Complex numbers – further maths worksheet with answers to download as PDF

Let \(z=3-2i\) and \(w=1+4i\). Write each answer in the form \(a+bi\) and show your working.

1. \(z+w\) and \(2z-3w\)
2. \(zw\)
3. \(\dfrac{z}{w}\), by multiplying top and bottom by the conjugate of \(w\)
4. \(zz^*\) and \(|z|\)

Solve each quadratic equation, giving the roots in the form \(p\pm qi\).

1. \(z^2-6z+25=0\)
2. \(z^2+2z+10=0\)
3. Find the quadratic equation with real coefficients that has \(2+i\) as a root.
4. For which real values of \(k\) does \(z^2+kz+9=0\) have non-real roots?

Write each number in the form \(r(\cos\theta+i\sin\theta)\) with \(-\pi\lt \theta\le\pi\). Sketch the point first to find the quadrant.

1. \(1+i\)
2. \(-\sqrt3+i\)
3. \(-2-2i\)
4. \(5-5\sqrt3\,i\)

Let \(z_1=2\left(\cos\tfrac\pi3+i\sin\tfrac\pi3\right)\) and \(z_2=4\left(\cos\tfrac\pi6+i\sin\tfrac\pi6\right)\).

1. Find \(|z_1z_2|\) and \(\arg(z_1z_2)\), and hence write \(z_1z_2\) in the form \(a+bi\).
2. Find \(\left|\dfrac{z_1}{z_2}\right|\) and \(\arg\dfrac{z_1}{z_2}\).
3. Find \(|z_1^2|\) and \(\arg(z_1^2)\).
4. Find \(\arg(z_1z_2^*)\).

Describe each locus in words, give its Cartesian equation where possible, and sketch it.

1. \(|z-2|=3\)
2. \(|z+1-i|\le2\)
3. \(\arg(z-1)=\tfrac\pi4\)
4. \(|z-1|=|z+3|\)

Find the two square roots of each number in the form \(a+bi\).

1. \(8-6i\)
2. \(-15+8i\)
3. Solve \(z^2=3+4i\).

In an alternating current circuit a resistor and an inductor in series have impedance \(Z=3+4i\) ohms: the real part is the resistance and the imaginary part is the reactance. A voltage \(V=20\) volts is applied, and the current is \(I=V/Z\).

1. Find \(|Z|\), and write the current \(I\) in the form \(a+bi\).
2. Find \(|I|\) amperes and the phase angle \(\arg I\), correct to 3 decimal places. Does the current lead or lag the voltage?
3. A second branch has impedance \(Z_2=6-8i\) and is connected in parallel with the first. The combined impedance satisfies \(\dfrac1{Z_T}=\dfrac1{Z}+\dfrac1{Z_2}\). Find \(Z_T\) in the form \(a+bi\), with \(a\) and \(b\) to 2 decimal places.

A wind turbine has three identical blades of length 45 m, spaced \(120^\circ\) apart. Model the hub as the origin of an Argand diagram with distances in metres. At one instant the tip of blade A is at \(z_A=45\left(\cos\tfrac\pi6+i\sin\tfrac\pi6\right)\).

1. Write \(z_A\) in the form \(a+bi\), giving exact values.
2. Explain why the tip of blade B is at \(z_B=z_A\left(\cos\tfrac{2\pi}3+i\sin\tfrac{2\pi}3\right)\) when it is the next blade anticlockwise. Find its modulus and argument.
3. Show that \(z_A+z_B+z_C=0\), where \(z_C\) is the third tip.
4. Find the exact distance between two blade tips.

A survey drone hovers at the point \(P\) with complex number \(-3+2i\) (distances in metres). A charging point is at \(C=1+i\). The drone is told to turn through \(90^\circ\) anticlockwise about \(C\), which sends \(P\) to \(P^\prime\) with \(p^\prime-c=i(p-c)\).

1. Find the complex number \(p^\prime\).
2. Find the distance \(PP^\prime\).
3. The drone then patrols the circle \(|z-C|=5\). Find the two points where the patrol circle crosses the real axis.

An electronic filter has complex gain \(G(\omega)=\dfrac1{1+0.01\,i\omega}\), where \(\omega\) is the angular frequency of the signal in rad s\(^{-1}\). The modulus \(|G|\) tells you how much the signal is reduced, and \(\arg G\) is the phase shift.

1. Complete the table, with \(G\) in the form \(a+bi\) and \(|G|\), \(\arg G\) to 3 decimal places.
2. Show that \(|G|=\dfrac1{\sqrt{1+10^{-4}\omega^2}}\) and explain why \(|G|\) decreases as \(\omega\) increases.
3. Find the frequency at which the signal is reduced to half its size.

On a chart, lighthouse \(L_1\) is at the origin and lighthouse \(L_2\) is at the point \(8+6i\) (distances in kilometres). A ship receives signals showing that it is the same distance from both lighthouses. The ship is also within the 10 km range of \(L_1\).

