Conjugate and Modulus

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JEE Main 2015

A complex number $z$ is said to be unimodular if $∣ z ∣ = 1$ . Suppose $z_{1}$ and $z_{2}$ are complex numbers such that $\frac{z _{1} - 2 z _{2}}{2 - z _{1} z ˉ _{2}}$ is unimodular and $z_{2}$ is not unimodular. Then the point $z_{1}$ lies on a

2

(C)

straight line parallel to x-axis

(D)

straight line parallel to y-axis.

JEE Main 2013

LEVELJEE Main

If $z$ is a complex number of unit modulus and argument $θ$ , then $a r g (\frac{1 + z}{1 + z ˉ})$ equals

(A)

- θ

(B)

π /2 - θ

(C)

θ

(D)

π - θ

JEE Main 2009

LEVELJEE Main

If $∣ z - 4/ z ∣ = 2$ , then the maximum value of $∣ Z ∣$ is equal to

(A)

5 + 1

3 + 1

JEE Advanced 2006

LEVELJEE Main

If $\frac{w - w ˉ z}{1 - z}$ is purely real where $w = α + i β, β \neq = 0$ and $z \neq = 1$ , then the set of the values of $z$ is

(A)

{z : ∣ z ∣ = 1}

(B)

{z : z = \overset{z}{ˉ}}

(C)

{z : z \neq = 1}

(D)

{z : ∣ z ∣ = 1, z \neq = 1}

JEE Advanced 2000S

LEVELJEE Main

If $z_{1}, z_{2}$ and $z_{3}$ are complex numbers such that $∣ z_{1} ∣ = ∣ z_{2} ∣ = ∣ z_{3} ∣ = ∣1/ z_{1} + 1/ z_{2} + 1/ z_{3} ∣ = 1$ , then $∣ z_{1} + z_{2} + z_{3} ∣$ is

Let $z$ and $ω$ be two complex numbers such that $∣ z ∣ \leq 1, ∣ ω ∣ \leq 1$ and $∣ z + iω ∣ = ∣ z - i \overset{ω}{ˉ} ∣ = 2$ then $z$ equals

If $z_{1}$ and $z_{2}$ are two complex numbers such taht $∣ z_{1} ∣ < 1 < ∣ z_{2} ∣$ then prove that $∣ \frac{1 - z _{1} z ˉ _{2}}{z _{1} - z _{2}} ∣ < 1$ .

JEE Advanced 1979

LEVELBoard

If $x + i y = \frac{a + ib}{c + i d}$ , prove that $(x^{2} + y^{2})^{2} = \frac{a ^{2} + b ^{2}}{c ^{2} + d ^{2}}$ .

JEE Main 2019 (09 April Shift 1)

LEVELJEE Main

All the points in the set $S = {\frac{α + i}{α - i} : α \in R} (i = - 1)$ lie on a

(A)

circle whose radius is 1.

(B)

straight line whose slope is 1.

(C)

straight line whose slope is -1

(D)

circle whose radius is

2

JEE Main 2019 (10 April Shift 1)

LEVELJEE Main

If $a > 0$ and $z = \frac{( 1 + i ) ^{2}}{a - i}$ , has magnitude $2/5$ , then $z$ is equal to :

(A)

\frac{3}{5} - \frac{1}{5} i

(B)

\frac{1}{5} + \frac{3}{5} i

(C)

- \frac{1}{5} + \frac{3}{5} i

(D)

\frac{1}{5} - \frac{3}{5} i

JEE Main 2021 (16 March Shift 1)

LEVELJEE Main

Let a complex number $z, ∣ z ∣ \neq = 1$ , satisfy $lo g_{\frac{1}{2}} (\frac{∣ z ∣ + 11}{( ∣ z ∣ - 1 ) ^{2}}) \leq 2$ . Then, the largest value of $∣ z ∣$ is equal to

JEE Main 2021 (16 March Shift 2)

LEVELJEE Main

The least value of $∣ z ∣$ where $z$ is complex number which satisfies the inequality $exp (\frac{( ∣ z ∣ + 3 ) ( ∣ z ∣ - 1 )}{∣∣ z ∣ + 1∣} lo g_{e} 2) \geq lo g_{2} ∣5 7 + 9 i ∣$ , $i = - 1$ , is equal to :

(A)

(B)

5

(C)

(D)

JEE Main 2022 (29 July Shift 2)

LEVELJEE Main

If $z \neq = 0$ be a complex number such that $∣ z - \frac{1}{z} ∣ = 2$ , then the maximum value of $∣ z ∣$ is:

(A)

2

(B)

(C)

2 - 1

(D)

2 + 1

JEE Main 2024 (30 Jan Shift 1)

LEVELBoard

If $z = x + i y, x y \neq = 0$ , satisfies the equation $z^{2} + i \overset{z}{ˉ} = 0$ , then $∣ z ∣^{2}$ is equal to :

JEE Main 2026 (24 January Shift 2)

LEVELBoard

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 .........

