If the difference between the roots of the equation x² + kx + 1 = 0 is strictly less than $\sqrt{5}, \mathrm{where} \left| \mathrm{k}\right| \ge 2$, then k can be any element of the interval

• $( - 3, - 2] \cup [2, 3)$

• ( - 3, 3)

• $\left[ - 3, - 2\right] \cup \left[2, 3\right]$

• None of the above

If the roots of the equation x² + px + q = 0 are in the same ratio as those of the equation x² + lx + m = 0, then which one of the following is correct ?

• p²m = l²q

• m²p = l²q

• m²p = q²l

• m²p² = l²q

23.

$\mathrm{The} \mathrm{value} \mathrm{of} {\left(\frac{ - 1 + \mathrm{i}\sqrt{3}}{2}\right)}^{\mathrm{n}} + {\left(\frac{ - 1 - \mathrm{i}\sqrt{3}}{2}\right)}^{\mathrm{n}} \mathrm{where} \mathrm{n} \mathrm{is} \mathrm{not} \mathrm{a} \mathrm{multiple} \mathrm{of} 3 \mathrm{and} \mathrm{i} = \sqrt{ - 1}, \mathrm{is}$

• 1

•  - 1

• i

•  - i

$\mathrm{If} 1, \mathrm{\omega }, {\mathrm{\omega }}^2 \mathrm{are} \mathrm{the} \mathrm{cube} \mathrm{roots} \mathrm{of} \mathrm{unity}, \mathrm{then} (1 + \mathrm{\omega })(1 + {\mathrm{\omega }}^2)(1 + {\mathrm{\omega }}^3)(1 + \mathrm{\omega } + {\mathrm{\omega }}^2) \mathrm{is} \mathrm{equal} \mathrm{to}$

•  - 2

•  - 1

• 0

• 2

If the graph of a quadratic polynomial lies entirely above x-axis, then which one of the following is correct ?

• Both the roots are real

• One root is real and the other is complex

• Both the roots are complex

• Cannot say

The modulus and principal argument of the complex number $\frac{1 + 2\mathrm{i}}{1 - {\left(1 - \mathrm{i}\right)}^2}$ are respectively

• 1, 0

• 1, 1

• 2, 0

• 2, 1

$\mathrm{If} \left|\mathrm{z} + 4\right| \le 3, \mathrm{then} \mathrm{maximum} \mathrm{value} \mathrm{of} \left|\mathrm{z} +1\right| \mathrm{is}$

• 0

• 4

• 6

• 10

The number of roots of the equation ${\mathrm{z}}^2 = 2\overline{\mathrm{z}}$ is

• 2

• 3

• 4

• zero

$$\begin{gathered} \mathrm{If} \cot \left(\mathrm{\alpha }\right) \mathrm{and} \cot \left(\mathrm{\beta }\right) \mathrm{are} \mathrm{the} \mathrm{roots} \mathrm{of} \mathrm{the} \mathrm{equation} {\mathrm{x}}^2 + \mathrm{bx} + \mathrm{c} = 0 \mathrm{with} \mathrm{b} \ne 0, \\ \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \cot (\mathrm{\alpha } + \mathrm{\beta }) \mathrm{is} \end{gathered}$$

• $\frac{\mathrm{c} - 1}{\mathrm{b}}$

• $\frac{1 - \mathrm{c}}{\mathrm{b}}$

• $\frac{\mathrm{b}}{\mathrm{c} - 1}$

• $\frac{\mathrm{b}}{1 - \mathrm{c}}$

The roots of the equation :

$\left(\mathrm{q} - \mathrm{r}\right){\mathrm{x}}^2 + \left(\mathrm{r} - \mathrm{p}\right)\mathrm{x} + \left(\mathrm{p} - \mathrm{q}\right) = 0 \mathrm{are}$

• $\frac{\left(\mathrm{r} - \mathrm{p}\right)}{\left(\mathrm{q} - \mathrm{r}\right)}, \frac{1}{2}$

• $\frac{\left(\mathrm{p} - \mathrm{q}\right)}{\left(\mathrm{q} - \mathrm{r}\right)}, 1$

• $\frac{\left(\mathrm{q} - \mathrm{r}\right)}{\left(\mathrm{p} - \mathrm{q}\right)}, 1$

• $\frac{\left(\mathrm{r} - \mathrm{p}\right)}{\left(\mathrm{p} - \mathrm{q}\right)}, \frac{1}{2}$