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:
(1 + i)(1 − i)−1
Advertisements
Solution
(1 + i)(1 − i)−1
= `(1 + "i")/(1 - "i")`
= `((1 + "i")(1 + "i"))/((1 - "i")(1 + "i")`
= `(1 + 2"i" + "i"^2)/(1 - "i"^2)`
= `(1 + 2"i" - 1)/(1 - (-1))` ...[∵ i2 = −1]
= `(2"i")/2`
= i
∴ (1 + i)(1 − i)−1 = 0 + i
∴ a = 0 and b = 1
APPEARS IN
RELATED QUESTIONS
Express the given complex number in the form a + ib: (1 – i)4
Evaluate the following:
\[\left( i^{41} + \frac{1}{i^{257}} \right)^9\]
Evaluate the following:
\[i^{30} + i^{40} + i^{60}\]
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:
\[\frac{(1 - i )^3}{1 - i^3}\]
Express the following complex number in the standard form a + i b:
\[(1 + 2i )^{- 3}\]
Express the following complex number in the standard form a + i b:
\[\left( \frac{1}{1 - 4i} - \frac{2}{1 + i} \right)\left( \frac{3 - 4i}{5 + i} \right)\]
Express the following complex number in the standard form a + i b:
\[\frac{5 + \sqrt{2}i}{1 - 2\sqrt{i}}\]
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`
Find the real value of x and y, if
\[(1 + i)(x + iy) = 2 - 5i\]
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 \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Re \[\left( \frac{z_1 z_2}{z_1} \right)\]
Find the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
If \[a = \cos\theta + i\sin\theta\], find the value of \[\frac{1 + a}{1 - a}\].
Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].
Solve the equation \[\left| z \right| = z + 1 + 2i\].
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 the value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\] .
Write −1 + i \[\sqrt{3}\] in polar form .
Write the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
Find z, if \[\left| z \right| = 4 \text { and }\arg(z) = \frac{5\pi}{6} .\]
The polar form of (i25)3 is
If a = cos θ + i sin θ, then \[\frac{1 + a}{1 - a} =\]
If z is a non-zero complex number, then \[\left| \frac{\left| z \right|^2}{zz} \right|\] is equal to
\[(\sqrt{- 2})(\sqrt{- 3})\] is equal to
The value of (i5 + i6 + i7 + i8 + i9) / (1 + i) is
Find a and b if (a – b) + (a + b)i = a + 5i
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")`
Show that `(-1 + sqrt(3)"i")^3` is a real number
Find the value of `(3 + 2/i) (i^6 - i^7) (1 + i^11)`.
Evaluate the following : i888
Evaluate the following : `1/"i"^58`
Match the statements of Column A and Column B.
| Column A | Column B |
| (a) The polar form of `i + sqrt(3)` is | (i) Perpendicular bisector of segment joining (–2, 0) and (2, 0). |
| (b) The amplitude of `-1 + sqrt(-3)` is | (ii) On or outside the circle having centre at (0, –4) and radius 3. |
| (c) If |z + 2| = |z − 2|, then locus of z is | (iii) `(2pi)/3` |
| (d) If |z + 2i| = |z − 2i|, then locus of z is | (iv) Perpendicular bisector of segment joining (0, –2) and (0, 2). |
| (e) Region represented by |z + 4i| ≥ 3 is | (v) `2(cos pi/6 + i sin pi/6)` |
| (f) Region represented by |z + 4| ≤ 3 is | (vi) On or inside the circle having centre (–4, 0) and radius 3 units. |
| (g) Conjugate of `(1 + 2i)/(1 - i)` lies in | (vii) First quadrant |
| (h) Reciprocal of 1 – i lies in | (viii) Third quadrant |
The real value of θ for which the expression `(1 + i cos theta)/(1 - 2i cos theta)` is a real number is ______.
