Math
ski
Worksheet
Questions
Solutions
Complex Numbers I (Non-Calculator)
IB Mathematics AA · HL · Shadow Worksheet
Practice
Name
Question 1
Sketch the following complex numbers on an Argand diagram:
(a)
\(z_1=3\)
(b)
\(z_2=2+3i\)
(c)
\(z_3=-2i\)
(d)
\(z_4=-3+i\)
Answer
Plot: \((3,0)\), \((2,3)\), \((0,-2)\), \((-3,1)\) on Re-Im axes.
Question 2
Calculate \(i^2,\;i^3,\;i^4\). What is \(i^{101}\)?
Powers of \(i\)
\(i^2=-1;\;i^3=-i;\;i^4=1\). Cycle of 4.
\(i^{101}=i^{4 imes25+1}\)
Answer
\(i^2=-1,\;i^3=-i,\;i^4=1,\;i^{101}=i\)
Question 3
Simplify the following:
(a)
\((2+3i)(1+2i)\)
(b)
\(z_2=(1-4i)(2+i)\)
(c)
\(z_3=\sqrt{-25}\)
(a)
FOIL
\(2+4i+3i+6i^2=2+7i-6\)
Answer
\(-4+7i\)
(b)
FOIL
\(2+i-8i-4i^2=2-7i+4\)
Answer
\(6-7i\)
(c)
Answer
\(5i\)
Question 4
Rationalise the denominator for the following:
(a)
\(\dfrac{3+i}{2-i}\)
(b)
\(\dfrac{1+2i}{3+4i}\)
(c)
\(\dfrac{2i}{1-3i}\)
(a)
Multiply by \(\overline{(2-i)}\)
\(\dfrac{(3+i)(2+i)}{5}=\dfrac{6+3i+2i-1}{5}=\dfrac{5+5i}{5}\)
Answer
\(1+i\)
(b)
Multiply by \((3-4i)\)
\(\dfrac{(1+2i)(3-4i)}{25}=\dfrac{3-4i+6i+8}{25}=\dfrac{11+2i}{25}\)
Answer
\(\dfrac{11}{25}+\dfrac{2}{25}i\)
(c)
Multiply by \((1+3i)\)
\(\dfrac{2i(1+3i)}{10}=\dfrac{2i-6}{10}\)
Answer
\(-\dfrac{3}{5}+\dfrac{1}{5}i\)
Question 5
Find the complex roots of the following quadratics:
(a)
\(z^2+4z+13=0\)
(b)
\(2z^2-2z+5=0\)
(a)
Quadratic formula
\(z=\dfrac{-4\pm\sqrt{16-52}}{2}=\dfrac{-4\pm\sqrt{-36}}{2}\)
Answer
\(z=-2\pm3i\)
(b)
Quadratic formula
\(z=\dfrac{2\pm\sqrt{4-40}}{4}=\dfrac{2\pm6i}{4}\)
Answer
\(z=\dfrac{1}{2}\pm\dfrac{3}{2}i\)
Generated by Mathski · mathski.io · IB Mathematics AA Shadow Worksheets