Q93 / 180 · KCET 2017 · Mathematics
If \( y=\left|\begin{array}{ccc}f(x) & g(x) & h(x) \\ l & m & n \\ a & b & c\end{array}\right| \), then \( \frac{d y}{d x} \) is equal to
- Correct answerA\( \left|\begin{array}{ccc}f^{\prime}(x) & g^{\prime}(x) & h^{\prime}(x) \\ l & m & n \\ a & b & c\end{array}\right| \)
- B\( \left|\begin{array}{ccc}l & m & n \\ f^{\prime}(x) & g^{\prime}(x) & h^{\prime}(x) \\ a & b & c\end{array}\right| \)
- C\( \left|\begin{array}{lll}f^{\prime}(x) & l & a \\ g^{\prime}(x) & m & b \\ h^{\prime}(x) & n & c\end{array}\right| \)
- D\( \left|\begin{array}{ccc}l & m & n \\ a & b & c \\ f^{\prime}(x) & g^{\prime}(x) & h^{\prime}(x)\end{array}\right| \)
Solution
Given that, \( y=\left|\begin{array}{ccc}f(x) & g(x) & h(x) \\ l & m & n \\ a & b & c\end{array}\right| \)
So,
\( \frac{d y}{d x}=\left|\begin{array}{ccc}f^{\prime}(x) & g^{\prime}(x) & h^{\prime}(x) \\ l & m & n \\ a & b & c\end{array}\right|+\left|\begin{array}{ccc}f(x) & g(x) & h(x) \\ 0 & 0 & 0 \\ a & b & c\end{array}\right|+ \)
\( \left|\begin{array}{ccc}f(x) & g(x) & h(x) \\ l & m & n \\ 0 & 0 & 0\end{array}\right| \)
\( =\left|\begin{array}{ccc}f^{\prime}(x) & g^{\prime}(x) & h^{\prime}(x) \\ l & m & n \\ a & b & c\end{array}\right| \)
So,
\( \frac{d y}{d x}=\left|\begin{array}{ccc}f^{\prime}(x) & g^{\prime}(x) & h^{\prime}(x) \\ l & m & n \\ a & b & c\end{array}\right|+\left|\begin{array}{ccc}f(x) & g(x) & h(x) \\ 0 & 0 & 0 \\ a & b & c\end{array}\right|+ \)
\( \left|\begin{array}{ccc}f(x) & g(x) & h(x) \\ l & m & n \\ 0 & 0 & 0\end{array}\right| \)
\( =\left|\begin{array}{ccc}f^{\prime}(x) & g^{\prime}(x) & h^{\prime}(x) \\ l & m & n \\ a & b & c\end{array}\right| \)
Hence, the correct answer is option: 'A'.