A) \[ad-bc>0\]
B) \[ad-bc<0\]
C) \[ab-cd>0\]
D) \[ab-cd<0\]
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 \[\frac{dy}{dx}<0\]. \[\frac{(c\sin x+d\cos x)(a\cos x-b\sin x)-(a\sin x+b\cos x)(c\cos x-d\sin x)}{{{(c\sin x+d\cos x)}^{2}}}<0\] Þ \[ac\sin x\cos x-bc{{\sin }^{2}}x+ad{{\cos }^{2}}x\] \[-bd\sin x\cos x-ac\sin x\cos x+ad{{\sin }^{2}}x\] \[-bc{{\cos }^{2}}x+bd\sin x\cos x<0\] Þ \[ad({{\sin }^{2}}x+{{\cos }^{2}}x)-bc({{\sin }^{2}}x+{{\cos }^{2}}x)<0\] Þ \[(ad-bc)<0\].
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