Question

If a² + b² + c² = – 2 and f(x) = `|(1 + a^2x, (1 + b^2)x, (1 + c^2)x),((1 + a^2)x, 1 + b^2x, (1 + c^2)x),((1 + a^2)x, (1 + b^2)x, 1 + c^2x)|` then f(x) is a polynomial of degree

Options

• 1

• 0

• 3

• 2

Answer 1

2

Explanation:

`|(1 + a^2x, (1 + b^2)x, (1 + c^2)x),((1 + a^2)x, 1 + b^2x, (1 + c^2)x),((1 + a^2)x, (1 + b^2)x, 1 + c^2x)|`

= `|(1 + 2x + (a^2 + b^2 + c^2)x, (1 + b^2)x, (1 + c^2)x),(1 + 2x + (a^2 + b^2 + c^2)x, 1 = b^2x, (1 + c^2)x),(1 + 2x + (a^2 + b^2 + c^2)x, (1 + b^2)x, 1 + c^2x)|`  ......[C₁ → C₁ + C₂ + C₃]

= `|(1, (1 + b^2)x, (1 + c^2)x),(1, 1 + b^2x, (1 + c^2)x),(1, (1 + b^2)x, 1 + c^2x)|`  ......[∵ a² + b² + c² = – 2]

= `|(1, (1 + b^2)x, (1 + c^2)x),(0, 1 - x, 0),(0, 0, 1 - x)|` ......[R₂ → R₂ – R₁; R₃ →R₃  – R₁]

= `(1 - x)^2`

∴ f(x) is a polynomial of degree 2.