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Complex numbers final test

  1. Express each of the following in the form of \(a+ib\)

    1. \((3i)(-4i)\)
    2. \((3+i)(2i)\)
    3. \((2+3i)(-1+i)\)
    4. \(i^{15}-\frac{3}{i^5}\)
    5. \((3-2i)^2\)

  2. Let \(z_1=5-4i\) and \(z_2=-2+3i\) be two complex numbers. Then find i. \(z_1+z_2\), ii. \(z_1-z_2\), iii. \(z_1\,z_2\), iv. \(\frac{z_1}{z_2}\), v. \(\frac{z_2}{z_1}\).

  3. Find the multiplicative inverse of the following:

    1. \(5-12i\)
    2. \(-3+6i\)
    3. \(-8i\)
    4. \(1+2i\)

  4. Express the following in the \(a+ib\) form:

    \[\frac{(3+2i)(-1+5i)}{(5-3i) + (1-5i)}\]

  5. Find the modulus and conjugate of the following:

    1. \(5-12i\)
    2. \(-3+6i\)
    3. \(-8i\)
    4. \(1+2i\)
    5. \(10\)

  6. Let \(z_1=3+4i\) and \(z_2=2-i\), then find

    1. \(\text{Re} \left( \frac{z_1}{\overline{z_2}} \right)\)
    2. \(\text{Im} \left( \frac{z_1}{\overline{z_2}} \right)\)

  7. If \(x+iy=\frac{1-i}{1+i} - \frac{1+i}{1-i}\), then find \(x^2+y^2\).

  8. (Bonus) If \((x+iy)^2=8-6i\) and \(x\) is positive, then find \(x\) and \(y\).

    Hint

    Use \((a+b)^2 - (a-b)^2 = 4ab\) to find \(x^2+y^2\) from given data.