Daily Question 108
$\textbf{Q108.}$ If $f(x)$ is a periodic function having period $7$ and $g(x)$ is periodic function having period $11$ then the period of $D(x)=\begin{vmatrix} f(x) & f\left(\dfrac{x}{3}\right)\\ g(x) & g\left(\dfrac{x}{5}\right)\\ \end{vmatrix}$ is
(A) $231$
(B) $385$
(C) $1155$
(D) $77$
$\textbf{Ans.} (C)$
$\textbf{Sol.}$ Given $D(x)=\begin{vmatrix} f(x) & f\left(\dfrac{x}{3}\right)\\ g(x) & g\left(\dfrac{x}{5}\right)\\ \end{vmatrix}$
$\implies D(x)=f(x)g\left(\dfrac{x}{5}\right)-g(x)f\left(\dfrac{x}{3}\right)$
Period of $f(x)g\left(\dfrac{x}{5}\right)$ is $7\times 55 = 385$
Period of $g(x)f\left(\dfrac{x}{3}\right)$ is $11\times 21 = 231$
Hence the period of $D(x) = \text{LCM}$ of $(385, 231)=1155$
(B) $385$
(C) $1155$
(D) $77$
$\textbf{Ans.} (C)$
$\textbf{Sol.}$ Given $D(x)=\begin{vmatrix} f(x) & f\left(\dfrac{x}{3}\right)\\ g(x) & g\left(\dfrac{x}{5}\right)\\ \end{vmatrix}$
$\implies D(x)=f(x)g\left(\dfrac{x}{5}\right)-g(x)f\left(\dfrac{x}{3}\right)$
Period of $f(x)g\left(\dfrac{x}{5}\right)$ is $7\times 55 = 385$
Period of $g(x)f\left(\dfrac{x}{3}\right)$ is $11\times 21 = 231$
Hence the period of $D(x) = \text{LCM}$ of $(385, 231)=1155$
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