Edexcel IAL P3 June 2023 mark scheme and question paper (WMA13/01)

3 P3 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. Functions · 5 marks

   Solve the inequality $|2x+1|\le |x-5|$.

   Worked solution

   1. M1 for squaring both sides to remove the modulus signs: $(2x+1{)}^2\le (x-5{)}^2$.
   2. M1 for expanding the brackets: $4x^2+4x+1\le x^2-10x+25$.
   3. M1 for rearranging to form a quadratic inequality: $3x^2+14x-24\le 0$.
   4. M1 for factorising to find the critical values: $(3x-4)(x+6)\le 0$, yielding critical values $-6$ and $\frac{4}{3}$.
   5. A1 for stating the correct closed interval $-6\le x\le \frac{4}{3}$.

2. Algebraic Methods · 4 marks

   Simplify fully $\frac{2x^2-x-1}{x^2-x}÷\frac{4x^2-1}{x}$.

   Worked solution

   1. M1 for rewriting the division as multiplication by the reciprocal: $\times \frac{x}{4x^2-1}$.
   2. M1 for factorising $2x^2-x-1$ to $(2x+1)(x-1)$ and $x^2-x$ to $x(x-1)$.
   3. M1 for factorising $4x^2-1$ to $(2x-1)(2x+1)$.
   4. A1 for cancelling all common factors to yield $\frac{1}{2x-1}$.

3. Trigonometric Functions · 6 marks

   Solve $3{\cot }^2\theta +5\text{cosec}\theta +1=0$ for $0^{\circ }\le \theta \le {360}^{\circ }$.

   Worked solution

   1. M1 for substituting ${\cot }^2\theta ={\text{cosec}}^2\theta -1$ to form $3({\text{cosec}}^2\theta -1)+5\text{cosec}\theta +1=0$.
   2. M1 for simplifying to the quadratic $3{\text{cosec}}^2\theta +5\text{cosec}\theta -2=0$.
   3. M1 for factorising to $(3\text{cosec}\theta -1)(\text{cosec}\theta +2)=0$.
   4. M1 for showing $\text{cosec}\theta =\frac{1}{3}⟹\sin \theta =3$ yields no real solutions.
   5. M1 for solving $\text{cosec}\theta =-2⟹\sin \theta =-0.5$.
   6. A1 for finding the two correct angles: ${210}^{\circ }$ and ${330}^{\circ }$.

More on these topics: Functions · Algebraic Methods · Trigonometric Functions

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 5 to 10 marks.
• Biggest question: Q9, 10 marks: double-angle identity and an equation.
• Marks by main topic: Further Differentiation 24, Functions 15, Exponentials and Logarithms 14, Trigonometric Addition Formulae 10, Trigonometric Functions 7, Numerical Methods 5.
• More Trigonometric Addition Formulae in other sittings: October 2023 Q8 (8 marks), January 2023 Q5 (9 marks), January 2024 Q9 (8 marks).
• Against the other 14 sittings we mapped: more marks on Further Differentiation (24 marks here, 19 on average); fewer on Algebraic Methods (0 marks here, 6 on average) and Further Integration (0 marks here, 5 on average).
• Not on this paper: Further Integration, Algebraic Methods.
• Calculator: allowed.

WMA13/01: question by question

Labels and topics are ours, read from the WMA13/01 paper (checked 2026-11-01); marks as printed. Tap a number to practise that topic.

1. Q1Root in an interval and an iteration (5 marks) · P3 Numerical Methods
2. Q2Linear graph of log T against log x (6 marks) · P3 Exponentials and Logarithms
3. Q3Differentiating ln sin and an integral (7 marks) · P3 Further Differentiation, P3 Further Integration
4. Q4Range and inverse of a quadratic function (6 marks) · P3 Functions
5. Q5Trig equations in radians (7 marks) · P3 Trigonometric Functions
6. Q6Modulus graph and its vertex (9 marks) · P3 Functions
7. Q7Two bacteria population models (8 marks) · P3 Exponentials and Logarithms
8. Q8Curve with (2x + 1)^3 e^x: stationary points (9 marks) · P3 Further Differentiation, P3 Exponentials and Logarithms
9. Q9Double-angle identity and an equation (10 marks) · P3 Trigonometric Addition Formulae
10. Q10dx/dy for a curve and tangents (8 marks) · P3 Further Differentiation

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