Class 11 : Maths (In English) – Lesson 4. Complex Numbers and Quadratic Equations
EXPLANATION & SUMMARY
1️⃣ Introduction
🔵 A quadratic equation a·x² + b·x + c = 0 has real roots only when the discriminant D = b² − 4ac ≥ 0.
🟢 When D < 0, roots are not real. To handle such cases, mathematicians introduced a new number i such that i² = −1.
💡 Using i, we form complex numbers which extend the real number system.
✨ Complex numbers help solve equations that cannot be solved in real numbers.
2️⃣ Imaginary Unit
📘 Definition: i = √(−1)
✔ i² = −1, i³ = −i, i⁴ = 1 (then repeats)
🧠 Examples:
🔹 √(−4) = 2i
🔹 √(−9) = 3i
🔹 √(−25) = 5i
3️⃣ Definition of Complex Number
🔵 A complex number is of the form z = a + i·b
➡ a = real part, Re(z)
➡ b = imaginary part, Im(z)
📌 Set of all complex numbers is denoted by C.
🧾 Examples:
1️⃣ 3 + 2i → Re = 3, Im = 2
2️⃣ −5i → Re = 0, Im = −5
3️⃣ 7 → purely real (Im = 0)
4️⃣ 4i → purely imaginary (Re = 0)
4️⃣ Equality of Complex Numbers
Two complex numbers a + i·b and c + i·d are equal if
✅ a = c
✅ b = d
Example: 2 + 3i = 2 + 3i ✔ but ≠ 3 + 2i ❌
5️⃣ Representation on Argand Plane
🧭 Complex number z = a + i·b is represented as point (a, b).
📈 x-axis → real part
📉 y-axis → imaginary part
💡 This plane is called the Argand Plane.
📍 Example: z = 3 + 4i → point (3, 4)
6️⃣ Modulus of a Complex Number
🔹 The modulus is the distance from origin (0, 0) to (a, b).
Formula: |z| = √(a² + b²)
Example: z = 3 + 4i → |z| = √(9 + 16) = 5
📏 Represents length of vector.
7️⃣ Argument (Amplitude)
🔸 Argument θ is the angle made by line joining origin to (a, b) with positive x-axis.
Formula: tan θ = b / a
🧭 Quadrant check:
• I: a > 0, b > 0
• II: a < 0, b > 0
• III: a < 0, b < 0
• IV: a > 0, b < 0
✏️ Example: z = 1 + √3·i → tan θ = √3 ⇒ θ = π/3
8️⃣ Conjugate of a Complex Number
🔹 Conjugate of z = a + i·b is z̄ = a − i·b
🧾 Properties:
1️⃣ z·z̄ = a² + b² = |z|²
2️⃣ (z₁ + z₂)̄ = z̄₁ + z̄₂
3️⃣ (z₁·z₂)̄ = z̄₁ · z̄₂
📘 Example: z = 3 + 4i → z̄ = 3 − 4i
✔ z·z̄ = 25
9️⃣ Operations on Complex Numbers
Let z₁ = a + i·b, z₂ = c + i·d
➕ Addition: z₁ + z₂ = (a + c) + i·(b + d)
➖ Subtraction: z₁ − z₂ = (a − c) + i·(b − d)
✖ Multiplication: z₁·z₂ = (a·c − b·d) + i·(a·d + b·c)
➗ Division: z₁ ÷ z₂ = [(a + i·b)(c − i·d)] / (c² + d²)
🧠 Example: (3 + 2i) ÷ (1 − i)
= [(3 + 2i)(1 + i)] / 2
= (1/2) + (5/2)i
🔟 Polar (Trigonometric) Form
🧭 Any complex number z = a + i·b can be written as
z = r (cos θ + i·sin θ)
where r = √(a² + b²), θ = argument
✏️ Example: z = 1 + i
r = √2, θ = π/4
So z = √2 (cos π/4 + i·sin π/4)
1️⃣1️⃣ Euler Form
Using e^(iθ) = cos θ + i·sin θ
✔ z = r·e^(iθ)
1️⃣2️⃣ Algebraic Properties
🟢 Commutative: z₁ + z₂ = z₂ + z₁, z₁·z₂ = z₂·z₁
🟡 Associative: (z₁ + z₂) + z₃ = z₁ + (z₂ + z₃)
🔵 Distributive: z₁·(z₂ + z₃) = z₁·z₂ + z₁·z₃
1️⃣3️⃣ Quadratic Equation
General: a·x² + b·x + c = 0 (a ≠ 0)
Discriminant D = b² − 4ac
📊 Nature of Roots:
• D > 0 → real, distinct
• D = 0 → real, equal
• D < 0 → complex conjugate
Roots: x = [−b ± √D] / (2a)
If D < 0 → √D = i√|D|
✏️ Example: x² + 4x + 8 = 0
D = 16 − 32 = −16
Roots: x = (−4 ± 4i) / 2 = −2 ± 2i
1️⃣4️⃣ Relations between Roots and Coefficients
If α, β are roots:
α + β = −b/a
α·β = c/a
1️⃣5️⃣ Formation of Quadratic Equation from Roots
If roots are α, β → x² − (α + β)x + αβ = 0
1️⃣6️⃣ Cube Roots of Unity
Numbers satisfying x³ = 1 → 1, ω, ω²
ω = (−1 + i√3)/2, ω² = (−1 − i√3)/2
Properties:
• ω³ = 1
• 1 + ω + ω² = 0
• ω ≠ 1
1️⃣7️⃣ Geometrical Interpretation
🧭 Each complex number = vector from origin to (a, b).
