The complex numbers sin(x) + i cos(2x) and cos(x) - i sin(2x) are conjugate to each other for

• x = $\mathrm{n\pi }$

• $\mathrm{x} = \left(\mathrm{n} + \frac{1}{2}\right)\mathrm{\pi }$

• x = 0

• No value of x

The locus of the points z which satisfy the condition $\arg \left(\frac{\mathrm{z} - 1}{\mathrm{z} + 1}\right) = \frac{\mathrm{\pi }}{3}$

• a straight line

• a circle

• a parabola

• None of the above

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

129.

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