Argand Plane and Polar Representation

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JEE Main 2004

Let $z$ and $w$ be complex numbers such that $\overset{z}{ˉ} + i \overset{w}{ˉ} = 0$ and $a r g z w = π$ . Then $a r g z$ equals

5 π /4

(B)

π /2

(C)

3 π /4

(D)

π /4

JEE Main 2003

LEVELJEE Main

If $z$ and $ω$ are two non-zero complex numbers such that $∣ z ω ∣ = 1$ and $A r g (z) - A r g (ω) = π /2$ , then $\overset{z}{ˉ} ω$ is equal to

$z$ and $w$ are two nonzero complex numbers such that $∣ z ∣ = ∣ w ∣$ and $A r g z + A r g w = π$ then $z$ equals

(A)

\overset{w}{ˉ}

(B)

- \overset{w}{ˉ}

(C)

w

(D)

- w

JEE Advanced 2013

LEVELJEE Advanced

Let $ω = \frac{3 + i}{2}$ and $P = {ω^{n} : n = 1, 2, 3, \dots}$ . Further $H_{1} = {z \in C : R ez > 1/2}$ and $H_{2} = {z \in C : R ez < - 1/2}$ , where $C$ is the set of all complex numbers. If $z_{1} \in P \cap H_{1}, z_{2} \in P \cap H_{2}$ and $O$ represents the origin, then $∠ z_{1} O z_{2} =$

* Multiple Correct Options

(A)

π /2

(B)

π /6

(C)

2 π /3

(D)

5 π /6

JEE Advanced 2000S

LEVELJEE Main

If $a r g (z) < 0$ , then $a r g (- z) - a r g (z) =$

(A)

π

(B)

- π

(C)

π /2

(D)

- π /2

JEE Advanced 1995S

LEVELJEE Main

Let $z$ and $ω$ be two non zero complex numbers such that $∣ z ∣ = ∣ ω ∣$ and $A r g z + A r g ω = π$ , then $z$ equals

(A)

ω

(B)

- ω

(C)

\overset{ω}{ˉ}

(D)

- \overset{ω}{ˉ}

JEE Main 2019 (10 January)

LEVELJEE Main

Let $z = (\frac{3}{2} + \frac{i}{2})^{5} + (\frac{3}{2} - \frac{i}{2})^{5}$ . If $R (z)$ and $I (z)$ respectively denote the real and imaginary parts of $z$ , then :

(A)

R(z) > 0 and I(z) > 0

(B)

R(z) < 0 and I(z) > 0

JEE Main 2019 (10 April Shift 2)

LEVELJEE Main

If $z$ and $w$ are two complex numbers such that $∣ z w ∣ = 1$ and $ar g (z) - ar g (w) = π /2$ , then :

(A)

\overset{z}{ˉ} w = i

(B)

z \overset{w}{ˉ} = - i

(C)

\overset{z}{ˉ} w = (1 - i) / 2

(D)

z \overset{w}{ˉ} = (- 1 + i) / 2

JEE Main 2021 (26 Aug Shift 2)

LEVELJEE Main

The least positive integer $n$ such that $\frac{( 2 i ) ^{n}}{( 1 - i ) ^{n - 2}}$ , $i = - 1$ is a positive integer, is.

JEE Main 2021 (20 July Shift 1)

LEVELJEE Main

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 $z$ )

(A)

\frac{π}{4}

(B)

- \frac{3 π}{4}

(C)

- \frac{π}{4}

(D)

\frac{3 π}{4}

JEE Main 2021 (February)

LEVELJEE Advanced

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 $\sum_{j = 0}^{n + 5} (j + 5)^{2} - \sum_{j = 0}^{n} (j + 5)$ is equal to

JEE Main 2022 (25 June Shift 2)

LEVELJEE Main

Let $z_{1}$ and $z_{2}$ be two complex numbers such that $\overset{z}{ˉ}_{1} = i \overset{z}{ˉ}_{2}$ and $ar g (\frac{z _{1}}{z ˉ _{2}}) = π$ . Then

(A)

ar g z_{2} = \frac{π}{4}

(B)

ar g z_{2} = - \frac{3 π}{4}

(C)

ar g z_{1} = \frac{π}{4}

(D)

ar g z_{1} = - \frac{3 π}{4}

JEE Main 2023 (31 January Shift 2)