• Length = |z|
• Angle = θ
✔ Multiplying by i rotates 90° anticlockwise.
Example: 1 × i = i
1️⃣8️⃣ Key Identities
📘 i² = −1, i³ = −i, i⁴ = 1
✔ z·z̄ = |z|²
✔ |z₁·z₂| = |z₁||z₂|
✔ arg(z₁·z₂) = arg(z₁) + arg(z₂)
✔ (z₁·z₂)̄ = z̄₁·z̄₂
1️⃣9️⃣ Applications
🧠 Used to solve equations with negative discriminant
⚡ Electrical circuits (AC)
📈 Vector rotations
🧪 Quantum mechanics
🔶 Summary (≈300 words)
• z = a + i·b
• i² = −1, i³ = −i, i⁴ = 1
• Re(z) = a, Im(z) = b
• |z| = √(a² + b²)
• z̄ = a − i·b
• Polar: r(cos θ + i·sin θ)
• Euler: r·e^(iθ)
• Argand plane → point (a, b)
Quadratic eq: a·x² + b·x + c = 0
D > 0 → real, distinct
D = 0 → real, equal
D < 0 → complex conjugates
Roots: (−b ± i√|D|)/2a
α + β = −b/a, αβ = c/a
Cube roots of unity: 1, ω, ω²
ω = (−1 + i√3)/2
ω³ = 1, 1 + ω + ω² = 0
📝 Quick Recap
✔ z = a + i·b
✔ i² = −1
✔ |z| = √(a² + b²)
✔ Conjugate = a − i·b
✔ Polar = r(cos θ + i·sin θ)
✔ D < 0 → complex roots
✔ Cube roots unity: 1, ω, ω²
✔ 1 + ω + ω² = 0
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QUESTIONS FROM TEXTBOOK
📗 Exercise 4.1
🔵 Question 1
(5i) (−3i/5). Express in the form a + ib.
🟢 Answer
➡️ (5i)(−3i/5) = (5·−3/5)·i²
➡️ = (−3)·(−1)
➡️ = 3
✔️ Result: 3 + 0i
🔵 Question 2
i⁹ + i¹⁹. Express in a + ib.
🟢 Answer
➡️ i⁴ = 1 ⇒ powers repeat every 4
➡️ i⁹ = i¹ (since 9 ≡ 1 mod 4) = i
➡️ i¹⁹ = i³ (since 19 ≡ 3 mod 4) = −i
➡️ i⁹ + i¹⁹ = i + (−i) = 0
✔️ Result: 0 + 0i
🔵 Question 3
i⁻³⁹. Express in a + ib.
🟢 Answer
➡️ i⁻³⁹ = 1 / i³⁹
➡️ i³⁹ = i³ (since 39 ≡ 3 mod 4) = −i
➡️ 1/(−i) = i
✔️ Result: 0 + 1i
🔵 Question 4
3(7 + 7i) + i(7 + 7i). Express in a + ib.
🟢 Answer
➡️ 3(7 + 7i) = 21 + 21i
➡️ i(7 + 7i) = 7i + 7i² = 7i − 7
➡️ Sum = (21 − 7) + (21i + 7i)
➡️ = 14 + 28i
✔️ Result: 14 + 28i
🔵 Question 5
(1 − i) − (−1 + 6i). Express in a + ib.
