If 3 is a root of x² + kx - 24 = 0. It is also a root of

• ${\mathrm{x}}^2 + 5\mathrm{x} + \mathrm{k} = 0$

• ${\mathrm{x}}^2 + \mathrm{kx} + 24 = 0$

• ${\mathrm{x}}^2 - \mathrm{kx} + 6 = 0$

• ${\mathrm{x}}^2 - 5\mathrm{x} + \mathrm{k} = 0$

To remove the second term of the equation x⁴ - 8x³ + x² - x + 3 = 0, diminish the roots of the equation by

• 1

• 2

• 3

• 4

The maximum possible number of real roots of the equation x⁵ - 6x² - 4x + 5 = 0 is

• 0

• 3

• 4

• 5

$$\begin{gathered} \mathrm{If} \mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma } \mathrm{are} \mathrm{the} \mathrm{roots} \mathrm{of} \mathrm{the} \mathrm{equation} {\mathrm{x}}^3 + {\mathrm{ax}}^2 + \mathrm{bx} + \mathrm{c} = 0, \\ \mathrm{then} \mathrm{\alpha }{ }^{- 1}{\mathrm{\beta }}^{ -1}\mathrm{\gamma }{ }^{-1} \mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• $\frac{\mathrm{a}}{\mathrm{c}}$

• $\frac{\mathrm{c}}{\mathrm{a}}$

• $- \frac{\mathrm{b}}{\mathrm{c}}$

• None of these

$$\begin{gathered} \mathrm{If} \frac{1 + \sqrt{3}\mathrm{i}}{2} \mathrm{is} \mathrm{a} \mathrm{root} \mathrm{of} \mathrm{the} \mathrm{equation} {\mathrm{x}}^4 - {\mathrm{x}}^2 + \mathrm{x} - 1 = 0. \\ \mathrm{Then}, \mathrm{its} \mathrm{real} \mathrm{roots} \mathrm{are} \end{gathered}$$

• 1, 1

• - 1, - 1

• 1, 2

• 1, - 1

$\mathrm{If} \mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma } \mathrm{are} \mathrm{the} \mathrm{roots} \mathrm{of} 2{\mathrm{x}}^3 - 2\mathrm{x} - 1 = 0, \mathrm{then} {\left(\underset{}{\sum \mathrm{\alpha \beta }}\right)}^2 \mathrm{is} \mathrm{equal} \mathrm{to}$

• - 1

• 1

• 2

• 3

$\mathrm{If} \mathrm{z}= \mathrm{x} +\mathrm{iy} \mathrm{is} \mathrm{a} \mathrm{complex} \mathrm{number} \mathrm{satisfying} {\left|\mathrm{z} + \frac{\mathrm{i}}{2}\right|}^2 = {\left|\mathrm{z} - \frac{\mathrm{i}}{2}\right|}^2, \mathrm{then} \mathrm{the} \mathrm{locus} \mathrm{of} \mathrm{z} \mathrm{is}$

• x-axis

• y-axis

• y = x

• 2y = x

If 1 - i is a root of the equation x² + 9x + b = 0, then b is equal to

• 1

• - 1

• - 2

• 2

If $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma }$ are the roots of the equation x³ + 4x + 1 = 0, then ${\left(\mathrm{\alpha } + \mathrm{\beta }\right)}^{-1} + {\left(\mathrm{\beta } + \mathrm{\gamma }\right)}^{-1} + {\left(\mathrm{\gamma } + \mathrm{\alpha }\right)}^{-1}$ is equal to

• 2

• 3

• 4

• 5

Let a $\ne$ 0 and p(x) be a polynomial of degree greater than 2. If p(x) leaves remainders a and - a when divided respectively by x + a and x - a, then the remainder when p(x) is divided by x² - a² is:

• x

• - x

• - 2x

• 2x