Complex Numbers and Quadratic Equations
Punjab Board · Class 11 · Mathematics
Flashcards for Complex Numbers and Quadratic Equations — Punjab Board Class 11 Mathematics. Quick Q&A cards covering key concepts, definitions, and formulas.
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Get startedFind x and y if 4x + i(3x - y) = 3 + i(-6), where x and y are real numbers.
Answer
Step 1: Equate real parts: 4x = 3 → x = 3/4 Step 2: Equate imaginary parts: 3x - y = -6 Step 3: Substitute x = 3/4 into 3x - y = -6 → 3(3/4) - y = -6 → 9/4 - y = -6 Step 4: Solve for y: -y = -6 - 9/4 …
Add the complex numbers: (2 + 3i) + (-6 + 5i). Express in the form a + ib.
Answer
Step 1: Identify real parts: 2 and -6 Step 2: Identify imaginary parts: 3 and 5 Step 3: Add real parts: 2 + (-6) = -4 Step 4: Add imaginary parts: 3 + 5 = 8 Step 5: Combine: -4 + 8i Answer: -4 + 8i Ke…
Subtract: (6 + 3i) - (2 - i). Express in the form a + ib.
Answer
Step 1: Rewrite subtraction as addition of negative: (6 + 3i) + (-(2 - i)) Step 2: Find negative of (2 - i): -(2 - i) = -2 + i Step 3: Add: (6 + 3i) + (-2 + i) Step 4: Add real parts: 6 + (-2) = 4 Ste…
Multiply: (3 + 5i)(2 + 6i). Express in the form a + ib.
Answer
Step 1: Use formula (a + ib)(c + id) = (ac - bd) + i(ad + bc) Step 2: Identify: a = 3, b = 5, c = 2, d = 6 Step 3: Calculate ac - bd: (3)(2) - (5)(6) = 6 - 30 = -24 Step 4: Calculate ad + bc: (3)(6) +…
Divide: (6 + 3i) ÷ (2 - i). Express in the form a + ib.
Answer
Step 1: Write as fraction: (6 + 3i)/(2 - i) Step 2: Multiply numerator and denominator by conjugate of denominator (2 + i): Step 3: Numerator: (6 + 3i)(2 + i) = 12 + 6i + 6i + 3i² = 12 + 12i - 3 = 9 +…
Calculate i⁹ + i¹⁹. Express in the form a + ib.
Answer
Step 1: Use the pattern: i¹ = i, i² = -1, i³ = -i, i⁴ = 1, then repeats every 4 powers Step 2: For i⁹: 9 = 4(2) + 1, so i⁹ = i¹ = i Step 3: For i¹⁹: 19 = 4(4) + 3, so i¹⁹ = i³ = -i Step 4: Add: i + (-…
Calculate i⁻³⁹. Express in the form a + ib.
Answer
Step 1: Recognize that i⁻³⁹ = 1/i³⁹ Step 2: Find i³⁹: 39 = 4(9) + 3, so i³⁹ = i³ = -i Step 3: Therefore i⁻³⁹ = 1/(-i) Step 4: Rationalize: 1/(-i) × (i/i) = i/(-i²) = i/(-(-1)) = i/1 = i Alternatively:…
Express (-5i)(1/8 i) in the form a + ib.
Answer
Step 1: Multiply: (-5i)(1/8 i) = (-5 × 1/8) × (i × i) Step 2: Simplify: = -5/8 × i² Step 3: Substitute i² = -1: = -5/8 × (-1) = 5/8 Step 4: Write in standard form: 5/8 + 0i Answer: 5/8 + 0i…
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