2.2 MCQs-Complex Numbers
Complex Numbers
Basic Concepts and Definitions
1. A complex number is generally expressed in the form:
a + ib, where a and b are real numbers
(a, b), where a and b are real numbers
reiθ, where r ≥ 0 and θ is real
All of the above
Show me the answer
Answer: 4. All of the above
Explanation:
Complex numbers can be represented in multiple equivalent forms:
Rectangular/Cartesian form: z = a + ib, where a = Re(z) and b = Im(z)
Ordered pair form: z = (a, b)
Polar form: z = r(cosθ + i sinθ) = reiθ, where r = |z| and θ = arg(z)
Each form has its advantages for different operations.
2. The imaginary unit i is defined as:
√(-1)
-√(-1)
Both 1 and 2
√(1)
Show me the answer
Answer: 1. √(-1)
Explanation:
The imaginary unit i satisfies i² = -1
By convention, i = √(-1)
Note that (-i)² = (-1)² × i² = 1 × (-1) = -1 as well, so -i is also a square root of -1
In complex numbers, every non-zero number has two square roots
3. The complex conjugate of z = 3 - 4i is:
3 + 4i
-3 + 4i
-3 - 4i
4 - 3i
Show me the answer
Answer: 1. 3 + 4i
Explanation:
The complex conjugate of z = a + ib is denoted as (\bar{z}) or z* and is defined as (\bar{z} = a - ib)
For z = 3 - 4i, the conjugate is (\bar{z} = 3 + 4i)
Geometrically, conjugation reflects the complex number across the real axis
Properties: (\overline{z + w} = \bar{z} + \bar{w}), (\overline{zw} = \bar{z} \cdot \bar{w}), (\overline{z/w} = \bar{z}/\bar{w})
4. The modulus of z = 3 + 4i is:
3
4
5
7
Show me the answer
Answer: 3. 5
Explanation:
The modulus (or absolute value) of z = a + ib is |z| = √(a² + b²)
For z = 3 + 4i: |z| = √(3² + 4²) = √(9 + 16) = √25 = 5
Geometrically, |z| represents the distance from the origin to the point representing z in the complex plane
Properties: |zw| = |z||w|, |z/w| = |z|/|w|, |z|² = z(\bar{z})
Operations on Complex Numbers
5. If z₁ = 2 + 3i and z₂ = 1 - 2i, then z₁ + z₂ equals:
3 + i
3 + 5i
1 + 5i
3 - i
Show me the answer
Answer: 1. 3 + i
Explanation:
Addition of complex numbers is done by adding real parts and imaginary parts separately
z₁ + z₂ = (2 + 3i) + (1 - 2i) = (2 + 1) + (3i - 2i) = 3 + i
Geometrically, addition corresponds to vector addition in the complex plane
Subtraction is similar: z₁ - z₂ = (2 - 1) + (3i - (-2i)) = 1 + 5i
6. If z₁ = 2 + 3i and z₂ = 1 - 2i, then z₁ × z₂ equals:
8 - i
8 + i
-4 + 7i
-4 - i
Show me the answer
Answer: 1. 8 - i
Explanation:
Multiplication follows distributive law: (a + ib)(c + id) = ac + iad + ibc + i²bd = (ac - bd) + i(ad + bc)
z₁ × z₂ = (2 + 3i)(1 - 2i) = 2×1 + 2×(-2i) + 3i×1 + 3i×(-2i)
= 2 - 4i + 3i - 6i² = 2 - i - 6(-1) = 2 - i + 6 = 8 - i
In polar form: multiplication multiplies moduli and adds arguments
7. The multiplicative inverse of z = 3 + 4i is:
3 - 4i
(3 - 4i)/25
(3 + 4i)/25
(-3 - 4i)/25
Show me the answer
Answer: 2. (3 - 4i)/25
Explanation:
The multiplicative inverse of z = a + ib is 1/z = (\bar{z})/|z|²
For z = 3 + 4i: |z|² = 3² + 4² = 9 + 16 = 25
(\bar{z}) = 3 - 4i
Therefore, 1/z = (3 - 4i)/25
Verification: (3 + 4i) × (3 - 4i)/25 = (9 + 16)/25 = 25/25 = 1
Polar Form and De Moivre's Theorem
8. The polar form of z = 1 + i√3 is:
2(cos(π/3) + i sin(π/3))
2(cos(π/6) + i sin(π/6))
√2(cos(π/3) + i sin(π/3))
√2(cos(π/6) + i sin(π/6))
Show me the answer
Answer: 1. 2(cos(π/3) + i sin(π/3))
Explanation:
