Euler's Form and De Moivre's Theorem

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JEE Advanced 2009

Let $z = cos θ + i sin θ$ . Then the value of $\sum_{m = 1}^{15} I m (z^{2 m - 1})$ at $θ = 2^{\circ}$ is

\frac{1}{s i n 2 ^{\circ}}

(B)

\frac{1}{3 s i n 2 ^{\circ}}

(C)

\frac{1}{2 s i n 2 ^{\circ}}

(D)

\frac{1}{4 s i n 2 ^{\circ}}

JEE Advanced 1982

LEVELJEE Main

If $z = (\frac{3}{2} + \frac{i}{2})^{5} + (\frac{3}{2} - \frac{i}{2})^{5}$ , then

Find all non-zero complex numbers $Z$ satisfying $\overset{ˉ}{Z} = i Z^{2}$ .

JEE Main 2018 (16 April Shift 1)

LEVELJEE Main

The least positive integer n for which $(\frac{1 + i 3}{1 - i 3})^{n} = 1$ , is :

JEE Main 2019 (08 April Shift 2)

LEVELJEE Main

If $z = \frac{3}{2} + \frac{i}{2}$ , then $(1 + i z + z^{5} + i z^{8})^{9}$ is equal to

(1 + 2 i)^{9}

JEE Main 2020 - 2 Sep (Morning)

LEVELJEE Main

The value of $(\frac{1 + s i n \frac{2 π}{9} + i c o s \frac{2 π}{9}}{1 + s i n \frac{2 π}{9} - i c o s \frac{2 π}{9}})^{3}$ is:

(A)

- (1 - i 3)

(B)

(1 - i 3)

(C)

(3 - i)

(D)

- (3 - i)

JEE Main 2020 - 5 Sep (Evening)

LEVELJEE Main

The value of $(\frac{- 1 + i 3}{1 - i})^{30}$ is:

(A)

- 2^{15}

(B)

- 2^{15} i

(C)

2^{15}

(D)

6^{5}

JEE Main 2021 (26 Aug Shift 2)

LEVELJEE Main

If $(3 + i)^{100} = 2^{99} (p + i q)$ , then $p$ and $q$ are roots of the equation :

(A)

x^{2} - (3 - 1) x - 3 = 0

(B)

x^{2} + (3 + 1) x + 3 = 0

(C)

x^{2} + (3 - 1) x - 3 = 0

(D)

x^{2} - (3 + 1) x + 3 = 0

JEE Main 2021 (26 Aug Shift 1)

LEVELJEE Advanced

Let $z = \frac{1 - i 3}{2}, i = - 1$ . Then the value of $21 + (z + \frac{1}{z})^{3} + (z^{2} + \frac{1}{z ^{2}})^{3} + (z^{3} + \frac{1}{z ^{3}})^{3} + \dots + (z^{21} + \frac{1}{z ^{21}})^{3}$ is.

JEE Main 2021 (31 Aug Shift 1)

LEVELJEE Main

If $a_{r} = cos \frac{2 r π}{9} + i sin \frac{2 r π}{9}, r = 1, 2, 3, \dots, i = - 1$ , then the determinant $a_{1} a_{4} a_{7} a_{2} a_{5} a_{8} a_{3} a_{6} a_{9}$ is equal to :

(A)

a_{2} a_{6} - a_{4} a_{8}

(B)

a_{9}

(C)

a_{1} a_{9} - a_{3} a_{7}

(D)

a_{5}

JEE Main 2023 (24 January Shift 2)

LEVELJEE Main

The value of $(\frac{1 + s i n \frac{2 π}{9} + i c o s \frac{2 π}{9}}{1 + s i n \frac{2 π}{9} - i c o s \frac{2 π}{9}})^{3}$ is

(A)

\frac{- 1}{2} (1 - i 3)

(B)

\frac{1}{2} (1 - i 3)

(C)

\frac{- 1}{2} (3 - i)

(D)

\frac{1}{2} (3 + i)

JEE Main 2023 (12 Apr Shift 1)

LEVELJEE Advanced

Let $α, β$ be the roots of the quadratic equation $x^{2} + 6 x + 3 = 0$ . Then $\frac{α ^{23} + β ^{23} + α ^{14} + β ^{14}}{α ^{15} + β ^{15} + α ^{10} + β ^{10}}$ is equal to

JEE Main 2023 (13 Apr Shift 2)

LEVELJEE Main

Let $α, β$ be the roots of the equation $x^{2} - 2 x + 2 = 0$ . Then $α^{14} + β^{14}$ is equal to

(A)

- 64

(B)

- 642

(C)

- 128

(D)

- 1282

JEE Main 2024 (29 Jan Shift 2)

LEVELJEE Main

Let $α, β$ be the roots of the equation $x^{2} - 6 x + 3 = 0$ such that $Im (α) > Im (β)$ . Let $a, b$ be integers not divisible by 3 and $n$ be a natural number such that $\frac{α ^{99}}{β} + α^{98} = 3^{n} (a + ib), i = - 1$ . Then $n + a + b$ is equal to

JEE Main 2026 (23 January Shift 2)

LEVELJEE Main

If $z = \frac{3}{2} + \frac{i}{2}, i = - 1$ , then $(z^{201} - i)^{8}$ is equal to

Let $\overset{z}{ˉ}$ denote the complex conjugate of a complex number $z$ . If $z$ is a non-zero complex number for which both real and imaginary parts of $(\overset{z}{ˉ})^{2} + \frac{1}{z ^{2}}$ are integers, then which of the following is/are possible value(s) of $∣ z ∣$ ?

