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^2)x)|` then f(x) is a polynomial of degree ______.

पर्याय

• 1

• 0

• 3

• 2

Answer 1

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^2)x)|` then f(x) is a polynomial of degree 2.

Explanation:

Applying, C₁→C₁ + C₂ + C₃, we get

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

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

Applying, R₂→R₂ – R₁, R₃→R₃ – R₁

∴ f(x) = `|(1, (1 + b^2)x, (1 + c^2)x),(0, 1 - x, 0),(0, 0, 1 - x)|`

f(x) = (x – 1)²

Hence degree = 2.