1sec ≫ First term ≫ Algebra ≫ Unit 1 ≫ (1-2 ) introduction to complex numbers :
(Q20 To Q25)
[21] Find the values of \( x \) and \( y \) that satisfy \[ \frac{(2+\mathrm{i})(2-\mathrm{i})}{3+4 \mathrm{i}} = x + y \mathrm{i} \]
The correct values are:
\[ x = \frac{3}{5}, \quad y = -\frac{4}{5} \][22] Find the result of each of the following in the simplest form: \[ (2+\sqrt{-9})(3-4\mathrm{i}) \]
The correct answer is:
\[ (2+3\mathrm{i})(3-4\mathrm{i}) \]
\[ = 6 - 8\mathrm{i} + 9\mathrm{i} - 12\mathrm{i}^2 \]
\[ = 6 + \mathrm{i} + 12 \]
\[ = 18 + \mathrm{i} \]
[23] Find the result of each of the following in the simplest form: \[ (1-\mathrm{i})^{10} \]
The correct answer is:
\[ -32\mathrm{i} \]
[24] Put \( \frac{26}{3-2\mathrm{i}}\) in the form \((a+b\mathrm{i})\) where \(a\) and \(b\) are real numbers:
The correct answer is:
\[ \frac{26}{3-2\mathrm{i}} \times \frac{3+2\mathrm{i}}{3+2\mathrm{i}} \]
\[ = \frac{26(3+2\mathrm{i})}{(3-2\mathrm{i})(3+2\mathrm{i})} \]
\[ = \frac{26(3+2\mathrm{i})}{9+4} \]
\[ = \frac{26(3+2\mathrm{i})}{13} \]
\[ = 2(3+2\mathrm{i}) = 6 + 4\mathrm{i} \]
[25] Put \( \frac{2-3\mathrm{i}}{3+\mathrm{i}} \) in the form \( (a+b\mathrm{i}) \) where \(a\) and \(b\) are real numbers:
The correct answer is:
\[ \frac{2-3\mathrm{i}}{3+\mathrm{i}} \times \frac{3-\mathrm{i}}{3-\mathrm{i}} \]
\[ = \frac{(2-3\mathrm{i})(3-\mathrm{i})}{(3+\mathrm{i})(3-\mathrm{i})} \]
\[ = \frac{6 - 2\mathrm{i} - 9\mathrm{i} + 3\mathrm{i}^2}{9 + 1} \]
\[ = \frac{6 - 11\mathrm{i} - 3}{10} \]
\[ = \frac{3 - 11\mathrm{i}}{10} = \frac{3}{10} -\frac{11}{10}\mathrm{i} \]
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