Quantitative Aptitude Ques 1679

Question: If $ p=a\sin x+b\cos x $ and $ q=a\cos x-b\sin x, $ then what is the value of $ p^{2}+q^{2}? $

Options:

A) $ a+b $

B) $ ab $

C) $ a^{2}+b^{2} $

D) $ a^{2}-b^{2} $

Show Answer

Answer:

Correct Answer: C

Solution:

  • Given, $ p=a\sin x+b\cos x $ …(i) and $ q=a\cos x-b\sin x $ …(ii) On squaring Eqs. (i) and (ii), we get $ p^{2}=a^{2}{{\sin }^{2}}x+b^{2}{{\cos }^{2}}x+2ab\sin x\cos x $ and $ q^{2}=a^{2}{{\cos }^{2}}x+b^{2}{{\sin }^{2}}x-2ab\sin x\cos x $ Now, $ p^{2}+q^{2}=a^{2}{{\sin }^{2}}x+b^{2}+{{\cos }^{2}}x+2ab\sin x\cos x $ $ +a^{2}{{\cos }^{2}}+b^{2}{{\sin }^{2}}x-2ab\sin x\cos x $ $ =a^{2}({{\sin }^{2}}x+{{\cos }^{2}}x)+b^{2}({{\cos }^{2}}x+{{\sin }^{2}}x) $ $ =a^{2}+b^{2} $
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