🟢 Answer
➡️ = 1 − i + 1 − 6i
➡️ = 2 − 7i
✔️ Result: 2 − 7i
🔵 Question 6
(1/5 + (2/5)i) − (4 + (5/2)i). Express in a + ib.
🟢 Answer
➡️ Real: 1/5 − 4 = −19/5
➡️ Imag: 2/5 − 5/2 = (4/10 − 25/10) = −21/10
✔️ Result: −19/5 − (21/10)i
🔵 Question 7
[(1/3 + (7/3)i) + (4 + (1/3)i)] − (−4/3 + i). Express in a + ib.
🟢 Answer
➡️ Inside sum: real = 1/3 + 4 = 13/3; imag = 7/3 + 1/3 = 8/3
➡️ Subtract: (13/3 − (−4/3)) + (8/3 − 1)i
➡️ = 17/3 + (5/3)i
✔️ Result: 17/3 + (5/3)i
🔵 Question 8
(1 − i)⁴. Express in a + ib.
🟢 Answer
➡️ (1 − i)² = 1 − 2i + i² = −2i
➡️ (−2i)² = 4i² = −4
✔️ Result: −4 + 0i
🔵 Question 9
(1/3 + 3i)³. Express in a + ib.
🟢 Answer
➡️ Let a = 1/3, b = 3i
➡️ (a + b)³ = a³ + 3a²b + 3ab² + b³
➡️ a³ = 1/27
➡️ 3a²b = 3·(1/9)·3i = i
➡️ 3ab² = 3·(1/3)·(3i)² = 1·9i² = −9
➡️ b³ = (3i)³ = 27i³ = −27i
➡️ Sum = (1/27 − 9) + (i − 27i)
➡️ = (−242/27) − 26i
✔️ Result: −242/27 − 26i
🔵 Question 10
(−2 − (1/3)i)³. Express in a + ib.
🟢 Answer
➡️ Let a = −2, b = −(1/3)i
➡️ (a + b)³ = a³ + 3a²b + 3ab² + b³
➡️ a³ = −8
➡️ 3a²b = 3·4·(−1/3)i = −4i
➡️ b² = (−1/3 i)² = −1/9
➡️ 3ab² = 3·(−2)·(−1/9) = 2/3
➡️ b³ = (−1/3 i)³ = (1/27)i
➡️ Sum = (−8 + 2/3) + (−4 + 1/27)i
➡️ Real = −22/3; Imag = (−108/27 + 1/27)i = (−107/27)i
✔️ Result: −22/3 − (107/27)i
🔵 Question 11
Find the multiplicative inverse of 4 − 3i.
🟢 Answer
➡️ 1/(4 − 3i) = (4 + 3i)/(4² + 3²)
➡️ = (4 + 3i)/25
✔️ Inverse: 4/25 + (3/25)i
🔵 Question 12
Find the multiplicative inverse of √5 + 3i.
🟢 Answer
➡️ 1/(√5 + 3i) = (√5 − 3i)/( (√5)² + 3² )
➡️ = (√5 − 3i)/(5 + 9)
➡️ = (√5 − 3i)/14
✔️ Inverse: (√5)/14 − (3/14)i
🔵 Question 13
Find the multiplicative inverse of −i.
🟢 Answer
➡️ 1/(−i) = i
✔️ Inverse: 0 + 1i
🔵 Question 14
Express in the form a + ib:
[(3 + i√5)(3 − i√5)] / [(√3 + √2 i) − (√3 − i√2)]
🟢 Answer
➡️ Numerator: (a + ib)(a − ib) = a² + b²
➡️ = 3² + (√5)² = 9 + 5 = 14
➡️ Denominator: (√3 + √2 i) − (√3 − √2 i)
➡️ = √3 − √3 + √2 i − (−√2 i) = 2√2 i
➡️ Fraction = 14 / (2√2 i) = 7/(√2 i)
➡️ 1/i = −i ⇒ 7/(√2 i) = −(7/√2) i
➡️ = −(7√2/2) i
✔️ Result: 0 − (7√2/2)i
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OTHER IMPORTANT QUESTIONS FOR EXAMS
(CBSE MODEL QUESTIONS PAPER)
ESPECIALLY MADE FROM THIS LESSON ONLY
🟦 Section A – MCQs (1 mark each)
🔵 Question 1:
The value of i² + i⁴ + i⁶ + i⁸ is
1️⃣ 4
2️⃣ 0
3️⃣ 2
4️⃣ –4
🟢 Answer: 2️⃣ 0
🔵 Question 2:
If z = 3 + 4i, then |z| equals
1️⃣ 5
2️⃣ 7
3️⃣ 1
4️⃣ 25
🟢 Answer: 1️⃣ 5
🔵 Question 3:
Conjugate of (2 – 5i) is
1️⃣ 2 + 5i
2️⃣ –2 – 5i
3️⃣ 2 – 5i
4️⃣ –2 + 5i
🟢 Answer: 1️⃣ 2 + 5i
🔵 Question 4:
The real part of (3i + 5) is
1️⃣ 3
2️⃣ 5
3️⃣ 8
4️⃣ 0
🟢 Answer: 2️⃣ 5
🔵 Question 5:
If z = 2 + i√3, then argument of z is
1️⃣ π/6
2️⃣ π/3
3️⃣ π/2
4️⃣ π
🟢 Answer: 2️⃣ π/3
🔵 Question 6:
For any complex number z, z·z̄ equals
1️⃣ |z|²
2️⃣ z²
3️⃣ z
4️⃣ |z|
🟢 Answer: 1️⃣ |z|²
🔵 Question 7:
If i = √(–1), then i⁵ equals
1️⃣ i
2️⃣ –1
3️⃣ –i
4️⃣ 1
🟢 Answer: 1️⃣ i
🔵 Question 8:
The modulus of 1 – i is
1️⃣ √2
2️⃣ 1
3️⃣ 2
4️⃣ 0
🟢 Answer: 1️⃣ √2
🔵 Question 9:
The value of i³ + i⁵ is
1️⃣ –i
2️⃣ 0
3️⃣ i
4️⃣ –2i
🟢 Answer: 2️⃣ 0
🔵 Question 10:
If z = 4i, then Re(z) is
1️⃣ 4
2️⃣ 0
3️⃣ i
4️⃣ –4
🟢 Answer: 2️⃣ 0
🔵 Question 11:
Cube roots of unity satisfy
1️⃣ x² + x + 1 = 0
2️⃣ x³ – 1 = 0
3️⃣ x³ + 1 = 0
4️⃣ x² – x + 1 = 0
🟢 Answer: 2️⃣ x³ – 1 = 0
🔵 Question 12:
If ω is cube root of unity, then ω² + ω + 1 =
1️⃣ 1
2️⃣ 0
3️⃣ –1
4️⃣ 2
🟢 Answer: 2️⃣ 0
🔵 Question 13:
If α, β are roots of x² + 3x + 2 = 0, then α + β =
1️⃣ 2
2️⃣ –2
3️⃣ –3
4️⃣ 3
🟢 Answer: 3️⃣ –3
🔵 Question 14:
If α, β are roots of x² + bx + c = 0, then αβ =
1️⃣ b
2️⃣ c
3️⃣ c/b
4️⃣ c/a
🟢 Answer: 4️⃣ c/a
🔵 Question 15:
If z = a + ib, then Im(z) is
1️⃣ a
2️⃣ b
3️⃣ i·b
4️⃣ a + b
🟢 Answer: 2️⃣ b
🔵 Question 16:
The polar form of 1 + i is
1️⃣ √2 (cos π/4 + i sin π/4)
2️⃣ 2 (cos π/2 + i sin π/2)
3️⃣ √2 (cos π/3 + i sin π/3)
4️⃣ 1 (cos π/4 + i sin π/4)
🟢 Answer: 1️⃣ √2 (cos π/4 + i sin π/4)
🔵 Question 17:
If z₁ = 2 + 3i, z₂ = 1 – 2i, then Re(z₁ + z₂) =
1️⃣ 2
2️⃣ 3
3️⃣ 1
4️⃣ 4
🟢 Answer: 4️⃣ 4
🔵 Question 18:
The cube roots of unity are
1️⃣ 1, ω, ω²
2️⃣ 1, –1, i
3️⃣ 1, i, –i
4️⃣ 1, –1, ω
🟢 Answer: 1️⃣ 1, ω, ω²
🟨 Section B – Very Short Answer (2 marks each)
🔵 Question 19: Find the modulus and argument of z = 1 – i.
🟢 Answer: |z| = √(1² + (–1)²) = √2, θ = tan⁻¹(–1) = –π/4
🔵 Question 20: Express (1 + i)/(1 – i) in the form a + ib.
🟢 Answer: Multiply by conjugate:
(1 + i)/(1 – i) × (1 + i)/(1 + i) = (1 + 2i + i²)/2 = i
🔵 Question 21: If z₁ = 3 + 4i and z₂ = 1 – 2i, find z₁·z₂.
🟢 Answer: (3 + 4i)(1 – 2i) = 3 – 6i + 4i – 8i² = 11 – 2i
🔵 Question 22: Write the conjugate of (2 – 3i)/(4 + i).
🟢 Answer: First simplify:
Multiply by conjugate (4 – i): (2 – 3i)(4 – i)/(16 + 1) = (8 – 2i – 12i + 3i²)/17 = (8 – 14i – 3)/17 = (5 – 14i)/17
Conjugate = (5 + 14i)/17
🔵 Question 23: Solve x² + x + 1 = 0.
🟢 Answer: D = 1 – 4 = –3
x = [–1 ± i√3]/2 = –½ ± (√3/2)i
🟧 Section C – Short Answer (3 marks each)
🔵 Question 24: Find modulus and argument of z = –1 + √3 i.
🟢 Answer:
|z| = √[ (–1)² + (√3)² ] = √4 = 2
θ = tan⁻¹(√3 / –1) → QII → π – π/3 = 2π/3
z = 2 (cos 2π/3 + i sin 2π/3)
🔵 Question 25: If z₁ = 3 + 4i, z₂ = 1 + 2i, find |z₁·z₂| and |z₁|·|z₂|.
🟢 Answer:
z₁·z₂ = (3 + 4i)(1 + 2i) = 3 + 6i + 4i + 8i² = –5 + 10i
|z₁·z₂| = √( (–5)² + 10² ) = √125 = 5√5
|z₁| = 5, |z₂| = √5 → |z₁|·|z₂| = 5√5
✔ Verified |z₁z₂| = |z₁||z₂|
🔵 Question 26: Express z = 1 – √3 i in polar form.
🟢 Answer:
r = √(1² + (–√3)²) = 2
θ = tan⁻¹(–√3/1) = –π/3
z = 2 (cos(–π/3) + i sin(–π/3))
🔵 Question 27: Show that (1 + i)⁴ = –4.
🟢 Answer:
(1 + i)² = 1 + 2i + i² = 2i
(1 + i)⁴ = (2i)² = 4i² = –4
🔵 Question 28: If roots of x² + 2x + 5 = 0 are α, β, verify α·β = c/a.
🟢 Answer:
D = 4 – 20 = –16
α = –1 + 2i, β = –1 – 2i
α·β = (–1)² – (2i)² = 1 – (–4) = 5 = c/a ✔
🟥 Section D – Long Answer (5 marks each)
🔵 Question 29: Solve x² + 4x + 13 = 0 and represent the roots on Argand plane.
🟢 Answer:
D = 16 – 52 = –36
Roots: x = [–4 ± i√36]/2 = –2 ± 3i
🧭 Points: (–2, 3) and (–2, –3)
🔵 Question 30: Find all cube roots of 8( cos 300° + i sin 300° ).
🟢 Answer:
r = 8, θ = 300°
Cube roots: √[3]{8} [cos((300° + 360°k)/3) + i sin((300° + 360°k)/3)]
= 2[cos(100°), cos(220°), cos(340°)] with corresponding sines.
🔵 Question 31: Show that ω and ω² are complex cube roots of unity and find 1 + ω + ω².
🟢 Answer:
ω = (–1 + i√3)/2, ω² = (–1 – i√3)/2
ω³ = 1
1 + ω + ω² = 0 ✔
🟫 Section E – Case/Application (5 marks each)
🔵 Question 32:
A quadratic equation represents motion with complex roots.
Given: x² + 6x + 13 = 0
(a) Find the roots.
(b) Interpret geometrically.
🟢 Answer:
D = 36 – 52 = –16
Roots = –3 ± 2i
Representation: points (–3, 2), (–3, –2) on Argand plane.
🔵 Question 33:
An alternating current is given by I = 10(cos ωt + i sin ωt).
Find |I| and interpret.
🟢 Answer:
|I| = 10 √(cos² ωt + sin² ωt) = 10
Interpretation: amplitude of current = 10 units.
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