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Complex Numbers and Quadratic Equations

Multiple Choice Questions

201.

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

$- \frac{n}{(1 - α)}$
$- \frac{n}{{(1 + α)}^{2}}$
$\frac{n}{(1 - α)}$
None of these

202.

If $|z| \geq 3$ , then the least value of $|z + \frac{1}{4}|$

$\frac{11}{2}$
$\frac{11}{4}$
3
$\frac{1}{4}$

203.

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

204.

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

205.

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

3x - 4
4x - 3
4x + 3
None of these

206.

If $|z + 4| \geq 3$ then the greatest and the least value of $|z + 1|$ are

- 1, 6
6, 0
6, 3
None of these

207.
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

Let z = x + iy, then

z + iz = x + iy + i(x + iy) = (x - y) + i(x + y)

and iz = i(x + iy) = - y + ix,

Then, the area of the triangle formed by these lines is

$∆ = \frac{1}{2} |\begin{matrix} x & y & 1 \\ (x - y) & (x + y) & 1 \\ - y & x & 1 \end{matrix}| Applying R_{2} > R_{2} - (R_{1} + R_{3}), ∆ = \frac{1}{2} |\begin{matrix} x & y & 1 \\ 0 & 0 & - 1 \\ - y & x & 1 \end{matrix}| = \frac{1}{2} (x^{2} + y^{2}) \Rightarrow \frac{1}{2} {|z|}^{2} = 200 [∵ as given] {|z|}^{2} = 400 \Rightarrow |z| = 20 ∴ 3 |z| = 3 \times 20 = 60$