Practice set fourth

In this practice set will try to solve some complicated question of complex analysis

1). Find the value of   \(i^{-31}\).

2). Find the multiplicative inverse of   \(i^{10}\).

3). Find the multiplicative inverse of    \(2-3i\).

4). Find the value of     \((2i^{-30}+5)(20+2i^{-100})\).

5). Find the value of    \(\displaystyle \left (\frac{1}{1-4i}-\frac{2}{1+i} \right) \left(\frac{3+4i}{5+i}\right)\) .

6). Evaluate \( \displaystyle \left[i^{18}-\left(\frac{1}{i}\right)^{25} \right]^{3}\).

7). Find the  multiplicative inverse of    \(2+4i\).

8).  Evaluate \((i+i^{41}+i^{23})(i-2i^8)\).

9).  Simplify  \(\displaystyle \left(\frac{1}{5}+i\frac{2}{5}\right)+\left(4+i\frac{5}{2}\right)\)

10). Simplify \(\displaystyle \left(\frac{1}{1-4 i}+\frac{2}{1+i}\right)\left(\frac{3-4 i}{5+i}\right)\).

11). Write the \((1-2i)^{-3}\) in standard form.

Answer :

1) \(i\) 2) \( -1\) 3) \(\displaystyle \frac{2}{13}-\frac{3 i}{13}\)

4) \(110+44 i\) 5) \(\displaystyle-\frac{661}{442}+\frac{127 i}{442}\)

6) \(2+2i\) 7) \(\displaystyle\frac{1}{10}-\frac{i}{5} \)

8)\(-1-2 i\) 9) \(\displaystyle\frac{21}{5}+\frac{29 i}{10}\)

10) \(\displaystyle -\frac{101}{442}-\frac{557 i}{442}\)

11). \(\displaystyle -\frac{11}{125}-\frac{2 i}{125}\)