The function a x + b x c x + d , x is decreasing, if - Vidyadip

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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