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

• $\frac{\sqrt{12}}{5}$

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

Find the values of 'a' for which the expression x² - (3a - 1)x + 2a² + 2a - 11 is always positive

The value of (1 - w + w²)⁵ + (1 + w - w²)⁵, where w and w² are the complex cube roots of unity, is

• 0

• 32w

• - 32

• 32

Let $\mathrm{\alpha }, \mathrm{\beta }$ be the roots of x² - 2x$\cos \left(\mathrm{\varphi }\right)$ + 1 = 0, then the equation whose roots are ${\mathrm{\alpha }}^{\mathrm{n}}, {\mathrm{\beta }}^{\mathrm{n}}$ is

• ${\mathrm{x}}^2 - 2\mathrm{xcos}\left(\mathrm{n\varphi }\right) - 1$ = 0

• ${\mathrm{x}}^2 - 2\mathrm{xcos}\left(\mathrm{n\varphi }\right) + 1 = 0$

• ${\mathrm{x}}^2 - 2\mathrm{xsin}\left(\mathrm{n\varphi }\right) + 1 = 0$

• ${\mathrm{x}}^2 + 2\mathrm{xsin}\left(\mathrm{n\varphi }\right) - 1 = 0$

189.

If one root of the equation x² + (1 - 3i)x - 2(1 + i) = 0 is - 1 + i, then the other root is

• - 1 - i

• $\frac{\left(- 1 - \mathrm{i}\right)}{2}$

• i

• 2i

The equation ${\mathrm{x}}^2 - 3\left|\mathrm{x}\right| + 2 = 0$ has

• no real root

• one real root

• two real root

• four real root