Complex Numbers PYQs Set 14 (PDF)

Set 14 of 14 · Questions 391-417 of 417

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Complex Numbers · Set 14 (Q391-417)
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391. $If 1, ω, ω^{2} are three cube roots of unity, then (1 - ω + ω^{2}) (1 + ω - ω^{2}) is$ (1) $2$ (2) $4$ (3) $1$ (4) $3$ 392. $Let z, w be complex numbers such that z ˉ + i w ˉ = 0 and ar g z w = π . Then ar g z equals:$ (1) $\frac{π}{4}$ (2) $\frac{π}{2}$ (3) $\frac{3 π}{4}$ (4) $\frac{5 π}{4}$ 393. $If Arg z_{1} and Arg z_{2} are \frac{π}{3} and \frac{π}{5} respectively then the value of Arg z_{1} + Arg z_{2} is$ (1) $\frac{11 π}{15}$ (2) $\frac{6 π}{15}$ (3) $\frac{2 π}{15}$ (4) $\frac{8 π}{15}$ 394. $If Z is a complex number such that Z = - Z ˉ, then$ (1) $Z is any complex number$ (2) $Real part of Z is the same as its imaginary part$ (3) $Z is purely real$ (4) $Z is purely imaginary$ 395. $If z and w are complex numbers such that z ˉ - i w ˉ = 0 and ar g (z w) = \frac{3 π}{4}, then ar g z =$ (1) $\frac{π}{16}$ (2) $\frac{π}{8}$ (3) $\frac{π}{4}$ (4) $\frac{3 π}{4}$ 396. $Let w = α + i β, β \neq = 0 and z \neq = 1 . If \frac{w - w ˉ z}{1 - z} is purely real, then the set of values of z is$ (1) ${z : \| z \| = 1}$ (2) ${z : z = z ˉ}$ (3) ${z : \| z \| \neq = 1}$ (4) ${z : \| z \| = 1, z \neq = 1}$ 397. $The value of \sum_{k = 1}^{10} (sin \frac{2 k π}{11} + i cos \frac{2 k π}{11}) is$ (1) $1$ (2) $- 1$ (3) $- i$ (4) $i$ 398. $If ω is a complex cube root of unity, then (\frac{1 - 3 i}{2})^{2020} + (\frac{1 + 3 i}{2})^{2026} + sin ((\sum_{j = 1}^{6} (j + ω) (j + ω^{2})) \frac{3 π}{152}) =$ (1) $-2$ (2) $2$ (3) $-1$ (4) $0$ 399. $If z + \frac{1}{z} = 1, then \frac{( z ^{20} + 1 ) ( z ^{40} + 1 ) ( z ^{60} + 1 )}{z ^{60}} =$ (1) $-2$ (2) $2$ (3) $1$ (4) $-1$ 400. $Let z_{k} = cos (\frac{2 kπ}{10}) + isin (\frac{2 kπ}{10}); k = 1, 2, \dots 9 . List I (P) For each z_{k} there exists a z_{j} such that z_{k} \cdot z_{j} = 1 (Q) There exists a k \in 1, 2, \dots \dots 9 such that z_{1.} . z = z_{k} has no solution z in the set of complex numbers. (R) \frac{( 1 - z _{1} ) ( 1 - z _{2} ) \dots .. ( 1 - z _{9} )}{10} equals (S) 1 - \sum cos (\frac{2 kπ}{10}) equals List II (I) True (II) False (III) 1 (IV) 2$ (1) $P -(I), Q - (II), R - (IV), S - (III)$ (2) $P -(II), Q - (I), R - (III), S - (IV)$ (3) $P -(I), Q - (II), R - (III), S - (IV)$ (4) $P -(II), Q - (I), R - (IV), S - (III)$ 401. $Let S be the set of all (α, β), π < α, β < 2 π, for which the complex number \frac{1 - i sin α}{1 + 2 i sin α} is purely imaginary and \frac{1 + i cos β}{1 - 2 i cos β} is purely real. Let Z_{αβ} = sin 2 α + i cos 2 β, (α, β) \in S Then \sum (i Z_{αβ} + \frac{1}{i Z _{αβ}}) is equal to:$ (1) $3$ (2) $3 i$ (3) $1$ (4) $2-i$ 402. $The imaginary part of i^{i} is$ (1) $0$ (2) $1$ (3) $2$ (4) $-1$ 403. $i^{2} + i^{3} + \dots + i^{4000} =$ (1) $1$ (2) $0$ (3) $i$ (4) $- i$ 404. $If z_{1}, z_{2}, z_{3} \in C are the vertices of an equilateral triangle, whose centroid is z_{0}, then \sum_{k = 1}^{3} (z_{k} - z_{0})^{2} is equal to$ (1) $0$ (2) $1$ (3) $i$ (4) $-i$ 405. $Let S_{1} = {z_{1} \in C : ∣ z_{1} - 3∣ = \frac{1}{2}} and S_{2} = {z_{2} \in C : ∣ z_{2} - ∣ z_{2} + 1∣∣ = ∣ z_{2} + ∣ z_{2} - 1∣∣} . Then, for z_{1} \in S_{1} and z_{2} \in S_{2}, the least value of ∣ z_{2} - z_{1} ∣ is:$ (1) $0$ (2) $\frac{1}{2}$ (3) $\frac{3}{2}$ (4) $\frac{5}{2}$ 406. $For a non-zero complex number z, let ar g (z) denote the principal argument with - π < ar g (z) \leq π . Then, which of the following statement(s) is (are) FALSE?