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Complex Numbers and Quadratic Equations

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Chapter Notes

Overview
This chapter introduces complex numbers as an extension of the real number system to solve equations like x² + 1 = 0, which have no real solutions. The central idea is that a complex number is of the form a + ib, where i = √(-1), and we can perform algebraic operations (addition, subtraction, multiplication, division) on them just like real numbers. Students must master the algebra of complex numbers, including conjugates, modulus, and polar representation, as these form the foundation for solving quadratic equations with negative discriminants and for higher-level mathematics.
Key Concepts (14)
Complex number
A number of the form a + ib, where a and b are real numbers, is defined to be a complex number.
e.g. 2 + i3, (-1) + 3i, (1/4) + i(-11)
Real part (Re z)
For the complex number z = a + ib, a is called the real part, denoted by Re z.
e.g. If z = 2 + i5, then Re z = 2.
Imaginary part (Im z)
For the complex number z = a + ib, b is called the imaginary part, denoted by Im z.
e.g. If z = 2 + i5, then Im z = 5.
i (iota)
The symbol i is defined such that i² = -1, and i = √(-1).
e.g. i is a solution of the equation x² + 1 = 0.
Equality of two complex numbers
Two complex numbers z₁ = a + ib and z₂ = c + id are equal if a = c and b = d.
e.g. If 4x + i(3x – y) = 3 + i(-6), then 4x = 3 and 3x – y = -6, giving x = 3/4 and y = 33/4.
Additive identity (zero complex number)
The complex number 0 + i0 (denoted as 0) is called the additive identity, such that for every complex number z, z + 0 = z.
e.g. (2 + i3) + (0 + i0) = 2 + i3.
Additive inverse (negative of z)
For every complex number z = a + ib, the complex number -a + i(-b) (denoted as -z) is called the additive inverse, such that z + (-z) = 0.
e.g. For z = 2 + i3, -z = -2 + i(-3) = -2 - 3i.
Multiplicative identity
The complex number 1 + i0 (denoted as 1) is called the multiplicative identity, such that for every complex number z, z · 1 = z.
e.g. (2 + i3) · (1 + i0) = 2 + i3.
Multiplicative inverse
For every non-zero complex number z = a + ib (a ≠ 0, b ≠ 0), the complex number a/(a²+b²) + i(-b/(a²+b²)) (denoted by 1/z or z⁻¹) is called the multiplicative inverse, such that z · (1/z) = 1.
e.g. For z = 2 - 3i, z⁻¹ = (2 + 3i)/13 = 2/13 + i(3/13).
Conjugate of a complex number (z̄)
The conjugate of the complex number z = a + ib, denoted by z̄, is the complex number a – ib.
e.g. For z = 3 + i, z̄ = 3 - i. For z = 2 - 5i, z̄ = 2 + 5i.
Modulus of a complex number (|z|)
The modulus of z = a + ib, denoted by |z|, is defined to be the non-negative real number √(a² + b²).
e.g. |3 + i| = √(3² + 1²) = √10. |2 - 5i| = √(2² + (-5)²) = √29.
Argand plane (complex plane)
The plane having a complex number assigned to each of its points is called the complex plane or the Argand plane. The x-axis is the real axis and the y-axis is the imaginary axis.
e.g. The complex number 2 + 4i corresponds to the point (2, 4) in the Argand plane.
Real axis
The x-axis in the Argand plane, which corresponds to complex numbers of the form a + i0.
e.g. The point (2, 0) represents the real number 2.
Imaginary axis
The y-axis in the Argand plane, which corresponds to complex numbers of the form 0 + ib.
e.g. The point (0, 3) represents the purely imaginary number 3i.
Formulas & Equations (18)
i² = -1
Reads as: i squared equals negative one
Variables: i is the imaginary unit
Use when: Simplifying powers of i or expressions involving √(-1).
⚠ Writing i² = 1 instead of -1.
