If f(α) = α & α \\ -α & α and if α, β and γ are angles of a triangle, then x is a solution to x^n = |f(α) × f(β) × f(γ)| where n = 3 . What are the va - Mathematics MCQ Mathematics Mock Test - 10 2026 | ExamTest.live

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If f (α) = [cos α - sin α sin α cos α] and if α, β and γ are angles of a triangle, then x is a solution to x^{n} = ∣ f (α) \cdot f (β) \cdot f (γ) ∣ where n = 3 . What are the values of x ?

Solution & Step-by-step Explanation

First, find the determinant of f (α) : ∣ f (α) ∣ = cos^{2} α - (- sin^{2} α) = cos^{2} α + sin^{2} α = 1 .Similarly, ∣ f (β) ∣ = 1 and ∣ f (γ) ∣ = 1 .The property of determinants states ∣ A BC ∣ = ∣ A ∣ \cdot ∣ B ∣ \cdot ∣ C ∣ .So, ∣ f (α) \cdot f (β) \cdot f (γ) ∣ = 1 \times 1 \times 1 = 1 .The equation is x^{n} = 1 with n = 3 . x^{3} = 1 The solutions to this equation are the cube roots of unity: 1, ω, ω^{2} .

Try it yourself before checking the explanation above.

If f (α) = [cos α - sin α sin α cos α] and if α, β and γ are angles of a triangle, then x is a solution to x^{n} = ∣ f (α) \cdot f (β) \cdot f (γ) ∣ where n = 3 . What are the values of x ?

1, ω, ω^{2}

- I_{2}

1, ω

sin α cos α \cdot I_{2}

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