Question

If a² + b² + c² = –2 and f(x) = ${\displaystyle \left|\begin{array}{ccc}1+a^{2}x& (1+b^2)x& (1+c^2)x\\ (1+a^2)x& 1+b^{2}x& (1+c^2)x\\ (1+a^2)x& (1+b^2)x& (1+c^2)x\end{array}\right|}$ then f(x) is a polynomial of degree ______.

Options

• 1

• 0

• 3

• 2

Answer 1

If a² + b² + c² = –2 and f(x) = ${\displaystyle \left|\begin{array}{ccc}1+a^{2}x& (1+b^2)x& (1+c^2)x\\ (1+a^2)x& 1+b^{2}x& (1+c^2)x\\ (1+a^2)x& (1+b^2)x& (1+c^2)x\end{array}\right|}$ then f(x) is a polynomial of degree 2.

Explanation:

Applying, C₁→C₁ + C₂ + C₃, we get

f(x) = ${\displaystyle \left|\begin{array}{ccc}1+(a^2+b^2+c^2+2)x& (1+b^2)x& (1+c^2)x\\ 1+(a^2+b^2+c^2+2)x& 1+b^{2}x& (1+c^2)x\\ 1+(a^2+b^2+c^2+2)x& (1+b^2)x& (1+c^2)x\end{array}\right|}$  ...[∵ a² + b² + c² = –2]

= ${\displaystyle \left|\begin{array}{ccc}1& (1+b^2)x& (1+c^2)x\\ 1& 1+b^{2}x& (1+c^2)x\\ 1& (1+b^2)x& (1+c^2)x\end{array}\right|}$

Applying, R₂→R₂ – R₁, R₃→R₃ – R₁

∴ f(x) = ${\displaystyle \left|\begin{array}{ccc}1& (1+b^2)x& (1+c^2)x\\ 0& 1-x& 0\\ 0& 0& 1-x\end{array}\right|}$

f(x) = (x – 1)²

Hence degree = 2.