z₁ + z₂ = (a + c) + i(b + d)
Reads as: z one plus z two equals a plus c plus i times b plus d
Variables: z₁ = a + ib, z₂ = c + id; a, b, c, d are real numbers
Use when: Adding two complex numbers.
⚠ Adding real and imaginary parts incorrectly, e.g., writing (a + ib) + (c + id) = (a + c) + i(b + d) but then misplacing signs.
z₁ z₂ = (ac – bd) + i(ad + bc)
Reads as: z one times z two equals a c minus b d plus i times a d plus b c
Variables: z₁ = a + ib, z₂ = c + id; a, b, c, d are real numbers
Use when: Multiplying two complex numbers.
⚠ Forgetting the minus sign in the real part: writing (ac + bd) instead of (ac – bd).
z⁻¹ = a/(a²+b²) + i(-b/(a²+b²))
Reads as: z inverse equals a over a squared plus b squared plus i times negative b over a squared plus b squared
Variables: z = a + ib, a ≠ 0, b ≠ 0
Use when: Finding the multiplicative inverse of a complex number.
⚠ Writing the imaginary part as +b/(a²+b²) instead of -b/(a²+b²).
i⁴ᵏ = 1, i⁴ᵏ⁺¹ = i, i⁴ᵏ⁺² = -1, i⁴ᵏ⁺³ = -i
Reads as: i to the power 4k equals 1, i to the power 4k plus 1 equals i, i to the power 4k plus 2 equals negative 1, i to the power 4k plus 3 equals negative i
Variables: k is any integer
Use when: Simplifying high powers of i, e.g., i³⁵, i⁻³⁵.
⚠ Confusing the cycle, e.g., writing i⁴ᵏ⁺² = i instead of -1.
√(-a) = i√a, for a > 0
Reads as: square root of negative a equals i times square root of a
Variables: a is a positive real number
Use when: Expressing square roots of negative real numbers.
⚠ Writing √(-a) = -i√a or forgetting the i.
|z| = √(a² + b²)
Reads as: modulus of z equals square root of a squared plus b squared
Variables: z = a + ib; a, b are real numbers
Use when: Finding the distance of the point (a, b) from the origin in the Argand plane.
⚠ Writing |z| = a² + b² (forgetting the square root).
z̄ = a – ib
Reads as: z bar equals a minus i b
Variables: z = a + ib; a, b are real numbers
Use when: Finding the conjugate of a complex number.
⚠ Writing z̄ = -a + ib instead of a – ib.
z · z̄ = |z|²
Reads as: z times z bar equals modulus of z squared
Variables: z = a + ib; z̄ = a – ib
Use when: Simplifying expressions involving conjugates and modulus, or finding multiplicative inverse.
⚠ Writing z · z̄ = |z| instead of |z|².
|z₁ z₂| = |z₁| |z₂|
Reads as: modulus of z one z two equals modulus of z one times modulus of z two
Variables: z₁, z₂ are complex numbers
Use when: Finding modulus of a product of complex numbers.
⚠ Writing |z₁ z₂| = |z₁| + |z₂|.
|z₁ / z₂| = |z₁| / |z₂|, provided z₂ ≠ 0
Reads as: modulus of z one over z two equals modulus of z one divided by modulus of z two
Variables: z₁, z₂ are complex numbers, z₂ ≠ 0
Use when: Finding modulus of a quotient of complex numbers.
⚠ Forgetting the condition z₂ ≠ 0.
z₁ z₂̄ = z̄₁ z̄₂
Reads as: conjugate of z one z two equals conjugate of z one times conjugate of z two
Variables: z₁, z₂ are complex numbers
Use when: Finding conjugate of a product.
⚠ Writing z₁ z₂̄ = z̄₁ + z̄₂.
z₁ ± z₂̄ = z̄₁ ± z̄₂
Reads as: conjugate of z one plus or minus z two equals conjugate of z one plus or minus conjugate of z two
Variables: z₁, z₂ are complex numbers
Use when: Finding conjugate of a sum or difference.
⚠ Writing z₁ ± z₂̄ = z̄₁ ∓ z̄₂.
(z₁ + z₂)² = z₁² + 2z₁z₂ + z₂²
Reads as: z one plus z two squared equals z one squared plus two z one z two plus z two squared
Variables: z₁, z₂ are complex numbers
Use when: Expanding the square of a sum of complex numbers.
⚠ Forgetting the cross term 2z₁z₂.
(z₁ – z₂)² = z₁² – 2z₁z₂ + z₂²
Reads as: z one minus z two squared equals z one squared minus two z one z two plus z two squared
Variables: z₁, z₂ are complex numbers
