If $\mathrm{\alpha }, \mathrm{\beta }$ be the roots of the quadratic equation x² + x + 1 = 0, then the equation whose roots are ${\mathrm{\alpha }}^{19}. {\mathrm{\beta }}^7$ f is

• x² - x + 1 = 0

• x² - x - 1 = 0

• x² + x - 1 = 0

• x² + x + 1 = 0

The roots of the quadratic equation ${\mathrm{x}}^2 - 2\sqrt{3}\mathrm{x} - 22 = 0$ are

• imaginary

• real, rational and equal

• real, irrational and unequal

• real, rational and unequal

The quadratic equation x² + 15$\left|\mathrm{x}\right|$ + 14 = 0 has

• only positive solutions

• only negative solutions

• no solution

• both positive and negative solution

If $\mathrm{z} = \frac{4}{1 - \mathrm{i}}$, then $\overline{\mathrm{z}}$ is (where $\overline{z}$ is complex conjugate of z)

• 2(1 + i)

• (1 + i)

• $\frac{2}{1 - \mathrm{i}}$

• $\frac{4}{1 + \mathrm{i}}$

For any complex number z, the minimum value of $\left|\mathrm{z}\right| + \left|\mathrm{z} - 1\right|$ is

• 0

• 1

• 2

• - 1

If a, b, c are real, then both the roots of the equation (x - b)(x - c) + (x - c)(x - a) + (x - a)(x - b) = 0

• positive

• negative

• real

• imaginary

The modulus of $\frac{1 - \mathrm{i}}{3 + \mathrm{i}} + \frac{4\mathrm{i}}{5}$ is

• $\sqrt{5}$ unit

• $\frac{\sqrt{11}}{5}$ unit

• $\frac{\sqrt{5}}{5}$ unit

• $\frac{\sqrt{12}}{5}$

98.

If $\mathrm{\alpha }, \mathrm{\beta }$ be the roots of x² - a(x - 1) + b = 0, then the value of $\frac{1}{{\mathrm{\alpha }}^2 - \mathrm{a\alpha }} + \frac{{\displaystyle 1}}{{\displaystyle {\mathrm{\beta }}^2 - \mathrm{a\beta }}} + \frac{2}{\mathrm{a} +\hspace{0.17em}\mathrm{b}}$ is

• $\frac{4}{\mathrm{a} + \mathrm{b}}$

• $\frac{1}{\mathrm{a} + \mathrm{b}}$

• 0

• - 1

The sum of all real roots of the equation ${\left|\mathrm{x} - 2\right|}^2 + \left|\mathrm{x} - 2\right| - 2 = 0$ is

• 7

• 4

• 1

• 5

The quadratic equation whose roots are three times the roots of 3ax² + 3bx + c = 0 is

• ax² + 3bx + 3c = 0

• ax² + 3bx + c

• 9ax² + 9bx + c

• ax² + bx + 3c