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$ab - cd < 0$
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$
==> $ad({\sin ^2}x + {\cos ^2}x) - bc({\sin ^2}x + {\cos ^2}x) < 0$
==> $(ad - bc) < 0$.
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
$\frac{d y}{d x}+12 y=\cos \left(\frac{\pi}{12} x\right), y(0)=0 .$
Then, which of the following statements is/are $TRUE$?
$ + sin^{-1} \left\{ \,\frac{1}{{\sqrt {13} }}(2\cos x + 3\sin x)\,\,\,\right\} $ w.r.t. at $x = \frac{3}{4}$ is :