Advertisements
Advertisements
Question
Find the multiplicative inverse of the following complex number:
\[(1 + i\sqrt{3} )^2\]
Solution
\[ z = \left( 1 + \sqrt{3}i \right)^2 \]
\[ = 1 + 3 i^2 + 2\sqrt{3}i\]
\[ = - 2 + 2\sqrt{3}i\]
\[\text { Then }, \frac{1}{z} = \frac{1}{- 2 + 2\sqrt{3}i} \times \frac{- 2 - 2\sqrt{3}i}{- 2 - 2\sqrt{3}i}\]
\[ = \frac{- 2 - 2\sqrt{3}i}{4 - 12 i^2}\]
\[ = \frac{- 2 - 2\sqrt{3}i}{16}\]
\[ = \frac{- 1}{8} - \frac{\sqrt{3}}{8}i\]
APPEARS IN
RELATED QUESTIONS
Let z1 = 2 – i, z2 = –2 + i. Find `"Im"(1/(z_1barz_1))`
Evaluate the following:
\[\frac{1}{i^{58}}\]
Evaluate the following:
\[i^{49} + i^{68} + i^{89} + i^{110}\]
Express the following complex number in the standard form a + i b:
\[\frac{1}{(2 + i )^2}\]
Express the following complex number in the standard form a + i b:
\[\frac{(1 + i)(1 + \sqrt{3}i)}{1 - i}\] .
Express the following complex number in the standard form a + i b:
\[(1 + 2i )^{- 3}\]
Find the multiplicative inverse of the following complex number:
1 − i
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Re \[\left( \frac{z_1 z_2}{z_1} \right)\]
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Im `(1/(z_1overlinez_1))`
Find the real values of θ for which the complex number \[\frac{1 + i cos\theta}{1 - 2i cos\theta}\] is purely real.
Evaluate the following:
\[x^4 - 4 x^3 + 4 x^2 + 8x + 44,\text { when } x = 3 + 2i\]
Find the number of solutions of \[z^2 + \left| z \right|^2 = 0\].
Express \[\sin\frac{\pi}{5} + i\left( 1 - \cos\frac{\pi}{5} \right)\] in polar form.
Write the value of \[\sqrt{- 25} \times \sqrt{- 9}\].
If \[\left| z + 4 \right| \leq 3\], then find the greatest and least values of \[\left| z + 1 \right|\].
If \[\left| z \right| = 2 \text { and } \arg\left( z \right) = \frac{\pi}{4}\],find z.
If `(3+2i sintheta)/(1-2 i sin theta)`is a real number and 0 < θ < 2π, then θ =
The polar form of (i25)3 is
If z is a non-zero complex number, then \[\left| \frac{\left| z \right|^2}{zz} \right|\] is equal to
\[\text { If } z = \frac{1}{(2 + 3i )^2}, \text { than } \left| z \right| =\]
\[\text { If } z = \frac{1}{(1 - i)(2 + 3i)}, \text { than } \left| z \right| =\]
\[\text { If }z = 1 - \text { cos }\theta + i \text { sin }\theta, \text { then } \left| z \right| =\]
If θ is the amplitude of \[\frac{a + ib}{a - ib}\] , than tan θ =
The amplitude of \[\frac{1}{i}\] is equal to
The argument of \[\frac{1 - i}{1 + i}\] is
The amplitude of \[\frac{1 + i\sqrt{3}}{\sqrt{3} + i}\] is
If \[z = a + ib\] lies in third quadrant, then \[\frac{\bar{z}}{z}\] also lies in third quadrant if
A real value of x satisfies the equation \[\frac{3 - 4ix}{3 + 4ix} = a - ib (a, b \in \mathbb{R}), if a^2 + b^2 =\]
The complex number z which satisfies the condition \[\left| \frac{i + z}{i - z} \right| = 1\] lies on
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
`((2 + "i"))/((3 - "i")(1 + 2"i"))`
Find the value of `(3 + 2/"i")("i"^6 - "i"^7)(1 + "i"^11)`
Evaluate the following : i403
Evaluate the following : `1/"i"^58`
Show that 1 + i10 + i20 + i30 is a real number
If z1 = 3 – 2i and z2 = –1 + 3i, then Im(z1z2) = ______.
If w is a complex cube-root of unity, then prove the following
(w2 + w − 1)3 = −8
Find the value of `(i^(592) + i^(590) + i^(588) + i^(586) + i^(584))/(i^(582) + i^(580) + i^(578) + i^(576) + i^(574))`
