31.

If P, Q and R are angles of $△$PQR, then the value of

$\left|\begin{array}{ccc}- 1& \cos \left(\mathrm{R}\right)& \cos \left(\mathrm{Q}\right)\\ \cos \left(\mathrm{R}\right)& - 1& \cos \left(\mathrm{P}\right)\\ \cos \left(\mathrm{Q}\right)& \cos \left(\mathrm{P}\right)& - 1\end{array}\right|$ is equal to

• - 1

• 0

• $\frac{1}{2}$

• 1

The number of real values of a for which the system of equations

x + 3y + 5z = $\mathrm{\alpha x}$

5x + y + 3z = $\mathrm{\alpha y}$

3x + 5y + z = $\mathrm{\alpha z}$

has infinite number of solutions is

• 1

• 2

• 4

• 6

If z = $\left|\begin{array}{ccc}1& 1 + 2\mathrm{i}& - 5\mathrm{i}\\ 1 - 2\mathrm{i}& - 3& 5 +\hspace{0.17em}3\mathrm{i}\\ 5\mathrm{i}& 5 - 3\mathrm{i}& 7\end{array}\right|$, then $\left(\mathrm{i} = \sqrt{- 1}\right)$

• z is purely real

• z is purely imaginary

• $\mathrm{z} + \overline{\mathrm{z}} = 0$

• $\left(\mathrm{z} - \overline{\mathrm{z}}\right)$

If one of the cube roots of 1 be w then

$\left|\begin{array}{ccc}1& 1 + {\mathrm{w}}^2& {\mathrm{w}}^2\\ 1 - \mathrm{i}& - 1& {\mathrm{w}}^2 - 1\\ - \mathrm{i}& - 1 + \mathrm{w}& - 1\end{array}\right|$ is equal to

• w

• i

• 1

• 0

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

• 0

• - 1

• 1

• 2

w is an imaginary cube root of unity and

$\left|\begin{array}{ccc}\mathrm{x} +\hspace{0.17em}{\mathrm{w}}^2& \mathrm{w}& 1\\ \mathrm{w}& {\mathrm{w}}^2& 1 +\hspace{0.17em}\mathrm{x}\\ 1& \mathrm{x} + \mathrm{w}& {\mathrm{w}}^2\end{array}\right|$ = 0, then one of the value of x is

• 1

• 0

• - 1

• 2

If A = $\left[\begin{array}{cc}1& 2\\ - 4& - 1\end{array}\right]$, then A^{- 1} is

• $\frac{1}{7}\left[\begin{array}{cc}- 1& - 2\\ 4& 1\end{array}\right]$

• $\frac{1}{7}\left[\begin{array}{cc}1& 2\\ - 4& - 1\end{array}\right]$

• $\frac{1}{7}\left[\begin{array}{cc}- 1& - 2\\ 4& - 1\end{array}\right]$

• does not exist

Let w be the complex number $\cos \left(\frac{2\mathrm{\pi }}{3}\right) + \mathrm{isin}\left(\frac{2\mathrm{\pi }}{3}\right)$ Then, the number of distinct complex number z satisfying

$\left|\begin{array}{ccc}\mathrm{z} + 1& \mathrm{w}& {\mathrm{w}}^2\\ \mathrm{w}& \mathrm{z} + {\mathrm{w}}^2& 1\\ {\mathrm{w}}^2& 1& \mathrm{z} + \mathrm{w}\end{array}\right|$ = o is equal to

• 1

• 0

• 2

• 3

Let a, b and c be positive real numbers. The following system of equations in x, y and z,

$$\begin{gathered} \frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} + \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2} - \frac{{\mathrm{z}}^2}{{\mathrm{c}}^2} = 1, \frac{{\displaystyle {\mathrm{x}}^2}}{{\displaystyle {\mathrm{a}}^2}} - \frac{{\displaystyle {\mathrm{y}}^2}}{{\displaystyle {\mathrm{b}}^2}} + \frac{{\displaystyle {\mathrm{z}}^2}}{{\displaystyle {\mathrm{c}}^2}} = 1 \\ \mathrm{and} \frac{{\displaystyle - {\mathrm{x}}^2}}{{\displaystyle {\mathrm{a}}^2}} + \frac{{\displaystyle {\mathrm{y}}^2}}{{\displaystyle {\mathrm{b}}^2}} + \frac{{\displaystyle {\mathrm{z}}^2}}{{\displaystyle {\mathrm{c}}^2}} = 1 \mathrm{has} \end{gathered}$$

• finitely many solutions

• no solution

• unique solution

• infinitely many solutions

If $∆\left(\mathrm{r}\right) = \left|\begin{array}{cc}\mathrm{r}& {\mathrm{r}}^3\\ 1& \mathrm{n}\left(\mathrm{n} + 1\right)\end{array}\right|$, then $\sum _{\mathrm{r} = 1}^{\mathrm{n}}∆\left(\mathrm{r}\right)$ is equal to

• $\sum _{\mathrm{r} = 1}^{\mathrm{n}}{\mathrm{r}}^2$

• $\sum _{\mathrm{r} = 1}^{\mathrm{n}}{\mathrm{r}}^3$

• $\sum _{\mathrm{r} = 1}^{\mathrm{n}}\mathrm{r}$

• $\sum _{\mathrm{r} = 1}^{\mathrm{n}}{\mathrm{r}}^4$