LEVELJEE Main

The complex number $z = \frac{i - 1}{c o s \frac{π}{3} + i s i n \frac{π}{3}}$ is equal to:

(A)

2 (cos \frac{5 π}{12} + i sin \frac{5 π}{12})

(B)

cos \frac{π}{12} - i sin \frac{π}{12}

(C)

2 (cos \frac{π}{12} + i sin \frac{π}{12})

(D)

2 i (cos \frac{5 π}{12} - i sin \frac{5 π}{12})

JEE Main 2024 (29 Jan Shift 2)

LEVELJEE Main

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

(A)

(2 sec \frac{3 π}{8}, \frac{3 π}{8})

(B)

(2 sec \frac{3 π}{8}, \frac{5 π}{8})

(C)

(2 sec \frac{5 π}{8}, \frac{3 π}{8})

(D)

(2 sec \frac{11 π}{8}, \frac{11 π}{8})

JEE Main 2025 (January)

LEVELJEE Main

Let $z_{1}, z_{2}$ and $z_{3}$ be three complex numbers on the circle $∣ z ∣ = 1$ with $a r g (z_{1}) = \frac{- π}{4}, a r g (z_{2}) = 0$ and $a r g (z_{3}) = \frac{π}{4}$ . If $∣ z_{1} \overline{z}_{2} + z_{2} \overline{z}_{3} + z_{3} \overline{z}_{1} ∣^{2} = α + β 2, α, β \in Z,$ then the value of $α^{2} + β^{2}$ is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let z and w be complex numbers such that zˉ+iwˉ=0 and argzw=π. Then argz equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If z and ω are two non-zero complex numbers such that ∣zω∣=1 and Arg(z)−Arg(ω)=π/2, then zˉω is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } z and w are two nonzero complex numbers such that ∣z∣=∣w∣ and Argz+Argw=π then z equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If arg(z)<0, then arg(−z)−arg(z)=

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let z and ω be two non zero complex numbers such that ∣z∣=∣ω∣ and Argz+Argω=π, then z equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let z=(23​​+2i​)5+(23​​−2i​)5. If R(z) and I(z) respectively denote the real and imaginary parts of z, then :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If z and w are two complex numbers such that ∣zw∣=1 and arg(z)−arg(w)=π/2, then :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The least positive integer n such that (1−i)n−2(2i)n​, i=−1​ is a positive integer, is.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let z1​ and z2​ be two complex numbers such that zˉ1​=izˉ2​ and arg(zˉ2​z1​​)=π. Then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The complex number z=cos3π​+isin3π​i−1​ is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let r and θ respectively be the modulus and amplitude of the complex number z=2−i(2tan85π​), then (r,θ) is equal to

Let $z$ and $w$ be complex numbers such that $\overset{z}{ˉ} + i \overset{w}{ˉ} = 0$ and $a r g z w = π$ . Then $a r g z$ equals

If $z$ and $ω$ are two non-zero complex numbers such that $∣ z ω ∣ = 1$ and $A r g (z) - A r g (ω) = π /2$ , then $\overset{z}{ˉ} ω$ is equal to

$z$ and $w$ are two nonzero complex numbers such that $∣ z ∣ = ∣ w ∣$ and $A r g z + A r g w = π$ then $z$ equals

If $a r g (z) < 0$ , then $a r g (- z) - a r g (z) =$

Let $z$ and $ω$ be two non zero complex numbers such that $∣ z ∣ = ∣ ω ∣$ and $A r g z + A r g ω = π$ , then $z$ equals

Let $z = (\frac{3}{2} + \frac{i}{2})^{5} + (\frac{3}{2} - \frac{i}{2})^{5}$ . If $R (z)$ and $I (z)$ respectively denote the real and imaginary parts of $z$ , then :

If $z$ and $w$ are two complex numbers such that $∣ z w ∣ = 1$ and $ar g (z) - ar g (w) = π /2$ , then :

The least positive integer $n$ such that $\frac{( 2 i ) ^{n}}{( 1 - i ) ^{n - 2}}$ , $i = - 1$ is a positive integer, is.

Let $z_{1}$ and $z_{2}$ be two complex numbers such that $\overset{z}{ˉ}_{1} = i \overset{z}{ˉ}_{2}$ and $ar g (\frac{z _{1}}{z ˉ _{2}}) = π$ . Then

The complex number $z = \frac{i - 1}{c o s \frac{π}{3} + i s i n \frac{π}{3}}$ is equal to:

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