Challenging +1.2 This is a 6-mark AS-level proof requiring binomial expansion and algebraic manipulation. Students must expand both terms, observe that odd powers cancel, combine even powers, and recognize the resulting expression is non-negative. While it requires careful algebra and insight to see the cancellation pattern, it's a structured problem with a clear path once the expansion is done. The inequality proof is straightforward after simplification. Slightly above average difficulty for AS level due to the multi-step algebraic reasoning required.
Question 7:
7 | Expands at least one bracket –
must reach a quartic | 3.1a | M1 | (2 + 3y)4 = 24 + 4×23×(3y) + 6×22×(3y)2
+ 4×2×(3y)3 + (3y)4
= 16 + 96y + 216y2 + 216y3 + 81y4
(2 – 3y)4 = 24 + 4×23×(–3y) +
6×22×(–3y)2 + 4×2×(–3y)3 + (–3y)4
= 16 - 96y + 216y2 - 216y3 + 81y4
(2 + 3y)4 + (2 – 3y)4
= 32 + 432y2 + 162y4
y2 ≥ 0 and y4 ≥ 0 for all y
(2 + 3y)4 + (2 – 3y)4 ≥ 32 for all y
Obtains at least one correct
expansion – not necessarily
simplified | 1.1b | A1
Obtains two correct expansions –
not necessarily simplified | 1.1b | A1
Combines their expansions to
obtain a sum containing only even
power terms. | 1.1a | M1
Explains that y2 and y4 are always
positive or zero (for )
Or finds minimum value using
calculus and justifies𝑦𝑦 th∈isℝ as not just
a local minimum | 2.4 | E1
Reaches correct conclusion. Sets
out a well-constructed mathematical
argument. R1 can be awarded if E1
not given. | 2.1 | R1
Total | 6
Q | Marking Instructions | AO | Marks | Typical Solution
Given that $y \in \mathbb{R}$, prove that
$$(2 + 3y)^4 + (2 - 3y)^4 \geq 32$$
Fully justify your answer.
[6 marks]
\hfill \mbox{\textit{AQA AS Paper 1 2019 Q7 [6]}}