Question

Assertion :$si{n}^{-1}tan(ta{n}^{-1}x+ta{n}^{-1}(2-x\left)\right)=\pi /2$ has no non-zero integral solution. Reason: The greatest and least value of $(si{n}^{-1}x{)}^3+(co{s}^{-1}x{)}^3$ are respectively $7{\pi }^3/8$ and ${\pi }^3/32$.

• 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

For statement one to be true
$ta{n}^{-1}\left(x\right)+ta{n}^{-1}(2-x)=\frac{\pi }{4}$
$ta{n}^{-1}\left(\frac{2}{1-2x+x^2}\right)=\frac{\pi }{4}$
$\frac{2}{(x-1{)}^2}=1$
$(x-1{)}^2=2$
$(x-1)=\pm \sqrt{2}$
$x=1\pm \sqrt{2}$
Hence it has non integral roots.
Thus the assertion is true.

Now $f\left(x\right)=si{n}^{-1}(x{)}^3+co{s}^{-1}(x{)}^3$
Minimum value of f(x) is at $x=\frac{1}{\sqrt{2}}$
$=\frac{{\pi }^3}{64}+\frac{{\pi }^3}{64}$

$=\frac{{\pi }^3}{32}$

And maximum is attained at $x=-1$
$=-\frac{{\pi }^3}{8}+{\pi }^3$

$=\frac{7{\pi }^3}{8}$

Thus the reason is correct, however it is not the correct explanation for the assertion.