CP-2 Complex numbers — 9FM0 spec coverage

CP-2.1

Solve any quadratic equation with real coefficients. Solve cubic or quartic equations with real coefficients.

The roots of a real quadratic $ax^2+bx+c=0$ are $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ ; a negative discriminant produces a complex-conjugate pair.
For a cubic, first look for a real linear factor using inspection, the factor theorem or a supplied root, then solve the remaining quadratic.
A useful quartic principle is to recognise a biquadratic or factor the expression into two real quadratics; for example, symmetric $x^3$ and $x$ coefficients can guide the signs of the linear terms.
Do not stop after finding one factor: solve every remaining quadratic and list every root, including non-real roots.

Tier 1 · Easy

3 marks

ORIGINAL

Solve $x^2-4x+13=0$ , giving the roots in the form $a+ib$ .

Tier 2 · Standard

4 marks

ORIGINAL

Given that $x=2$ is a root of $x^3-2x^2+5x-10=0$ , solve the equation completely.

Tier 3 · Hard

5 marks

ORIGINAL

Given that $x^2+3x+6$ is a factor of $x^4+3x^2+36$ , solve $x^4+3x^2+36=0$ completely.

CP-2.2

Add, subtract, multiply and divide complex numbers in the form x + iy with x and y real. Understand and use the terms 'real part' and 'imaginary part'.

For $z=x+iy$ , the real and imaginary parts are $\operatorname{Re}(z)=x$ and $\operatorname{Im}(z)=y$ ; the imaginary part is the real coefficient $y$ , not $iy$ .
Add or subtract real parts together and imaginary parts together; when multiplying, expand and replace $i^2$ by $-1$ .
To divide Cartesian complex numbers, multiply numerator and denominator by the conjugate of the denominator so that the denominator becomes real.
A common error is to use $i^2=1$ or to leave a complex number unsimplified instead of collecting it into the form $x+iy$ .

Tier 1 · Easy

2 marks

ORIGINAL

Let $z=(3-2i)+(5+7i)$ . Find $z$ , $\operatorname{Re}(z)$ and $\operatorname{Im}(z)$ .

Tier 2 · Standard

3 marks

ORIGINAL

Express $\dfrac{4+i}{2-i}$ in the form $x+iy$ .

Tier 3 · Hard

5 marks

ORIGINAL

Express $\dfrac{(2+3i)(1-4i)}{3+i}$ in the form $x+iy$ .

CP-2.3

Understand and use the complex conjugate. Know that non-real roots of polynomial equations with real coefficients occur in conjugate pairs.

The conjugate of $z=x+iy$ is $\overline z=x-iy$ , and $z\overline z=x^2+y^2=|z|^2$ is real.
Changing $i$ to $-i$ throughout an expression with real coefficients shows that a non-real root $a+ib$ forces $a-ib$ to be a root as well.
A conjugate pair produces the real quadratic factor $[x-(a+ib)][x-(a-ib)]=(x-a)^2+b^2$ .
The conjugate-root rule requires real polynomial coefficients; applying it without checking this condition is a common error.

Tier 1 · Easy

2 marks

ORIGINAL

For $z=5-4i$ , write down $\overline z$ and calculate $z\overline z$ .

Tier 2 · Standard

4 marks

ORIGINAL

The polynomial $p(x)=x^3-x^2-7x+15$ has real coefficients, and $2+i$ is a root. Solve $p(x)=0$ completely.

Tier 3 · Hard

6 marks

ORIGINAL

A monic quartic polynomial has real coefficients. Two of its roots are $1+2i$ and $-2+i$ . Determine the polynomial in expanded form.

CP-2.4

Use and interpret Argand diagrams.

On an Argand diagram, $z=x+iy$ is represented by the point $(x,y)$ , with the real axis horizontal and the imaginary axis vertical.
Differences of affixes represent displacement vectors, so $|z_2-z_1|$ is the distance between the corresponding points.
Coordinate geometry applies directly: use gradients and scalar products to establish angles, and split polygons into triangles with base-and-height reasoning to establish areas.
Do not interchange the axes or plot $x+iy$ as $(y,x)$ ; label relevant points and exact coordinates on any sketch.

Tier 1 · Easy

2 marks

ORIGINAL

The complex number $z=-2+3i$ is represented by $P$ on an Argand diagram. State the coordinates of $P$ and the quadrant in which it lies.

Tier 2 · Standard

3 marks

ORIGINAL

Points $A$ , $B$ and $C$ have affixes $1+i$ , $5+i$ and $1+4i$ . Determine the exact area of triangle $ABC$ .

