AS 91577 · 2025 paper · two points on an Argand diagram, a fifth power, a modulus, comparing parts, and an argument
Question 2(a) Achieved
Practises Argand diagram & polar form
Five separate parts: a combination of two points read off an Argand diagram, a power of a number in polar form, an equation with a modulus in it, two unknowns found by comparing real and imaginary parts, and an unknown found from an argument.
On the paper’s Argand diagram, u is the point (4, 2) and w is the point (−2, −3).
If z = 2u + 3w, find z, and show it clearly on an Argand diagram.
| u | = | 4 + 2i |
|---|---|---|
| w | = | −2 − 3i |
Work it out
Try it on paper first. Stuck? Take one hint at a time.
-
1 Read u and w off the diagram as complex numbers.
u is at (4, 2), so u = 4 + 2i. w is at (−2, −3), so w = −2 − 3i. -
2 Multiply each by its number, then add the real parts and the imaginary parts separately.
2u = 8 + 4i and 3w = −6 − 9i
z = (8 − 6) + (4 − 9)i = 2 − 5i -
3 Where does it go on the diagram?
At the point (2, −5): 2 along the real axis, then 5 down.
What the marker wanted
- Achieved z worked out, and its point marked on the diagram: both are needed.
In our words, from NZQA’s 2025 marking schedule.
How did it go?
Got it moves it further out. Review brings it back tomorrow.