Edexcel IAL P4 October 2023 mark scheme and question paper (WMA14/01)

3 P4 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. Parametric Equations · 8 marks

   A curve $C$ has parametric equations $x=t^2$, $y=t^3-3t$.
   (a) Find ${\displaystyle \frac{dy}{dx}}$ in terms of $t$. (3)
   (b) Find the coordinates of the points on $C$ where the tangent is horizontal. (3)
   (c) Find the Cartesian equation of the tangent at $t=2$. (2)

   Worked solution

   (a) $\frac{dx}{dt}=2t$, $\frac{dy}{dt}=3t^2-3$   M1 A1. $\frac{dy}{dx}=\frac{3t^2-3}{2t}=\frac{3(t^2-1)}{2t}$   A1.
   (b) Horizontal iff $\frac{dy}{dt}=0$: $3t^2=3\Rightarrow t=\pm 1$   M1. $t=1$: $(1,-2)$; $t=-1$: $(1,2)$   A1 A1.
   (c) At $t=2$: $(x,y)=(4,2)$; gradient $=\frac{3\cdot 3}{4}=\frac{9}{4}$; equation: $y-2=\frac{9}{4}(x-4)$, i.e. $y=\frac{9}{4}x-7$   M1 A1.

2. The Binomial Expansion · 5 marks

   Expand $(2-x{)}^{-2}$ up to the term in $x^2$. Use your expansion to find an approximate value for $\frac{1}{{1.98}^2}$.

   Worked solution

   1. M1 for rewriting as $2^{-2}(1-\frac{x}{2}{)}^{-2}=\frac{1}{4}(1-\frac{x}{2}{)}^{-2}$.
   2. M1 for expanding correctly: $\frac{1}{4}(1+2(\frac{x}{2})+3(\frac{x}{2}{)}^2)=\frac{1}{4}+\frac{1}{4}x+\frac{3}{16}{x}^2$.
   3. M1 for deducing the appropriate value of $x$ to substitute is $x=0.02$.
   4. M1 for substituting $x=0.02$ into the expansion: $\frac{1}{4}+\frac{1}{4}(0.02)+\frac{3}{16}(0.0004)$.
   5. A1 for arriving at the correct approximation: $0.25+0.005+0.000075=0.255075$.

3. Integration · 6 marks

   Find the exact value of ${\int }_0^{\pi }{x}^2\sin xdx$.

   Worked solution

   1. M1 for setting up the first integration by parts with $u=x^2$ and $dv=\sin xdx$.
   2. M1 for obtaining $-x^2\cos x+\int 2x\cos xdx$.
   3. M1 for applying integration by parts a second time on $\int 2x\cos xdx$ with $u=2x$ and $dv=\cos xdx$.
   4. A1 for the fully integrated expression $-x^2\cos x+2x\sin x+2\cos x$.
   5. M1 for substituting the limits $0$ and $\pi$: $(-{\pi }^2(-1)+0+2(-1))-(0+0+2)$.
   6. A1 for calculating the exact final answer ${\pi }^2-4$.

More on these topics: Parametric Equations · The Binomial Expansion · Integration

What's on this paper

What this paper is like

• Shape: 8 questions, 75 marks in 1 h 30 min (about 1.2 minutes a mark); questions from 5 to 14 marks.
• Biggest question: Q8, 14 marks: parametric curve: area under it.
• Marks by main topic: Differentiation 17, Parametric Equations 14, Integration 12, Differential Equations 12, Vectors 10, The Binomial Expansion 5, Proof 5.
• More Parametric Equations in other sittings: January 2024 Q9 (12 marks), June 2023 Q8 (12 marks), June 2024 Q5 (13 marks).
• Against the other 14 sittings we mapped: more marks on Differential Equations (12 marks here, 6 on average) and Differentiation (17 marks here, 13 on average); fewer on Partial Fractions (0 marks here, 6 on average).
• Not on this paper: Partial Fractions, Coordinate Geometry (parametric).
• Calculator: allowed.

WMA14/01: question by question

Labels and topics are ours, read from the WMA14/01 paper (checked 2026-11-01); marks as printed. Tap a number to practise that topic.

1. Q1Binomial expansion, first four terms (5 marks) · P4 The Binomial Expansion
2. Q2Rates of change for a growing cube (7 marks) · P4 Differentiation
3. Q3Integration by parts and by substitution (12 marks) · P4 Integration
4. Q4Proof by contradiction of an inequality (5 marks) · P4 Proof
5. Q5Implicit curve with a constant k (10 marks) · P4 Differentiation
6. Q6Vector equations of two lines (10 marks) · P4 Vectors
7. Q7Goat population model (12 marks) · P4 Differential Equations
8. Q8Parametric curve: area under it (14 marks) · P4 Parametric Equations, P4 Integration

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