Use when: Expanding the square of a difference of complex numbers.
⚠ Writing the cross term as +2z₁z₂ instead of -2z₁z₂.
(z₁ + z₂)³ = z₁³ + 3z₁²z₂ + 3z₁z₂² + z₂³
Reads as: z one plus z two cubed equals z one cubed plus three z one squared z two plus three z one z two squared plus z two cubed
Variables: z₁, z₂ are complex numbers
Use when: Expanding the cube of a sum of complex numbers.
⚠ Forgetting the binomial coefficients (3).
(z₁ – z₂)³ = z₁³ – 3z₁²z₂ + 3z₁z₂² – z₂³
Reads as: z one minus z two cubed equals z one cubed minus three z one squared z two plus three z one z two squared minus z two cubed
Variables: z₁, z₂ are complex numbers
Use when: Expanding the cube of a difference of complex numbers.
⚠ Sign errors in alternating terms.
z₁² – z₂² = (z₁ – z₂)(z₁ + z₂)
Reads as: z one squared minus z two squared equals z one minus z two times z one plus z two
Variables: z₁, z₂ are complex numbers
Use when: Factorising difference of squares of complex numbers.
⚠ Writing z₁² – z₂² = (z₁ + z₂)².
How It Works
Addition of two complex numbers
  1. Step 1: Identify the real parts a and c of z₁ = a + ib and z₂ = c + id.
  2. Step 2: Add the real parts: a + c.
  3. Step 3: Add the imaginary parts: b + d.
  4. Step 4: Write the sum as (a + c) + i(b + d).
💡 Addition is component-wise; never mix real and imaginary parts.
Multiplication of two complex numbers
  1. Step 1: Write z₁ = a + ib and z₂ = c + id.
  2. Step 2: Multiply using distributive law: (a + ib)(c + id) = ac + iad + ibc + i²bd.
  3. Step 3: Replace i² with -1: ac + iad + ibc - bd.
  4. Step 4: Combine real parts: ac - bd.
  5. Step 5: Combine imaginary parts: i(ad + bc).
  6. Step 6: Write the product as (ac - bd) + i(ad + bc).
💡 The real part has a minus sign: ac - bd, not ac + bd.
Finding the multiplicative inverse of a complex number
  1. Step 1: Given z = a + ib (non-zero), write 1/z.
  2. Step 2: Multiply numerator and denominator by the conjugate z̄ = a - ib: 1/(a+ib) = (a-ib)/((a+ib)(a-ib)).
  3. Step 3: Simplify denominator: (a+ib)(a-ib) = a² - i²b² = a² + b².
  4. Step 4: Write the result: 1/z = a/(a²+b²) + i(-b/(a²+b²)).
💡 The imaginary part has a minus sign: -b/(a²+b²), not +b/(a²+b²).
Simplifying powers of i
  1. Step 1: Write the exponent n in the form 4k + r, where r = 0, 1, 2, or 3.
  2. Step 2: Use the cycle: i⁴ᵏ = 1, i⁴ᵏ⁺¹ = i, i⁴ᵏ⁺² = -1, i⁴ᵏ⁺³ = -i.
  3. Step 3: For negative exponents, use i⁻ⁿ = 1/iⁿ and rationalise if needed.
💡 The cycle repeats every 4 powers; i⁰ = 1.
Expressing square roots of negative real numbers
  1. Step 1: For a > 0, write √(-a) = √(a × (-1)).
  2. Step 2: Write √(-1) = i, so √(-a) = i√a.
  3. Step 3: Note that both i√a and -i√a are square roots of -a.
💡 √(-a) × √(-b) ≠ √(ab) when both a and b are positive; instead, √(-a) × √(-b) = i√a × i√b = -√(ab).
Finding the conjugate of a complex number
  1. Step 1: Given z = a + ib, identify the real part a and imaginary part b.
  2. Step 2: Change the sign of the imaginary part: z̄ = a - ib.
💡 Only the sign of the imaginary part changes; the real part remains the same.
Finding the modulus of a complex number
  1. Step 1: Given z = a + ib, identify a and b.
  2. Step 2: Compute a² + b².
  3. Step 3: Take the non-negative square root: |z| = √(a² + b²).
💡 The modulus is always a non-negative real number.
Representing a complex number in the Argand plane
  1. Step 1: Given z = x + iy, identify the ordered pair (x, y).
  2. Step 2: Plot the point P(x, y) in the XY-plane.
  3. Step 3: The x-coordinate is the real part, the y-coordinate is the imaginary part.
