If $\mathrm{\alpha }$ is an nth root of unity, then $1 + 2\mathrm{\alpha } + 3{\mathrm{\alpha }}^2 + ... + {\mathrm{n\alpha }}^{\mathrm{n} - 1}$equals

• $- \frac{\mathrm{n}}{\left(1 - \mathrm{\alpha }\right)}$

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

• $\frac{\mathrm{n}}{\left(1 - \mathrm{\alpha }\right)}$

• None of these

If $\left|\mathrm{z}\right| \ge 3$, then the least value of $\left|\mathrm{z} + \frac{1}{4}\right|$

• $\frac{11}{2}$

• $\frac{11}{4}$

• 3

• $\frac{1}{4}$

If the complex number z lies on a circle with centre at the origin and radius = $\frac{1}{4}$, then the 4 complex number - 1 + 8z lies on a circle with radius

• 4

• 1

• 3

• 2

If x² + x + 1 = 0, then the value of $\sum _{\mathrm{n} = 1}^6{\left({\mathrm{x}}^{\mathrm{n}} + \frac{1}{{\mathrm{x}}^{\mathrm{n}}}\right)}^2$ is

• 13

• 12

• 9

• 14

If x + iy = $\frac{3}{2 + \cos \left(\mathrm{\theta }\right) + \mathrm{isin}\left(\mathrm{\theta }\right)}$,  then x² + y² is equal to

• 3x - 4

• 4x - 3

• 4x + 3

• None of these

If $\left|\mathrm{z} + 4\right| \ge 3$ then the greatest and the least value of $\left|\mathrm{z} + 1\right|$ are

• - 1, 6

• 6, 0

• 6, 3

• None of these

If the area of the triangle on the complex plane formed by the points z, z + iz and iz is 200, then the value of 3lz I must be equal to

• 20

• 40

• 60

• 80

A cmplex number z is such that arg$\left(\frac{\mathrm{z} - 2}{\mathrm{z} + 2}\right) = \frac{\mathrm{z}}{3}$. The points representing this complex number will lie on

• an ellipse

• a parabola

• a circle

• a straight line

209.

If $\left|{\mathrm{x}}^2 - \mathrm{x} - 6\right| = \mathrm{x} + 2$, then the values of x are

• - 2, 2, - 4

• - 2, 2, 4

• 3, 2, - 2

• 4, 4, 3

If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are the roots of x² - ax + b = 0 and if ${\mathrm{\alpha }}^{\mathrm{n}} \mathrm{and} {\mathrm{\beta }}^{\mathrm{n}}$ = Vⁿ, then

• V_{n + 1} = aV_n + bV_{n - 1}

• V_{n + 1} = aV_n + aV_{n - 1}

• V_{n + 1} = aV_n - bV_{n - 1}

• V_{n + 1} = aV_{n - 1} + bV_n