If $\left|\begin{array}{ccc}\mathrm{x}& 2& 8\\ 2& 8& \mathrm{x}\\ 8& \mathrm{x}& 2\end{array}\right| = \left|\begin{array}{ccc}3& \mathrm{x}& 7\\ \mathrm{x}& 7& 3\\ 7& 3& \mathrm{x}\end{array}\right| = \left|\begin{array}{ccc}5& 5& \mathrm{x}\\ 5& \mathrm{x}& 5\\ \mathrm{x}& 5& 5\end{array}\right|$ = 0, then x is equal to

• 0

• - 10

• 3

• None of these

If the determinant $∆ = \left|\begin{array}{ccc}3& - 2& \sin \left(3\mathrm{\theta }\right)\\ - 7& 8& \cos \left(2\mathrm{\theta }\right)\\ - 11& 14& 2\end{array}\right|$ = 0, then the value of $\sin \left(\mathrm{\theta }\right)$ is

• $\frac{1}{3}\mathrm{or} 1$

• $\frac{1}{\sqrt{2}} \mathrm{or} \frac{\sqrt{3}}{2}$

• $0 \mathrm{or} \frac{1}{2}$

• None of these

If in a $∆\mathrm{ABC}$, ${\sin }^3\left(\mathrm{A}\right) + {\sin }^3\left(\mathrm{B}\right) + {\sin }^3\left(\mathrm{C}\right) = 3\sin \left(\mathrm{A}\right)\sin \left(\mathrm{B}\right)\sin \left(\mathrm{C}\right)$, then the value of determinant $\left|\begin{array}{ccc}\mathrm{a}& \mathrm{b}& \mathrm{c}\\ \mathrm{b}& \mathrm{c}& \mathrm{a}\\ \mathrm{c}& \mathrm{a}& \mathrm{b}\end{array}\right|$ is equal to

• 0

• 1

• 2

• 3

${∆}_1 = \left|\begin{array}{ccc}\mathrm{x}& \mathrm{b}& \mathrm{b}\\ \mathrm{a}& \mathrm{x}& \mathrm{b}\\ \mathrm{a}& \mathrm{a}& \mathrm{x}\end{array}\right| \mathrm{and} {∆}_2 = \left|\begin{array}{cc}\mathrm{x}& \mathrm{b}\\ \mathrm{a}& \mathrm{x}\end{array}\right|$ are the given determinants, then

• ${∆}_1 = 3{\left({∆}_2\right)}^2$

• $\frac{\mathrm{d}}{\mathrm{dx}}\left({∆}_1\right) = 3{∆}_2$

• $\frac{\mathrm{d}}{\mathrm{dx}}\left({∆}_1\right) = 3{\left({∆}_2\right)}^2$

• ${∆}_1 = 3{{∆}_2}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

For the equations x + 2y + 3z = 1, 2x + y + 3z = 2 and 5x + 5y + 9z = 4

• there is only one solution

• there exists infinitely many solution

• there is no solution

• None of the above

If $\left|\begin{array}{ccc}\mathrm{x}& {\mathrm{x}}^2& 1 +\hspace{0.17em}{\mathrm{x}}^3\\ \mathrm{y}& {\mathrm{y}}^2& 1 + {\mathrm{y}}^3\\ \mathrm{z}& {\mathrm{z}}^2& 1 + {\mathrm{z}}^3\end{array}\right| = 0$ and x, y, z are all distinct, then xyz is equal to

• - 1

• 1

• 0

• 3

The determinant $\left|\begin{array}{ccc}\mathrm{a}& \mathrm{b}& \mathrm{a\alpha } + \mathrm{b}\\ \mathrm{b}& \mathrm{c}& \mathrm{b\alpha } + \mathrm{c}\\ \mathrm{a\alpha } + \mathrm{b}& \mathrm{b\alpha } + \mathrm{c}& 0\end{array}\right| = 0$, if a, b, c are in

• AP

• GP

• HP

• None of these

If A = $\left[\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right]$, then ${\mathrm{A}}^{-1}$ is

• - A

• A

• 1

• None of these

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)

100.

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