Tier 3 · Hard

5 marks

ORIGINAL

Points $A$ and $B$ have affixes $3+i$ and $-1+3i$ , and $O$ is the origin. Prove that triangle $OAB$ is right-angled and isosceles, and find its exact area.

CP-2.5

Convert between the Cartesian form and the modulus-argument form of a complex number.

For $z=x+iy$ , the modulus is $r=|z|=\sqrt{x^2+y^2}$ and an argument $\theta$ satisfies $\tan\theta=y/x$ with the quadrant checked.
The modulus-argument forms are $r(\cos\theta+i\sin\theta)$ and, using Euler notation, $re^{i\theta}$ .
To return to Cartesian form, calculate $x=r\cos\theta$ and $y=r\sin\theta$ ; exact special-angle values should remain exact.
Using $\tan^{-1}(y/x)$ without a quadrant check can give an argument differing by $\pi$ ; state the principal argument when requested.

Tier 1 · Easy

2 marks

ORIGINAL

Express $1+\sqrt3i$ in the form $r(\cos\theta+i\sin\theta)$ , where $r>0$ and $-\pi<\theta\leq\pi$ .

Tier 2 · Standard

3 marks

ORIGINAL

Express $-2-2i$ in modulus-argument form using its principal argument.

Tier 3 · Hard

4 marks

ORIGINAL

The complex number $z=a+ib$ has modulus $10$ , positive real part and principal argument $-\tan^{-1}(3/4)$ . Determine $z$ in Cartesian form.

CP-2.6

Multiply and divide complex numbers in modulus-argument form.

When multiplying complex numbers in modulus-argument form, multiply their moduli and add their arguments.
When dividing, divide the moduli and subtract the denominator's argument from the numerator's argument.
After calculating, add or subtract $2\pi$ if a principal argument in $(-\pi,\pi]$ is required.
Do not add moduli or multiply arguments; those operations do not correspond to complex multiplication.

Tier 1 · Easy

2 marks

ORIGINAL

Express $2e^{i\pi/6}\times3e^{i\pi/4}$ in the form $re^{i\theta}$ .

Tier 2 · Standard

3 marks

ORIGINAL

Express $\dfrac{12e^{5\pi i/6}}{3e^{-\pi i/4}}$ in the form $re^{i\theta}$ using the principal argument.

Tier 3 · Hard

4 marks

ORIGINAL

Given that $z(2e^{i\pi/7})^3=16e^{-2\pi i/7}$ , determine $z$ in the form $re^{i\theta}$ , where $-\pi<\theta\leq\pi$ .

CP-2.7

Construct and interpret simple loci in the Argand diagram such as |z - a| > r and arg(z - a) = theta.

The condition $|z-a|=r$ is a circle of radius $r$ centred at the point with affix $a$ ; $<$ selects the interior and $>$ the exterior.
The condition $\arg(z-a)=\theta$ is a ray starting at $a$ and making directed angle $\theta$ with the positive real axis.
Translate $z=x+iy$ when an equation is needed: modulus conditions become distance equations and argument conditions give a line plus a direction restriction.
Strict inequalities exclude their boundary, and the point $z=a$ is excluded from an argument locus because the argument of zero is undefined.

Tier 1 · Easy

2 marks

ORIGINAL

Describe geometrically the locus $|z-(2-i)|>3$ , stating whether its boundary is included.

Tier 2 · Standard

3 marks

ORIGINAL

Let $z=x+iy$ . Find a Cartesian description of the locus $\arg(z+1)=\pi/4$ , including the required restriction on $x$ .

Tier 3 · Hard

5 marks

ORIGINAL

A region is defined by $|z-(1-i)|<4$ and $0<\arg(z-(1-i))<\pi/3$ . Sketch the region and determine its exact area.

CP-2.8

Understand de Moivre's theorem and use it to find multiple angle formulae and sums of series.

De Moivre's theorem states $[\cos\theta+i\sin\theta]^n=\cos(n\theta)+i\sin(n\theta)$ for integer $n$ .
To derive a multiple-angle identity, expand the left-hand side and equate real parts for cosine or imaginary parts for sine.
For trigonometric sums, write the terms as real or imaginary parts of a geometric series in $e^{i\theta}$ , sum it, then take the required part.
A common error is to equate the whole binomial expansion with only a real or only an imaginary target; separate the two parts first.

