Advertisements
Advertisements
Question
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"))`
Advertisements
Solution 1
`((2 + "i"))/((3 - "i")(1 + 2"i"))`
`= (2 + "i")/(3 + 6"i" - "i" - 2"i"^2)`
= `(2 + "i")/(3 + 5"i" - 2(-1))` ...[∵ i2 = – 1]
= `(2 + "i")/(5 + 5"i")`
= `(2 + "i")/(5(1 + "i"))`
= `((2 + "i")(1 - "i"))/(5(1 + "i")(1 - "i"))`
= `(2 - 2"i" + "i" - "i"^2)/(5(1 - "i"^2)`
= `(2 - "i" - (-1))/(5[1 - (-1)]` ...[∵ i2 = – 1]
= `(3- "i")/10`
∴ `(2 + "i")/((3 - "i")(1 + 2"i")) = 3/10 - 1/10"i"`
∴ a = `3/10` and b = `(-1)/10`
Solution 2
`((2 + "i"))/((3 - "i")(1 + 2"i"))`
`= (2 + "i")/(3 + 6"i" - "i" - 2"i"^2)`
= `(2 + "i")/(3 + 5"i" - 2(-1))` ...[∵ i2 = – 1]
= `(2 + "i")/(3 + 5"i" + 2)`
= `(2 + "i")/(5 + 5"i")`
= `((2 + "i").(5 - 5"i"))/((5 + 5"i").(5 - 5"i"))`
= `(10 - 10"i" + 5"i" - 5"i"^2)/(5^2 - 5"i"^2)`
= `(10 - 10"i" + 5"i" - 5(-1))/(5^2 - 5"i"^2)` ...[∵ i2 = – 1]
= `(10 - 10"i" + 5"i" + 5)/(5^2 - 5"i"^2)`
= `(15 - 5"i")/(25 - 25(-1))`
= `(15 - 5"i")/(25 +25)`
= `(15 - 5"i")/(50)`
= `15/50 - (5"i")/50`
= `3/10 - (1"i")/10`
∴ write in a + ib form
∴ a = `3/10` and b = `(-1)/10`
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: 3(7 + i7) + i(7 + i7)
Express the given complex number in the form a + ib: `(-2 - 1/3 i)^3`
Let z1 = 2 – i, z2 = –2 + i. Find Re`((z_1z_2)/barz_1)`
Find the value of the following expression:
i49 + i68 + i89 + i110
Find the value of the following expression:
(1 + i)6 + (1 − i)3
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:
\[\frac{(1 + i)(1 + \sqrt{3}i)}{1 - i}\] .
Express the following complex number in the standard form a + i b:
\[\frac{3 - 4i}{(4 - 2i)(1 + i)}\]
Find the real value of x and y, if `((1+i)x-2i)/(3+i) + ((2-3i)y+i)/(3-i) = i, xy ∈ R, i = sqrt-1`
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Im `(1/(z_1overlinez_1))`
If \[x + iy = \frac{a + ib}{a - ib}\] prove that x2 + y2 = 1.
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.
If \[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy\] find the value of \[x^2 + y^2\].
Write the value of \[\sqrt{- 25} \times \sqrt{- 9}\].
Find the real value of a for which \[3 i^3 - 2a i^2 + (1 - a)i + 5\] is real.
If\[z = \cos\frac{\pi}{4} + i \sin\frac{\pi}{6}\], then
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
The least positive integer n such that \[\left( \frac{2i}{1 + i} \right)^n\] is a positive integer, is.
If (x + iy)1/3 = a + ib, then \[\frac{x}{a} + \frac{y}{b} =\]
\[(\sqrt{- 2})(\sqrt{- 3})\] is equal to
The argument of \[\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\] is
If \[z = \left( \frac{1 + i}{1 - i} \right)\] then z4 equals
If \[z = \frac{1}{1 - cos\theta - i sin\theta}\] then Re (z) =
\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals
Which of the following is correct for any two complex numbers z1 and z2?
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")/(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:
(1 + i)−3
Evaluate the following : i93
Evaluate the following : i403
If z1 = 3 – 2i and z2 = –1 + 3i, then Im(z1z2) = ______.
Show that `(-1 + sqrt3 "i")^3` is a real number.
Find the value of `(i^(592) + i^(590) + i^(588) + i^(586) + i^(584))/(i^(582) + i^(580) + i^(578) + i^(576) + i^(574))`
