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 sum of the roots of the equation x² + bx + c = 0 (where b and c are non-zero) is equal to the sum of the  reciprocals of their squares. Then $\frac{1}{\mathrm{c}}, \mathrm{b}, \frac{\mathrm{c}}{\mathrm{b}}$ are in

• AP

• GP

• HP

• None of the above

The sum of the roots of the equation ax² + x + c = 0 (where a and c are non-zero) is equal to the sum of the reciprocals of their squares. Then a, ca², c² are in

• AP

• GP

• HP

• None of the above

$\mathrm{The} \mathrm{value} \mathrm{of} \left[\mathrm{C}\left(7, 0\right) + \mathrm{C}\left(7, 1\right)\right] + \left[\mathrm{C}\left(7, 1\right) + \mathrm{C}\left(7, 2\right)\right] + ... + \left[\mathrm{C}\left(7, 6\right) + \mathrm{C}\left(7, 7\right)\right] \mathrm{is}$

• 254

• 255

• 256

• 257

19.

The number of different words (eight-letter words) ending and beginning with a consonant which can bemade out of the letters of the word 'EQUATlON' is

• 5200

• 4320

• 3000

• 2160

The fifth term of an AP of 11 terms, whose sum is n² - 2n, is

• 5

• 7

• 8

• 15