Tier 1 · Easy

1 mark

ORIGINAL

Use de Moivre's theorem to write $(\cos\theta+i\sin\theta)^5$ in modulus-argument form.

Tier 2 · Standard

4 marks

ORIGINAL

Starting from de Moivre's theorem, establish the identity $\sin3\theta=3\sin\theta-4\sin^3\theta$ .

Tier 3 · Hard

5 marks

ORIGINAL

Using a complex geometric series, evaluate exactly $S=\displaystyle\sum_{r=0}^{5}\cos\left(\dfrac{r\pi}{4}\right)$ .

CP-2.9

Know and use the definition e^(i theta) = cos theta + i sin theta and the form z = r e^(i theta).

Euler's definition is $e^{i\theta}=\cos\theta+i\sin\theta$ , so $re^{i\theta}$ has modulus $r$ and argument $\theta$ .
The identities $e^{-i\theta}=\cos\theta-i\sin\theta$ , $2\cos\theta=e^{i\theta}+e^{-i\theta}$ and $2i\sin\theta=e^{i\theta}-e^{-i\theta}$ follow immediately.
Exponential form makes rotations and trigonometric rearrangements concise; for example, factor out the exponential with the mean of two arguments.
Do not interpret $e^{i\theta}$ as a real exponential: its modulus is $1$ , and its argument is periodic modulo $2\pi$ .

Tier 1 · Easy

1 mark

ORIGINAL

Express $4\left(\cos\dfrac{2\pi}{5}+i\sin\dfrac{2\pi}{5}\right)$ in exponential form.

Tier 2 · Standard

3 marks

ORIGINAL

Using Euler's definition, show that $e^{2i\theta}+e^{-2i\theta}=2\cos2\theta$ .

Tier 3 · Hard

4 marks

ORIGINAL

For $-\pi<\theta<\pi$ , prove that $1+e^{i\theta}=2\cos(\theta/2)e^{i\theta/2}$ .

CP-2.10

Find the n distinct nth roots of r e^(i theta) for r != 0 and know that they form the vertices of a regular n-gon in the Argand diagram.

If $z^n=re^{i\theta}$ with $r>0$ , the roots are $z=r^{1/n}e^{i(\theta+2k\pi)/n}$ for $k=0,1,\ldots,n-1$ .
There are exactly $n$ distinct roots because their arguments differ by $2\pi/n$ before the pattern repeats.
On an Argand diagram the roots lie on the circle of radius $r^{1/n}$ and form a regular $n$ -gon centred at the origin.
Using only the principal value of the argument gives just one root; include the $2k\pi$ term before dividing by $n$ .

Tier 1 · Easy

3 marks

ORIGINAL

Find the three cube roots of $8$ , giving them in exponential form.

Tier 2 · Standard

4 marks

ORIGINAL

Find the four roots of $z^4=16e^{i\pi/3}$ , giving their arguments in the interval $0\leq\theta<2\pi$ .

Tier 3 · Hard

6 marks

ORIGINAL

The roots of $z^5=32e^{-i\pi/2}$ are plotted on an Argand diagram. Find all five roots in exponential form and determine the exact area of the regular pentagon they form.

CP-2.11

Use complex roots of unity to solve geometric problems.

The $n$ th roots of unity are $1,\omega,\omega^2,\ldots,\omega^{n-1}$ , where $\omega=e^{2\pi i/n}$ , and they are vertices of a regular $n$ -gon.
Because $\omega^n=1$ and $\omega\ne1$ , the geometric sum gives $1+\omega+\cdots+\omega^{n-1}=0$ .
Distances are moduli of differences, while multiplying every affix by $\omega$ rotates the entire figure through $2\pi/n$ without changing lengths.
Do not cancel $\omega-1$ in $\omega^n-1=(\omega-1)(1+\cdots+\omega^{n-1})$ without first stating that the chosen root is not $1$ .

Tier 1 · Easy

2 marks

ORIGINAL

Let $\omega=e^{2\pi i/3}$ . Find the exact distance between the points with affixes $1$ and $\omega$ .

Tier 2 · Standard

4 marks

ORIGINAL

Let $\omega=e^{2\pi i/3}$ . Points $A$ , $B$ and $C$ have affixes $1$ , $\omega$ and $\omega^2$ . Prove that triangle $ABC$ is equilateral.

Tier 3 · Hard

7 marks

ORIGINAL

The fifth roots of unity are the vertices of a regular pentagon. Prove that the ratio of a diagonal to a side is $\dfrac{1+\sqrt5}{2}$ .