Edexcel IAL P2 January 2024 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. Further Trigonometry · 5 marks

   Solve the equation $2{\cos }^{2}x-\sin x-1=0$ for $0^{\circ }\le x\le {360}^{\circ }$.

   Worked solution

   1. M1 for using the identity ${\cos }^{2}x=1-{\sin }^{2}x$ to rewrite the equation: $2(1-{\sin }^{2}x)-\sin x-1=0$.
   2. M1 for simplifying to a quadratic in $\sin x$: $2{\sin }^{2}x+\sin x-1=0$.
   3. M1 for factorising the quadratic to $(2\sin x-1)(\sin x+1)=0$.
   4. A1 for finding the solutions from $\sin x=0.5$, which are $x={30}^{\circ },{150}^{\circ }$.
   5. A1 for finding the solution from $\sin x=-1$, which is $x={270}^{\circ }$.

2. 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 $p=-1$.

3. Proof · 5 marks

   (a) Prove by exhaustion that the square of any integer leaves a remainder of 0, 1 or 4 when it is divided by 5. [4]

   (b) Hence show that the last digit of a square number is never 2, 3, 7 or 8. [1]

   Worked solution

   (a) Considers every case: $n=5k,5k\pm 1,5k\pm 2$ (or $5k,5k+1,5k+2,5k+3,5k+4$), $k$ an integer M1

   Squares at least two cases correctly, e.g. $(5k\pm 1{)}^2=25k^2\pm 10k+1$ M1

   All cases correct: $(5k{)}^2=5(5k^2)$; $(5k\pm 1{)}^2=5(5k^2\pm 2k)+1$; $(5k\pm 2{)}^2=5(5k^2\pm 4k)+4$ A1

   Conclusion: every integer is in one of the cases, so the remainder is always 0, 1 or 4 A1 (cso)

   (b) A whole number ending in 2 or 7 leaves remainder 2, and one ending in 3 or 8 leaves remainder 3, on division by 5; by (a) a square leaves remainder 0, 1 or 4, so it cannot end in 2, 3, 7 or 8 B1

More on these topics: Further Trigonometry · Algebraic Methods · Proof

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 3 to 14 marks.
• Biggest question: Q9, 14 marks: trig equations in degrees.
• Marks by main topic: Logarithms and Exponentials 17, Sequences and Series 17, Further Trigonometry 14, Differentiation (stationary points) 9, Circles 6, Proof 6, Algebraic Methods 3, The Binomial Expansion 3.
• More Further Trigonometry in other sittings: October 2023 Q3 (7 marks), June 2024 Q8 (12 marks), June 2023 Q9 (7 marks).
• Against the other 16 sittings we mapped: more marks on Logarithms and Exponentials (17 marks here, 10 on average) and Further Trigonometry (14 marks here, 8 on average); fewer on Integration (definite integrals and area) (0 marks here, 9 on average) and Algebraic Methods (3 marks here, 7 on average).
• Not on this paper: Integration (definite integrals and area).
• 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. Q1Remainder theorem with a constant (3 marks) · P2 Algebraic Methods
2. Q2Coefficient in a binomial expansion (3 marks) · P2 The Binomial Expansion
3. Q3Circle through a point and a tangent (6 marks) · P2 Circles
4. Q4Exponential graph with an asymptote (9 marks) · P2 Logarithms and Exponentials
5. Q5Sigma notation and a recurrence (8 marks) · P2 Sequences and Series
6. Q6Log equation reduced to a cubic (8 marks) · P2 Logarithms and Exponentials
7. Q7Wheat yield as a geometric series (9 marks) · P2 Sequences and Series
8. Q8Counter-example and an algebraic proof (6 marks) · P2 Proof
9. Q9Trig equations in degrees (14 marks) · P2 Further Trigonometry
10. Q10Tangent to a curve and an enclosed area (9 marks) · P2 Differentiation (stationary points), P2 Integration (definite integrals and area)

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