Complex numbers practice · Easy

The modulus of $\sqrt{2{i}}-\sqrt{-2 {i}}$ is:

The value of $\frac{(\cos 2 \theta-i \sin 2 \theta)^7(\cos 3 \theta+i \sin 3 \theta)^{-5}}{(\cos 4 \theta+i \sin 4 \theta)^{12}(\cos 5 \theta+i \sin 5 \theta)^{-6}}$ is

Let $Z=\left[\begin{array}{ll}7+5 i & 3+3 i \\ 8+5 i & 5+2 i\end{array}\right]$ then $\operatorname{det}(Z \bar{Z})$, (where $\bar{Z}$ is the matrix obtained by taking conjugate of each element of $Z$ ) is equal to

If $x + iy = \frac{3}{{2 + \cos \mathrm{\theta } + i\sin \mathrm{\theta }}},\;\,\mathrm{then}\,\;{x^2} + {y^2}$ is equal to

If $x+i y=\frac{3}{2+\cos \theta+i \sin \theta}$ then $x^2+y^2=$

${i^2} + {i^4} + {i^6} + ...\mathrm{upto}\;\left( {2k + 1} \right)\mathrm{terms},\;k \in N$ is

The modulus and amplitude of ${\left( {1 + i\sqrt 3 } \right)^8}$ are respectively

If $x = {\log _b}a,\;y = {\log _c}b,\;z = {\log _a}c$, then $xyz$ is

Let ${z_1}{\text{ }}\,\,{\text{and}}\,\,{\text{ }}{z_2}$ be two complex numbers such that $\overline {{z_1}} = i\,\overline {{z_2}} {\text{ }}\,\,{\text{and }}\,\,\arg \left( {\frac{{{z_1}}}{{\overline {{z_2}} }}} \right) = \pi .$ then

Consider the following statements.

I. Additive inverse of $(1-i)$ is equal to $-1+i$.

II. If ${z}_1$ and ${z}_2$ are two complex numbers, then ${z}_1-{z}_2$ represents a complex number which is sum of $z_1$ and additive inverse of $z_2$.
III. Simplest form of $\frac{5+\sqrt{2} i}{1-\sqrt{2} i}$ is $1+2 \sqrt{2}i.$

Choose the correct option.

The inequality $a+i b > c+i d$ is true when

If $\omega$ is a complex cube root of unity, then $\sin \left\{ {\left( {{\omega ^{10}} + {\omega ^{23}}\pi - \frac{\pi }{4}} \right)} \right\}$ is equal to

The locus of point $z$ satisfying ${\mathop\mathrm{Re}\nolimits} \left( {\frac{1}{z}} \right) = k$, where $k$ is a non-zero real number, is

If $\frac{3}{{2 + \cos \theta + i\sin \theta }} = a + ib$, then $\left[ {{{\left( {a - 2} \right)}^2} + {b^2}} \right]$ is equal to

If $1,\;\omega ,{\omega ^2}$ are the cube roots of unity, then $\Delta = \left| {\begin{array}{cccccccccccccccccccc} 1&{{\omega ^n}}&{{\omega ^{2n}}}\\ {{\omega ^n}}&{{\omega ^{2n}}}&1\\ {{\omega ^{2n}}}&1&{{\omega ^n}} \end{array}} \right|$ is equal to

If $z=5 i\left(\frac{-3}{5} i\right)$, then $z$ is equal to $3+bi.$ The value of '${b}$' is

If $z=\frac{\sqrt{ } 3-i}{2}$, where $i=\sqrt{-1}$, then $\left(i^{101}+z^{101}\right)^{103}$ equals to

The curve represented by $\operatorname{Re}\left(z^2\right)=4$ is

The perpendicular distance of line $(1-i) z+(1+i) \bar{z}+3=0$ from $(3+2 i)$ will be

If $z=1+i$, then the multiplicative inverse of $z^2$ is

The number of complex numbers satisfying $\bar{z}=i z^2$ is

If $z$ is a complex number such that Re $\left( z \right) = \mathrm{Im}\;\left( z \right)$, then

If $\alpha=\cos \left(\frac{8 \pi}{11}\right)+i \sin \left(\frac{8 \pi}{11}\right)$, then $\operatorname{Re}\left(\alpha+\alpha^2+\alpha^3+\alpha^4+\alpha^5\right)$ is equal to

If $1,\;\omega ,{\omega ^2}$ are the cube roots of unity, then ${\omega ^2}{\left( {1 + \omega } \right)^3} - \left( {1 + {\omega ^2}} \right)\omega$ is equal to

The value of the expression
$\left( {2 - \omega } \right)\left( {2 - {\omega ^2}} \right) + 2.\left( {3 - \omega } \right)\left( {3 - {\omega ^2}} \right) + ... \ldots + \left( {n - 1} \right).\left( {n - \omega } \right)\left( {n - {\omega ^2}} \right)$
Where $\omega$ is an imaginary cube root of unity, is

The area of the triangle whose vertices are represented by the complex number $0,z,z{e^{i\alpha }},(0 < \mathrm{\alpha } < \pi )$ equals

If $z$ satisfies the relation
$|z-i| z 1|=| z+i|z 1|,$ (1)
then

Which of the following is correct?

If $\Delta=\left|\begin{array}{ccc}6 i & -3 i & 1 \\ 4 & 3 i & -1 \\ 20 & 3 & i\end{array}\right|=x+i y$, then

If $\omega(\neq 1)$ is a cube root of unity, then the value of $\tan \left[\left(\omega^{2017}+\omega^{2225}\right) \pi-\pi / 3\right]$