  4. Step 4: The distance from the origin to P is the modulus |z| = √(x² + y²).
💡 The point (x, -y) is the mirror image of (x, y) on the real axis, representing the conjugate.
Exam Traps — Don't Make These Mistakes
⚠ Confusing i² with √(-1)
✗ i = √(-1) and i² = 1
→ i is defined as √(-1), so i² = (√(-1))² = -1, not 1.
✓ i = √(-1) and i² = -1
⚠ Mistake in multiplication real part sign
✗ (a + ib)(c + id) = (ac + bd) + i(ad + bc)
→ The term i²bd = (-1)bd = -bd, so the real part is ac - bd, not ac + bd.
✓ (a + ib)(c + id) = (ac - bd) + i(ad + bc)
⚠ Incorrect multiplicative inverse sign
✗ 1/(a + ib) = a/(a²+b²) + i(b/(a²+b²))
→ The conjugate is a - ib, so the numerator becomes a - ib, giving imaginary part -b/(a²+b²).
✓ 1/(a + ib) = a/(a²+b²) + i(-b/(a²+b²))
⚠ Assuming √(ab) = √a × √b for negative numbers
✗ √(-4) × √(-9) = √(36) = 6
→ The property √(ab) = √a × √b holds only when both a and b are non-negative. For negative numbers, √(-4) = 2i and √(-9) = 3i, so product = 6i² = -6.
✓ √(-4) × √(-9) = (2i)(3i) = 6i² = -6
⚠ Forgetting the modulus is a distance
✗ |3 + 4i| = 3 + 4 = 7
→ The modulus is √(a² + b²), not a + b. |3 + 4i| = √(3² + 4²) = √25 = 5.
✓ |3 + 4i| = √(3² + 4²) = 5
⚠ Confusing conjugate with negative
✗ Conjugate of 2 - 3i is -2 + 3i
→ The conjugate only changes the sign of the imaginary part, not the real part. For 2 - 3i, the imaginary part is -3, so its conjugate is 2 + 3i.
✓ Conjugate of 2 - 3i is 2 + 3i
Compare & Contrast
Definition in a complex number z = a + ib
Real part (Re z)
Imaginary part (Im z)
Re z = a is the real number coefficient, while Im z = b is the real number coefficient of i (the imaginary unit).
Operation on a complex number z = a + ib
Conjugate (z̄)
Negative (-z)
z̄ = a - ib changes only the sign of the imaginary part, while -z = -a - ib changes the sign of both real and imaginary parts.
Geometric representation in Argand plane
Modulus (|z|)
Argument (not in this chapter)
|z| = √(a² + b²) is the distance from the origin to the point (a, b), always a non-negative real number.
Exam Focus
Marks pattern: 1 MCQ (1 mark), 1 SA-I (2 marks), 1 SA-II (3 marks), 1 LA (5 marks) — total approximately 11 marks
Most asked: Algebra of complex numbers (addition, multiplication, division) · Finding multiplicative inverse · Simplifying powers of i · Conjugate and modulus properties · Expressing complex numbers in a + ib form · Solving equations involving complex numbers
Don't skip: Express a given complex expression in the form a + ib (5 marks); Find the multiplicative inverse of a complex number (2-3 marks); Prove identities involving modulus and conjugate (3-5 marks); Simplify powers of i and express in a + ib form (2-3 marks)
One-Line Revision (read aloud before the exam)
  1. A complex number is of the form a + ib, where a and b are real numbers and i = √(-1).
  2. i² = -1, and powers of i repeat every 4: i⁴ᵏ = 1, i⁴ᵏ⁺¹ = i, i⁴ᵏ⁺² = -1, i⁴ᵏ⁺³ = -i.
  3. Two complex numbers are equal if and only if their real parts and imaginary parts are equal.
  4. Addition: (a + ib) + (c + id) = (a + c) + i(b + d).
  5. Multiplication: (a + ib)(c + id) = (ac - bd) + i(ad + bc).
  6. The multiplicative inverse of a + ib is a/(a²+b²) + i(-b/(a²+b²)).
  7. The conjugate of z = a + ib is z̄ = a - ib.
  8. The modulus of z = a + ib is |z| = √(a² + b²).
  9. z · z̄ = |z|².
  10. |z₁ z₂| = |z₁| |z₂| and |z₁/z₂| = |z₁|/|z₂| (z₂ ≠ 0).
  11. √(-a) = i√a for a > 0.
  12. √(-a) × √(-b) = -√(ab) when a, b > 0.
  13. In the Argand plane, the x-axis is the real axis and the y-axis is the imaginary axis.
  14. The point (x, y) represents the complex number x + iy in the Argand plane.