For z = 1 + i√3: a = 1, b = √3
Modulus: r = √(1² + (√3)²) = √(1 + 3) = √4 = 2
Argument: θ = tan⁻¹(b/a) = tan⁻¹(√3/1) = tan⁻¹(√3) = π/3 (since point is in first quadrant)
Polar form: z = r(cosθ + i sinθ) = 2(cos(π/3) + i sin(π/3))
Exponential form: z = 2e^(iπ/3)
9. De Moivre's theorem states that for any integer n:
(cosθ + i sinθ)^n = cos(nθ) + i sin(nθ)
(cosθ + i sinθ)^n = n(cosθ + i sinθ)
(cosθ + i sinθ)^n = cosθ + i sin(nθ)
(cosθ + i sinθ)^n = cos(nθ) + sin(nθ)
Show me the answer
Answer: 1. (cosθ + i sinθ)^n = cos(nθ) + i sin(nθ)
Explanation:
De Moivre's theorem: (cosθ + i sinθ)^n = cos(nθ) + i sin(nθ) for any integer n
This follows from Euler's formula: e^(iθ) = cosθ + i sinθ, so (e^(iθ))^n = e^(inθ) = cos(nθ) + i sin(nθ)
The theorem is useful for:
Finding powers of complex numbers
Finding roots of complex numbers
Expressing cos(nθ) and sin(nθ) in terms of powers of cosθ and sinθ
10. The cube roots of unity are:
1, ω, ω² where ω = (-1 + i√3)/2
1, -1, i
1, i, -i
1, ω, ω² where ω = (1 + i√3)/2
Show me the answer
Answer: 1. 1, ω, ω² where ω = (-1 + i√3)/2
Explanation:
The cube roots of unity satisfy z³ = 1
They are: 1, ω = e^(2πi/3) = cos(2π/3) + i sin(2π/3) = (-1 + i√3)/2
and ω² = e^(4πi/3) = cos(4π/3) + i sin(4π/3) = (-1 - i√3)/2
Properties: 1 + ω + ω² = 0, ω³ = 1, ω² = (\bar{ω})
The roots are equally spaced on the unit circle at angles 0°, 120°, 240°
Roots of Complex Numbers
11. The square roots of -4 are:
±2i
±2
±4i
±√2(1 + i)
Show me the answer
Answer: 1. ±2i
Explanation:
To find square roots of -4: solve z² = -4
Let z = a + ib, then (a + ib)² = a² - b² + 2iab = -4
Equating real and imaginary parts: a² - b² = -4 and 2ab = 0
From 2ab = 0: either a = 0 or b = 0
If b = 0: a² = -4 ⇒ no real solution
If a = 0: -b² = -4 ⇒ b² = 4 ⇒ b = ±2
Therefore z = ±2i
Verification: (2i)² = 4i² = 4(-1) = -4, (-2i)² = 4i² = -4
12. The fourth roots of 16 are:
±2, ±2i
±4, ±4i
2, 2i, -2, -2i
4, 4i, -4, -4i
Show me the answer
Answer: 3. 2, 2i, -2, -2i
Explanation:
Solve z⁴ = 16
In polar form: 16 = 16e^(i·0) = 16e^(i·2πk) for any integer k
Using De Moivre: z = 16^(1/4) e^(i·2πk/4) = 2 e^(i·πk/2) for k = 0, 1, 2, 3
k = 0: z = 2e^0 = 2
k = 1: z = 2e^(iπ/2) = 2(cos(π/2) + i sin(π/2)) = 2i
k = 2: z = 2e^(iπ) = 2(cosπ + i sinπ) = -2
k = 3: z = 2e^(i3π/2) = 2(cos(3π/2) + i sin(3π/2)) = -2i
The roots are equally spaced on a circle of radius 2 at 90° intervals
Complex Plane and Geometry
13. In the complex plane, |z - z₀| = r represents:
A straight line
A circle with center z₀ and radius r
An ellipse with foci at 0 and z₀
A parabola
Show me the answer
Answer: 2. A circle with center z₀ and radius r
Explanation:
|z - z₀| represents the distance between complex numbers z and z₀
The equation |z - z₀| = r means: "all points z whose distance from z₀ is r"
This is exactly the definition of a circle with center z₀ and radius r
Example: |z - (3 + 4i)| = 5 is a circle centered at 3 + 4i with radius 5
14. The equation |z - 1| = |z + i| represents:
A circle
A straight line
An ellipse
A hyperbola
Show me the answer
Answer: 2. A straight line
Explanation:
|z - 1| = distance from z to 1 (point (1, 0) on real axis)
|z + i| = |z - (-i)| = distance from z to -i (point (0, -1) on imaginary axis)
The equation |z - 1| = |z + i| means: "all points equidistant from (1, 0) and (0, -1)"
