Let $\mathrm{\alpha } \ne 1$ be a real root of the equation x³ - ax² + ax - 1 = 0, where $\mathrm{a} \ne - 1$ is a real number. Then, a root of this equation, among the following, is

• ${\mathrm{\alpha }}^2$

• $- \frac{1}{\mathrm{\alpha }}$

• $\frac{1}{\mathrm{\alpha }}$

• $- \frac{1}{{\mathrm{\alpha }}^2}$

$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = \left|\begin{array}{ccc}2\cos \left(\mathrm{x}\right)& 1& 0\\ \mathrm{x} - \frac{\mathrm{\pi }}{2}& 2\cos \left(\mathrm{x}\right)& 1\\ 0& 1& 2\cos \left(\mathrm{x}\right)\end{array}\right|, \mathrm{then} \mathrm{f}\text{'}\left(\mathrm{\pi }\right) = ?$

• 0

• 2

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi } - 6$

If $$\begin{gathered} \mathrm{z} = 1 + \mathrm{i}\sqrt{3}, \\ \mathrm{then} \left|\mathrm{Arg} \mathrm{z}\right| + \left|\mathrm{Arg} \overline{\mathrm{z}}\right| = ? \end{gathered}$$

• 0

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{2\mathrm{\pi }}{3}$

If ω is a complex cube root of unity, (x + 1) (x + ω)(x - ω - 1) is equal to

• x³ - 1

• x³ + 1

• x³ + 2

• x³ - 2

${\left(\sqrt{3} + \mathrm{i}\right)}^7 + {\left(\sqrt{3} - \mathrm{i}\right)}^7 = ?$

• $128\sqrt{3}$

• $256\sqrt{3}$

• $- 128\sqrt{3}$

• $- 256\sqrt{3}$

If a > 0 and b² - 4ac = 0, then the curve y = ax² + bx + c

• cuts the x-axis

• touches the x-axis and lies below it

• lies entirely above the x-axis

• touches the x-axis and lies above it

If tan(A) and tan(B) are the roots of the quadratic equation x² - px + q = 0, then sin²(A + B) is equal to

• $\frac{{\mathrm{p}}^2}{{\mathrm{p}}^2 + {\mathrm{q}}^2}$

• $\frac{{\mathrm{p}}^2}{{\mathrm{p}}^2 +\hspace{0.17em}{\mathrm{q}}^2}$

• $1 - \frac{ \mathrm{p}}{{\left(1 - \mathrm{q}\right)}^2}$

• $\frac{{\mathrm{p}}^2}{{\mathrm{p}}^2 + {\left(1 - \mathrm{q}\right)}^2}$

The value of a for which the equations x³ + ax + 1 = 0 and x⁴ + ax² + 1 = 0 have acommon root is

• - 2

• - 1

• 1

• 2

$\mathrm{Let} \mathrm{z} = \mathrm{a} - \frac{\mathrm{i}}{2}; \mathrm{a} \in \mathrm{R}. \mathrm{Then} {\left|\mathrm{i} + \mathrm{z}\right|}^2 - {\left|\mathrm{i} - \mathrm{z}\right|}^2 = ?$

• 2

• - 2

• 4

• - 4

$\mathrm{The} \mathrm{locus} \mathrm{of} \mathrm{the} \mathrm{complex} \mathrm{number} \mathrm{z} \mathrm{such} \mathrm{that} \arg \left(\frac{\mathrm{z} - 2}{\mathrm{z} +\hspace{0.17em}2}\right) = \frac{\mathrm{\pi }}{3} \mathrm{is}$

• a circle 

• a straight line 

• a parabola 

• an ellipse