Complex numbers and quadratic, cubic and quartic equationsEdexcel A-Level Further Maths: Subtopic test

1. 1
   The complex numbers $z_1=3+4i$ and $z_2=1-2i$ are given.

   (a)

   Find $z_1z_2$.

   [1 mark]

   • A$-5-2i$
   • B$3-8i$
   • C$11-2i$
   • D$11+10i$

   (b)

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

   [1 mark]

   • A$-1+2i$
   • B$3-2i$
   • C$\frac{11}{5}+2i$
   • D$-5+10i$

   (c)

   Find the exact value of $|z_1z_2|$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The complex number $z=x+iy$, where $x$ and $y$ are real, satisfies $z+2z^*=9-2i$.

   (a)

   What is the imaginary part of $z+2z^*$ in terms of $x$ and $y$?

   [1 mark]

   • A$y$
   • B$-y$
   • C$-iy$
   • D$3y$

   (b)

   Find $z$.

   [1 mark]

   • A$3-2i$
   • B$9-2i$
   • C$9+2i$
   • D$3+2i$

   (c)

   Hence find the modulus of $z$ and its argument, in radians to 2 decimal places.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The quartic equation $z^4-8z^3+27z^2-50z+50=0$ has $1+2i$ as one root. All of its coefficients are real.

   (a)

   Write down another root of the equation and hence find a quadratic factor with real coefficients.

   [3 marks]

   (b)

   Hence solve the quartic equation completely.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Throughout this question, $x$, $y$, $a$ and $b$ are real numbers and $\mathrm{i}^2=-1$.

   (a)

   The complex number $z=x+iy$ satisfies $(3+i)z+2z^*=7+5i$. Find $z$.

   [6 marks]

   (b)

   The cubic equation $z^3+az^2+17z+b=0$ has $2+3i$ as a root. Find the values of $a$ and $b$ and the other two roots.

   [6 marks]

   Total for question 4: 12 marks

End of questions