Edexcel IAL P4 October 2022 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 · 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.

3. Integration · 5 marks

   Find $\int \frac{x^2+1}{x(x-1)}dx$.

   Worked solution

   1. M1 for recognising the improper fraction and performing division to get $1+\frac{x+1}{x(x-1)}$.
   2. M1 for setting up partial fractions for the remainder: $\frac{x+1}{x(x-1)}=\frac{A}{x}+\frac{B}{x-1}$.
   3. A1 for determining the constants $A=-1$ and $B=2$.
   4. M1 for formulating the complete integral: $\int (1-\frac{1}{x}+\frac{2}{x-1})dx$.
   5. A1 for integrating the terms correctly to $x-\ln |x|+2\ln |x-1|+c$.

More on these topics: Vectors · Parametric Equations · Integration

What's on this paper

What this paper is like

• Shape: 11 questions, 75 marks in 1 h 30 min (about 1.2 minutes a mark); questions from 3 to 12 marks.
• Biggest question: Q7, 12 marks: integration by substitution with e^x.
• Marks by main topic: Integration 18, Parametric Equations 11, Vectors 10, Differentiation 9, The Binomial Expansion 8, Differential Equations 8, Partial Fractions 7, Proof 4.
• More Integration in other sittings: January 2023 Q4 (9 marks), June 2022 Q8 (10 marks), June 2023 Q5 (10 marks).
• Against the other 14 sittings we mapped: fewer on Differentiation (9 marks here, 13 on average).
• Not on this paper: 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. Q1Parametric curve to Cartesian form (3 marks) · P4 Parametric Equations
2. Q2Partial fractions, then an exact integral (7 marks) · P4 Partial Fractions, P4 Integration
3. Q3Vectors in a triangle and an angle (5 marks) · P4 Vectors
4. Q4Binomial expansion of a root function (8 marks) · P4 The Binomial Expansion
5. Q5Area of a region with a constant k (6 marks) · P4 Integration
6. Q6Parametric curve with tan: tangent (8 marks) · P4 Parametric Equations
7. Q7Integration by substitution with e^x (12 marks) · P4 Integration
8. Q8Finding the error in a proof by contradiction (4 marks) · P4 Proof
9. Q9Two lines: intersect, parallel or skew (5 marks) · P4 Vectors
10. Q10Melting ball of ice: a differential equation (8 marks) · P4 Differential Equations
11. Q11Implicit curve: points with a given gradient (9 marks) · P4 Differentiation

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