Advertisements
Advertisements
प्रश्न
Express the following complex number in the standard form a + i b:
\[(1 + 2i )^{- 3}\]
Advertisements
उत्तर
\[( 1 + 2i )^{- 3} \]
\[ = \frac{1}{\left( 1 + 2i \right)^3}\]
\[ = \frac{1}{1 + 8 i^3 + 6i + 12 i^2}\]
\[ = \frac{1}{1 - 8i + 6i - 12} \left( \because i^2 = - 1 \text { & } i^3 = - i \right)\]
\[ = \frac{1}{- 2i - 11}\]
\[ = \frac{1}{- 2i - 11} \times \frac{- 2i + 11}{- 2i + 11}\]
\[ = \frac{- 2i + 11}{4 i^2 - 121}\]
\[ = \frac{- 2i + 11}{- 4 - 121}\]
\[ = \frac{- 2i + 11}{- 125}\]
\[ = - \frac{11}{125} + \frac{2i}{125}\]
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: (1 – i) – (–1 + i6)
Express the given complex number in the form a + ib: `(1/5 + i 2/5) - (4 + i 5/2)`
Express the given complex number in the form a + ib: `(1/3 + 3i)^3`
Evaluate the following:
(ii) i528
Evaluate the following:
\[i^{37} + \frac{1}{i^{67}}\].
Evaluate the following:
\[i^{30} + i^{40} + i^{60}\]
Express the following complex number in the standard form a + i b:
\[\frac{1 - i}{1 + i}\]
Express the following complex number in the standard form a + i b:
\[\frac{(1 - i )^3}{1 - i^3}\]
If \[z_1 = 2 - i, z_2 = 1 + i,\text { find } \left| \frac{z_1 + z_2 + 1}{z_1 - z_2 + i} \right|\]
Evaluate the following:
\[2 x^3 + 2 x^2 - 7x + 72, \text { when } x = \frac{3 - 5i}{2}\]
Evaluate the following:
\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]
Evaluate the following:
\[x^6 + x^4 + x^2 + 1, \text { when }x = \frac{1 + i}{\sqrt{2}}\]
For a positive integer n, find the value of \[(1 - i )^n \left( 1 - \frac{1}{i} \right)^n\].
If \[\left( 1 + i \right)z = \left( 1 - i \right) \bar{z}\],then show that \[z = - i \bar{z}\].
If \[\frac{z - 1}{z + 1}\] is purely imaginary number (\[z \neq - 1\]), find the value of \[\left| z \right|\].
What is the smallest positive integer n for which \[\left( 1 + i \right)^{2n} = \left( 1 - i \right)^{2n}\] ?
If z1, z2, z3 are complex numbers such that \[\left| z_1 \right| = \left| z_2 \right| = \left| z_3 \right| = \left| \frac{1}{z_1} + \frac{1}{z_2} + \frac{1}{z_3} \right| = 1\] then find the value of \[\left| z_1 + z_2 + z_3 \right|\] .
Write (i25)3 in polar form.
Express the following complex in the form r(cos θ + i sin θ):
1 + i tan α
Write −1 + i \[\sqrt{3}\] in polar form .
Find the principal argument of \[\left( 1 + i\sqrt{3} \right)^2\] .
Find z, if \[\left| z \right| = 4 \text { and }\arg(z) = \frac{5\pi}{6} .\]
The value of \[(1 + i)(1 + i^2 )(1 + i^3 )(1 + i^4 )\] is.
\[\text { If } z = \frac{1}{(2 + 3i )^2}, \text { than } \left| z \right| =\]
\[\text { If } z = \frac{1}{(1 - i)(2 + 3i)}, \text { than } \left| z \right| =\]
If \[z = \frac{1 + 7i}{(2 - i )^2}\] , then
If \[z = a + ib\] lies in third quadrant, then \[\frac{\bar{z}}{z}\] also lies in third quadrant if
The complex number z which satisfies the condition \[\left| \frac{i + z}{i - z} \right| = 1\] lies on
Find a and b if `1/("a" + "ib")` = 3 – 2i
Find a and b if (a + ib) (1 + i) = 2 + i
Evaluate the following : i–888
State True or False for the following:
The order relation is defined on the set of complex numbers.
State True or False for the following:
2 is not a complex number.
Show that `(-1+sqrt3i)^3` is a real number.
