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TG EAMCET 2 May 2025 Shift 1 Paper

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Question : 23 of 160

Marks: +1, -0

The period of the function

f(x)=\frac{2\;\sin\left(\frac{\pi x}{3}\right)\cos\left(\frac{2\pi x}{5}\right)}{3\;\tan\left(\frac{7\pi x}{2}\right)-5\;\sec\left(\frac{5\pi x}{3}\right)}

30
60
300
150

Solution:

Given,

f(x)=\frac{2\;\sin\left(\frac{\pi x}{3}\right)\cos\left(\frac{2\pi x}{5}\right)}{3\;\tan\left(\frac{7\pi x}{2}\right)-5\;\sec\left(\frac{5\pi x}{3}\right)}

We know that

\;\sin\;(k x) \to \;\text{ period }\;=\;\frac{2\pi}{k}

\;\cos(k x) \to \;\text{ period }\;=\;\frac{2\pi}{k}

\;\tan(k x) \to \;\text{ period }\;=\;\frac{\pi}{k}

\;\sec(k x) \to \;\text{ period }\;=\;\frac{2\pi}{k}

So, for

\sin\;\left(\frac{\pi x}{3}\right)

, period

=\;\frac{2\pi}{\frac{\pi}{3}}=6

\;\cos\;\left(\frac{2\pi x}{5}\right)\;\text{ period }\;=\;\frac{2\pi}{\frac{2\pi}{5}}=5

\;\tan\;\left(\frac{7\pi x}{2}\right)\;\text{ period }\;=\;\frac{\pi}{\frac{7\pi}{2}}\;=\;\frac{2}{7}

\;\sec\;\left(\frac{5\pi x}{3}\right)\;\text{ period }\;=\;\frac{2\pi}{\frac{5\pi}{3}}\;=\;\frac{6}{5}

Now, the LCM of

6,5,\;\frac{2}{7}

and

\frac{6}{5}

is as follows

\text{LCM}

of

\left\{\;\frac{6}{1},\;\frac{5}{1},\;\frac{2}{7},\;\frac{6}{5}\right\} =\;\frac{\text{LCM}\;\text{ of }(6,5,2,6)}{\;\text{ HCF of }(1,1,7,5)}

\Rightarrow \;\frac{30}{1}=30

So, the period of

f(x)

is 30 .

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