AQA A Level Mathematics Paper 2
Question 1: Binomial Expansion with Unknown Coefficients
The expansion of (1 + 𝑎𝑥)𝑛 , where 𝑎 > 0 and 𝑛 ∈ ℕ, has a coefficient of 60𝑥 2 and a coefficient
of 240𝑥 3 .
(a) Show that 3𝑎(𝑛 − 1) = 4.
(b) Given 𝑎 = 2, find the value of 𝑛.
(c) Determine the coefficient of 𝑥 4 in the expansion.
Answer:
Question 2: Integration with Logarithmic Substitution
ln(𝑥 2 +1)
Evaluate ∫ 𝑑𝑥 using the substitution 𝑢 = 𝑥 2 + 1.
𝑥
Answer:
, Question 3: Parametric Equations & Area
A curve is defined parametrically by:
𝑥 = 𝑡 2 − 1, 𝑦 = 𝑡 3 − 3𝑡 (−2 ≤ 𝑡 ≤ 2).
(a) Find the area enclosed by the loop of the curve.
(b) Determine the coordinates of the points where the curve intersects itself.
Answer
Question 4: Proof by Induction (Inequality)
Prove by induction that for all integers 𝑛 ≥ 1:
𝑛
1
∑ > 2√𝑛 + 1 − 2.
𝑟=1
√ 𝑟
Answer:
Question 1: Binomial Expansion with Unknown Coefficients
The expansion of (1 + 𝑎𝑥)𝑛 , where 𝑎 > 0 and 𝑛 ∈ ℕ, has a coefficient of 60𝑥 2 and a coefficient
of 240𝑥 3 .
(a) Show that 3𝑎(𝑛 − 1) = 4.
(b) Given 𝑎 = 2, find the value of 𝑛.
(c) Determine the coefficient of 𝑥 4 in the expansion.
Answer:
Question 2: Integration with Logarithmic Substitution
ln(𝑥 2 +1)
Evaluate ∫ 𝑑𝑥 using the substitution 𝑢 = 𝑥 2 + 1.
𝑥
Answer:
, Question 3: Parametric Equations & Area
A curve is defined parametrically by:
𝑥 = 𝑡 2 − 1, 𝑦 = 𝑡 3 − 3𝑡 (−2 ≤ 𝑡 ≤ 2).
(a) Find the area enclosed by the loop of the curve.
(b) Determine the coordinates of the points where the curve intersects itself.
Answer
Question 4: Proof by Induction (Inequality)
Prove by induction that for all integers 𝑛 ≥ 1:
𝑛
1
∑ > 2√𝑛 + 1 − 2.
𝑟=1
√ 𝑟
Answer:
