then ar g (\frac{1 - 2 z ˉ ω}{1 + 3 z ˉ ω}) is : (Here ar g (z) denotes the principal argument of complex number z)$ (1) $\frac{π}{4}$ (2) $- \frac{3 π}{4}$ (3) $- \frac{π}{4}$ (4) $\frac{3 π}{4}$ 381. $If α and β are different complex numbers with ∣ β ∣ = 1, then \frac{β - α}{1 - α ˉ β} is equal to$ (1) $2$ (2) $1/2$ (3) $1$ (4) $1/3$ 382. $If α is a root of the equation x^{2} + x + 1 = 0 and \sum_{k = 1}^{n} (α^{k} + \frac{1}{α ^{k}})^{2} = 20, then n is equal to ______.$ 383. $The value of \sum_{r = 1}^{8} (sin \frac{2 r π}{9} + i cos \frac{2 r π}{9}) is N . Find modulus of N .$ 384. $Let α, β be the roots of the equation x^{2} - a x - b = 0 with Im (α) < Im (β) . Let P_{n} = α^{n} - β^{n} . If P_{3} = - 57 i, P_{4} = - 37 i, P_{5} = 117 i and P_{6} = 457 i, then (α^{4} + β^{4}) is equal to$ 385. $If α satisfies the equation x^{2} + x + 1 = 0 and (1 + α)^{7} = A + B α + C α^{2}, A, B, C \geq 0, then 5 (3 A - 2 B - C) is equal to______$ 386. $For any integer k, let α_{k} = cos (\frac{k π}{7}) + i sin (\frac{k π}{7}), where i = - 1 . The value of the expression \frac{\sum _{k = 1}^{12} ∣ α _{k + 1} - α _{k} ∣}{\sum _{k = 1}^{3} ∣ α _{4 k - 1} - α _{4 k - 2} ∣} is$ 387. $If z^{2} + z + 1 = 0, z \in C, then \sum_{n = 1}^{15} (z^{n} + (- 1)^{n} \frac{1}{z ^{n}})^{2} is equal to______.$ 388. $Let α = 8 - 14 i, A = {z \in C : \frac{α z - α ˉ z ˉ}{z ^{2} - ( z ˉ ) ^{2} - 112 i} = 1} and B = {z \in C : ∣ z + 3 i ∣ = 4} . Then \sum_{z \in A \cap B} (Re z - Im z) is equal to$ 389. $Let ω \neq = 1 be a cube root of unity. Then the minimum of the set {∣ a + bω + c ω^{2} ∣^{2} : a, b, c are distinct non-zero integers} equals ____ . (JEE Advanced 2019)$ 390. $Let the complex numbers α and \frac{1}{α} lie on the circles ∣ z - z_{0} ∣^{2} = 4 and ∣ z - z_{0} ∣^{2} = 16 respectively, where z_{0} = 1 + i . Then, the value of 100∣ α ∣^{2} is.$