MCQs — 10 Questions with Answers

Q1. If z = 3 + 4i, what is the real part and imaginary part of z?

  • A. Real part = 3, Imaginary part = 4i
  • B. Real part = 3, Imaginary part = 4 ✓
  • C. Real part = 4, Imaginary part = 3
  • D. Real part = 3i, Imaginary part = 4

Answer: B — The imaginary part is the coefficient b in a + ib, which is 4 (not 4i); the real part is 3.

Q2. Which of the following is the multiplicative identity for complex numbers?

  • A. 0 + i0
  • B. 1 + i0 ✓
  • C. i
  • D. –1 + i0

Answer: B — The multiplicative identity is 1 + i0 (or simply 1) because z × 1 = z for any complex number z.

Q3. What is i²⁰²⁵?

  • A. i ✓
  • B. –1
  • C. 1
  • D. –i

Answer: A — 2025 = 4 × 506 + 1, so 2025 ≡ 1 (mod 4); therefore i²⁰²⁵ = i⁴·⁵⁰⁶⁺¹ = i.

Q4. If (2x + 3) + i(y – 1) = 5 + i2, where x and y are real, find x + y.

  • A. 4 ✓
  • B. 5
  • C. 6
  • D. 7

Answer: A — Equate real parts: 2x + 3 = 5 → x = 1; equate imaginary parts: y – 1 = 2 → y = 3; thus x + y = 1 + 3 = 4.

Q5. Simplify (3 + 2i) + (4 – 3i).

  • A. 7 – i ✓
  • B. 7 + i
  • C. 1 + 5i
  • D. 7 – 5i

Answer: A — Add real parts: 3 + 4 = 7; add imaginary parts: 2i – 3i = –i; result is 7 – i.

Q6. Simplify (2 + i)(3 – 2i).

  • A. 6 – 2i
  • B. 8 – i ✓
  • C. 6 + 4i
  • D. 8 + i

Answer: B — Using (a+ib)(c+id) = (ac–bd) + i(ad+bc): (2×3 – 1×(–2)) + i(2×(–2) + 1×3) = (6+2) + i(–4+3) = 8 – i.

Q7. Which statement is NOT correct about complex numbers?

  • A. The sum of two complex numbers is always a complex number
  • B. √(–2) × √(–3) = √6 ✓
  • C. Every complex number has a multiplicative inverse except zero
  • D. Complex numbers satisfy the commutative law of multiplication

Answer: B — √(–2) × √(–3) = i√2 × i√3 = i²√6 = –√6, not √6; the formula √a√b = √(ab) fails when both a and b are negative.

