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CDS Model Paper 3 Mathematics

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Question : 52 of 100

Marks: +1, -0

If

\sin x+\cos x=p

\sin^{3}x+\cos^{3}x=q

p^{3}-3p

equal to?

0
$-2q$
$2q$
$4q$

Solution:

Let

\sin x+\cos x=p

...(i)

\sin^{3}x+\cos^{3}x=q

...(ii)

On cubing Eq. (i) both sides

\sin^{3}x+\cos^{3}x+3\sin x\cos x(\sin x+\cos x)=p^{3}

Put

\sin^{3}x+\cos^{3}x=q

from equation (ii)

⇒

q+3\sin x\cos x(p)=p^{3}

...(iii)

On squaring Eq. (i) both sides, we get

\sin^{2}x+\cos^{2}x+2\sin x\cos x=p^{2}

⇒

\sin x\cos x=\frac{p^{2}-1}{2}

[\because \sin^{2}x+\cos^{2}x=1]

From Eq. (iii),

q+\frac{3(p^{2}-1)p}{2}=p^{3}

⇒

2q+3p^{3}-3p=2p^{3}\Rightarrow p^{3}-3p=-2q

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