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3 and 4 .Determinants and Matrices
normal
Let $[.]$ , $ \{.\} $ and $sgn$$(.)$ denotes greatest integer function, fractional part function and signum function respectively, then value of determinant
$\left| {\begin{array}{*{20}{c}}
{\left[ \pi \right]}&{amp(1 + i\sqrt 3 )}&1 \\
1&0&2 \\
{\operatorname{sgn} ({{\cot }^{ - 1}}x)}&1&{\{ \pi \} }
\end{array}} \right|$ is-
A
$ - 6 + \frac{{5\pi }}{3} - \frac{{{\pi ^2}}}{3}$
B
$\frac{{5\pi }}{3} - \frac{{{\pi ^2}}}{3} - 5$
C
$\frac{{5\pi }}{3} + \frac{{{\pi ^2}}}{3} + 6$
D
$ - 5 + \frac{{{\pi ^3}}}{3} - \frac{{5{\pi ^2}}}{3}$
Solution
$\left|\begin{array}{ccc}{3} & {\pi / 3} & {1} \\ {1} & {0} & {2} \\ {1} & {1} & {\pi-3}\end{array}\right|=-6+\frac{\pi}{3}(2-\pi+3)+1$
$=-\frac{\pi^{2}}{3}+\frac{5 \pi}{3}-5$
Standard 12
Mathematics
