Advertisements
Advertisements
Question
Let z1 = 2 – i, z2 = –2 + i. Find `"Im"(1/(z_1barz_1))`
Advertisements
Solution
`1/(z_1z_1) = 1/((2 - i)(2-i)) = 1/((2- i)(2 + i))`
= `1/(4 - i^2) = 1/5`
∴ `"Im"(1/(z_1barz_1))` = 0
APPEARS IN
RELATED QUESTIONS
Express the given complex number in the form a + ib: `(5i) (- 3/5 i)`
Express the given complex number in the form a + ib: i–39
Let z1 = 2 – i, z2 = –2 + i. Find Re`((z_1z_2)/barz_1)`
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:
\[\frac{2 + 3i}{4 + 5i}\]
Express the following complex number in the standard form a + i b:
\[\frac{(1 - i )^3}{1 - i^3}\]
If \[z_1 = 2 - i, z_2 = 1 + i,\text { find } \left| \frac{z_1 + z_2 + 1}{z_1 - z_2 + i} \right|\]
If \[x + iy = \frac{a + ib}{a - ib}\] prove that x2 + y2 = 1.
Find the smallest positive integer value of m for which \[\frac{(1 + i )^n}{(1 - i )^{n - 2}}\] is a real number.
If \[\left( \frac{1 - i}{1 + i} \right)^{100} = a + ib\] find (a, b).
If \[a = \cos\theta + i\sin\theta\], find the value of \[\frac{1 + a}{1 - a}\].
Evaluate the following:
\[x^4 - 4 x^3 + 4 x^2 + 8x + 44,\text { when } x = 3 + 2i\]
If \[\frac{z - 1}{z + 1}\] is purely imaginary number (\[z \neq - 1\]), find the value of \[\left| z \right|\].
Solve the equation \[\left| z \right| = z + 1 + 2i\].
What is the smallest positive integer n for which \[\left( 1 + i \right)^{2n} = \left( 1 - i \right)^{2n}\] ?
Write the value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\] .
If \[\left| z - 5i \right| = \left| z + 5i \right|\] , then find the locus of z.
Write the sum of the series \[i + i^2 + i^3 + . . . .\] upto 1000 terms.
Find the real value of a for which \[3 i^3 - 2a i^2 + (1 - a)i + 5\] is real.
If \[z = \frac{- 2}{1 + i\sqrt{3}}\],then the value of arg (z) is
The principal value of the amplitude of (1 + i) is
\[(\sqrt{- 2})(\sqrt{- 3})\] is equal to
If \[z = \frac{1 + 2i}{1 - (1 - i )^2}\], then arg (z) equal
If \[z = \frac{1 + 7i}{(2 - i )^2}\] , then
The amplitude of \[\frac{1}{i}\] is equal to
The amplitude of \[\frac{1 + i\sqrt{3}}{\sqrt{3} + i}\] is
The value of (i5 + i6 + i7 + i8 + i9) / (1 + i) is
\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals
The value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}} - 1\] is
A real value of x satisfies the equation \[\frac{3 - 4ix}{3 + 4ix} = a - ib (a, b \in \mathbb{R}), if a^2 + b^2 =\]
Find a and b if `1/("a" + "ib")` = 3 – 2i
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
`((1 + "i")/(1 - "i"))^2`
Express the following in the form of a + ib, a, b ∈ R i = `sqrt(−1)`. State the values of a and b:
`(- sqrt(5) + 2sqrt(-4)) + (1 -sqrt(-9)) + (2 + 3"i")(2 - 3"i")`
Evaluate the following : i30 + i40 + i50 + i60
If z1 and z2 both satisfy `z + barz = 2|z - 1|` arg`(z_1 - z_2) = pi/4`, then find `"Im" (z_1 + z_2)`.
State True or False for the following:
The order relation is defined on the set of complex numbers.
Show that `(-1 + sqrt3 "i")^3` is a real number.
Show that `(-1+sqrt3i)^3` is a real number.
