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STD 12 - 6. Application of derivatives — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 6. Application of derivatives4 Marks
MCQ
The function ${{a\sin x + b\cos x} \over {c\sin x + d\,\cos x}}$ is decreasing, if
  • ✓
    $ad - bc < 0$
  • B
    $ad - bc > 0$
  • C
    $ab - cd > 0$
  • D
    $ab - cd < 0$
✓

Answer

Correct option: A.
$ad - bc < 0$
a
(b) Let $y = \frac{{a\sin x + b\cos x}}{{c\sin x + d\cos x}}$

The function will be decreasing, when $\frac{{dy}}{{dx}} < 0$.

$\frac{{(c\sin x + d\cos x)(a\cos x - b\sin x) - (a\sin x + b\cos x)(c\cos x - d\sin x)}}{{{{(c\sin x + d\cos x)}^2}}} < 0$

==> $ac\sin x\cos x - bc{\sin ^2}x + ad{\cos ^2}x$

$ - bd\sin x\cos x - ac\sin x\cos x + ad{\sin ^2}x$

$ - bc{\cos ^2}x + bd\sin x\cos x < 0$

==> $ad({\sin ^2}x + {\cos ^2}x) - bc({\sin ^2}x + {\cos ^2}x) < 0$

==> $(ad - bc) < 0$.

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