FP1: Complex numbersEdexcel International A Level Further Maths: Topic test

1. 1
   The complex number $z=8-6\mathrm{i}$.

   (a)

   Find $|z|$.

   [1 mark]

   • A$2$
   • B$10$
   • C$14$
   • D$100$

   (b)

   Find the real part of $z^2$.

   [1 mark]

   • A$100$
   • B$-28$
   • C$28$
   • D$-96$

   (c)

   Find $\arg z$ in radians to 3 significant figures.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The complex numbers $z_1=1+5\mathrm{i}$ and $z_2=4-\mathrm{i}$.

   (a)

   Find $z_1z_2$.

   [1 mark]

   • A$9+19\mathrm{i}$
   • B$-1+19\mathrm{i}$
   • C$4-5\mathrm{i}$
   • D$9-19\mathrm{i}$

   (b)

   Find $\dfrac{z_1}{z_2}$.

   [1 mark]

   • A$\dfrac{9+19\mathrm{i}}{17}$
   • B$\dfrac{-1+21\mathrm{i}}{15}$
   • C$\dfrac{-1-21\mathrm{i}}{17}$
   • D$\dfrac{-1+21\mathrm{i}}{17}$

   (c)

   Given that $z_1$ is a root of $z^2+pz+q=0$, where $p$ and $q$ are real constants, find $p$ and $q$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The equation $2z^2+4z+7=0$ has two complex roots.

   (a)

   Solve the equation, giving the roots in the form $a+b\mathrm{i}$ with exact values of $a$ and $b$.

   [3 marks]

   (b)

   The roots are represented by the points $A$ and $B$ on an Argand diagram, and $O$ is the origin. Show that $|z|=\sqrt{\frac72}$ for each root and find the exact area of triangle $OAB$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The cubic function $\mathrm{f}(z)=z^3-7z^2+az+b$, where $a$ and $b$ are real constants, has a root $2+3\mathrm{i}$.

   (a)

   Find the other two roots of $\mathrm{f}(z)=0$ and the values of $a$ and $b$.

   [6 marks]

   (b)

   (i) Find the modulus and the argument of the root $2+3\mathrm{i}$, giving the argument in radians to 3 significant figures.
   (ii) Find $\dfrac{3}{2+3\mathrm{i}}$ in the form $x+y\mathrm{i}$, where $x$ and $y$ are rational.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The quartic equation $z^4-2z^3+6z^2-8z+8=0$ has a root $z=2\mathrm{i}$.

   (a)

   Which quadratic is a factor of $z^4-2z^3+6z^2-8z+8$?

   [1 mark]

   • A$z^2-4$
   • B$z^2-4z+4$
   • C$z^2+2z+2$
   • D$z^2+4$

   (b)

   Dividing by the factor $z^2+4$ leaves $z^2-2z+2$. Which of these is another root of the quartic?

   [1 mark]

   • A$1+\mathrm{i}$
   • B$2+\mathrm{i}$
   • C$-1+\mathrm{i}$
   • D$1+2\mathrm{i}$

   (c)

   Find the sum of the moduli of the four roots of the equation.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The complex numbers $p=7+\mathrm{i}$ and $q=1-\mathrm{i}$.

   (a)

   Find $pq$.

   [1 mark]

   • A$6-6\mathrm{i}$
   • B$8-6\mathrm{i}$
   • C$7-\mathrm{i}$
   • D$8+6\mathrm{i}$

   (b)

   Find $\dfrac{p}{q}$.

   [1 mark]

   • A$4-3\mathrm{i}$
   • B$6+8\mathrm{i}$
   • C$3+4\mathrm{i}$
   • D$3-4\mathrm{i}$

   (c)

   Verify that $\left|\dfrac{p}{q}\right|=\dfrac{|p|}{|q|}$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The quadratic equation $z^2+bz+c=0$, where $b$ and $c$ are real constants, has a root $-1+4\mathrm{i}$.

   (a)

   Find the values of $b$ and $c$.

   [3 marks]

   (b)

   The two roots are $z_1=-1+4\mathrm{i}$ and $z_2=-1-4\mathrm{i}$. Show that $\dfrac{z_1}{z_2}=\dfrac{-15-8\mathrm{i}}{17}$ and hence find $\left|\dfrac{z_1}{z_2}\right|$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The cubic function $\mathrm{f}(z)=z^3+pz^2+qz+r$, where $p$, $q$ and $r$ are real constants, has roots $4$ and $w$, where $|w|=2$ and $\arg w=\frac{\pi}{3}$.

   (a)

   Find $p$, $q$ and $r$.

   [6 marks]

   (b)

   The three roots of $\mathrm{f}(z)=0$ are represented by the points $A$, $B$ and $C$ on an Argand diagram. Show that triangle $ABC$ is equilateral and find its exact area.

   [6 marks]

   Total for question 8: 12 marks

End of questions