If ${\log }_{\mathrm{e}}\left({\mathrm{x}}^2 - 16\right) \le {\log }_{\mathrm{e}}\left(4\mathrm{x} - 11\right)$, then

• $4 < \mathrm{x} \le 5$

• x < - 4 or x > 4

• $- 1 < \mathrm{x} \le 5$

• x < - 1 or x > 5

Determine the sum of imaginary roots of the equation (2x + x - 1) ( 4x² + 2x - 3) = 6

93.

Let a, b, c be three real numbers, such that a + 2b + 4c = 0, Then, the equation ax² + bx + c = 0

• has both the roots complex

• has its roots lying within - 1 < x < 0

• has one of roots equal to $\frac{1}{2}$

• has its roots lying within 2 < x < 6

If the ratio of the roots of the equation px² + qx + r = 0 is a : b, then $\frac{\mathrm{ab}}{{\left(\mathrm{a} + \mathrm{b}\right)}^2}$

• $\frac{{\mathrm{p}}^2}{\mathrm{qr}}$

• $\frac{\mathrm{pr}}{{\mathrm{q}}^2}$

• $\frac{{\mathrm{q}}^2}{\mathrm{pr}}$

• $\frac{\mathrm{pq}}{{\mathrm{r}}^2}$

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

• x² - x - 1 = 0

• x² - x + 1 = 0

• x² + x - 1 = 0

• x² + x + 1 = 0

For the real parameter t, the locus of the complex number $\mathrm{z} = \left(1 - {\mathrm{t}}^2\right) + \mathrm{i}\sqrt{1 + {\mathrm{t}}^2}$ in the complex plane is

• an ellipse

• a parabola

• a circle

• a hyperbola

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)

• $\frac{2}{1 - \mathrm{i}}$

• $\frac{4}{1 + \mathrm{i}}$