Question

Assertion :$3si{n}^{-1}\left(\frac{1}{3}\right)+si{n}^{-1}\left(\frac{3}{5}\right)<2\pi /3$ and $ta{n}^{-1}(2\sqrt{2}-1)>\pi /3$ Reason: If $3si{n}^{-1}x=\pi /6,$ then $6x-8x^3-1=0$

• 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

$3si{n}^{-1}\left(\frac{1}{3}\right)+si{n}^{-1}\left(\frac{3}{5}\right)$
$=si{n}^{-1}(3\times \frac{1}{3}-4(1/3{)}^3)+si{n}^{-1}\left(\frac{3}{5}\right)$
$=si{n}^{-1}(1-\frac{4}{27})+si{n}^{-1}\left(\frac{3}{5}\right)$
$=si{n}^{-1}\frac{23}{27}+si{n}^{-1}\frac{3}{5}$
$<si{n}^{-1}\frac{\sqrt{3}}{2}+si{n}^{-1}\frac{\sqrt{3}}{2}$
$[\because \frac{23}{27}=0.85,\frac{3}{5}=0.6and\frac{\sqrt{3}}{2}=0.86]$
$=\frac{\pi }{3}+\frac{\pi }{3}=\frac{2\pi }{3}$
and $ta{n}^{-1}(2\sqrt{2}-1)>ta{n}^{-1}\sqrt{3}=\frac{\pi }{3}$
$[\because 2\sqrt{2}-1=1.8,\sqrt{3}=1.7]$
so statement-1 is true
In statement-2, $33si{n}^{-1}x=\frac{\pi }{6}$
$\Rightarrow sin\left(3si{n}^{-1}x\right)=\frac{1}{2}$
$\Rightarrow 3x-4x^3=1/2$
$\Rightarrow 6x-8x^3-1=0$
Showing that statement-2 is also true but does not lead to statemen-1.