If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are roots of ax + bx + c = 0, then the equation whose roots are ${\mathrm{\alpha }}^2 \mathrm{and} {\mathrm{\beta }}^2$ is

• a²x² - (b² - 2ac)x + c² = 0

• a²x² + (b² - ac)x + c² = 0

• a²x² + (b² + ac)x + c² = 0

• a²x² + (b² + 2ac)x + c² = 0

If the equation x² + y² - 10x + 21 = 0 has real roots $\mathrm{x} = \mathrm{\alpha } \mathrm{and} \mathrm{y} = \mathrm{\beta }$, then

• $3 \le \mathrm{x} \le 7$

• $3 \le \mathrm{y} \le 7$

• $- 2 \le \mathrm{x} \le 2$

• $- 2 \le \mathrm{x} \le 2$

If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta } \mathrm{are} \mathrm{the} \mathrm{roots} \mathrm{of} {\mathrm{x}}^2 - \mathrm{px} +\hspace{0.17em}1 = 0 \mathrm{and} \mathrm{\gamma } \mathrm{is} \mathrm{aroot} \mathrm{of} {\mathrm{x}}^2 + \mathrm{px} +\hspace{0.17em}1 = 0, \mathrm{then} \left(\mathrm{\alpha } + \mathrm{\gamma }\right)\left(\mathrm{\beta } + \mathrm{\gamma }\right)$ is

• 0

• 1

• - 1

• $\mathrm{\rho }$

The quadratic expression ${\left(2\mathrm{x} +1\right)}^2 - \mathrm{px} + \mathrm{q} \ne 0$ for any real x, if

• p² - 16p - 8q < 0

• p² - 8p + 16q < 0

• p² - 8p - 16q < 0

• p² - 16p + 8q < 0

Let f: $\mathrm{R} \to \mathrm{R} \mathrm{be} \mathrm{defmed} \mathrm{as} \mathrm{f}\left(\mathrm{x}\right) = \frac{{\mathrm{x}}^2 - \mathrm{x} + 4}{{\mathrm{x}}^2 + \mathrm{x} + 4}$ Then, range of the function f(x) is

• $\left[\frac{3}{5}, \frac{5}{3}\right]$

• $\left(\frac{3}{5}, \frac{5}{3}\right)$

• $\left(- \infty , \frac{3}{5}\right) \cup \left(\frac{5}{3}, \infty \right)$

• $\left[- \frac{5}{3}, - \frac{3}{5}\right]$

The least value of 2x² + y² + 2xy + 2x -3y + 8 for real numbers x and y, is

• 2

• 8

• 3

• - 1/2

Find the maximum value of $\left|z\right|$ when $\left|\mathrm{z} - \frac{3}{\mathrm{z}}\right| = 2$, where z being a complex number.

• 1 + $\sqrt{3}$

• 3

• $1 + \sqrt{2}$

• 1

138.

Given that, x is a real number satisfying $\frac{5{\mathrm{x}}^2 - 26\mathrm{x} + 5}{3{\mathrm{x}}^2 - 10\mathrm{x} + 3} < 0$, then

• $\mathrm{x} < \frac{1}{5}$

• $\frac{1}{5} < \mathrm{x} < 3$

• x > 5

• $\frac{1}{5} < \mathrm{x} < \frac{1}{3} \mathrm{or} 3 < \mathrm{x} < 5$

If (2 + i) and $\left(\sqrt{5} - 2\mathrm{i}\right)$ are the roots of the equation (x² + ax + b )(x² + ex + d) = 0, where a, b, c and d are real constants, then product of all the roots of the equation is

• 40

• 9$\sqrt{5}$

• 45

• 35

Which of the following is /are always false?

• A quadratic equation with rational coefficients has zero or two irrational roots

• A quadratic equation with real coefficients has zero or two non-real roots

• A quadratic equation with irrational coefficients has zero or two irrational roots

• A quadratic equation with integer coefficients has zero or two irrational roots