Q8. If z = 1 + 2i, then z⁻¹ equals:

  • A. (1 – 2i)/5 ✓
  • B. (1 + 2i)/5
  • C. (2 – i)/5
  • D. (–1 + 2i)/5

Answer: A — For z = a + ib, z⁻¹ = (a – ib)/(a² + b²); here z⁻¹ = (1 – 2i)/(1² + 2²) = (1 – 2i)/5.

Q9. Assertion: i³ = –i. Reason: i² = –1 and i³ = i² × i = (–1) × i = –i.

  • A. Both assertion and reason are true; reason explains assertion ✓
  • B. Both assertion and reason are true; reason does not explain assertion
  • C. Assertion is true; reason is false
  • D. Assertion is false; reason is true

Answer: A — i³ = –i is correct, and the reasoning using i² = –1 directly derives i³ = i × i² = i × (–1) = –i, so reason fully explains assertion.

Q10. If (x + iy)² = –5 + 12i where x, y ∈ ℝ, then x² + y² equals:

  • A. 5
  • B. 13 ✓
  • C. 17
  • D. 25

Answer: B — Expand (x + iy)² = x² – y² + 2ixy = –5 + 12i; equate: x² – y² = –5 and 2xy = 12 (so xy = 6); solving gives x = 3, y = 2 (or x = –3, y = –2); thus x² + y² = 9 + 4 = 13.

Flashcards

Define a complex number and its real and imaginary parts.

A complex number z = a + ib where a is the real part (Re z) and b is the imaginary part (Im z), both real numbers.

When are two complex numbers z₁ = a + ib and z₂ = c + id equal?

They are equal if and only if a = c and b = d; both real and imaginary parts must match separately.

What is the formula for multiplication of complex numbers z₁ = a + ib and z₂ = c + id?

z₁z₂ = (ac – bd) + i(ad + bc), obtained by expanding and using i² = –1.

State the cyclic pattern of powers of i.

i⁴ᵏ = 1, i⁴ᵏ⁺¹ = i, i⁴ᵏ⁺² = –1, i⁴ᵏ⁺³ = –i; the pattern repeats every 4 powers.

What is the multiplicative inverse of a non-zero complex number z = a + ib?

z⁻¹ = (a – ib)/(a² + b²), which equals the conjugate divided by the square of the modulus.

When is the property √a × √b = √(ab) valid for negative numbers?

Only when both a and b are positive or one is zero; it fails when both a < 0 and b < 0 because it leads to –1 = 1.

How do you divide two complex numbers z₁ ÷ z₂ where z₂ ≠ 0?

z₁/z₂ = z₁ × z₂⁻¹, meaning multiply z₁ by the multiplicative inverse of z₂.

Express √(–a) where a > 0 in terms of the imaginary unit.

√(–a) = i√a, which is purely imaginary with no real part.

Prove the identity (z₁ + z₂)² = z₁² + 2z₁z₂ + z₂² for complex numbers.

Expand (z₁ + z₂)(z₁ + z₂) using the distributive law twice, rearrange using commutativity of multiplication, and collect like terms.

If 4x + i(3x – y) = 3 + i(–6) where x, y are real, what is the method to find x and y?

Equate real parts: 4x = 3; equate imaginary parts: 3x – y = –6; solve the resulting system of two equations.

Important Board Questions

If z₁ = 2 + 3i and z₂ = 1 – 2i, find z₁ + z₂ and z₁ – z₂. Also state which law of addition validates the property z₁ + z₂ = z₂ + z₁. [2 marks]

Add real parts and imaginary parts separately for addition and subtraction; the commutative law of addition states z₁ + z₂ = z₂ + z₁ for all complex numbers.

