KCET · Maths · Continuity and Differentiability
If \(f(x)=\left|\begin{array}{ccc}x^{3}-x & a+x & b+x \\ x-a & x^{2}-x & c+x \\ x-b & x-c & 0\end{array}\right|\), then
- A \(f(\mathrm{l})=0\)
- B \(f(2)=0\)
- C \(f(0)=0\)
- D \(f(-1)=0\)
Answer & Solution
Correct Answer
(C) \(f(0)=0\)
Step-by-step Solution
Detailed explanation
We have,
\(\begin{aligned}
f(x) &=\left|\begin{array}{ccc}
x^{3}-x & a+x & b+x \\
x-a & x^{2}-x & c+x \\
x-b & x-c & 0
\end{array}\right| \\
f(0) &=\left|\begin{array}{ccc}
0 & a & b \\
-a & 0 & c \\
-b & -c & 0
\end{array}\right| \\
f(0)=0 &
\end{aligned}\)
\(f(0)\) is skew symmetric matrix of order \(3 .\)
\(\begin{aligned}
f(x) &=\left|\begin{array}{ccc}
x^{3}-x & a+x & b+x \\
x-a & x^{2}-x & c+x \\
x-b & x-c & 0
\end{array}\right| \\
f(0) &=\left|\begin{array}{ccc}
0 & a & b \\
-a & 0 & c \\
-b & -c & 0
\end{array}\right| \\
f(0)=0 &
\end{aligned}\)
\(f(0)\) is skew symmetric matrix of order \(3 .\)
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