Edexcel IAL P4 January 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. Vectors · 5 marks

   Two lines have vector equations $l_1:\mathbf{r}=(2\mathbf{i}+3\mathbf{j})+t(\mathbf{i}-\mathbf{j}+\mathbf{k})$ and $l_2:\mathbf{r}=(-3\mathbf{j}-2\mathbf{k})+s(\mathbf{i}+\mathbf{j}+\mathbf{k})$. Prove that the lines intersect and find the position vector of their point of intersection.

   Worked solution

   1. M1 for setting up equations for the components: $2+t=s$, $3-t=-3+s$, and $t=-2+s$.
   2. M1 for solving the first two equations simultaneously. From $s=2+t$, substituting into the second gives $3-t=-3+(2+t)$.
   3. M1 for solving to find $2t=4⟹t=2$, which means $s=4$.
   4. M1 for checking consistency in the third (z) equation: $t=2$, and $-2+s=-2+4=2$. Since $2=2$, the lines intersect.
   5. A1 for substituting $t=2$ or $s=4$ back to find the point of intersection: $4\mathbf{i}+\mathbf{j}+2\mathbf{k}$.

2. Parametric Equations · 7 marks

   A curve has parametric equations $x=t+1$, $y=t^3-12t$.
   (a) Find the values of $t$ at the stationary points on the curve. (3)
   (b) Find the coordinates of the stationary points. (2)
   (c) Determine the nature of each stationary point. (2)

   Worked solution

   (a) $\frac{dy}{dx}=\frac{3t^2-12}{1}=3t^2-12$   M1 A1. Set $=0$: $t^2=4\Rightarrow t=\pm 2$   A1.
   (b) $t=2$: $(3,-16)$; $t=-2$: $(-1,16)$   A1 A1.
   (c) $\frac{d^{2}y}{d{x}^2}=6t\cdot \frac{dt}{dx}=6t$ (since $\frac{dt}{dx}=1$)   M1. $t=2$: $12>0$ ⇒ minimum; $t=-2$: $-12<0$ ⇒ maximum   A1.

3. Differential Equations · 4 marks

   Use the method of separation of variables to find the general solution to the differential equation: $$\frac{dy}{dx}=\frac{x}{y}$$ Express your answer in the form $y^2=f(x)$.

   Worked solution

   Separate the variables by multiplying both sides by $y$ and $dx$: M1
   $ydy=xdx$.
   Integrate both sides: $\int ydy=\int xdx$. M1
   $\frac{y^2}{2}=\frac{x^2}{2}+C$. A1
   Multiply entirely by 2 to isolate $y^2$:
   $y^2=x^2+2C$. (Note: $2C$ can just be written as a new constant $K$). A1

More on these topics: Vectors · Parametric Equations · Differential Equations

What's on this paper

What this paper is like

• Shape: 9 questions, 75 marks in 1 h 30 min (about 1.2 minutes a mark); questions from 5 to 12 marks.
• Biggest question: Q7, 12 marks: rates of change for a balloon.
• Marks by main topic: Differentiation 19, Parametric Equations 17, Integration 14, Partial Fractions 9, Vectors 8, Proof 8.
• More Differentiation in other sittings: October 2022 Q11 (9 marks), June 2023 Q2 (10 marks), June 2022 Q3 (8 marks).
• Against the other 14 sittings we mapped: more marks on Differentiation (19 marks here, 13 on average) and Proof (8 marks here, 4 on average); fewer on The Binomial Expansion (0 marks here, 7 on average) and Differential Equations (0 marks here, 7 on average).
• Not on this paper: Differential Equations, The Binomial Expansion, 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. Q1Partial fractions, then a binomial expansion (9 marks) · P4 Partial Fractions, P4 The Binomial Expansion
2. Q2Parametric equations to a Cartesian equation (6 marks) · P4 Parametric Equations
3. Q3Area of a region by integration (5 marks) · P4 Integration
4. Q4Substitution with an exponential (9 marks) · P4 Integration
5. Q5Implicit curve: vertical tangents (7 marks) · P4 Differentiation
6. Q6Position vectors and a line (8 marks) · P4 Vectors
7. Q7Rates of change for a balloon (12 marks) · P4 Differentiation, P4 Differential Equations
8. Q8Parametric curve: tangent and area (11 marks) · P4 Parametric Equations, P4 Integration
9. Q9Proof by contradiction about multiples of 3 (8 marks) · P4 Proof

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