Edexcel IAL P2 June 2023 mark scheme and question paper (WMA12/01)

3 P2 questions like these, with worked solutions

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

1. Algebraic Methods · 5 marks

   When $f(x)=2x^3+p{x}^2+qx+1$ is divided by $(2x-1)$ the remainder is 2. When $f(x)$ is divided by $(x+1)$ the remainder is -4. Find the values of $p$ and $q$.

   Worked solution

   1. M1 for applying the Remainder Theorem with $x=0.5$: $f(0.5)=2⟹2(\frac{1}{8})+\frac{p}{4}+\frac{q}{2}+1=2$.
   2. M1 for simplifying the first equation: $\frac{1}{4}+\frac{p}{4}+\frac{2q}{4}=1⟹p+2q=3$.
   3. M1 for applying the Remainder Theorem with $x=-1$: $f(-1)=-4⟹-2+p-q+1=-4⟹p-q=-3$.
   4. M1 for solving the simultaneous equations (e.g., subtracting them gives $3q=6$).
   5. A1 for stating $q=2$ and $p=-1$.

2. The Binomial Expansion · 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 $b=5$.

3. Integration (definite integrals and area) · 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: $\frac{5}{12}+\frac{8}{3}=\frac{37}{12}$.

More on these topics: Algebraic Methods · The Binomial Expansion · Integration (definite integrals and area)

What's on this paper

What this paper is like

• Shape: 11 questions, 75 marks in 1 h 30 min (about 1.2 minutes a mark); questions from 3 to 10 marks.
• Biggest question: Q10, 10 marks: curve with a constant k: stationary point and area.
• Marks by main topic: Differentiation (stationary points) 18, Sequences and Series 17, Circles 7, The Binomial Expansion 7, Further Trigonometry 7, Algebraic Methods 6, Logarithms and Exponentials 5, Proof 5, Integration (definite integrals and area) 3.
• More Differentiation (stationary points) in other sittings: October 2023 Q9 (12 marks), January 2023 Q2 (9 marks), January 2024 Q10 (9 marks).
• Against the other 16 sittings we mapped: more marks on Differentiation (stationary points) (18 marks here, 10 on average) and Sequences and Series (17 marks here, 13 on average); fewer on Integration (definite integrals and area) (3 marks here, 9 on average) and Logarithms and Exponentials (5 marks here, 11 on average).
• Coverage: every topic on the list came up.
• Calculator: allowed.

WMA12/01: question by question

Labels and topics are ours, read from the WMA12/01 paper (checked 2026-11-01); marks as printed. Tap a number to practise that topic.

1. Q1Trapezium rule from a table (3 marks) · P2 Integration (definite integrals and area)
2. Q2Factor theorem with a constant a (6 marks) · P2 Algebraic Methods
3. Q3Circle through a point and a tangent (7 marks) · P2 Circles
4. Q4Binomial expansion with a constant p (7 marks) · P2 The Binomial Expansion
5. Q5Log equation reduced to a quadratic (5 marks) · P2 Logarithms and Exponentials
6. Q6App downloads as a geometric series (9 marks) · P2 Sequences and Series, P2 Logarithms and Exponentials
7. Q7Model for a river's height and its maximum (8 marks) · P2 Differentiation (stationary points)
8. Q8Disproving a claim about consecutive primes (5 marks) · P2 Proof
9. Q9Trig identity, then an equation (7 marks) · P2 Further Trigonometry
10. Q10Curve with a constant k: stationary point and area (10 marks) · P2 Differentiation (stationary points), P2 Integration (definite integrals and area)
11. Q11Recurrence relation with constants a and b (8 marks) · P2 Sequences and Series

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