Advertisements
Advertisements
Question
If `((1 - i)/(1 + i))^100` = a + ib, then find (a, b).
Advertisements
Solution
We have `((1 - i)/(1 + i))^100` = a + bi
⇒ `((1 - i)/(1 + i) xx (1 - i)/(1 - i))^100` = a + bi
⇒ `((1 + i^2 - 2i)/(1 - i^2))^100` = a + bi
⇒ `((1 - 1 - 2i)/(1 + 1))^100` = a + bi
⇒ `((-2i)/2)^100` = a + bi
⇒ (–i)100 = a + bi
⇒ i100 = a + bi
⇒ (i4)25 = a + bi
⇒ (1)25 = a + bi
⇒ 1 = a + bi
⇒ 1 + 0i = a + bi
Comparing the real and imaginary parts,
We have a = 1, b = 0
Hence (a, b) = (1, 0)
APPEARS IN
RELATED QUESTIONS
Express the given complex number in the form a + ib: i–39
Express the given complex number in the form a + ib: 3(7 + i7) + i(7 + i7)
Evaluate the following:
\[\frac{1}{i^{58}}\]
Evaluate the following:
\[i^{49} + i^{68} + i^{89} + i^{110}\]
Find the value of the following expression:
i + i2 + i3 + i4
Find the value of the following expression:
1+ i2 + i4 + i6 + i8 + ... + i20
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:
\[\frac{2 + 3i}{4 + 5i}\]
Find the real value of x and y, if
\[(x + iy)(2 - 3i) = 4 + i\]
Find the real value of x and y, if
\[(3x - 2iy)(2 + i )^2 = 10(1 + i)\]
Find the real value of x and y, if
\[(1 + i)(x + iy) = 2 - 5i\]
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Re \[\left( \frac{z_1 z_2}{z_1} \right)\]
If \[a = \cos\theta + i\sin\theta\], find the value of \[\frac{1 + a}{1 - a}\].
Evaluate the following:
\[2 x^4 + 5 x^3 + 7 x^2 - x + 41, \text { when } x = - 2 - \sqrt{3}i\]
Write (i25)3 in polar form.
Express the following complex in the form r(cos θ + i sin θ):
1 + i tan α
Express the following complex in the form r(cos θ + i sin θ):
1 − sin α + i cos α
Write the value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\] .
For any two complex numbers z1 and z2 and any two real numbers a, b, find the value of \[\left| a z_1 - b z_2 \right|^2 + \left| a z_2 + b z_1 \right|^2\].
If n ∈ \[\mathbb{N}\] then find the value of \[i^n + i^{n + 1} + i^{n + 2} + i^{n + 3}\] .
Write the argument of \[\left( 1 + i\sqrt{3} \right)\left( 1 + i \right)\left( \cos\theta + i\sin\theta \right)\].
Disclaimer: There is a misprinting in the question. It should be \[\left( 1 + i\sqrt{3} \right)\] instead of \[\left( 1 + \sqrt{3} \right)\].
The value of \[(1 + i)(1 + i^2 )(1 + i^3 )(1 + i^4 )\] is.
If `(3+2i sintheta)/(1-2 i sin theta)`is a real number and 0 < θ < 2π, then θ =
\[\text { If }z = 1 - \text { cos }\theta + i \text { sin }\theta, \text { then } \left| z \right| =\]
If \[x + iy = \frac{3 + 5i}{7 - 6i},\] then y =
If θ is the amplitude of \[\frac{a + ib}{a - ib}\] , than tan θ =
The amplitude of \[\frac{1 + i\sqrt{3}}{\sqrt{3} + i}\] is
Which of the following is correct for any two complex numbers z1 and z2?
Find a and b if (a + ib) (1 + i) = 2 + i
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
(1 + 2i)(– 2 + i)
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
`("i"(4 + 3"i"))/((1 - "i"))`
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
(1 + i)−3
Evaluate the following : i888
Evaluate the following : `1/"i"^58`
State True or False for the following:
The order relation is defined on the set of complex numbers.
