The value of $\left|\begin{array}{ccc}1& 1& 1\\ \mathrm{bc}& \mathrm{ca}& \mathrm{ab}\\ \mathrm{b} + \mathrm{c}& \mathrm{c} + \mathrm{a}& \mathrm{a} + \mathrm{b}\end{array}\right|$ is

• 1

• 0

• (a - b)(b - c)(c - a)

• (a + b)(b + c)(c + a)

The value of $\left|\begin{array}{ccc}441& 442& 443\\ 445& 446& 447\\ 449& 450& 451\end{array}\right|$ is :

• $441 \times 446 \times 4510$

• 0

• - 1

• 1

433.

Inverse of the matrix $\left[\begin{array}{cc}1& - 2\\ 3& 4\end{array}\right]$ is

• $\frac{1}{10}\left[\begin{array}{cc}1& - 2\\ 3& 4\end{array}\right]$

• $\frac{1}{10}\left[\begin{array}{cc}4& 2\\ - 3& 1\end{array}\right]$

• $\left[\begin{array}{cc}4& 2\\ - 3& 1\end{array}\right]$

• $\frac{1}{10}\left[\begin{array}{cc}4& - 2\\ - 3& 1\end{array}\right]$

$\left|\begin{array}{ccc}{\mathrm{b}}^2{\mathrm{c}}^2& \mathrm{bc}& \mathrm{b} +\hspace{0.17em}\mathrm{c}\\ {\mathrm{c}}^2{\mathrm{a}}^2& \mathrm{ca}& \mathrm{c} + \mathrm{a}\\ {\mathrm{a}}^2{\mathrm{b}}^2& \mathrm{a} \mathrm{b}& \mathrm{a} + \mathrm{b}\end{array}\right|$ is equal to

• $\frac{1}{\mathrm{abc}}\left(\mathrm{ab} +\mathrm{bc} + \mathrm{ca}\right)$

• ab  + bc + ca

• 0

• a + b + c

If w = $- \frac{1}{2} + \frac{\sqrt{3}\mathrm{i}}{2}$, the value of the determinant $\left|\begin{array}{ccc}1& \mathrm{w}& {\mathrm{w}}^2\\ \mathrm{w}& {\mathrm{w}}^2& 1\\ {\mathrm{w}}^2& 1& \mathrm{w}\end{array}\right|$ is

• zero

• 3

• - 1

• 1

If f(x) = $\left|\begin{array}{ccc}\sin \left(\mathrm{x}\right)& \cos \left(\mathrm{x}\right)& \tan \left(\mathrm{x}\right)\\ {\mathrm{x}}^3& {\mathrm{x}}^2& \mathrm{x}\\ 2\mathrm{x}& 1& 1\end{array}\right|$, then $\underset{\mathrm{x}\to 0}{\lim }\frac{\mathrm{f}\left(\mathrm{x}\right)}{{\mathrm{x}}^2}$ is

• - 1

• 3

• 1

• zero

If for a triangle ABC, $\left|\begin{array}{ccc}1& \mathrm{a}& \mathrm{b}\\ 1& \mathrm{b}& \mathrm{a}\\ 1& \mathrm{c}& \mathrm{c}\end{array}\right| = 0$, then the value of ${\sin }^2\left(\mathrm{A}\right) + {\sin }^2\left(\mathrm{B}\right) + {\sin }^2\left(\mathrm{C}\right)$ is

• $\frac{4}{9}$

• $\frac{9}{4}$

• 1

• $\frac{3\sqrt{3}}{2}$

The inverse of the matrix $\left[\begin{array}{cc}5& - 2\\ 3& 1\end{array}\right]$ is

• $\frac{1}{11}\left[\begin{array}{cc}1& 2\\ - 3& 5\end{array}\right]$

• $\left[\begin{array}{cc}1& 2\\ - 3& 5\end{array}\right]$

• $\frac{1}{13}\left[\begin{array}{cc}- 2& 5\\ 1& 3\end{array}\right]$

• $\left[\begin{array}{cc}1& 3\\ - 2& 5\end{array}\right]$

If A = $\left[\begin{array}{ccc}1& - 1& 1\\ 2& 1& - 3\\ 1& 1& 1\end{array}\right], 10\mathrm{B} = \left[\begin{array}{ccc}4& 2& 2\\ - 5& 0& \mathrm{\alpha }\\ 1& - 2& 3\end{array}\right]$ and B is the inverse of A, then the value of $\mathrm{\alpha }$ is

• 2

• 0

• 5

• 4

If A = $\left[\begin{array}{ccc}1& - 2& 2\\ 0& 2& - 3\\ 3& - 2& 4\end{array}\right]$, then A . adj (A) is equal to

• $\left[\begin{array}{ccc}5& 1& 1\\ 1& 5& 1\\ 1& 1& 5\end{array}\right]$

• $\left[\begin{array}{ccc}5& 0& 0\\ 0& 5& 0\\ 0& 0& 5\end{array}\right]$

• $\left[\begin{array}{ccc}8& 0& 0\\ 0& 8& 0\\ 0& 0& 8\end{array}\right]$

• $\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$