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Principle: Cube roots of unity (1 + ω + ω² = 0, ω³ = 1)
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Set · 2 questions
Let
Z
1
Z_1
Z
1
and
Z
2
Z_2
Z
2
be any two complex numbers such that
Z
1
2
+
Z
2
2
+
Z
1
Z
2
=
0
Z_1^2+Z_2^2+Z_1 Z_2=0
Z
1
2
+
Z
2
2
+
Z
1
Z
2
=
0
.
Q26
#26
NDA → Mathematics → Complex Numbers → Modulus, Argument, and Conjugate
·
Moderate
What is the value of
∣
Z
1
Z
2
∣
\left|\dfrac{Z_1}{Z_2}\right|
Z
2
Z
1
?
Add
Lever: Cube roots of unity (1 + ω + ω² = 0, ω³ = 1)
A
1
1
1
B
2
2
2
C
3
3
3
D
4
4
4
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[Q41 · Sep · 2024]
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Q27
#27
NDA → Mathematics → Complex Numbers → Modulus, Argument, and Conjugate
·
Moderate
What is the value of
1
2
+
Re
(
Z
1
Z
2
)
\dfrac{1}{2}+\text{Re}\!\left(\dfrac{Z_1}{Z_2}\right)
2
1
+
Re
(
Z
2
Z
1
)
?
Add
Lever: Cube roots of unity (1 + ω + ω² = 0, ω³ = 1)
A
−
1
-1
−
1
B
0
0
0
C
1
1
1
D
2
2
2
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[Q42 · Sep · 2024]
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Q28
#28
NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
·
Moderate
If
ω
\omega
ω
is a non-real cube root of unity, then what is a root of the following equation?
∣
x
+
1
ω
ω
2
ω
x
+
ω
2
1
ω
2
1
x
+
ω
∣
=
0
\begin{vmatrix} x+1 & \omega & \omega^2 \\ \omega & x+\omega^2 & 1 \\ \omega^2 & 1 & x+\omega \end{vmatrix} = 0
x
+
1
ω
ω
2
ω
x
+
ω
2
1
ω
2
1
x
+
ω
=
0
Add
Lever: Cube roots of unity (1 + ω + ω² = 0, ω³ = 1)
Concept: Cube-root-of-unity determinants
A
x
=
0
x=0
x
=
0
B
x
=
1
x=1
x
=
1
C
x
=
ω
x=\omega
x
=
ω
D
x
=
ω
2
x=\omega^2
x
=
ω
2
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[Q12 · Apr · 2025]
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Q29
#29
NDA → Mathematics → Complex Numbers → Cube Roots of Unity
·
Hard
If
x
2
−
x
+
1
=
0
x^2 - x + 1 = 0
x
2
−
x
+
1
=
0
, then what is
(
x
−
1
x
)
2
+
(
x
−
1
x
4
)
+
(
x
−
1
x
8
)
\left(x - \frac{1}{x}\right)^2 + \left(x - \frac{1}{x^4}\right) + \left(x - \frac{1}{x^8}\right)
(
x
−
x
1
)
2
+
(
x
−
x
4
1
)
+
(
x
−
x
8
1
)
equal to?
Add
Lever: Cube roots of unity (1 + ω + ω² = 0, ω³ = 1)
A
81
81
81
B
85
85
85
C
87
87
87
D
90
90
90
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[Q14 · Apr · 2025]
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