(A)

(\frac{43 + 3 205}{2})^{1/4}

(B)

(\frac{7 + 33}{4})^{1/4}

(C)

(\frac{9 + 65}{4})^{1/4}

(D)

(\frac{7 + 13}{6})^{1/4}

JEE Advanced 2024

LEVELJEE Advanced

Let $S = {a + b 2 : a, b \in Z}$ , $T_{1} = {(- 1 + 2)^{n} : n \in Z}$ , and $T_{2} = {(1 + 2)^{n} : n \in N}$ . Then which of the following statements is(are) TRUE ?

* Multiple Correct Options

(A)

Z \cup T_{1} \cup T_{2} \subset S

(B)

T_{1} \cap (0, \frac{1}{2024}) = ϕ

, where

ϕ

denotes the empty set

(C)

T_{2} \cap (2024, \infty) \neq = ϕ

(D)

For any given

a, b \in Z

cos (π (a + b 2)) + i sin (π (a + b 2)) \in Z

if and only if

b = 0

, where

i = - 1

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let z=cosθ+isinθ. Then the value of ∑m=115​Im(z2m−1) at θ=2∘ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If z=(23​​+2i​)5+(23​​−2i​)5, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find all non-zero complex numbers Z satisfying Zˉ=iZ2.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The least positive integer n for which (1−i3​1+i3​​)n=1, is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If z=23​​+2i​, then (1+iz+z5+iz8)9 is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of (1+sin92π​−icos92π​1+sin92π​+icos92π​​)3 is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of (1−i−1+i3​​)30 is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If (3​+i)100=299(p+iq), then p and q are roots of the equation :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let z=21−i3​​,i=−1​. Then the value of 21+(z+z1​)3+(z2+z21​)3+(z3+z31​)3+⋯+(z21+z211​)3 is.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ar​=cos92rπ​+isin92rπ​,r=1,2,3,…,i=−1​, then the determinant ​a1​a4​a7​​a2​a5​a8​​a3​a6​a9​​​ is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of (1+sin92π​−icos92π​1+sin92π​+icos92π​​)3 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let α,β be the roots of the quadratic equation x2+6​x+3=0. Then α15+β15+α10+β10α23+β23+α14+β14​ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let α,β be the roots of the equation x2−2​x+2=0. Then α14+β14 is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If z=23​​+2i​,i=−1​, then (z201−i)8 is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let S={a+b2​:a,b∈Z}, T1​={(−1+2​)n:n∈Z}, and T2​={(1+2​)n:n∈N}. Then which of the following statements is(are) TRUE ?

Let $z = cos θ + i sin θ$ . Then the value of $\sum_{m = 1}^{15} I m (z^{2 m - 1})$ at $θ = 2^{\circ}$ is

If $z = (\frac{3}{2} + \frac{i}{2})^{5} + (\frac{3}{2} - \frac{i}{2})^{5}$ , then

Find all non-zero complex numbers $Z$ satisfying $\overset{ˉ}{Z} = i Z^{2}$ .

The least positive integer n for which $(\frac{1 + i 3}{1 - i 3})^{n} = 1$ , is :

If $z = \frac{3}{2} + \frac{i}{2}$ , then $(1 + i z + z^{5} + i z^{8})^{9}$ is equal to

The value of $(\frac{1 + s i n \frac{2 π}{9} + i c o s \frac{2 π}{9}}{1 + s i n \frac{2 π}{9} - i c o s \frac{2 π}{9}})^{3}$ is:

The value of $(\frac{- 1 + i 3}{1 - i})^{30}$ is:

If $(3 + i)^{100} = 2^{99} (p + i q)$ , then $p$ and $q$ are roots of the equation :

Let $z = \frac{1 - i 3}{2}, i = - 1$ . Then the value of $21 + (z + \frac{1}{z})^{3} + (z^{2} + \frac{1}{z ^{2}})^{3} + (z^{3} + \frac{1}{z ^{3}})^{3} + \dots + (z^{21} + \frac{1}{z ^{21}})^{3}$ is.

If $a_{r} = cos \frac{2 r π}{9} + i sin \frac{2 r π}{9}, r = 1, 2, 3, \dots, i = - 1$ , then the determinant $a_{1} a_{4} a_{7} a_{2} a_{5} a_{8} a_{3} a_{6} a_{9}$ is equal to :

The value of $(\frac{1 + s i n \frac{2 π}{9} + i c o s \frac{2 π}{9}}{1 + s i n \frac{2 π}{9} - i c o s \frac{2 π}{9}})^{3}$ is

Let $α, β$ be the roots of the quadratic equation $x^{2} + 6 x + 3 = 0$ . Then $\frac{α ^{23} + β ^{23} + α ^{14} + β ^{14}}{α ^{15} + β ^{15} + α ^{10} + β ^{10}}$ is equal to

Let $α, β$ be the roots of the equation $x^{2} - 2 x + 2 = 0$ . Then $α^{14} + β^{14}$ is equal to

If $z = \frac{3}{2} + \frac{i}{2}, i = - 1$ , then $(z^{201} - i)^{8}$ is equal to

Let $S = {a + b 2 : a, b \in Z}$ , $T_{1} = {(- 1 + 2)^{n} : n \in Z}$ , and $T_{2} = {(1 + 2)^{n} : n \in N}$ . Then which of the following statements is(are) TRUE ?