Q41 / 160 · AP EAMCET 2024 (22 May Shift 2) · Mathematics

The range of the real valued function $f(x)=\frac{15}{3 \sin x+4 \cos x+10}$ is

  • A
    $[0,3]$
  • B
    $[-1,3]$
  • Correct answer
    C
    $[1,3]$
  • D
    $[-1,1]$

Solution

$\begin{aligned} & -\sqrt{9+16} \leq 3 \sin x+4 \cos x \leq \sqrt{9+16} \\ \Rightarrow & 5 \leq 3 \sin x+4 \cos x+10 \leq 15 \\ \Rightarrow & \frac{1}{15} \leq \frac{1}{3 \sin x+4 \cos x+10} \leq \frac{1}{5} \\ \Rightarrow & 1 \leq \frac{15}{3 \sin x+4 \cos x+10} \leq 3 \\ \therefore & \text { Required range is }[1,3]\end{aligned}$

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

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