Complex number arithmeticAQA A-Level Further Maths: Subtopic test

1. 1
   The complex numbers $z$ and $w$ are given by $z=3-2i$ and $w=1+4i$.

   (a)

   What is the imaginary part of $z$?

   [1 mark]

   • A$2$
   • B$-2i$
   • C$-2$
   • D$3$

   (b)

   Find $zw$.

   [1 mark]

   • A$3-8i$
   • B$11+10i$
   • C$-5+10i$
   • D$11-10i$

   (c)

   Find $\frac{z}{w}$ in the form $a+bi$, where $a$ and $b$ are real.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The quadratic equation $z^2-6z+13=0$ has roots $\alpha$ and $\beta$, where $\alpha$ has positive imaginary part.

   (a)

   What is the value of the discriminant $b^2-4ac$?

   [1 mark]

   • A$16$
   • B$88$
   • C$-52$
   • D$-16$

   (b)

   Find $\alpha$.

   [1 mark]

   • A$3+2i$
   • B$3+4i$
   • C$-3+2i$
   • D$5$

   (c)

   Find the value of $\alpha^2+\beta^2$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The complex number $z=x+iy$, where $x$ and $y$ are real, satisfies $(2+i)z=7+i$.

   (a)

   Find $z$ in the form $x+iy$.

   [3 marks]

   (b)

   Find the real numbers $p$ and $q$ such that $z^2+pz+q=0$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The quadratic equation $z^2-4z+13=0$ has roots $z_1$ and $z_2$, where $z_1$ has positive imaginary part.

   (a)

   (i) Solve the equation, showing that $z_1=2+3i$.
   (ii) Find $z_1^2$ in the form $a+bi$.
   (iii) Show that $z_1z_2=13$.

   [6 marks]

   (b)

   (i) Show that $\frac{z_1}{z_2}=\frac{-5+12i}{13}$.
   (ii) Find the real number $k$ for which $z_1+kz_2$ is purely imaginary, and state the value of $z_1+kz_2$ for this $k$.

   [6 marks]

   Total for question 4: 12 marks

End of questions