Advertisements
Advertisements
Question
Express the following complex number in the standard form a + i b:
\[\frac{2 + 3i}{4 + 5i}\]
Advertisements
Solution
\[ \frac{2 + 3i}{4 + 5i}\]
\[ = \frac{2 + 3i}{4 + 5i} \times \frac{4 - 5i}{4 - 5i}\]
\[ = \frac{8 - 10i + 12i - 15 i^2}{16 - 25 i^2} \left( \because i^2 = - 1 \right)\]
\[ = \frac{23 + 2i}{16 + 25}\]
\[ = \frac{23}{41} + \frac{2}{41}i\]
APPEARS IN
RELATED QUESTIONS
Express the given complex number in the form a + ib: i–39
Evaluate the following:
i457
Evaluate the following:
(ii) i528
Evaluate the following:
\[\frac{1}{i^{58}}\]
Evaluate the following:
\[\left( i^{41} + \frac{1}{i^{257}} \right)^9\]
Evaluate the following:
\[i^{49} + i^{68} + i^{89} + i^{110}\]
Show that 1 + i10 + i20 + i30 is a real number.
Find the value of the following expression:
\[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\]
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:
\[\frac{1}{(2 + i )^2}\]
Express the following complex number in the standard form a + ib:
\[\frac{(2 + i )^3}{2 + 3i}\]
Express the following complex number in the standard form a + i b:
\[\frac{3 - 4i}{(4 - 2i)(1 + i)}\]
Find the multiplicative inverse of the following complex number:
1 − i
If \[\frac{\left( 1 + i \right)^2}{2 - i} = x + iy\] find x + y.
Find the number of solutions of \[z^2 + \left| z \right|^2 = 0\].
Write the argument of −i.
Find z, if \[\left| z \right| = 4 \text { and }\arg(z) = \frac{5\pi}{6} .\]
If \[\left| z - 5i \right| = \left| z + 5i \right|\] , then find the locus of z.
Write the value of \[\arg\left( z \right) + \arg\left( \bar{z} \right)\].
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 \[\left( 1 + \sqrt{3} \right)\].
If a = cos θ + i sin θ, then \[\frac{1 + a}{1 - a} =\]
The principal value of the amplitude of (1 + i) is
\[\text { If } z = \frac{1}{(1 - i)(2 + 3i)}, \text { than } \left| z \right| =\]
If \[z = \frac{1}{1 - cos\theta - i sin\theta}\] then Re (z) =
If θ is the amplitude of \[\frac{a + ib}{a - ib}\] , than tan θ =
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
\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals
Find a and b if abi = 3a − b + 12i
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:
`(2 + sqrt(-3))/(4 + sqrt(-3))`
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 : i116
Evaluate the following : i403
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 |
Find the value of `(i^(592) + i^(590) + i^(588) + i^(586) + i^(584))/(i^(582) + i^(580) + i^(578) + i^(576) + i^(574))`
