Question

Assertion :The value of the determinant $\left|\begin{array}{ccc}{\tan }^{-1}x& {\cot }^{-1}x& \pi /2\\ {\sin }^{-1}\left(4/5\right)& {\sin }^{-1}\left(3/5\right)& {\sin }^{-1}1\\ {\cos }^{-1}\left(3/5\right)& {\cos }^{-1}\left(4/5\right)& 1\end{array}\right|$ is equal to zero for all values of $x$. Reason: $2{\cos }^{-1}x={\cos }^{-1}(2x^2-1)$ if $-1\le x\le 1$

• 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

Statement-2 is true if $0\le x\le 1$
For $-1\le x<0,2{\cos }^{-1}x=\pi -{\cos }^{-1}(2x^2-1)$
so statement-2 is false.
If $\Delta$ denotes the given determinant
Then
$\Delta =\left|\begin{array}{ccc}{\tan }^{-1}x+{\cot }^{-1}x& {\cot }^{-1}x& \pi /2\\ {\sin }^{-1}\left(4/5\right)+{\sin }^{-1}\left(3/5\right)& {\sin }^{-1}\left(3/5\right)& {\sin }^{-1}1\\ {\cos }^{-1}\left(3/5\right)+{\cos }^{-1}\left(4/5\right)& {\cos }^{-1}\left(4/5\right)& 1\end{array}\right|$
(Applying $C_1\to C_1+C_2$)
$\left|\begin{array}{ccc}\pi /2& {\cot }^{-1}x& \pi /2\\ {\sin }^{-1}[\frac{4}{5}\sqrt{1-\frac{9}{25}}+\frac{3}{5}\sqrt{1-\frac{16}{25}}]& {\sin }^{-1}\left(3/5\right)& {\sin }^{-1}1\\ {\cos }^{-1}[\frac{3}{5}\times \frac{4}{5}-\sqrt{1-\frac{9}{25}}\sqrt{1-\frac{16}{25}}]& {\cos }^{-1}\left(4/5\right)& 1\end{array}\right|$
$\left|\begin{array}{ccc}\pi /2& {\cot }^{-1}x& \pi /2\\ {\sin }^{-1}1& {\sin }^{-1}\left(3/5\right)& {\sin }^{-1}1\\ {\cos }^{-1}0& {\cos }^{-1}\left(4/5\right)& 1\end{array}\right|=0$
(because ${\cos }^{-1}0=1$)
($C_1$ and $C_2$ are identical)
So statement-1 is true.