Question

Assertion :Let $f\left(\theta \right)=sin\theta .sin(\pi /3+\theta ).sin(\pi /3-\theta )$

$f\left(\theta \right)\le 1/4$

Reason: $f\left(\theta \right)=\left(1/4\right)sin2\theta$

• 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 C Assertion is correct but Reason is incorrect

$f\left(\theta \right)$
$=sin\theta .sin(\frac{\pi }{3}+\theta ).sin(\frac{\pi }{3}-\theta )$
$=\frac{sin\theta }{2}.2.sin(\frac{\pi }{3}+\theta ).sin(\frac{\pi }{3}-\theta )$
$=\frac{sin\theta }{2}[-cos(\frac{2\pi }{3})+cos(2\theta \left)\right]$
$=\frac{sin\theta }{2}\left[cos\right(2\theta )+\frac{1}{2}]$
$=\frac{sin\theta }{4}+\frac{2sin\theta .cos2\theta }{4}$
$=\frac{sin\theta +2sin\theta -4si{n}^3\theta }{4}$
$=\frac{3sin\theta -4si{n}^3\theta }{4}$
$=\frac{sin3\theta }{4}$
Now maximum value of $f\left(\theta \right)$ will be when $sin3\theta =1$
That is $\theta =\frac{\pi }{6}$
Hence
$f\left(\theta \right)ϵ[-\frac{1}{4},\frac{1}{4}]$
Hence reason is false.