Question

Assertion :If $x{\sin }^3\theta +y{\cos }^3\theta =\sin \theta \cos \theta$ and $x\sin \theta =y\cos \theta$, then $x^2+y^2=1$. Reason: $4{\sin }^3\theta =3\sin \theta -\sin 3\theta$ and $4{\cos }^3\theta =3\cos \theta +\cos 3\theta$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Reason:
$\sin 3\theta =3\sin \theta -4{\sin }^3\theta \Rightarrow 4{\sin }^3\theta =3\sin \theta -\sin 3\theta \cos 3\theta =4{\cos }^3\theta -3\cos \theta \Rightarrow 4{\cos }^3\theta =3\cos \theta +\cos 3\theta$
Assertion:
$x{\sin }^3\theta +y{\cos }^3\theta =\sin \theta \cos \theta \Rightarrow x\sin \theta \cdot {\sin }^2\theta +y\cos \theta \cdot {\cos }^2\theta =\sin \theta \cos \theta$
$x\sin \theta =y\cos \theta$
We get
$x\sin \theta \cdot {\sin }^2\theta +x\sin \theta \cdot {\cos }^2\theta =\sin \theta \cos \theta \Rightarrow x\sin \theta =\sin \theta \cos \theta \Rightarrow x=\cos \theta \Rightarrow y=\sin \theta \Rightarrow x^2+y^2=1$
Hence assertion and reason both are true but reason is not correct explanation of Assertion