AQA A-Level Mathematics – Paper 1 (2025)
– Complete Exam Paper with Questions
Contents
A-Level Maths Paper 1 — Study Guide + Fully Worked Solutions .................................................. 7
PART 1 — STUDY GUIDE ............................................................................................................. 8
1. Differentiation ....................................................................................................................... 8
2. Logarithms............................................................................................................................. 8
3. Sequences and Series ............................................................................................................ 8
4. Binomial Expansion .............................................................................................................. 9
5. Integration ............................................................................................................................. 9
6. Graph Transformations & Sketching ................................................................................. 9
7. Trigonometry......................................................................................................................... 9
8. Parametric Equations ......................................................................................................... 10
9. Newton–Raphson Method .................................................................................................. 10
10. Modelling with Exponentials ........................................................................................... 10
PART 2 — FULL WORKED SOLUTIONS ...................................................................................... 10
1.
A curve is defined by 𝑦 = 3𝑒 𝑥 .
𝑑𝑦
Write down the derivative 𝑑𝑥 .
2.
,A sequence is given by
1
𝑥𝑛+1 = − 4 𝑥𝑛 with 𝑥1 = 32.
The first four terms are:
32, −8,2, −0.5.
3.
Rewrite the expression
log3 (2𝑥) − log3 (𝑥)
as a single logarithm.
4.
The first three terms (in increasing powers of 𝑥) of the binomial expansion of
(1 − 8𝑥)1/2
are
1 + 𝑛𝑥 − 8𝑥 2 ,
where 𝑛is a constant.
(a) Identify the interval of 𝑥-values for which this expansion is valid.
Pick the correct choice.
(b) State the value of 𝑛.
5.
Evaluate
∫ (4𝑥 − 3 + 𝑥 −1/2 ) 𝑑𝑥.
6.
An arithmetic sequence has first term 3 and common difference −0.5.
Find the sum of the first 250 terms.
, 7.
You are told that 0 < 𝑎 < 1.
Sketch the graph of 𝑦 = 𝑎 𝑥 on the axes provided.
8.
For small values of 𝑥, show that the function
4 + sin(5𝑥) − 3cos(2𝑥)
𝑦=
1 + 2tan 𝑥
can be approximated by a straight line of the form
𝑦 = 𝑎𝑥 + 1.
Find the value of the constant 𝑎.
9. (a)
A geometric series 𝑆has second term 60 and common ratio 0.2.
Find the exact sum of the first five terms.
9 (b) A different geometric series 𝑇has second term 60 and a positive common ratio 𝑟. The sum
to infinity is denoted by 𝑇∞ .
(i) Show that
60
𝑇∞ = .
𝑟 − 𝑟2
(ii) Determine the maximum value of 𝑟 − 𝑟 2.
(iii) Hence state the possible range of values for 𝑇∞ .
Give full reasoning.
10.
– Complete Exam Paper with Questions
Contents
A-Level Maths Paper 1 — Study Guide + Fully Worked Solutions .................................................. 7
PART 1 — STUDY GUIDE ............................................................................................................. 8
1. Differentiation ....................................................................................................................... 8
2. Logarithms............................................................................................................................. 8
3. Sequences and Series ............................................................................................................ 8
4. Binomial Expansion .............................................................................................................. 9
5. Integration ............................................................................................................................. 9
6. Graph Transformations & Sketching ................................................................................. 9
7. Trigonometry......................................................................................................................... 9
8. Parametric Equations ......................................................................................................... 10
9. Newton–Raphson Method .................................................................................................. 10
10. Modelling with Exponentials ........................................................................................... 10
PART 2 — FULL WORKED SOLUTIONS ...................................................................................... 10
1.
A curve is defined by 𝑦 = 3𝑒 𝑥 .
𝑑𝑦
Write down the derivative 𝑑𝑥 .
2.
,A sequence is given by
1
𝑥𝑛+1 = − 4 𝑥𝑛 with 𝑥1 = 32.
The first four terms are:
32, −8,2, −0.5.
3.
Rewrite the expression
log3 (2𝑥) − log3 (𝑥)
as a single logarithm.
4.
The first three terms (in increasing powers of 𝑥) of the binomial expansion of
(1 − 8𝑥)1/2
are
1 + 𝑛𝑥 − 8𝑥 2 ,
where 𝑛is a constant.
(a) Identify the interval of 𝑥-values for which this expansion is valid.
Pick the correct choice.
(b) State the value of 𝑛.
5.
Evaluate
∫ (4𝑥 − 3 + 𝑥 −1/2 ) 𝑑𝑥.
6.
An arithmetic sequence has first term 3 and common difference −0.5.
Find the sum of the first 250 terms.
, 7.
You are told that 0 < 𝑎 < 1.
Sketch the graph of 𝑦 = 𝑎 𝑥 on the axes provided.
8.
For small values of 𝑥, show that the function
4 + sin(5𝑥) − 3cos(2𝑥)
𝑦=
1 + 2tan 𝑥
can be approximated by a straight line of the form
𝑦 = 𝑎𝑥 + 1.
Find the value of the constant 𝑎.
9. (a)
A geometric series 𝑆has second term 60 and common ratio 0.2.
Find the exact sum of the first five terms.
9 (b) A different geometric series 𝑇has second term 60 and a positive common ratio 𝑟. The sum
to infinity is denoted by 𝑇∞ .
(i) Show that
60
𝑇∞ = .
𝑟 − 𝑟2
(ii) Determine the maximum value of 𝑟 − 𝑟 2.
(iii) Hence state the possible range of values for 𝑇∞ .
Give full reasoning.
10.
