Advertisements
Advertisements
Question
Let z1 = 2 – i, z2 = –2 + i. Find Re`((z_1z_2)/barz_1)`
Advertisements
Solution
z1 = 2 – i, z2 = –2 + i
`((z_1z_2)/barz_1) = ((2 - i)(-2 +i))/(2 -i) = (-(2 - i)(2 -i))/(2 + i)`
= `- (2-i)^2/(2 + i) = (- (4 + i^2 - 4i))/(2 + i)`
= `(-(4 - 1 - 4i))/((2 + i)) = -(3 - 4i)/(2 + i)`
= `-(3 - 4i)/(2 + i) xx (2 - i)/(2 - i)`
= `(- 6 - 4i^2 + 3i + 8i)/(4 - i^2) = (- 6 + 4 + 11i)/(4 + 1)`
= `(- 2 + 11i)/5 = - 2/5 + 11/5 i`
Re`((z_1z_2)/barz_1) = - 2/5`
APPEARS IN
RELATED QUESTIONS
Express the given complex number in the form a + ib: i9 + i19
Express the given complex number in the form a + ib: `(1/3 + 3i)^3`
Find the value of the following expression:
i + i2 + i3 + i4
Find the value of the following expression:
i5 + i10 + i15
Find the value of the following expression:
\[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\]
Express the following complex number in the standard form a + i b:
\[\frac{(1 - i )^3}{1 - i^3}\]
Find the real value of x and y, if
\[(3x - 2iy)(2 + i )^2 = 10(1 + i)\]
Find the multiplicative inverse of the following complex number:
\[(1 + i\sqrt{3} )^2\]
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 \[\left( \frac{1 - i}{1 + i} \right)^{100} = a + ib\] find (a, b).
If \[a = \cos\theta + i\sin\theta\], find the value of \[\frac{1 + a}{1 - a}\].
Evaluate the following:
\[x^6 + x^4 + x^2 + 1, \text { when }x = \frac{1 + i}{\sqrt{2}}\]
For a positive integer n, find the value of \[(1 - i )^n \left( 1 - \frac{1}{i} \right)^n\].
Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].
If \[\left| z + 1 \right| = z + 2\left( 1 + i \right)\],find z.
Write (i25)3 in polar form.
Express the following complex in the form r(cos θ + i sin θ):
1 + i tan α
Write the value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\] .
Find z, if \[\left| z \right| = 4 \text { and }\arg(z) = \frac{5\pi}{6} .\]
Write the value of \[\arg\left( z \right) + \arg\left( \bar{z} \right)\].
The polar form of (i25)3 is
If i2 = −1, then the sum i + i2 + i3 +... upto 1000 terms is equal to
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
If \[z = \left( \frac{1 + i}{1 - i} \right)\] then z4 equals
If \[z = \frac{1 + 7i}{(2 - i )^2}\] , then
The amplitude of \[\frac{1}{i}\] is equal to
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 abi = 3a − b + 12i
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"))`
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")`
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
`(4"i"^8 - 3"i"^9 + 3)/(3"i"^11 - 4"i"^10 - 2)`
Show that `(-1 + sqrt(3)"i")^3` is a real number
Evaluate the following : i35
Evaluate the following : i403
State True or False for the following:
The order relation is defined on the set of complex numbers.
Find the value of `(i^(592) + i^(590) + i^(588) + i^(586) + i^(584))/(i^(582) + i^(580) + i^(578) + i^(576) + i^(574))`
