Complex Numbers
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See all →- The value of \((1+i)^8\) is Answer: C. 16 (1+i)²=2i⇒(1+i)⁸=(2i)⁴=16 . 1
- If \(\omega\) is a complex cube root of unity, then \((1+\omega-\omega^2)^3=\) Answer: D. −8 1+ω=-ω²⇒(-2ω²)³=-8ω⁶=-8 . 1
- z – z¯ = 6i ( z – z¯ = 6i) z – z¯ = (2 + 3i) – (2 – 3i) = 2 + 3i – 2 + 3i = 6i 1
- (1 + 2i) 2 (1 – i) Let z = (1 + 2i)² (1 – i) = (1 + 4i + 4i²) (1 – i) = [1 + 4i + 4(-1)] (1 – i) ….[∵ i² = -1] = (-3 + 4i) (1 – i) = -3 + 3i + 4i – 4i² = -3 +... 1
- (1 + ω)(1 + ω2)(1 + ω4)(1 + ω8) ω is the complex cube root of unity. ω³ = 1 and 1 + ω + ω² = 0 Also, 1 + ω² = -ω, 1 + ω = -ω² and ω + ω² = -1 (1 + ω)(1 + ω²)(1 + ω⁴)(1 + ω⁸... 1
- (1 – ω – ω2)3 + (1 – ω + ω2)3 ω is the complex cube root of unity. ω³ = 1 and 1 + ω + ω² = 0 Also, 1 + ω² = -ω, 1 + ω = -ω² and ω + ω² = -1 (1 – ω – ω²)³ + (1 – ω + ω²)³... 1