Complex numbers practice · Easy

$(1+i)^6+(1-i)^6=$

Given, $z=f(x)+i g(x)$, where $i=\sqrt{-1}$ and $f, g:(0,1) \rightarrow(0,1)$ are real-valued functions, which of the following hold good?

$\left(\frac{1+i}{1-i}\right)^2+\left(\frac{1-i}{1+i}\right)^2$ is equal to

If $\left| {\frac{{z - 25}}{{z - 1}}} \right| = 5$, find the value of $\left| z \right|$

If $\frac{{5{z^2}}}{{11{z_1}}}$ is purely imaginary, then the value of $\left| {\frac{{2{z_1} + 3{z_2}}}{{2{z_1} - 3{z_2}}}} \right|$ is

Consider the following statements.

I. If ${z}, {z}_1, {z}_2$ be three complex numbers then $z \bar{z}=|{z}|^2$

II. The modulus of a complex number $z=a+i b$ is defined as $|z|=\sqrt{a^2+b^2}$.

III. Multiplicative inverse of $z=3-2 {i}$ is $\frac{3}{13}+\frac{2}{13} {i}$

Choose the correct option.

If ${\alpha _1},{\alpha _2},\;{\alpha _3}$ respectively denote the moduli of the complex numbers $- i,\frac{1}{3}\left( {1 + i} \right)$ and $- 1 + i$, then their increasing order is

The region of the Argand diagram defined by $\left| {z - 1} \right| + \left| {z + 1} \right| \le 4$ is

If $x_r=\cos \frac{\pi}{2^r}+i \sin \frac{\pi}{2^r}$, then the value of $x_1 \times x_2 \times x_3 \times \ldots \times \infty$ is

If $x^2+y^2=1$ and $x \neq-1$ then $\frac{1+y+i x}{1+y-i x}$ equals

Suppose $z_1, z_2, z_3$ are vertices of an equilateral triangle whose circumcentre $-3+4 i$, then $z_1+z_2+$ $z_3$ is equal to

If ${\mathop\mathrm{Im}\nolimits} \left( {\frac{{z - 1}}{{2z + 1}}} \right) = - 4$, then locus of $z$ is

Let $z=a\left(\cos \frac{\pi}{5}+i \sin \frac{\pi}{5}\right), a \in \mathbf{R},|a|<1$, then $S=z^{2015}+z^{2016}+z^{2017}+\ldots$ equals

If $\frac{{2{z_1}}}{{3{z_2}}}$ is purely imaginary, then $\left| {\frac{{{z_1} - {z_2}}}{{{z_1} + {z_2}}}} \right|$ is

The number of solutions of the system of equations $\mathrm{Re\;}\left( {{z^2}} \right) = 0;\left| \mathrm{z} \right| = 2$ is

If $z=i^{-39}$, then simplest form of $z$ is equal to $a+i$. The value of &#39;$a$&#39; is

The continued product of the four values of $\left[\cos \left(\frac{\pi}{3}\right)+i \sin \left(\frac{\pi}{3}\right)\right]^{3 / 4}$ is

If ${z_r} = \cos \frac{{r\alpha }}{{{n^2}}} + i\sin \frac{{r\alpha }}{{{n^2}}},\mathrm{\;where}\;r = 1,\;2,\;3 \ldots ,\;n,\;\;\mathrm{then}\;\lim\limits_{n \to \infty } {z_1}\;{z_2}\;{z_3} \ldots {z_n}\;$ is equal to

The solution set of ${x^2} + 2 \le 3x \le 2{x^2} - 5,$ is

If ${\left( {\sqrt 5 + \sqrt 3 i} \right)^{33}} = {2^{49}}\;z$, then modulus of the complex number $z$ is equal to

The complex number $\mathrm{z}=1+\mathrm{i}$ is rotated through an angle $\frac{3 \pi}{2}$ in anticlockwise direction about the origin and stretched by additional $\sqrt{2}$ unit, then the new complex number is:

If $z=5+t+i \sqrt{25-t^2}$, $(-5 \leq t \leq 5)$, then locus of $z$ is a curve which passes through

The point (4, 1) undergoes the following three transformations successively (i) Refletion about the line $y = x$
(ii) Translation through a distance of 2 unit along the positive direction of $x$-axis
(iii) Rotation through an angle of $\frac{\pi }{4}$ about the origin in the anti-clockwise direction

The final position of the point is

The value of $\frac{{\left[ {(\cos 20^\circ + i\sin 20^\circ )\left( {\cos 75^\circ + i\sin 75^\circ } \right)\left( {\cos 10^\circ + i\sin 10^\circ } \right)} \right]}}{{\sin 15^\circ - i\cos 15^\circ }}$ is

If $\cos \mathrm{\alpha } + \cos \mathrm{\beta } + \cos \mathrm{\gamma } = \sin \mathrm{\alpha } + \sin \mathrm{\beta } + \sin \mathrm{\gamma } = 0,\mathrm{\;then}\cos 3\mathrm{\alpha } + \cos 3\mathrm{\beta } + \cos 3\mathrm{\gamma \;}$ equals

If ${z_1}$ and ${z_2}$ are two complex numbers such that $\left| {\frac{{{z_1} - {z_2}}}{{{z_1} + {z_2}}}} \right| = 1,$ then

The complex number $z=1+i$ is rotated through an angle $3 \pi / 2$ in anti-clockwise direction about the origin and stretched by additional $\sqrt{2}$ units, then the new complex number is

Which term of the sequence

$\left( { - 8 + 18i} \right),\;\left( { - 6 + 15i} \right),\left( { - 4 + 12i} \right),\;$ … is purely imaginary?

If $\left|z^2-1\right|=|z|^2+1$, then ${P}(z)$ lies on

The smallest positive integer $n$ for which ${\left( {1 + i} \right)^{2n}} = {\left( {1 - i} \right)^{2n}}$ is