AS 91577 · 2024 paper · a surd fraction, a square on the Argand diagram, a conjugate equation, fourth roots, and a cubic identity
Question 3(a) Achieved
Practises Arithmetic & the conjugate
Five separate parts: a fraction of surds simplified, a complex number squared and plotted, an equation in a number and its conjugate, the fourth roots of a negative number, and a cube of x + 1/x.
Write √2p/(√2p − √p) in the form a + √a, where p and a are real constants.
Work it out
Try it on paper first. Stuck? Take one hint at a time.
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1 Every term has a √p in it. Take it out first.
√2p = √2 √p, so
(√2 √p)/(√2 √p − √p) = (√2 √p)/(√p (√2 − 1)) = √2/(√2 − 1) -
2 Rationalise: multiply top and bottom by the conjugate of √2 − 1.
(√2(√2 + 1))/((√2 − 1)(√2 + 1)) = (2 + √2)/(2 − 1) = 2 + √2 -
3 Match it to a + √a.
a = 2, and 2 + √2 is exactly that shape. The p has gone completely.
What the marker wanted
- Achieved The correct simplified form.
In our words, from NZQA’s 2024 marking schedule.
✕Common mistake
Getting the rationalising wrong. NZQA’s 2024 report, in its general comments on the paper, lists not rationalising a denominator correctly among the things Not Achieved scripts had in common.How did it go?
Got it moves it further out. Review brings it back tomorrow.