Q62 / 160 · TS EAMCET 2024 (09 May Shift 1) · Mathematics

If the period of the function $f(x)=2 \cos (3 x+4)-3 \tan (2 x-3)+5 \sin (5 x)-7$ is $k$, then

  • A
    $\sin \frac{k}{8}=\frac{1}{2}$
  • B
    $\cos \frac{k}{6}=\frac{1}{\sqrt{2}}$
  • Correct answer
    C
    $\tan \frac{k}{3}=-\sqrt{3}$
  • D
    $\sec \frac{k}{2}=2$

Solution

Period of $\cos (3 x+4)$ is $\frac{2 x}{3}$ Period of $\tan (2 x-3)$ is $\frac{\pi}{2}$ Period of $\sin 5 x$ is $\frac{2 \pi}{5}$ Period of $f(x)$ is L.C.M. of $\frac{2 \pi}{3}, \frac{\pi}{2}, \frac{2 \pi}{5} \Rightarrow k=2 \pi$ $\tan \left(\frac{K}{3}\right)=\tan \left(\frac{2 \pi}{3}\right)=-\sqrt{3}$.

Hence, the correct answer is option: 'C'.

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