If $\mathrm{M}_1$ and $\mathrm{M}_2$ are the maximum values of $\frac{1}{11 \cos 2 x+60 \sin 2 x+69}$ and $3…

If $\mathrm{M}_1$ and $\mathrm{M}_2$ are the maximum values of $\frac{1}{11 \cos 2 x+60 \sin 2 x+69}$ and $3 \cos ^2 5 x+4 \sin ^2 5 x$ respectively, then $\frac{M_1}{M_2}=$
  1. $\frac{65}{2}$
  2. $\frac{1}{32}$
  3. $\frac{8}{3}$
  4. 2

Solution

$\begin{aligned} & -\sqrt{11^2+60^2} \leq 11 \cos 2 x+60 \sin 2 x \leq \sqrt{11^2+60^2} \\ & \Rightarrow-61 \leq 11 \cos 2 x+60 \sin 2 x \leq 61 \\ & \Rightarrow M_1=\frac{1}{69-61}=\frac{1}{8} \\ & \Rightarrow 3 \cos ^2 5 x+4 \sin ^2 5 x=3+\sin ^2 5 x \\ & \Rightarrow M_2=3+1=4 \Rightarrow \frac{M_1}{M_2}=\frac{1}{32} .\end{aligned}$

Asked in: AP EAMCET 2024 (22 May Shift 1)

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