$\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma }$ are the roots of the equation x³ - 10x² + 7x + 8=0. Match the following and choose the correct answer.

$$\begin{gathered} A. \mathrm{\alpha } + \mathrm{\beta } + \mathrm{\gamma } \left(1\right) - \frac{43}{7} \\ \mathrm{B}. {\mathrm{\alpha }}^2 + {\mathrm{\beta }}^2 + {\mathrm{\gamma }}^2 \left(2\right) - \frac{7}{8} \\ \mathrm{C}. \frac{1}{\mathrm{\alpha }} + \frac{1}{\mathrm{\beta }} + \frac{1}{\mathrm{\gamma }} \left(3\right) 86 \\ \mathrm{D}. \frac{\mathrm{\alpha }}{\mathrm{\beta \gamma }} + \frac{\mathrm{\beta }}{\mathrm{\gamma \alpha }} + \frac{\mathrm{\gamma }}{\mathrm{\alpha \beta }} \left(4\right) 0 \\ \left(10\right) \end{gathered}$$

If f(x) is apolynomial ofdegreen with rational coefficients and 1 + 2i , $2 - \sqrt{3}$ and 5 are three roots of f(x) = 0 , then the least value of n is

• 5

• 4

• 3

• 6

Match the following elements of $\left[\begin{array}{ccc}1& - 1& 0\\ 0& 4& 2\\ 3& - 4& 6\end{array}\right]$ with their co-factors and choose the correct answer

$$\begin{gathered} \mathrm{Element} \mathrm{Co}-\mathrm{factor} \\ \mathrm{A} - 1 \left(1\right) - 2 \\ \mathrm{B} 1 \left(2\right) 32 \\ \mathrm{C} 3 \left(3\right) 4 \\ \mathrm{D} 6 \left(4\right) 6 \\ \left(5\right) - 6 \end{gathered}$$

If z₁, z₂ are two complex numbers satisfying $\left|\frac{{\mathrm{z}}_1 - 3{\mathrm{z}}_2}{3 - {\mathrm{z}}_1\overline{{\mathrm{z}}_2}}\right| = 1, \left|{\mathrm{z}}_1\right| \ne 3$, then $\left|{\mathrm{z}}_2\right|$ is equal to

• 1

• 2

• 3

• 4

The value of $\sum _{\mathrm{n} = 0}^{\infty }{\left(\frac{2\mathrm{i}}{3}\right)}^{\mathrm{n}}$ is

• $\frac{9 + 6\mathrm{i}}{13}$

• $\frac{9 - 6\mathrm{i}}{13}$

• 9 + 6i

• 9 - 6i

The roots of the equation x - 3x - 2 = 0 are

• - 1, - 1, 2

• - 1, 1, - 2

• - 1, 2, - 3

• - 1, - 1, - 2

$\mathrm{If} \mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma } \mathrm{are} \mathrm{the} \mathrm{roots} \mathrm{of} {\mathrm{x}}^3 + 2{\mathrm{x}}^2 - 3\mathrm{x} - 1 = 0 \mathrm{then} {\mathrm{\alpha }}^{- 2} + {\mathrm{\beta }}^{- 2} + {\mathrm{\gamma }}^{- 2} =$

• 12

• 13

• 14

• 15

$\mathrm{If} \mathrm{\alpha } \mathrm{is} \mathrm{a} \mathrm{non}-\mathrm{real} \mathrm{root} \mathrm{of} {\mathrm{x}}^6 = 1, \mathrm{then} \frac{{\mathrm{\alpha }}^5 + {\mathrm{\alpha }}^{3 } + \mathrm{\alpha } + 1}{{\mathrm{\alpha }}^2 + 1} \mathrm{is} \mathrm{equal} \mathrm{to},$

• ${\mathrm{\alpha }}^2$

• 0

• $- {\mathrm{\alpha }}^2$

• $\mathrm{\alpha }$

199.

The difference between two roots of the equation x³ - 13x² + 15x + 189 = 0 is 2. Then  the roots of the equation are :

• - 3, 5, 7

• - 3, - 7, - 9

• 3, - 5, 7

• - 3, - 7, 9

$$\begin{gathered} \mathrm{If} \mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma } \mathrm{are} \mathrm{the} \mathrm{roots} \mathrm{of} \mathrm{the} \mathrm{equation} {\mathrm{x}}^3 - 6{\mathrm{x}}^2 + 11\mathrm{x} + 6 = 0, \\ \mathrm{then} \sum _{}{\mathrm{\alpha }}^2\mathrm{\beta } + \sum _{}{\mathrm{\alpha \beta }}^2 \mathrm{is} \mathrm{equal} \mathrm{to}\hspace{0.17em}: \end{gathered}$$

• 80

• 84

• 90

• - 84