Determinants

$\left|\begin{array}{ccc}\mathrm{a} - \mathrm{b} - \mathrm{c}& 2\mathrm{a}& 2\mathrm{a}\\ 2\mathrm{b}& \mathrm{b} - \mathrm{c} - \mathrm{a}& 2\mathrm{b}\\ 2\mathrm{c}& 2\mathrm{c}& \mathrm{c} - \mathrm{a} - \mathrm{b}\end{array}\right|$ is equal to

• 0

• a + b + c

• (a + b + c)²

• (a + b + c)³

$\mathrm{The} \mathrm{inverse} \mathrm{of} \mathrm{the} \mathrm{matrix} \left[\begin{array}{ccc}7& - 3& - 3\\ - 1& 1& 0\\ - 1& 0& 1\end{array}\right] \mathrm{is}$

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

• $\left[\begin{array}{ccc}1& 3& 1\\ 4& 3& 8\\ 3& 4& 1\end{array}\right]$

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

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

If x, y, z are all positive and are the pth, qth and rth terms of a geometric progression respectively, then the value of the determinant

$\left|\begin{array}{ccc}\log \left(\mathrm{x}\right)& \mathrm{p}& 1\\ \log \left(\mathrm{y}\right)& \mathrm{q}& 1\\ \log \left(\mathrm{z}\right)& \mathrm{r}& 1\end{array}\right|$ equals

• log(xyz)

• (p - 1)(q - 1)(r - 1)

• pqr

• 0

If $\left[\begin{array}{ccc}1& - 1& \mathrm{x}\\ 1& \mathrm{x}& 1\\ \mathrm{x}& - 1& 1\end{array}\right]$ has no inverse, then the real value of x is

• 2

• 3

• 0

• 1

$$\begin{gathered} \mathrm{If} \mathrm{f}\left(\mathrm{x}\right) \\ \left|\begin{array}{ccc}1& \mathrm{x}& \mathrm{x} + 1\\ 2\mathrm{x}& \mathrm{x}\left(\mathrm{x} - 1\right)& \mathrm{x}\left(\mathrm{x} + 1\right)\\ 3\mathrm{x}\left(\mathrm{x} - 1\right)& \mathrm{x}\left(\mathrm{x} - 1\right)\left(\mathrm{x} - 2\right)& \left(\mathrm{x} - 1\right)\mathrm{x}\left(\mathrm{x} + 1\right)\end{array}\right| \\ \mathrm{then} \mathrm{f}\left(2012\right) = ? \end{gathered}$$

• 0

• 1

• - 500

• 500

$\left|\begin{array}{ccc}\mathrm{x} + 2& \mathrm{x} + 3& \mathrm{x} + 5\\ \mathrm{x} + 4& \mathrm{x} + 6& \mathrm{x} + 9\\ \mathrm{x} + 8& \mathrm{x} + 11& \mathrm{x} + 15\end{array}\right| = ?$

• 3x² + 4x + 5

• x³ + 8x + 2

• 0

• - 2 

${\left(\frac{1 + \mathrm{i}}{1 - \mathrm{i}}\right)}^4 + {\left(\frac{1 - \mathrm{i}}{1 - \mathrm{i}}\right)}^4 = ?$

• 0

• 1

• 2

• 4

The value of the determinant $\left|\begin{array}{ccc}{\mathrm{b}}^2 - \mathrm{ab}& \mathrm{b} - \mathrm{c}& \mathrm{bc} - \mathrm{ac}\\ \mathrm{ab} - {\mathrm{a}}^2& \mathrm{a} - \mathrm{b}& {\mathrm{b}}^2 - \mathrm{ab}\\ \mathrm{bc} - \mathrm{ac}& \mathrm{c} - \mathrm{a}& \mathrm{ab} - {\mathrm{a}}^2\end{array}\right| \mathrm{is}$

• abc

• a + b + c

• 0

• ab + bc + ca

If the lines x + 2ay + a = 0, x + 3by + b = 0, 32x + 4cy + c = 0 are concurrent, then a, b and c are in

• arithmetic progression

• geometric progression

• harmonc progression

• arithmetico-geometric progression

If a, b, c are distinct positive real numbers, then the value of the determinant $\left|\begin{array}{ccc}\mathrm{a}& \mathrm{b}& \mathrm{c}\\ \mathrm{b}& \mathrm{c}& \mathrm{a}\\ \mathrm{c}& \mathrm{a}& \mathrm{b}\end{array}\right| \mathrm{is}$

• $< 0$

• $> 0$

• 0

• $\ge 0$