Algebraic Operations on Complex Numbers

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JEE Main 2016

A value of $θ$ for which $\frac{2 + 3 i s i n θ}{1 - 2 i s i n θ}$ is purely imaginary, is

sin^{- 1} (3 /4)

(B)

sin^{- 1} (1/ 3)

(C)

π /3

(D)

π /6

JEE Main 2008

LEVELBoard

The conjugate of a complex number is $\frac{1}{i - 1}$ then that complex number is

(A)

- 1/ (i - 1)

(B)

1/ (i + 1)

(C)

- 1/ (i + 1)

(D)

1/ (i - 1)

JEE Main 2004

LEVELBoard

If $z = x - i y$ and $z^{1/3} = p + i q$ , then $(x / p + y / q) / (p^{2} + q^{2})$ is equal to

If $(\frac{1 + i}{1 - i})^{x} = 1$ then

(A)

x = 2 n + 1

, where

n

is any positive integer

(B)

x = 4 n

, where

n

is any positive integer

(C)

x = 2 n

, where

n

is any positive integer

(D)

x = 4 n + 1

, where

n

is any positive integer

JEE Advanced 2012

LEVELJEE Main

Let $z$ be a complex number such that the imaginary part of $z$ is non-zero and $a = z^{2} + z + 1$ is real. Then $a$ cannot take the value

If $∣ z ∣ = 1$ and $ω = \frac{z - 1}{z + 1}$ (where $z \neq = - 1$ ), then $R e (ω)$ is

(A)

(B)

\frac{1}{∣ z + 1 ∣ ^{2}}

(C)

\frac{z}{z + 1} \cdot \frac{1}{∣ z + 1 ∣ ^{2}}

(D)

\frac{2}{∣ z + 1 ∣ ^{2}}

JEE Advanced 1998

LEVELJEE Main

The value of the sum $\sum_{n = 1}^{13} (i^{n} + i^{n + 1})$ , where $i = - 1$ , equals

For positive integers $n_{1}, n_{2}$ , the value of the expression $(1 + i)^{n_{1}} + (1 + i^{3})^{n_{1}} + (1 + i^{5})^{n_{2}} + (1 + i^{7})^{n_{2}}$ , where $i = - 1$ is a real number if and only if

(A)

n_{1} = n_{2} + 1

(B)

n_{1} = n_{2} - 1

(C)

n_{1} = n_{2}

(D)

n_{1} > 0, n_{2} > 0

JEE Advanced 1980

LEVELJEE Main

The smallest positive integer $n$ for which $(\frac{1 + i}{1 - i})^{n} = 1$ is

(A)

n = 8

(B)

n = 16

(C)

n = 12

Let $ω = e^{iπ /3}$ , and $a, b, c, x, y, z$ be non-zero complex numbers such that $a + b + c = x, a + bω + c ω^{2} = y, a + b ω^{2} + c ω = z$ . Then the value of $\frac{∣ x ∣ ^{2} + ∣ y ∣ ^{2} + ∣ z ∣ ^{2}}{∣ a ∣ ^{2} + ∣ b ∣ ^{2} + ∣ c ∣ ^{2}}$ is .........

JEE Advanced 1988

LEVELBoard

For any two complex numbers $z_{1}, z_{2}$ and any real number $a$ and $b$ , $∣ a z_{1} - b z_{2} ∣^{2} + ∣ b z_{1} + a z_{2} ∣^{2} = \dots$

JEE Advanced 1987

LEVELJEE Main

If the expression $[\frac{sin ( \frac{x}{2} ) + cos ( \frac{x}{2} ) + i tan ( x )}{1 + 2 i sin ( \frac{x}{2} )}]$ is real, then the set of all possible values of $x$ is .........

JEE Advanced 1980

LEVELJEE Main

Find the real values of $x$ and $y$ for which the following equation is satisfied $\frac{( 1 + i ) x - 2 i}{3 + i} + \frac{( 2 - 3 i ) y + i}{3 - i} = i$ .

JEE Advanced 1978

LEVELJEE Main

Express $\frac{1}{1 - cos θ + 2 i sin θ}$ in the form $x + i y$ .

JEE Advanced 1999

LEVELJEE Advanced

For complex numbers $z$ and $w$ , prove that $∣ z ∣^{2} w - ∣ w ∣^{2} z = z - w$ if and only if $z = w$ or $z \overset{w}{ˉ} = 1$ .

JEE Advanced 1995

LEVELJEE Main

If $i z^{3} + z^{2} - z + i = 0$ , then show that $∣ z ∣ = 1$ .

