Pure Mathematics — First: Algebra · Mathematics. These are 5 questions from the Magmo3 bank on this lesson, from easy to hard. Solve each one, then open "Show answer and explanation". Magmo3 has 37 questions on this lesson, each with an explanation.
If ω is a complex cubic root of unity, then 1 + ω + ω² = ?
Correct answer: (C) 0
Explanation: Step 1: the cubic roots of unity are the roots of z³ - 1 = (z - 1)(z² + z + 1) = 0 Step 2: ω satisfies ω² + ω + 1 = 0 ∴ 1 + ω + ω² = 0
If 1, ω, ω² are the cubic roots of unity, then the conjugate of ω² is ..........
Correct answer: (B) ω
Explanation: Step 1: ω = −1/2 ± (√3/2)i and ω² is the other complex root: ω² = −1/2 ∓ (√3/2)i. Step 2: The two complex roots are conjugates of each other. Step 3: So the conjugate of ω² is ω.
If ω is a complex cubic root of unity, which of the following equals ω² ?
Correct answer: (D) -1/2 - (√3/2)i
Explanation: Step 1: the non-real cubic roots of unity are ω = -1/2 + (√3/2)i and ω² Step 2: ω² is the conjugate of ω , so ω² = -1/2 - (√3/2)i ∴ ω² = -1/2 - (√3/2)i
If x = (−1 − √3 i)/2, then x⁷ + x⁵ = ..........
Correct answer: (B) −1
Explanation: Step 1: x = −1/2 − (√3/2)i is a complex cubic root of unity, so x³ = 1 and 1 + x + x² = 0. Step 2: x⁷ = (x³)² · x = x and x⁵ = x³ · x² = x². Step 3: x⁷ + x⁵ = x + x². Step 4: = −1.
If ω is a non-real cubic root of unity, find (1 − ω)(1 − ω²)(1 − ω⁴)(1 − ω⁵).
Correct answer: (B) 9
Explanation: Step 1: ω⁴ = ω and ω⁵ = ω², so the product = [(1 − ω)(1 − ω²)]². Step 2: (1 − ω)(1 − ω²) = 1 − ω² − ω + ω³ = 2 − (ω + ω²) = 2 + 1 = 3, so the product = 3² = 9. ∴ 9
The first 10 questions in every subject are free, each with an explanation.