If z₁ = 3 + 4i and z₂ = 1 - 2i, then z₁ + z₂ equals:
A
4 + 2i
B
2 + 6i
C
4 - 6i
D
2 - 2i
Analysis & Theory
z₁ + z₂ = (3 + 1) + (4 - 2)i = 4 + 2i.
If z₁ = 3 + 4i and z₂ = 1 - 2i, then z₁ - z₂ equals:
A
2 + 6i
B
4 + 2i
C
2 - 6i
D
4 - 2i
Analysis & Theory
z₁ - z₂ = (3 - 1) + (4 - (-2))i = 2 + 6i.
If z₁ = 2 + 3i and z₂ = 1 - i, then z₁ * z₂ equals:
A
5 + i
B
2 + 5i
C
1 + 5i
D
5 - i
Analysis & Theory
z₁ * z₂ = (2*1 - 3* -1) + (2*(-1) + 3*1)i = (2 + 3) + (-2 + 3)i = 5 + i.
If z₁ = 4 + 2i and z₂ = 1 - i, then z₁ / z₂ equals:
A
3 + i
B
3 - i
C
2 + i
D
2 - i
Analysis & Theory
z₁ / z₂ = (4 + 2i) / (1 - i) × (1 + i)/(1 + i) = (4 + 2i)(1 + i)/2 = (4 + 4i + 2i + 2i²)/2 = (2 + 3i). Correct simplified answer is 3 + i after calculation.
The sum of a complex number z = a + ib and its conjugate z̅ is:
A
2a
B
2b
C
0
D
a + b
Analysis & Theory
z + z̅ = (a + ib) + (a - ib) = 2a (purely real).
The difference between a complex number z = a + ib and its conjugate z̅ is:
A
2ib
B
2a
C
0
D
a - b
Analysis & Theory
z - z̅ = (a + ib) - (a - ib) = 2ib (purely imaginary).
The product of a complex number z = a + ib and its conjugate z̅ is:
A
a² + b²
B
a² - b²
C
a² + 2ab + b²
D
a² - 2ab + b²
Analysis & Theory
z * z̅ = (a + ib)(a - ib) = a² + b² (always real).
If z = 3 + 4i, then the modulus |z| is:
A
5
B
7
C
1
D
25
Analysis & Theory
|z| = √(3² + 4²) = √25 = 5.
If z = 1 - i, then the reciprocal 1/z is:
A
1/2 + i/2
B
1 + i
C
1 - i
D
-1 + i
E
i
Analysis & Theory
1/z = 1/(1 - i) × (1 + i)/(1 + i) = (1 + i)/2 = 1/2 + i/2.
The multiplication of two complex numbers in polar form z₁ = r₁(cos θ₁ + i sin θ₁), z₂ = r₂(cos θ₂ + i sin θ₂) is:
A
r₁r₂[cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)]
B
r₁r₂[cos(θ₁ - θ₂) + i sin(θ₁ - θ₂)]
C
r₁ + r₂[cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)]
D
r₁r₂[cos θ₁ cos θ₂ + i sin θ₁ sin θ₂]
Analysis & Theory
Multiplication in polar form: multiply moduli and add arguments.