Q60 / 160 · TS EAMCET 2020 (14 Sep Shift 2) · Mathematics

If $\cos x-\sin x=\sqrt{a} \sin x$, then $a \sin x+\cos x-\sin x=$

  • A
    $-\sqrt{a} \sin x$
  • Correct answer
    B
    $\sqrt{a} \cos x$
  • C
    $(\sqrt{a}-1) \sin x$
  • D
    $-\sqrt{a} \cos x$

Solution

We have, $\begin{aligned} & \cos x-\sin x=\sqrt{a} \sin x \Rightarrow \cos x=\sin x(\sqrt{a}+1) \\ & \cos x=\frac{\sin x(\sqrt{a}+1)(\sqrt{a}-1)}{(\sqrt{a}-1)} \\ & (\sqrt{a}-1) \cos x=a \sin x-\sin x \\ & \sqrt{a} \cos x=a \sin x+\cos x-\sin x \\ \therefore \quad & a \sin x+\cos x-\sin x=\sqrt{a} \cos x \end{aligned}$

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

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