Express (5 + 4i)(2 – 3i) in the form a + ib. Verify your answer by showing that the real and imaginary parts are computed using the formula (a+ib)(c+id) = (ac–bd) + i(ad+bc), and explain why the product of two non-zero complex numbers is also a complex number (closure property). [5 marks]

Multiply using (a+ib)(c+id) = (ac–bd) + i(ad+bc) formula: real part = 5×2 – 4×(–3), imaginary part = 5×(–3) + 4×2; closure property means the result is always a complex number of form p + iq.

Prove that the following identity holds for all complex numbers z₁ and z₂: (z₁ + z₂)² = z₁² + 2z₁z₂ + z₂². Also demonstrate this identity with z₁ = 3 + i and z₂ = 1 – 2i by computing both sides separately and verifying they are equal. Explain which fundamental properties of complex number operations (such as distributive law and commutativity) are used in the proof. [6 marks]

Proof: expand (z₁ + z₂)(z₁ + z₂) using distributive law twice, apply commutativity of multiplication to rearrange, and collect like terms; numerical verification: compute LHS directly and RHS as z₁² + 2z₁z₂ + z₂², both should equal –3 + 4i after simplification.

Assertion-Reason Questions for Complex Numbers and Quadratic Equations

Q1. Assertion (A):

The number i is a solution of the equation x² + 1 = 0.

Reason (R):

i² = -1 by definition.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. The reason directly explains the assertion: since i² = -1, substituting x = i into x² + 1 gives i² + 1 = -1 + 1 = 0, confirming i is a solution.

Q2. Assertion (A):

Complex number addition satisfies the closure property.

Reason (R):

The sum of two complex numbers may or may not be a complex number.

Answer: C — A is true but R is false — The assertion is true: the sum of any two complex numbers z₁ = a + ib and z₂ = c + id is (a + c) + i(b + d), which is also a complex number. However, the reason is false—the sum is always a complex number, not sometimes.

Q3. Assertion (A):

For all complex numbers z, the additive inverse -z satisfies z + (-z) = 0.

Reason (R):

Every complex number has a unique additive identity.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — Both statements are true. The assertion correctly describes the additive inverse property. However, the reason does not explain the assertion—it states a related but different fact. The reason talks about the additive identity (which is 0), not the additive inverse.

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Case-Based Questions (CBQ) for Complex Numbers and Quadratic Equations

Case Study:

An electronics engineer is designing an AC circuit that requires calculating impedance (resistance to AC current flow). The circuit has two impedance components: Z₁ = 4 + 3i ohms and Z₂ = 2 - i ohms. When components are connected in series, the total impedance is Z_total = Z₁ + Z₂. When the engineer needs to find the equivalent impedance of parallel components, they must calculate 1/Z_total = 1/(Z₁ + Z₂). The engineer wants to express all results in the standard form a + ib to implement them in circuit simulations.

Q1. What is the total impedance Z_total when Z₁ and Z₂ are connected in series?

  • A. 6 + 2i ohms ✓
  • B. 6 + 4i ohms
  • C. 2 + 2i ohms
  • D. -2 + 4i ohms

Answer: A

Q2. If the engineer needs the admittance (reciprocal of impedance) for parallel connection analysis, what is 1/Z_total in the form a + ib?

  • A. (6 - 2i)/40 ✓
  • B. (6 + 2i)/40
  • C. 1/(6 + 2i)
  • D. (6 - 2i)/8

Answer: A

Q3. What is the real part (resistance) of 1/Z_total?

  • A. -2/40 ohms⁻¹
  • B. 2/40 = 1/20 ohms⁻¹
  • C. 3/20 ohms
  • D. 6/40 = 3/20 ohms⁻¹ ✓

Answer: D

Q4. If the engineer multiplies Z_total by its complex conjugate (Z_total)*, what is the result?

  • A. |Z_total|² = 40
  • B. A complex number 36 - 4i
  • C. The magnitude squared = (6² + 2²) = 40
  • D. Both A and C are correct ✓

Answer: D

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Class 11 Maths Formula Sheet

Every formula from all 14 chapters on one page, checked against the NCERT textbook and the CBSE 2026-27 syllabus. Free PDF download.

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