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Question
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"))`
Solution
`("i"(4 + 3"i"))/((1 - "i")) = (4"i" + 3"i"^2)/(1 - "i")`
= `(4"i" - 3)/(1 - "i")` ...[∵ i2 = – 1]
= `(4"i" - 3)/(1 - "i") xx (1 + "i")/(1 + "i")`
= `(4"i" + 4"i"^2 - 3 - 3"i")/(1 - "i"^2)`
= `(4"i" - 4 - 3 -3"i")/(1 + 1)` ...[∵ i2 = – 1]
= `(-7 + "i")/2`
= `(-7)/2 + 1/2"i"`
This is of the form a + bi, where a = `(-7)/2` and b = `1/2`.
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| (b) The amplitude of `-1 + sqrt(-3)` is | (ii) On or outside the circle having centre at (0, –4) and radius 3. |
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| (g) Conjugate of `(1 + 2i)/(1 - i)` lies in | (vii) First quadrant |
| (h) Reciprocal of 1 – i lies in | (viii) Third quadrant |
