If {f_n}(x),{g_n}(x),{h_n}(x) for n = 1, 2, 3 | vedclass

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(A) Given $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

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Vedclass · Answer

(A) Given $F(x) = \left| \begin{matrix} {f_1}(x) & {f_2}(x) & {f_3}(x) \ {g_1}(x) & {g_2}(x) & {g_3}(x) \ {h_1}(x) & {h_2}(x) & {h_3}(x) \end{matrix} ight|$.At $x = a$,the determinant becomes:$F(a) = \left| \begin{matrix} {f_1}(a) & {f_2}(a) & {f_3}(a) \ {g_1}(a) & {g_2}(a) & {g_3}(a) \ {h_1}(a) & {h_2}(a) & {h_3}(a) \end{matrix} ight|$.Since it is given that ${f_n}(a) = {g_n}(a) = {h_n}(a)$ for $n = 1, 2, 3$,let $k_n = {f_n}(a) = {g_n}(a) = {h_n}(a)$.Then the determinant becomes:$F(a) = \left| \begin{matrix} k_1 & k_2 & k_3 \ k_1 & k_2 & k_3 \ k_1 & k_2 & k_3 \end{matrix} ight|$.Since all three rows are identical,the value of the determinant is $0$.

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