Edexcel A Level Maths June 2023 Paper 2: question paper and mark scheme (9MA0/02)

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. Algebra and functions · 5 marks

   Let $f(x)=x^3+c{x}^2+dx-4$. Given $f(x)$ has a factor $(x+2)$ and leaves a remainder of $-10$ when divided by $(x-3)$, find the values of $c$ and $d$.

   Worked solution

   1. M1 for applying the Factor Theorem for $(x+2)$: $f(-2)=0⟹-8+4c-2d-4=0⟹4c-2d=12$.
   2. M1 for applying the Remainder Theorem for $(x-3)$: $f(3)=-10⟹27+9c+3d-4=-10⟹9c+3d=-33$.
   3. M1 for simplifying both equations (e.g., $2c-d=6$ and $3c+d=-11$).
   4. M1 for solving the simultaneous equations (e.g., adding them gives $5c=-5$).
   5. A1 for finding the correct values $c=-1$ and $d=-8$.

2. Trigonometry · 5 marks

   Solve $2{\sin }^{2}x+3\cos x-3=0$ for $0^{\circ }\le x\le {360}^{\circ }$.

   Worked solution

   1. M1 for substituting ${\sin }^{2}x$ with $1-{\cos }^{2}x$ to get $2(1-{\cos }^{2}x)+3\cos x-3=0$.
   2. M1 for expanding and rearranging to form the quadratic $2{\cos }^{2}x-3\cos x+1=0$.
   3. M1 for factorising the quadratic as $(2\cos x-1)(\cos x-1)=0$.
   4. A1 for finding the solutions from $\cos x=0.5$ giving $x={60}^{\circ }$ and ${300}^{\circ }$.
   5. A1 for finding the solutions from $\cos x=1$ giving $x=0^{\circ }$ and ${360}^{\circ }$.

3. Differentiation · 5 marks

   Find the equation of the normal to the curve $y=x^2{e}^{-x}$ at the point where $x=1$. Give your answer in the form $y=mx+c$.

   Worked solution

   1. M1 for finding the y-coordinate at $x=1$: $y=e^{-1}$.
   2. M1 for applying the product rule to find $\frac{dy}{dx}=2x{e}^{-x}-x^2{e}^{-x}$.
   3. M1 for evaluating the gradient at $x=1$: $m_t=2e^{-1}-e^{-1}=e^{-1}$.
   4. M1 for finding the perpendicular gradient of the normal: $m_n=-e$.
   5. A1 for forming the equation $y-e^{-1}=-e(x-1)$ and rearranging to $y=-ex+e+e^{-1}$.

More on these topics: Algebra and functions · Trigonometry · Differentiation

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: Q13, 13 marks: binomial expansion and an approximation.
• Marks by main topic: Sequences and series 19, Exponentials and logarithms 19, Integration 15, Trigonometry 13, Differentiation 11, Coordinate geometry 7, Algebra and functions 7, Vectors 6, Proof 3.
• More Sequences and series in other sittings: June 2022 Q15 (10 marks), June 2024 Q2 (5 marks), November 2021 Q1 (4 marks).
• Against the other 7 sittings we mapped: more marks on Exponentials and logarithms (19 marks here, 7 on average) and Sequences and series (19 marks here, 9 on average); fewer on Algebra and functions (7 marks here, 20 on average) and Differentiation (11 marks here, 22 on average).
• Not on this paper: Numerical methods.
• Calculator: allowed.

9MA0/02: question by question

Labels and topics are ours, read from the 9MA0/02 paper (checked 2026-11-01); marks as printed. Tap a number to practise that topic.

1. Q1Second derivative and where a curve is concave (4 marks) · Differentiation
2. Q2Recurrence with a cosine term (6 marks) · Sequences and series, Trigonometry
3. Q3Log equation reduced to a quadratic (5 marks) · Exponentials and logarithms
4. Q4Exponential cooling model for coffee (4 marks) · Exponentials and logarithms
5. Q5Curve from its derivative and a turning point (5 marks) · Integration
6. Q6Vectors and a ratio of lengths (6 marks) · Vectors
7. Q7Implicit differentiation and a tangent (7 marks) · Differentiation
8. Q8R cos(x - a) form and its greatest value (6 marks) · Trigonometry
9. Q9Parametric curve with ln to Cartesian (7 marks) · Coordinate geometry, Exponentials and logarithms
10. Q10Partial fractions, then integration (7 marks) · Algebra and functions, Integration
11. Q11Rates of change in a filling tank (10 marks) · Integration, Differentiation
12. Q12Two subscriber growth models (10 marks) · Exponentials and logarithms
13. Q13Binomial expansion and an approximation (13 marks) · Sequences and series
14. Q14Tan identity used to solve an equation (7 marks) · Trigonometry
15. Q15Spotting the error in a proof by contradiction (3 marks) · Proof