Edexcel IAL P2 January 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. Logarithms and Exponentials · 4 marks

   Given that ${\log }_{2}x=p$ and ${\log }_{2}y=q$, express ${\log }_2(8x^2{y}^3)$ in terms of $p$ and $q$.

   Worked solution

   1. M1 for splitting the logarithm into a sum: ${\log }_{2}8+{\log }_2{x}^2+{\log }_2{y}^3$.
   2. B1 for evaluating ${\log }_{2}8=3$.
   3. M1 for applying the power law: $2{\log }_{2}x+3{\log }_{2}y$.
   4. A1 for the final expression $3+2p+3q$.

2. Proof · 6 marks

   Prove that $n^4-n^2$ is a multiple of 12 for every integer $n$.

   Worked solution

   Factorises: $n^4-n^2=n^2(n-1)(n+1)$ M1

   $(n-1)n(n+1)$ is a product of three consecutive integers, so one factor is a multiple of 3 B1

   Considers $n$ even and $n$ odd M1

   $n$ even: $n=2k$, so $n^2=4k^2$ is a multiple of 4 A1

   $n$ odd: $n-1$ and $n+1$ are both even, so $(n-1)(n+1)$ is a multiple of 4 A1

   The number is a multiple of 3 and of 4, which have no common factor, so it is a multiple of 12 A1 (cso)

3. Integration (definite integrals and area) · 4 marks

   Find the exact value of ${\int }_4^9(\sqrt{x}+\frac{1}{\sqrt{x}})dx$.

   Worked solution

   1. M1 for writing as fractional indices: $\int (x^{\frac{1}{2}}+x^{-\frac{1}{2}})dx$.
   2. M1 for integrating to get $[\frac{2}{3}{x}^{\frac{3}{2}}+2x^{\frac{1}{2}}]$.
   3. M1 for substituting the limits: $(\frac{2}{3}(27)+2(3))-(\frac{2}{3}(8)+2(2))$.
   4. A1 for evaluating to get exactly $14\frac{2}{3}$ (or $\frac{44}{3}$).

More on these topics: Logarithms and Exponentials · Proof · Integration (definite integrals and area)

What's on this paper

What this paper is like

• Shape: 10 questions, 75 marks in 1 h 30 min (about 1.2 minutes a mark); questions from 4 to 10 marks.
• Biggest question: Q6, 10 marks: circle centre, radius and a tangent.
• Marks by main topic: Integration (definite integrals and area) 14, Circles 10, Differentiation (stationary points) 9, Algebraic Methods 9, Further Trigonometry 9, The Binomial Expansion 7, Sequences and Series 7, Logarithms and Exponentials 6, Proof 4.
• More Circles in other sittings: June 2023 Q3 (7 marks), June 2022 Q10 (12 marks), October 2023 Q8 (7 marks).
• Against the other 16 sittings we mapped: more marks on Integration (definite integrals and area) (14 marks here, 8 on average); fewer on Sequences and Series (7 marks here, 13 on average) and Logarithms and Exponentials (6 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 (6 marks) · P2 Integration (definite integrals and area)
2. Q2Minimising the surface area of a brick (9 marks) · P2 Differentiation (stationary points)
3. Q3Binomial expansion with a constant k (7 marks) · P2 The Binomial Expansion
4. Q4Log equation and an exponential equation (6 marks) · P2 Logarithms and Exponentials
5. Q5Factor theorem with constants p and q (9 marks) · P2 Algebraic Methods
6. Q6Circle centre, radius and a tangent (10 marks) · P2 Circles
7. Q7Geometric sequence with r > 0 (7 marks) · P2 Sequences and Series
8. Q8Trig equations in radians (9 marks) · P2 Further Trigonometry
9. Q9Areas between a line and a curve (8 marks) · P2 Integration (definite integrals and area)
10. Q10Proof by exhaustion (4 marks) · P2 Proof

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