Edexcel A Level Maths June 2023 Paper 1: question paper and mark scheme (9MA0/01)

3 A Level questions like these, with worked solutions

Written and checked by us, not taken from the Pearson Edexcel paper. Try each one, then tap "Worked solution" to see every step and its marks.

1. Differentiation · 5 marks

   A curve has the equation $y=e^{-x}\sin x$. Find the exact x-coordinate of the first positive turning point of the curve.

   Worked solution

   1. M1 for using the product rule to differentiate: $\frac{dy}{dx}=-e^{-x}\sin x+e^{-x}\cos x$.
   2. M1 for setting the derivative equal to 0 to find turning points: $e^{-x}(\cos x-\sin x)=0$.
   3. M1 for dividing by $e^{-x}$ (since $e^{-x}\ne 0$) to get $\cos x=\sin x$.
   4. M1 for dividing by $\cos x$ to obtain $\tan x=1$.
   5. A1 for identifying the first positive solution as $x=\frac{\pi }{4}$.

2. Integration · 7 marks

   Find the exact total area of the finite regions bounded by the curves $y=x^3+x^2$ and $y=2x^2+2x$.

   Worked solution

   1. M1 for setting $x^3+x^2=2x^2+2x⟹x^3-x^2-2x=0$.
   2. M1 for factoring $x(x-2)(x+1)=0$ to find the intersection points $x=-1,0,2$.
   3. M1 for recognising two distinct regions and setting up integrals: ${\int }_{-1}^0(x^3-x^2-2x)dx$ and ${\int }_0^2(2x^2+2x-x^3-x^2)dx$.
   4. M1 for integrating the generic polynomial to $\pm [\frac{x^4}{4}-\frac{x^3}{3}-x^2]$.
   5. M1 for evaluating the area of the first region to $\frac{5}{12}$.
   6. M1 for evaluating the area of the second region to $\frac{8}{3}$.
   7. A1 for calculating the combined total area: $\frac{5}{12}+\frac{8}{3}=\frac{37}{12}$.

3. Sequences and series · 5 marks

   The binomial expansion of $(1+ax{)}^b$, where $b$ is a positive integer, starts with $1-15x+90x^2$. Find the values of $a$ and $b$.

   Worked solution

   1. M1 for equating the $x$ coefficient: $ab=-15$.
   2. M1 for equating the $x^2$ coefficient: $\frac{b(b-1)}{2}{a}^2=90⟹b(b-1)a^2=180$.
   3. M1 for substituting $(ab)=-15$ or expanding to $a^2{b}^2-a^{2}b=180$ and substituting $a^2{b}^2=225$.
   4. M1 for solving for $a$: $225-a(-15)=180⟹15a=-45⟹a=-3$.
   5. A1 for finding the corresponding value $b=5$.

More on these topics: Differentiation · Integration · Sequences and series

What this paper covers

What this paper is like

• Shape: 15 questions, 100 marks in 2 h (about 1.2 minutes a mark); questions from 3 to 13 marks.
• Biggest question: Q15, 13 marks: quotient rule with exponentials and an iteration.
• Marks by main topic: Trigonometry 22, Differentiation 18, Algebra and functions 14, Exponentials and logarithms 13, Coordinate geometry 9, Sequences and series 7, Numerical methods 6, Integration 4, Proof 4, Vectors 3.
• More Differentiation in other sittings: June 2022 Q8 (8 marks), June 2024 Q5 (6 marks), November 2021 Q4 (9 marks).
• Against the other 7 sittings we mapped: more marks on Trigonometry (22 marks here, 14 on average) and Exponentials and logarithms (13 marks here, 7 on average); fewer on Integration (4 marks here, 16 on average).
• Coverage: every topic on the list came up.
• Calculator: allowed.

9MA0/01: question by question

Labels and topics are ours, read from the 9MA0/01 paper (checked 2026-11-01); marks as printed. Tap a number to practise that topic.

1. Q1Integrating a polynomial (4 marks) · Integration
2. Q2Factor theorem with a constant a (6 marks) · Algebra and functions
3. Q33D vectors and a distance (3 marks) · Vectors
4. Q4Small-angle approximation for a stationary point (5 marks) · Trigonometry, Differentiation
5. Q5Trapezium rule from a table of values (6 marks) · Numerical methods
6. Q6Laws of logarithms in terms of a and b (6 marks) · Exponentials and logarithms
7. Q7Range, inverse and composite functions (8 marks) · Algebra and functions
8. Q8Radians: a stage design with sectors (10 marks) · Trigonometry
9. Q9Geometric sequence with an unknown k (7 marks) · Sequences and series
10. Q10Circle with an unknown k and a tangent (9 marks) · Coordinate geometry
11. Q11Log-linear model for a phone's value (7 marks) · Exponentials and logarithms
12. Q12Differentiating sin x from first principles (5 marks) · Differentiation, Trigonometry
13. Q13Roller coaster height modelled with a trig function (7 marks) · Trigonometry
14. Q14Proof that an expression is odd (4 marks) · Proof
15. Q15Quotient rule with exponentials and an iteration (13 marks) · Differentiation, Numerical methods