81.

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

• $4 \le \mathrm{a} < 8$

• $\mathrm{a} \ge 8$

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

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