JEE Advanced 1992

LEVELJEE Main

$z \neq = 0$ is a complex number

List-I

(P)

R ez = 0

(Q)

A r g z = \frac{π}{4}

List-II

(1)

R e z^{2} = 0

(2)

I m z^{2} = 0

(3)

R e z^{2} = I m z^{2}

JEE Main 2018 (15 April Shift 1)

LEVELJEE Main

The set of all $α \in R$ , for which $w = \frac{1 + ( 1 - 8 α ) z}{1 - z}$ is a purely imaginary number, for all $z \in C$ satisfying $∣ z ∣ = 1$ and $Re z \neq = 1$ , is :

{0, \frac{1}{4}, - \frac{1}{4}}

JEE Main 2019 (9 January)

LEVELJEE Main

Let $α$ and $β$ be two roots of the equation $x^{2} + 2 x + 2 = 0$ , then $α^{15} + β^{15}$ is equal to :

JEE Main 2019 (9 January)

LEVELJEE Main

Let $A = {θ \in (- π /2, π) : \frac{3 + 2 i s i n θ}{1 - 2 i s i n θ} is purely imaginary}$ . Then the sum of the elements in A is :

(A)

\frac{5 π}{6}

(B)

\frac{2 π}{3}

(C)

\frac{3 π}{4}

(D)

π

JEE Main 2019 (10 January Shift 1)

LEVELJEE Main

Let $z_{1}$ and $z_{2}$ be any two non-zero complex numbers such that $3∣ z_{1} ∣ = 4∣ z_{2} ∣$ . If $z = \frac{3 z _{1}}{2 z _{2}} + \frac{2 z _{2}}{3 z _{1}}$ then :

(A)

|z| = \frac{1}{2}\sqrt{\frac{17}{2}}

(B)

Re(z) = 0

(C)

|z| = \sqrt{\frac{5}{2}}

(D)

Im(z) = 0

JEE Main 2019 (11 January)

LEVELJEE Main

Let $(\frac{- 2}{3} + \frac{1}{3} i)^{3} = \frac{x + i y}{27} (i = - 1)$ , where $x$ and $y$ are real numbers, then $y - x$ equals :

JEE Main 2019 (8 April Shift 1)

LEVELJEE Main

If $α$ and $β$ be the roots of the equation $x^{2} - 2 x + 2 = 0$ , then the least value of $n$ for which $(\frac{α}{β})^{n} = 1$ is :

(A)

2

(B)

3

(C)

4

(D)

5

JEE Main 2019 (12 April)

LEVELJEE Main

Let $z \in C$ with $I m (z) = 10$ and it satisfies $\frac{2 z - n}{2 z + n} = 2 i - 1$ for some natural number n. Then :

(A)

n = 20 and Re(z) = -10

(B)

n = 20 and Re(z) = 10

(C)

n = 40 and Re(z) = -10

(D)

n = 40 and Re(z) = 10

JEE Main 11 Jan 2019 (Evening)

LEVELJEE Main

Let $z$ be a complex number such that $∣ z ∣ + z = 3 + i$ (where $i = - 1$ ). Then $∣ z ∣$ is equal to:

(A)

5/3

(B)

5/4

(C)

41 /4

(D)

34 /3

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A value of θ for which 1−2isinθ2+3isinθ​ is purely imaginary, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The conjugate of a complex number is i−11​ then that complex number is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If z=x−iy and z1/3=p+iq, then (x/p+y/q)/(p2+q2) is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If (1−i1+i​)x=1 then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let z be a complex number such that the imaginary part of z is non-zero and a=z2+z+1 is real. Then a cannot take the value

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ∣z∣=1 and ω=z+1z−1​ (where z=−1), then Re(ω) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of the sum ∑n=113​(in+in+1), where i=−1​, equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For positive integers n1​,n2​, the value of the expression (1+i)n1​+(1+i3)n1​+(1+i5)n2​+(1+i7)n2​, where i=−1​ is a real number if and only if

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The smallest positive integer n for which (1−i1+i​)n=1 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For any two complex numbers z1​,z2​ and any real number a and b, ∣az1​−bz2​∣2+∣bz1​+az2​∣2=…

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the expression [1+2isin(2x​)sin(2x​)+cos(2x​)+itan(x)​] is real, then the set of all possible values of x is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find the real values of x and y for which the following equation is satisfied 3+i(1+i)x−2i​+3−i(2−3i)y+i​=i.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Express 1−cosθ+2isinθ1​ in the form x+iy.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For complex numbers z and w, prove that ∣z∣2w−∣w∣2z=z−w if and only if z=w or zwˉ=1.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If iz3+z2−z+i=0, then show that ∣z∣=1.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } z=0 is a complex number

