If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$  B are the roots of the quadratic equation is, x² + ax + b = 0, (b $\ne$ 0), then the quadratic equation whose roots are $\mathrm{\alpha } - \frac{1}{\mathrm{\beta }}, \mathrm{\beta } - \frac{1}{\mathrm{\alpha }}$, is

• ax² + a(b - 1)x + (a - 1)² = 0

• bx² + a(b - 1)x + (b - 1)² = 0

• x² + ax + b = 0

• abx² + bx + a = 0

If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are the roots of the quadratic equation ax² + bx + c = 0 and 3b² = 16ac, then

• $\mathrm{\alpha } = 4\mathrm{\beta } \mathrm{or} \mathrm{\beta } = 4\mathrm{\alpha }$

• $\mathrm{\alpha } = - 4\mathrm{\beta } \mathrm{or} \mathrm{\beta } = - 4\mathrm{\alpha }$

• $\mathrm{\alpha } = 3\mathrm{\beta } \mathrm{or} \mathrm{\beta } = 3\mathrm{\alpha }$

• $\mathrm{\alpha } = - 3\mathrm{\beta } \mathrm{or} \mathrm{\beta } = - 3\mathrm{\alpha }$

The number of solutions of the equation

$\frac{1}{2}{\log }_{\sqrt{3}}\left(\frac{\mathrm{x} + 1}{\mathrm{x} +\hspace{0.17em}5}\right) + {\log }_{\mathrm{e}}{\left(\mathrm{x} + 5\right)}^2$ = 1

• 0

• 1

• 2

• infinite

If $\sin \left(\mathrm{\alpha }\right), \cos \left(\mathrm{\alpha }\right)$  be the roots of the equation x² - bx + c = 0, Then, which of the following statements is/are correct?

• $\mathrm{c} \le \frac{1}{2}$

• $\mathrm{b} \le \sqrt{2}$

• $\mathrm{c} > \frac{1}{2}$

• b $> \sqrt{2}$

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

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

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

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

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

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

• 0

• 1

• 2

• 3

77.

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

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

• $\left|\mathrm{z}\right| = 1$

• $\left|\mathrm{z}\right| = 2$

• $\left|\mathrm{z}\right| = 3$