Applications Of Derivatives Question 249
Question: The function $ \frac{a\sin x+b\cos x}{c\sin x+d,\cos x} $ is decreasing, if
[RPET 1999]
Options:
A) $ ad-bc>0 $
B) $ ad-bc<0 $
C) $ ab-cd>0 $
D) $ ab-cd<0 $
Show Answer
Answer:
Correct Answer: B
Solution:
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 $
therefore $ 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 $
therefore $ ad({{\sin }^{2}}x+{{\cos }^{2}}x)-bc({{\sin }^{2}}x+{{\cos }^{2}}x)<0 $
therefore $ (ad-bc)<0 $ .
BETA
