Q42 / 160 · TS EAMCET 2020 (14 Sep Shift 2) · Mathematics

Domain of $\cos ^{-1}\left[\log _5\left(x^2+7 x+15\right)\right]$ is

  • A
    The set of all real numbers
  • Correct answer
    B
    $(-\infty,-5] \cup[-2, \infty)$
  • C
    $R-\{-5,-2\}$, where $R$ is the set of real numbers
  • D
    $[-5,-2]$

Solution

We have, $f(x)=\cos ^{-1}\left[\log _5\left(x^2+7 x+15\right)\right]$ $f(x)$ is defined of $\begin{array}{ll} & \log _5\left(x^2+7 x+15\right) \in[-1,1] \\ & -1 \leq \log _5\left(x^2+7 x+15\right) \leq 1 \\ & \frac{1}{5} \leq x^2+7 x+15 \leq 5 \\ \because \quad & x^2+7 x+15 \leq 5 \Rightarrow x^2+7 x+10 \leq 0 \\ & (x+5)(x+2) \leq 0 \Rightarrow x \in[-5,-2] \end{array}$

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

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