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The set of all α∈R, for which w=1−z1+(1−8α)z​ is a purely imaginary number, for all z∈C satisfying ∣z∣=1 and Re z=1, is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let α and β be two roots of the equation x2+2x+2=0, then α15+β15 is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A={θ∈(−π/2,π):1−2isinθ3+2isinθ​ is purely imaginary}. Then the sum of the elements in A is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let z1​ and z2​ be any two non-zero complex numbers such that 3∣z1​∣=4∣z2​∣. If z=2z2​3z1​​+3z1​2z2​​ then :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let (3−2​+31​i)3=27x+iy​(i=−1​), where x and y are real numbers, then y−x equals :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If α and β be the roots of the equation x2−2x+2=0, then the least value of n for which (βα​)n=1 is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let z∈C with Im(z)=10 and it satisfies 2z+n2z−n​=2i−1 for some natural number n. Then :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let z be a complex number such that ∣z∣+z=3+i (where i=−1​). Then ∣z∣ is equal to:

A value of $θ$ for which $\frac{2 + 3 i s i n θ}{1 - 2 i s i n θ}$ is purely imaginary, is

The conjugate of a complex number is $\frac{1}{i - 1}$ then that complex number is

If $z = x - i y$ and $z^{1/3} = p + i q$ , then $(x / p + y / q) / (p^{2} + q^{2})$ is equal to

If $(\frac{1 + i}{1 - i})^{x} = 1$ then

Let $z$ be a complex number such that the imaginary part of $z$ is non-zero and $a = z^{2} + z + 1$ is real. Then $a$ cannot take the value

If $∣ z ∣ = 1$ and $ω = \frac{z - 1}{z + 1}$ (where $z \neq = - 1$ ), then $R e (ω)$ is

The value of the sum $\sum_{n = 1}^{13} (i^{n} + i^{n + 1})$ , where $i = - 1$ , equals

For positive integers $n_{1}, n_{2}$ , the value of the expression $(1 + i)^{n_{1}} + (1 + i^{3})^{n_{1}} + (1 + i^{5})^{n_{2}} + (1 + i^{7})^{n_{2}}$ , where $i = - 1$ is a real number if and only if

The smallest positive integer $n$ for which $(\frac{1 + i}{1 - i})^{n} = 1$ is

For any two complex numbers $z_{1}, z_{2}$ and any real number $a$ and $b$ , $∣ a z_{1} - b z_{2} ∣^{2} + ∣ b z_{1} + a z_{2} ∣^{2} = \dots$

If the expression $[\frac{sin ( \frac{x}{2} ) + cos ( \frac{x}{2} ) + i tan ( x )}{1 + 2 i sin ( \frac{x}{2} )}]$ is real, then the set of all possible values of $x$ is .........

Find the real values of $x$ and $y$ for which the following equation is satisfied $\frac{( 1 + i ) x - 2 i}{3 + i} + \frac{( 2 - 3 i ) y + i}{3 - i} = i$ .

Express $\frac{1}{1 - cos θ + 2 i sin θ}$ in the form $x + i y$ .

For complex numbers $z$ and $w$ , prove that $∣ z ∣^{2} w - ∣ w ∣^{2} z = z - w$ if and only if $z = w$ or $z \overset{w}{ˉ} = 1$ .

If $i z^{3} + z^{2} - z + i = 0$ , then show that $∣ z ∣ = 1$ .

$z \neq = 0$ is a complex number

The set of all $α \in R$ , for which $w = \frac{1 + ( 1 - 8 α ) z}{1 - z}$ is a purely imaginary number, for all $z \in C$ satisfying $∣ z ∣ = 1$ and $Re z \neq = 1$ , is :

Let $α$ and $β$ be two roots of the equation $x^{2} + 2 x + 2 = 0$ , then $α^{15} + β^{15}$ is equal to :

Let $A = {θ \in (- π /2, π) : \frac{3 + 2 i s i n θ}{1 - 2 i s i n θ} is purely imaginary}$ . Then the sum of the elements in A is :

Let $z_{1}$ and $z_{2}$ be any two non-zero complex numbers such that $3∣ z_{1} ∣ = 4∣ z_{2} ∣$ . If $z = \frac{3 z _{1}}{2 z _{2}} + \frac{2 z _{2}}{3 z _{1}}$ then :

Let $(\frac{- 2}{3} + \frac{1}{3} i)^{3} = \frac{x + i y}{27} (i = - 1)$ , where $x$ and $y$ are real numbers, then $y - x$ equals :

If $α$ and $β$ be the roots of the equation $x^{2} - 2 x + 2 = 0$ , then the least value of $n$ for which $(\frac{α}{β})^{n} = 1$ is :

Let $z \in C$ with $I m (z) = 10$ and it satisfies $\frac{2 z - n}{2 z + n} = 2 i - 1$ for some natural number n. Then :

Let $z$ be a complex number such that $∣ z ∣ + z = 3 + i$ (where $i = - 1$ ). Then $∣ z ∣$ is equal to: