Complex Numbers and Quadratic Equations — Class 11 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 11 Mathematics chapter "Complex Numbers and Quadratic Equations" — 8 important questions with detailed answers for CBSE board exam preparation.
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TL;DR: Free step-by-step NCERT solutions for Class 11 Mathematics chapter "Complex Numbers and Quadratic Equations" — 8 important questions with detailed ans…
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Key Questions Covered:
- Express (3 + 2i) + (1 - 4i) and (3 + 2i) - (1 - 4i) in the form a + bi.
- Multiply (2 + 3i)(1 - i) and express the result in the form a + bi.
- Find the complex conjugate and modulus of z = 3 - 4i. Also find z · z̄.
- Divide (5 + 2i) by (1 + i) and express the result in the form a + bi.
- Solve the quadratic equation x² - 4x + 5 = 0 using the quadratic formula.
- Find i⁴⁵ and simplify your answer.
- + 2 more questions in the full chapter
Solutions Summary:
| Question | Status |
|---|---|
| Express (3 + 2i) + (1 - 4i) and (3 + 2i) - (1 - 4i) in th… | ✓ Solved |
| Multiply (2 + 3i)(1 - i) and express the result in the fo… | ✓ Solved |
| Find the complex conjugate and modulus of z = 3 - 4i. Als… | ✓ Solved |
| Divide (5 + 2i) by (1 + i) and express the result in the … | ✓ Solved |
| Solve the quadratic equation x² - 4x + 5 = 0 using the qu… | ✓ Solved |
| Find i⁴⁵ and simplify your answer. | ✓ Solved |
Showing 6 of 8 questions
Q1: Express (3 + 2i) + (1 - 4i) and (3 + 2i) - (1 - 4i) in the form a + bi.
Step 1: Add the complex numbers.
(3 + 2i) + (1 - 4i)
= 3 + 1 + 2i - 4i
= 4 + (2 - 4)i
= 4 - 2i
Step 2: Subtract the complex numbers.
(3 + 2i) - (1 - 4i)
= 3 - 1 + 2i - (-4i)
= 2 + 2i + 4i
= 2 + 6i
Step 3: Verify using the general rule.
For addition: (a + bi) + (c + di) = (a + c) + (b + d)i
For subtraction: (a + bi) - (c + di) = (a - c) + (b - d)i
Final Answer: (3 + 2i) + (1 - 4i) = 4 - 2i; (3 + 2i) - (1 - 4i) = 2 + 6i
Q2: Multiply (2 + 3i)(1 - i) and express the result in the form a + bi.
Step 1: Use the distributive property (FOIL).
(2 + 3i)(1 - i)
= 2(1) + 2(-i) + 3i(1) + 3i(-i)
= 2 - 2i + 3i - 3i²
Step 2: Use i² = -1.
= 2 - 2i + 3i - 3(-1)
= 2 - 2i + 3i + 3
= (2 + 3) + (-2 + 3)i
= 5 + i
Step 3: Verify by expanding carefully.
(2 + 3i)(1 - i) = 2·1 + 2·(-i) + 3i·1 + 3i·(-i)
= 2 - 2i + 3i + 3
= 5 + i ✓
Final Answer: (2 + 3i)(1 - i) = 5 + i
Q3: Find the complex conjugate and modulus of z = 3 - 4i. Also find z · z̄.
Step 1: Find the complex conjugate z̄.
If z = 3 - 4i, then z̄ = 3 + 4i
Step 2: Find the modulus |z|.
|z| = √(3² + (-4)²) = √(9 + 16) = √25 = 5
Step 3: Calculate z · z̄.
z · z̄ = (3 - 4i)(3 + 4i)
= 3² - (4i)²
= 9 - 16i²
= 9 - 16(-1)
= 9 + 16
= 25
Step 4: Verify the relationship.
Notice that z · z̄ = |z|²
25 = 5² ✓
Final Answer: z̄ = 3 + 4i, |z| = 5, z · z̄ = 25
Q4: Divide (5 + 2i) by (1 + i) and express the result in the form a + bi.
Step 1: Write the division as a fraction.
(5 + 2i) / (1 + i)
Step 2: Multiply numerator and denominator by the conjugate of the denominator.
The conjugate of (1 + i) is (1 - i).
= (5 + 2i)(1 - i) / [(1 + i)(1 - i)]
Step 3: Calculate the denominator.
(1 + i)(1 - i) = 1² - i² = 1 - (-1) = 1 + 1 = 2
Step 4: Calculate the numerator.
(5 + 2i)(1 - i)
= 5(1) + 5(-i) + 2i(1) + 2i(-i)
= 5 - 5i + 2i - 2i²
= 5 - 5i + 2i + 2
= 7 - 3i
Step 5: Divide.
(7 - 3i) / 2 = 7/2 - 3i/2 = 7/2 - (3/2)i
Final Answer...
Q5: Solve the quadratic equation x² - 4x + 5 = 0 using the quadratic formula.
Step 1: Identify coefficients in ax² + bx + c = 0.
a = 1, b = -4, c = 5
Step 2: Calculate the discriminant.
Δ = b² - 4ac = (-4)² - 4(1)(5) = 16 - 20 = -4
Step 3: Since Δ < 0, the solutions are complex.
Step 4: Apply the quadratic formula.
x = (-b ± √Δ) / (2a)
x = (-(-4) ± √(-4)) / (2·1)
x = (4 ± √(-4)) / 2
x = (4 ± 2i) / 2
x = 2 ± i
Step 5: Write the two solutions.
x₁ = 2 + i
x₂ = 2 - i
Step 6: Verify by substitution.
For x = 2 + i:
(2 + i)² - 4(2 + i) + 5
= 4 + 4i + i² - 8 - 4i + 5
= 4 ...
Q6: Find i⁴⁵ and simplify your answer.
Step 1: Identify the pattern of powers of i.
i¹ = i
i² = -1
i³ = i² · i = -1 · i = -i
i⁴ = i² · i² = (-1)(-1) = 1
i⁵ = i⁴ · i = 1 · i = i
The pattern repeats every 4 powers: i, -1, -i, 1, i, -1, -i, 1, ...
Step 2: Find the remainder when 45 is divided by 4.
45 = 4 × 11 + 1
45 ≡ 1 (mod 4)
Step 3: Use the pattern.
Since the remainder is 1:
i⁴⁵ = i¹ = i
Final Answer: i⁴⁵ = i
Showing 6 of 8 questions. Visit the full page for complete solutions.
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