1. Write the ship's condition as \(|z|=|z-8-6i|\) and show that its Cartesian equation is \(4x+3y=25\).
2. Write down the inequality for the range of \(L_1\) and find where the line \(4x+3y=25\) meets the circle \(|z|=10\). Give exact coordinates.
3. Hence find the length of ship track that lies inside the range of \(L_1\).

Computer graphics programs draw fractals by repeating \(z_{n+1}=z_n^2+c\) from \(z_0=0\), where \(c\) is a complex number. If \(|z_n|\) ever exceeds 2, the sequence escapes to infinity and the point \(c\) is coloured as outside the set.

1. Take \(c=i\). Find \(z_1,z_2,z_3,z_4\).
2. What happens next? Explain why \(c=i\) is inside the set.
3. Take \(c=1+i\) and find \(z_1,z_2,z_3\). Show that \(|z_3|>2\).

Engineers testing the vibrations of a footbridge find that its response depends on the roots of the characteristic cubic \(z^3+6z^2+21z+26=0\). The bridge is stable only when every root has a negative real part.

1. Show that \(z=-2\) is a root.
2. Find the other two roots in the form \(p\pm qi\).
3. Is the bridge stable? Explain how the roots show it.
4. Sketch the three roots on an Argand diagram and state the sum of the roots.

Two tidal effects at a harbour have heights (in metres) \(h_1=3\cos\omega t\) and \(h_2=2\cos\left(\omega t+\tfrac\pi3\right)\). Oceanographers add them by treating each as the real part of a complex number, \(3e^{i\omega t}\) and \(2e^{i(\omega t+\pi/3)}\).

1. Write the sum of the amplitudes \(3+2\left(\cos\tfrac\pi3+i\sin\tfrac\pi3\right)\) in the form \(a+bi\).
2. Find the modulus and argument of this number to 3 decimal places.
3. Hence write \(h_1+h_2\) in the form \(R\cos(\omega t+\phi)\), and state the greatest height.

The quartic polynomial \(f(z)=z^4-6z^3+18z^2-30z+25\) has real coefficients.

1. Show that \(z=2+i\) is a root of \(f(z)=0\).
2. Write down a second root, and hence find a quadratic factor of \(f(z)\) with real coefficients.
3. Factorise \(f(z)\) completely over the real numbers and solve \(f(z)=0\).
4. Plot the four roots on an Argand diagram. What shape do they make and what is its area?

Let \(z_1\) and \(z_2\) be any complex numbers.

1. Use \(|z|^2=zz^*\) to prove that \(|z_1+z_2|^2+|z_1-z_2|^2=2\left(|z_1|^2+|z_2|^2\right)\).
2. Check the result for \(z_1=3+i\) and \(z_2=1-2i\).
3. Interpret the result geometrically for the parallelogram with vertices \(0,z_1,z_1+z_2,z_2\).
4. Prove that \(z+\dfrac1z\) is real if and only if \(z\) is real or \(|z|=1\).

The region \(R\) is defined by \(|z-3-4i|\le5\).

1. Show that the origin lies on the boundary of \(R\) and find the Cartesian equation of the boundary.
2. Find where the boundary crosses the imaginary axis.
3. Find the greatest and least values of \(|z|\) for points in \(R\), and the complex number at which the greatest value occurs.
4. The tangent to the circle at the origin has gradient \(-\tfrac34\). Find the greatest and least values of \(\arg z\) for points on the boundary other than the origin, to 3 decimal places.
5. Find the area of the part of \(R\) with \(\operatorname{Re}(z)\ge3\).

Three drones fly in an equilateral triangle formation. In an Argand diagram representing the sky, the drones are at the points \(0\), \(z_1\) and \(z_2\).

1. Suppose \(z_2=z_1\left(\cos\tfrac\pi3+i\sin\tfrac\pi3\right)\). Explain why the triangle is equilateral.
2. Show that \(z_1^2+z_2^2=z_1z_2\).
3. Take \(z_1=6\). Find \(z_2\) in the form \(a+bi\) and verify the identity numerically.
4. Find the position of the centroid \(\tfrac13(0+z_1+z_2)\) and show that it is the same distance from all three drones.

The cubic equation \(z^3+pz^2+qz+r=0\) has real coefficients \(p,q,r\). Two of its roots are \(3+2i\) and \(3-2i\), and the third root is the real number \(\alpha\). The product of all three roots is 13.

1. Explain why \(3-2i\) must be a root.
2. Find \(\alpha\).
3. Find \(p\), \(q\) and \(r\).
4. Verify your cubic by substituting \(z=3+2i\).
5. Write the cubic as the product of a linear and a real quadratic factor.

Let \(z=\cos\theta+i\sin\theta\) with \(-\pi\lt \theta\lt \pi\), so that \(|z|=1\).

1. Show that \(1+z=2\cos\tfrac\theta2\left(\cos\tfrac\theta2+i\sin\tfrac\theta2\right)\) and find \(|1+z|\) and \(\arg(1+z)\).
2. Hence show that \(\dfrac{z-1}{z+1}\) is purely imaginary, and find it in terms of \(\theta\).
3. Check your answer for \(z=i\).
4. Explain geometrically why the points \(0\), \(-1\) and \(1\) give a right angle at \(z\) when \(z\) is on the unit circle.