JEE Advanced 2023

LEVELJEE Main

Let $z$ be complex number satisfying $∣ z ∣^{3} + 2 z^{2} + 4 \overset{z}{ˉ} - 8 = 0$ , where $\overset{z}{ˉ}$ denotes the complex conjugate of $z$ . Let the imaginary part of $z$ be nonzero. Match each entry in List-I to the correct entries in List-II.

List-I

(P)

∣ z ∣^{2}

is equal to

(Q)

∣ z - \overset{z}{ˉ} ∣^{2}

is equal to

(R)

∣ z ∣^{2} + ∣ z - \overset{z}{ˉ} ∣^{2}

is equal to

(S)

∣ z + 1 ∣^{2}

is equal to

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If z is a complex number of unit modulus and argument θ, then arg(1+zˉ1+z​) equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ∣z−4/z∣=2, then the maximum value of ∣Z∣ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If 1−zw−wˉz​ is purely real where w=α+iβ,β=0 and z=1, then the set of the values of z is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If z1​,z2​ and z3​ are complex numbers such that ∣z1​∣=∣z2​∣=∣z3​∣=∣1/z1​+1/z2​+1/z3​∣=1, then ∣z1​+z2​+z3​∣ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let z and ω be two complex numbers such that ∣z∣≤1,∣ω∣≤1 and ∣z+iω∣=∣z−iωˉ∣=2 then z equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If z1​ and z2​ are two complex numbers such taht ∣z1​∣<1<∣z2​∣ then prove that ∣z1​−z2​1−z1​zˉ2​​∣<1.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If x+iy=c+ida+ib​​, prove that (x2+y2)2=c2+d2a2+b2​.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } All the points in the set S={α−iα+i​:α∈R}(i=−1​) lie on a

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a>0 and z=a−i(1+i)2​, has magnitude 2/5​, then z is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a complex number z,∣z∣=1, satisfy log2​1​​((∣z∣−1)2∣z∣+11​)≤2. Then, the largest value of ∣z∣ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If z=0 be a complex number such that ∣z−z1​∣=2, then the maximum value of ∣z∣ is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If z=x+iy,xy=0, satisfies the equation z2+izˉ=0, then ∣z∣2 is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let z=(1+i)(1+2i)(1+3i)…(1+ni), where i=−1​. If ∣z∣2=44200, then n is equal to .........

List-I

List-II

If $z$ is a complex number of unit modulus and argument $θ$ , then $a r g (\frac{1 + z}{1 + z ˉ})$ equals

If $∣ z - 4/ z ∣ = 2$ , then the maximum value of $∣ Z ∣$ is equal to

If $\frac{w - w ˉ z}{1 - z}$ is purely real where $w = α + i β, β \neq = 0$ and $z \neq = 1$ , then the set of the values of $z$ is

If $z_{1}, z_{2}$ and $z_{3}$ are complex numbers such that $∣ z_{1} ∣ = ∣ z_{2} ∣ = ∣ z_{3} ∣ = ∣1/ z_{1} + 1/ z_{2} + 1/ z_{3} ∣ = 1$ , then $∣ z_{1} + z_{2} + z_{3} ∣$ is

Let $z$ and $ω$ be two complex numbers such that $∣ z ∣ \leq 1, ∣ ω ∣ \leq 1$ and $∣ z + iω ∣ = ∣ z - i \overset{ω}{ˉ} ∣ = 2$ then $z$ equals

If $z_{1}$ and $z_{2}$ are two complex numbers such taht $∣ z_{1} ∣ < 1 < ∣ z_{2} ∣$ then prove that $∣ \frac{1 - z _{1} z ˉ _{2}}{z _{1} - z _{2}} ∣ < 1$ .

If $x + i y = \frac{a + ib}{c + i d}$ , prove that $(x^{2} + y^{2})^{2} = \frac{a ^{2} + b ^{2}}{c ^{2} + d ^{2}}$ .

All the points in the set $S = {\frac{α + i}{α - i} : α \in R} (i = - 1)$ lie on a

If $a > 0$ and $z = \frac{( 1 + i ) ^{2}}{a - i}$ , has magnitude $2/5$ , then $z$ is equal to :

Let a complex number $z, ∣ z ∣ \neq = 1$ , satisfy $lo g_{\frac{1}{2}} (\frac{∣ z ∣ + 11}{( ∣ z ∣ - 1 ) ^{2}}) \leq 2$ . Then, the largest value of $∣ z ∣$ is equal to

If $z \neq = 0$ be a complex number such that $∣ z - \frac{1}{z} ∣ = 2$ , then the maximum value of $∣ z ∣$ is:

If $z = x + i y, x y \neq = 0$ , satisfies the equation $z^{2} + i \overset{z}{ˉ} = 0$ , then $∣ z ∣^{2}$ is equal to :

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 .........