This is the perpendicular bisector of the segment joining (1, 0) and (0, -1)
In coordinate form: √((x-1)² + y²) = √(x² + (y+1)²)
Squaring: (x-1)² + y² = x² + (y+1)²
Expanding: x² - 2x + 1 + y² = x² + y² + 2y + 1
Simplifying: -2x = 2y ⇒ y = -x, which is a straight line
Euler's Formula and Exponential Form
15. Euler's formula states that:
e^(iθ) = cosθ + i sinθ
e^(iθ) = cosθ - i sinθ
e^(θ) = cosθ + i sinθ
e^(iθ) = sinθ + i cosθ
Show me the answer
Answer: 1. e^(iθ) = cosθ + i sinθ
Explanation:
Euler's formula: e^(iθ) = cosθ + i sinθ
This is one of the most important formulas in mathematics
Consequences:
e^(iπ) = cosπ + i sinπ = -1 (Euler's identity: e^(iπ) + 1 = 0)
e^(iπ/2) = i
e^(2πi) = 1
The formula connects exponential, trigonometric, and complex number theories
16. The exponential form of z = 2(cos(π/4) + i sin(π/4)) is:
2e^(iπ/4)
e^(2iπ/4)
2e^(π/4)
e^(2π/4)
Show me the answer
Answer: 1. 2e^(iπ/4)
Explanation:
For z = r(cosθ + i sinθ), the exponential form is z = re^(iθ)
Here r = 2, θ = π/4
Therefore, z = 2e^(iπ/4)
The exponential form is particularly useful for multiplication, division, and exponentiation:
Multiplication: r₁e^(iθ₁) × r₂e^(iθ₂) = (r₁r₂)e^(i(θ₁+θ₂))
Division: (r₁e^(iθ₁))/(r₂e^(iθ₂)) = (r₁/r₂)e^(i(θ₁-θ₂))
Exponentiation: (re^(iθ))^n = r^n e^(inθ)
Applications and Properties
17. The real part of (1 + i)^4 is:
0
-4
4
8
Show me the answer
Answer: 2. -4
Explanation:
Method 1: Direct expansion (1 + i)² = 1 + 2i + i² = 1 + 2i - 1 = 2i (1 + i)⁴ = [(1 + i)²]² = (2i)² = 4i² = 4(-1) = -4 So Re((1 + i)⁴) = -4
Method 2: Polar form 1 + i = √2 e^(iπ/4) (1 + i)⁴ = (√2)^4 e^(iπ) = 4 e^(iπ) = 4(cosπ + i sinπ) = 4(-1 + 0i) = -4
18. If z is a complex number such that |z| = 1, then 1/z equals:
z
-z
(\bar{z})
-(\bar{z})
Show me the answer
Answer: 3. (\bar{z})
Explanation:
For any complex number z, we have z(\bar{z}) = |z|²
If |z| = 1, then z(\bar{z}) = 1
Therefore, 1/z = (\bar{z})
Example: If z = cosθ + i sinθ = e^(iθ), then |z| = 1
(\bar{z}) = cosθ - i sinθ = e^(-iθ) = 1/z
This property is useful in many trigonometric identities and signal processing applications
19. The locus of points satisfying arg(z) = π/4 is:
A circle
A straight line
A ray from the origin
A line segment
Show me the answer
Answer: 3. A ray from the origin
Explanation:
arg(z) = θ means the argument (angle) of z is θ
In polar coordinates: z = re^(iθ) with r > 0, θ fixed
This represents all points on the ray starting at the origin and making angle θ with the positive real axis
Specifically, arg(z) = π/4 represents the ray y = x with x > 0 (first quadrant)
Note: The origin itself is excluded since arg(0) is undefined
20. The triangle inequality for complex numbers states:
|z₁ + z₂| ≤ |z₁| + |z₂|
|z₁ + z₂| ≥ |z₁| + |z₂|
|z₁ + z₂| = |z₁| + |z₂|
|z₁ + z₂| ≤ ||z₁| - |z₂||
Show me the answer
Answer: 1. |z₁ + z₂| ≤ |z₁| + |z₂|
Explanation:
Triangle inequality: |z₁ + z₂| ≤ |z₁| + |z₂| for all complex numbers z₁, z₂
Geometrically: In any triangle, the length of one side is less than or equal to the sum of the lengths of the other two sides
Equality occurs when z₁ and z₂ have the same argument (are in the same direction)
Reverse triangle inequality: |z₁ - z₂| ≥ ||z₁| - |z₂||
These inequalities are fundamental in analysis and have many applications
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