MATHEMATICS 2 (MA10193) EXAMPLES SHEET 3

I will look at work given to me after Easter. If you do not have a copy of this sheet, you can find one at

http://www.bath.ac.uk/~masgks/MA10193/sheet3.ps (or .dvi or .pdf).

1. Calculate (7 − 2i) + (8 + 5i); (3 − i)(6 + 5i); (3 − i)/(6 + 5i); (2 + i)/i; \sqrt{−i}.

2. Find the inverse of the matrix A given by

A = ( \begin{matrix}0 & 2 & 1 \\ −1 & 3 & 2 \\ 2 & 1 & −1 \end{matrix} ) .

3. Let z = 3 + i. Find z^{2} in Cartesian form. Plot z and z^{2} on an Argand diagram.

4. Let z_{1} = 1 − 2i and let z_{2} = 4 + yi for some real number y. Determine the value of y if the real part of z_{1}z_{2} is zero. Determine the value of y if \frac{z_{1}}{z_{2}} is a real number.

5. Consider the complex number z = \sqrt{3} + i. Calculate |z|, and arg z (in radians). What is the smallest positive integer n such that z^{n} is a real number?

6. Solve the equation x^{2}−5x+32 = 0. Hence find the eigenvalues of the matrix ( \begin{matrix}3 & 13 \\ −2 & 2 \end{matrix} ).

GKS, 18/03/05