If $\left(\mathrm{\alpha } + \sqrt{\mathrm{\beta }}\right) \mathrm{and} \left(\mathrm{\alpha } - \sqrt{\mathrm{\beta }}\right)$ are the roots of the equation x² + px + q = 0, where $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{p} \mathrm{and} \mathrm{q}$ are real, then the roots of the equation (p² - 4q)(p²x² + 4px) - 16q = 0 are

• $\left(\frac{1}{\mathrm{\alpha }} + \frac{1}{\sqrt{\mathrm{\beta }}}\right) \mathrm{and} \left(\frac{1}{\mathrm{\alpha }} - \frac{1}{\sqrt{\mathrm{\beta }}}\right)$

• $\left(\frac{1}{\sqrt{\mathrm{\alpha }}} + \frac{1}{\mathrm{\beta }}\right) \mathrm{and} \left(\frac{1}{\sqrt{\mathrm{\alpha }}} - \frac{1}{\mathrm{\beta }}\right)$

• $\left(\frac{1}{\sqrt{\mathrm{\alpha }}} + \frac{1}{\sqrt{\mathrm{\beta }}}\right) \mathrm{and} \left(\frac{1}{\sqrt{\mathrm{\alpha }}} - \frac{1}{\sqrt{\mathrm{\beta }}}\right)$

• $\left(\sqrt{\mathrm{\alpha }} + \sqrt{\mathrm{\beta }}\right) \mathrm{and} \left(\sqrt{\mathrm{\alpha }} - \sqrt{\mathrm{\beta }}\right)$

82.

The number of solutions of the equation ${\log }_2\left({\mathrm{x}}^2 + 2\mathrm{x} - 1\right) = 1$ is

• 0

• 1

• 2

• 3

Let R be the set of real numbers and the functions f : R ➔ R and g : R ➔ R be defined by f(x) = x² + 2x - 3 and g(x) = x + 1. Then, the value of x for which f(g(x)) = g(f(x)) is

• - 1

• 0

• 1

• 2

The maximum value of $\left|\mathrm{z}\right|$, when the complex number z satisfies the condition $\left|\mathrm{z} + \frac{2}{\mathrm{z}}\right|$ = 2 is

• $\sqrt{3}$

• $\sqrt{3} + \sqrt{2}$

• $\sqrt{3} + 1$

• $\sqrt{3} - 1$

If ${\left(\frac{3}{2} + \mathrm{i}\frac{\sqrt{3}}{2}\right)}^{50} = 3^{25}\left(\mathrm{x} +\hspace{0.17em}\mathrm{iy}\right)$, where x and y are real, then the ordered pair (x, y) is

• $\left(- 3, 0\right)$

• $\left(0, 3\right)$

• $\left(0, - 3\right)$

• $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$

If $\frac{\mathrm{z} - 1}{\mathrm{z} + 1}$ is pure imaginary, then

• $\left|\mathrm{z}\right| = \frac{1}{2}$

• $\left|\mathrm{z}\right| = 1$

• $\left|\mathrm{z}\right| = 2$

• $\left|\mathrm{z}\right| = 3$

Let f(x) = ax² + bx + c, g(x) = px² + qx + r such that f(1) = g(1), f(2) = g(2) and f(3) - g(3) = 2. Then, f(4) - g(4) is

• 4

• 5

• 6

• 7

The equations x² + x + a= 0 and x² + ax + 1 = 0 have a common real root

• for no value of a

• for exactly one value of a

• for exactly two value of a

• for exactly three value of a

The points representing the complex number z for which arg $\left(\frac{\mathrm{z} - 2}{\mathrm{z} + 2}\right) = \frac{\mathrm{\pi }}{3}$ lie on

• a circle

• a straight line

• an ellipse

• a parabola

The quadratic equation 2x² - (a³ + 8a - 1)x + a² - 4a = 0 posses roots of opposite sign. Then,

• a $\le$ 0

• 0 < a < 4

• $4 \le \mathrm{a} < 8$

• $\mathrm{a} \ge 8$