$ (1) $Arg (- 1 - i) = \frac{π}{4}, where i = - 1$ (2) $The function f : R \to (- π, π], defined by f (t) = ar g (- 1 + i t) for all t \in R, is continuous at all points of R, where i = - 1$ (3) $For any two non-zero complex numbers z_{1} and z_{2}, arg (\frac{z _{1}}{z _{2}}) - arg (z_{1}) + arg (z_{2}) is an integer multiple of 2π$ (4) $For any three given distinct complex numbers z_{1}, z_{2} and z_{3}, the locus of the point z satisfying the condition arg (\frac{( z - z _{1} ) ( z _{2} - z _{3} )}{( z - z _{3} ) ( z _{2} - z _{1} )}) = π, lies on a straight line.$ 407. $If z_{1}, z_{2} are two distinct complex number such that \frac{z _{1} - 2 z _{2}}{\frac{1}{2} - z _{1} z _{2}} = 2, then$ (1) $z_{1} lies on a circle of radius \frac{1}{2} and z_{2} lies on a circle of radius 1 .$ (2) $both z_{1} and z_{2} lie on the same circle.$ (3) $either z_{1} lies on a circle of radius \frac{1}{2} or z_{2} lies on a circle of radius 1 .$ (4) $either z_{1} lies on a circle of radius 1 or z_{2} lies on a circle of radius \frac{1}{2} .$ 408. $For any complex number w = c + id, let arg (w) \in (- π, π], where i = - 1 . Let α and β be real numbers such that for all complex numbers z = x + iy satisfying arg (\frac{z + α}{z + β}) = \frac{π}{4} the ordered pair (x, y) lies on the circle x^{2} + y^{2} + 5 x - 3 y + 4 = 0 . Then which of the following statements is (are) TRUE ? (JEE Advanced 2021)$ (1) $α = - 1$ (2) $αβ = 4$ (3) $αβ = - 4$ (4) $β = 4$ 409. $For any complex number w = c + id, let arg (w) \in (- π, π], where i = - 1 . Let α and β be real numbers such that for all complex numbers z = x + iy satisfying arg (\frac{z + α}{z + β}) = \frac{π}{4}, the ordered pair (x, y) lies on the circle x^{2} + y^{2} + 5 x - 3 y + 4 = 0 Then, which of the following statements is (are) true? [2021 Adv.]$ (1) $α = - 1$ (2) $αβ = 4$ (3) $αβ = - 4$ (4) $β = 4$ 410. $Let θ_{1}, θ_{2}, \dots, θ_{10} be positive valued angles (in radian) such that θ_{1} + θ_{2} + \dots + θ_{10} = 2 π . Define the complex numbers z_{1} = e^{i θ_{1}}, z_{k} = z_{k - 1} e^{i θ_{k}} for k = 2, 3, \dots, 10, where i = - 1 . Consider the Statements P and Q given below: P : (z_{2} - z_{1}) + (z_{3} - z_{2}) + \dots + (z_{10} - z_{9}) + (z_{1} - z_{10}) \leq 2 π Q : (z_{2}^{2} - z_{1}^{2}) + (z_{3}^{2} - z_{2}^{2}) + \dots + (z_{10}^{2} - z_{9}^{2}) + (z_{1}^{2} - z_{10}^{2}) \leq 4 π Then, [2021 Adv.]$ (1) $P is true and Q is false$ (2) $Q is true and P is false$ (3) $Both P and Q are true$ (4) $Both P and Q are false$ 411. $The multiplicative inverse of z is$ (1) $\frac{1}{z + z ˉ}$ (2) $\frac{z}{∣ z ˉ ∣}$ (3) $\frac{z ˉ}{∣ z ∣ ^{2}}$ (4) $\frac{1}{z ˉ}$ 412. $Let z_{1} = 2 + 3 i and z_{2} = 3 + 4 i . The set S = {z \in C : ∣ z - z_{1} ∣^{2} - ∣ z - z_{2} ∣^{2} = ∣ z_{1} - z_{2} ∣^{2}} represents a$ (1) $Straight line with sum of its intercepts on the coordinate axes equals 14$ (2) $Hyperbola with the length of the transverse axis 7$ (3) $Straight line with the sum of its intercepts on the coordinate axes equals - 18$ (4) $Hyperbola with eccentricity 2$ 413. $If log_{2} (9^{x - 1} + 7) - log_{2} (3^{x - 1} + 1) = 2, then values of x are$ (1) $1, 2$ (2) $0, 2$ (3) $0, 1$ (4) $1, 4$ 414. $Let arg (z) represent the principal argument of the complex number z. The, \|z\| = 3 and arg (z - 1) - arg (z + 1) = \frac{π}{4} intersect:$ (1) $Exactly at one point$ (2) $Exactly at two points$ (3) $Nowhere$ (4) $At infinitely many points.$ 415. $Let z, w be complex numbers such that z ˉ + i w ˉ = 0 and ar g (z w) = π . Then ar g z in degree equals$ 416. $If the real part of the complex number z = \frac{3 + 2 i c o s θ}{1 - 3 i c o s θ}, θ \in (0, \frac{π}{2}) is zero, then the value of sin^{2} 3 θ + cos^{2} θ is equal to$ 417. $Let S = {z \in C : ∣ z - 3∣ \leq 1 and z (4 + 3 i) + z ˉ (4 - 3 i) \leq 24} . If α + i β is the point in S which is closest to 4 i, then 25 (α + β) is equal to_____.$