$\mathrm{In} ∆\mathrm{ABC} \mathrm{if} \left|\begin{array}{ccc}1& \mathrm{a}& \mathrm{b}\\ 1& \mathrm{c}& \mathrm{a}\\ 1& \mathrm{b}& \mathrm{c}\end{array}\right|$ = 0, if then sin²(A) + sin²(B) + sin²(C) is equal to

• $\frac{4}{9}$

• $\frac{9}{4}$

• $3\sqrt{3}$

• 1

402.

Assuming that the sums and products given below are defined which ofthe following is not true for matrices?

• AB = AC does not imply B = C

• A + B = B + A

• (AB)' = B'A'

• AB = 0 implies A = 0 or B = 0

If 5 is one root of the equations $\left|\begin{array}{ccc}\mathrm{x}& 3& 7\\ 2& \mathrm{x}& - 2\\ 7& 8& \mathrm{x}\end{array}\right| = 0$, then the other two roots of the equation are

• - 2, - 7

• - 2, 7

• 2, - 7

• 2, 7

For how many values of x in the closed interval [- 4, - 1] the matrix $\left[\begin{array}{ccc}3& \mathrm{x} - 1& 2\\ 3& - 1& \mathrm{x} + 2\\ \mathrm{x} + 3& - 1& 2\end{array}\right]$ is singular

• zero

• 2

• 1

• 3

G = $\left\{\left(\begin{array}{cc}\cos \left(\mathrm{\theta }\right)& \sin \left(\mathrm{\theta }\right)\\ - \sin \left(\mathrm{\theta }\right)& \cos \left(\mathrm{\theta }\right)\end{array}\right) : \mathrm{\theta } \in \mathrm{R}\right\}$is a group under matrix multiplication. Then, which one ofthe following statements in respect of G is ture

• $\left(\begin{array}{cc}- 1& 0\\ 0& - 1\end{array}\right)$ is inverse of itself

• G is finite group

• $\left(\begin{array}{cc}\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.& \raisebox{1ex}{$- \sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ \raisebox{1ex}{$\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.& \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\end{array}\right)$ is not an element of G

• $\left(\begin{array}{cc}1& 1\\ - 1& 1\end{array}\right)$ is an element of G

If two circles (x - 1)² + (y - 3)² = r² and x² + y² - 8x + 2y + 8 = 0 intersect in two distinct points, then

• r < 2

• 8 < r < 10

• r = 2

• 2 < r < 8

If w is a complex cube-root of unity, then $\left|\begin{array}{ccc}1& \mathrm{w}& {\mathrm{w}}^2\\ \mathrm{w}& {\mathrm{w}}^2& 1\\ {\mathrm{w}}^2& 1& \mathrm{w}\end{array}\right|$ is equal to

• - 1

• 1

• 0

• w

The solutions of the equation $\left|\begin{array}{ccc}\mathrm{x}& 2& - 1\\ 2& 5& \mathrm{x}\\ - 1& 2& \mathrm{x}\end{array}\right|$ = 0 are

• 3, - 1

• - 3, 1

• 3, 1

• - 3, - 1

If A = $\left[\begin{array}{cc}3& 5\\ 2& 0\end{array}\right] \mathrm{and} \mathrm{B} = \left[\begin{array}{cc}1& 17\\ 0& - 10\end{array}\right]$, then $\left|\mathrm{AB}\right|$ is equal to

• 80

• 100

• - 110

• 92

The identity element in the group $\mathrm{M} = \left\{\left|\begin{array}{cc}\mathrm{x}& \mathrm{x}\\ \mathrm{x}& \mathrm{x}\end{array}\right|, \mathrm{x} \in \mathrm{R}; \mathrm{x} \ne 0\right\}$with respect to matrix multiplication is

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

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

• $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

• $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$