Use this as a self-guided practice sequence or return directly to a topic.

1. Attempt the question in your notebook.
2. Open a hint only if you need one.
3. Compare with the worked solution.
4. Correct and explain the first step you would change.

For principal arguments use −π < Arg(z) ≤ π. GDC and your usual formula reference are allowed.

Use your own D3 working to decide.

Did you find both roots, −1 ± 2i, and explain that y = (x + 1)² + 4 is always positive?

If either step needed the worked solution, choose the quadratic route. If both were independent and correct, choose consolidation.

1. Find the vertex of the real graph.
2. Solve y = 0 over the complex numbers.
3. Plot both solutions on an Argand diagram.

GDC: inspect the vertex on the real graph. Use a complex-capable solver to check the roots, or substitute your algebraic answers.

Hodder 6A, pp.195–196; related practice Q22–24, p.198.

Find both solutions and give their Argand coordinates.

Check the sign of the discriminant before using your GDC.

Formula: x = [−b ± √(b² − 4ac)] / (2a)

Find a + b and ab. Explain why the methods differ.

Then revisit D1 or D2 and correct your first incorrect step.

Related comparison: Hodder 6B Q20, p.207. If these are secure, continue to the sine-and-baseline task.

For two signals with the same angular frequency, what can you factor out?

Find the fixed sum first:

1. Predict the quadrant of the resultant.
2. Write f(t) as one cosine with positive amplitude.
3. Give its period and check the value at t = 0.

Useful form: Z = 2e^πi/3 + 4e^πi

Write this in the form C + R sin(2t + φ), with R positive.

What changes about the amplitude? What happens to the baseline and angular frequency?

Use the imaginary part for sine. Baselines add separately. Related transfer: Hodder 6B Q29 and Q32, p.208.

1. Sketch both signals for 0 ≤ t ≤ 0.5.
2. Write the total in the form R cos(30t + φ), with R positive.

Record the fixed complex sum before converting it.

Hodder 6B Q31, p.208. Radians throughout. Take t in seconds and voltage in volts for the interpretation that follows.

Use your unrounded amplitude and phase from the previous task.

1. Find the period.
2. Find the first time t ≥ 0 when the total voltage is maximum.
3. Check using your GDC. Explain both coordinates of the maximum.

Extension based on Hodder 6B Q31. For a maximum, 30t + φ = 2kπ.

Give exponential form with a principal argument, then exact Cartesian form.

Keep the modulus and angle steps visible.

−π < Arg(w) ≤ π. For nonzero z = re^iθ, zⁿ = rⁿe^inθ.

Compare one completed solution with the worked answer.

1. Underline the first step that needs changing.
2. Rewrite that step and finish the calculation.
3. Explain the check that now supports your answer.

Without calculating an arctangent, find the resultant amplitude.

What can you say about its phase?

Write this as one cosine with positive amplitude. State its period.

Show the fixed phasors and their sum. Work independently before checking.

Choose a topic to revisit, or work through all three.

• Compound growth and a whole-year threshold
• Every solution of a sinusoidal equation
• Angles and areas using 3D vectors

A balance of RM6000 earns 4% compound interest once per year. There are no deposits or withdrawals.

Let n count completed years, so n is a nonnegative integer.

1. Find the balance after 8 years.
2. Find the first end-of-year balance that is at least RM9000. State the year number.

Model: A(n) = P(1 + r)ⁿ. You may use logs, a numerical solver or a graph.

A model for water height is

t is hours after midnight and h is measured in metres.

1. State the amplitude, midline and period.
2. Find every time in this interval when h(t) = 4.5 m.

Check radian mode and the full domain. A graph can check the number of crossings.

Coordinates are measured in metres.

1. Find the vectors AB and AC.
2. Find angle BAC in degrees.
3. Find the area of triangle ABC.

u · v = |u||v| cos θ
Triangle area = ½|u × v|

Choose one answer that needed a correction.

1. Show the first incorrect step.
2. Write the corrected step and finish it.
3. Name the reason: model/setup, method, GDC/mode or interpretation.

Choose the skill that needed most attention in your own work.

• Growth: you rounded the threshold incorrectly or forgot the whole-year domain.
• Trigonometry: you found only one solution or missed the interval.
• Vectors: your vectors did not start at the same point.

If all three were secure, choose any one, or attempt all three.

n counts completed years.

Find the first whole year for which A(n) is at least 7500.

Find every solution. Give answers in radians.

Find angle BAC.

Keep one independent answer and one explained correction in your notes.

Record which method you can now use without help and which needs another attempt.

When you return, try the relevant question with its solution closed, then compare your working.