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Question: If \(f_{r}(x),g_{r}(x),h_{r}(x),r = 1,2,3\) are polynomials in x such that \(f_{r}(a) = g_{r}(a) = h...

If fr(x),gr(x),hr(x),r=1,2,3f_{r}(x),g_{r}(x),h_{r}(x),r = 1,2,3 are polynomials in x such that fr(a)=gr(a)=hr(a),r=1,2,3f_{r}(a) = g_{r}(a) = h_{r}(a),r = 1,2,3 and

F(x)=∣f1(x)f2(x)f3(x)g1(x)g2(x)g3(x)h1(x)h2(x)h3(x)∣F(x) = \left| \begin{matrix} f_{1}(x) & f_{2}(x) & f_{3}(x) \\ g_{1}(x) & g_{2}(x) & g_{3}(x) \\ h_{1}(x) & h_{2}(x) & h_{3}(x) \end{matrix} \right|, then find F′(x)F'(x) at x=ax = a

A

0

B

f1(a)g2(a)h3(a)f_{1}(a)g_{2}(a)h_{3}(a)

C

1

D

None of these

Answer

0

Explanation

Solution

f_{1}^{'}(x) & f_{2}^{'}(x) & f_{3}^{'}(x) \\ g_{1}(x) & g_{2}(x) & g_{3}(x) \\ h_{1}(x) & h_{2}(x) & h_{3}(x) \end{matrix} \right| + \left| \begin{matrix} f_{1}(x) & f_{2}(x) & f_{3}(x) \\ g_{1}^{'}(x) & g_{2}^{'}(x) & g_{3}^{'}(x) \\ h_{1}(x) & h_{2}(x) & h_{3}(x) \end{matrix} \right|$$ \+$\left| \begin{matrix} f_{1}(x) & f_{2}(x) & f_{3}(x) \\ g_{1}(x) & g_{2}(x) & g_{3}(x) \\ h_{1}^{'}(x) & h_{2}^{'}(x) & h_{3}^{'}(x) \end{matrix} \right|$ **∴** $\mathbf{F}^{\mathbf{'}}\mathbf{(a) =}\left| \begin{matrix} \mathbf{f}_{\mathbf{1}}^{\mathbf{'}}\mathbf{(a)} & \mathbf{f}_{\mathbf{2}}^{\mathbf{'}}\mathbf{(a)} & \mathbf{f}_{\mathbf{3}}^{\mathbf{'}}\mathbf{(a)} \\ \mathbf{g}_{\mathbf{1}}\mathbf{(a)} & \mathbf{g}_{\mathbf{2}}\mathbf{(a)} & \mathbf{g}_{\mathbf{3}}\mathbf{(a)} \\ \mathbf{h}_{\mathbf{1}}\mathbf{(a)} & \mathbf{h}_{\mathbf{2}}\mathbf{(a)} & \mathbf{h}_{\mathbf{3}}\mathbf{(a)} \end{matrix} \right|$ + $\left| \begin{matrix} \mathbf{f}_{\mathbf{1}}\mathbf{(a)} & \mathbf{f}_{\mathbf{2}}\mathbf{(a)} & \mathbf{f}_{\mathbf{3}}\mathbf{(a)} \\ \mathbf{g}_{\mathbf{1}}^{\mathbf{'}}\mathbf{(a)} & \mathbf{g}_{\mathbf{2}}^{\mathbf{'}}\mathbf{(a)} & \mathbf{g}_{\mathbf{3}}^{\mathbf{'}}\mathbf{(a)} \\ \mathbf{h}_{\mathbf{1}}\mathbf{(a)} & \mathbf{h}_{\mathbf{2}}\mathbf{(a)} & \mathbf{h}_{\mathbf{3}}\mathbf{(a)} \end{matrix} \right|$ **+**$\left| \begin{matrix} \mathbf{f}_{\mathbf{1}}\mathbf{(a)} & \mathbf{f}_{\mathbf{2}}\mathbf{(a)} & \mathbf{f}_{\mathbf{3}}\mathbf{(a)} \\ \mathbf{g}_{\mathbf{1}}\mathbf{(a)} & \mathbf{g}_{\mathbf{2}}\mathbf{(a)} & \mathbf{g}_{\mathbf{3}}\mathbf{(a)} \\ \mathbf{h}_{\mathbf{1}}^{\mathbf{'}}\mathbf{(a)} & \mathbf{h}_{\mathbf{2}}^{\mathbf{'}}\mathbf{(a)} & \mathbf{h}_{\mathbf{3}}^{\mathbf{'}}\mathbf{(a)} \end{matrix} \right|$ **=.**$\mathbf{0 + 0 + 0 = 0}\mathbf{\lbrack}\mathbf{\because}\mathbf{f}_{\mathbf{r}}\mathbf{(a) =}\mathbf{g}_{\mathbf{r}}\mathbf{(a) =}\mathbf{h}_{\mathbf{r}}\mathbf{(a),r = 1,2,3\rbrack}$