Edexcel IAL P4 June 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 · 8 marks

   Consider the following two lines in 3D space: $$L_1:r=\left(\begin{array}{c}1\\ -1\\ 2\end{array}\right)+\lambda \left(\begin{array}{c}2\\ 1\\ -3\end{array}\right)\text{and}{L}_2:r=\left(\begin{array}{c}4\\ 2\\ -1\end{array}\right)+\mu \left(\begin{array}{c}-1\\ 2\\ 1\end{array}\right)$$ Show algebraically that these two lines are skew (they are neither parallel nor do they intersect).

   Worked solution

   Set the components equal:
   $1+2\lambda =4-\mu ⟹2\lambda +\mu =3$ (Eq 1). M1
   $-1+\lambda =2+2\mu ⟹\lambda -2\mu =3$ (Eq 2). M1
   $2-3\lambda =-1+\mu ⟹-3\lambda -\mu =-3$ (Eq 3). M1
   Solve Eq 1 and Eq 2: $2(3+2\mu )+\mu =3⟹6+5\mu =3⟹\mu =-0.6$. M1
   Substitute $\mu =-0.6⟹\lambda =1.8$. A1
   Check in Eq 3: $-3(1.8)-(-0.6)=-5.4+0.6=-4.8$. M1
   However, the RHS of Eq 3 is $-3$. Since $-4.8\ne -3$, there is no consistent solution. B1
   Therefore, the lines do not intersect. Since their direction vectors are not scalar multiples, they are skew. A1

2. Partial Fractions · 7 marks

   Consider the rational function $f(x)=\frac{3x}{(x-1)(x+2)}$.

   1. Express $f(x)$ in partial fractions.
   2. Hence, evaluate exactly the definite integral ${\int }_2^{3}f(x)dx$, giving your answer in the form $\ln (k)$ where $k$ is a rational number.
   Worked solution

   1. M1 for setting up the partial fraction decomposition: $\frac{3x}{(x-1)(x+2)}=\frac{A}{x-1}+\frac{B}{x+2}$.
      M1 for clearing the denominators: $3x=A(x+2)+B(x-1)$.
      M1 for substituting strategic values of $x$ (e.g., $x=1$ and $x=-2$) to solve for constants.
      A1 for finding $A=1$ and $B=2$, yielding $\frac{1}{x-1}+\frac{2}{x+2}$.
   2. M1 for setting up the integral and determining the anti-derivatives: ${[\ln |x-1|+2\ln |x+2|]}_2^3$.
      M1 for substituting the limits: $(\ln 2+2\ln 5)-(\ln 1+2\ln 4)$.
      A1 for using logarithm laws to combine into a single logarithm: $\ln 2+\ln 25-0-\ln 16=\ln \left(\frac{50}{16}\right)=\ln \left(\frac{25}{8}\right)$.

3. Coordinate Geometry (parametric) · 5 marks

   A curve has parametric equations $x=t^2-t$ and $y=t^3-3t$. Find the exact coordinates of the points where the curve crosses the x-axis.

   Worked solution

   1. M1 for setting $y=0$ to find x-axis intersections: $t^3-3t=0$.
   2. M1 for factorising $t(t^2-3)=0$.
   3. A1 for finding the parameter values $t=0,\sqrt{3},-\sqrt{3}$.
   4. M1 for substituting these values of $t$ back into $x=t^2-t$.
   5. A1 for the exact coordinates: $(0,0)$, $(3-\sqrt{3},0)$, and $(3+\sqrt{3},0)$.

More on these topics: Vectors · Partial Fractions · Coordinate Geometry (parametric)

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 4 to 12 marks.
• Biggest question: Q8, 12 marks: parametric curve: tangent and area.
• Marks by main topic: Parametric Equations 12, Partial Fractions 11, Differentiation 10, Vectors 10, Integration 10, The Binomial Expansion 9, Differential Equations 9, Proof 4.
• More Parametric Equations in other sittings: October 2023 Q8 (14 marks), January 2023 Q8 (11 marks), January 2024 Q9 (12 marks).
• Against the other 14 sittings we mapped: more marks on Partial Fractions (11 marks here, 5 on average); fewer on Integration (10 marks here, 16 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. Q1Binomial expansion and its validity (9 marks) · P4 The Binomial Expansion
2. Q2Implicit curve: tangent at a point (10 marks) · P4 Differentiation
3. Q3Partial fractions and an exact integral (11 marks) · P4 Partial Fractions, P4 Integration
4. Q4Position vectors and lines (10 marks) · P4 Vectors
5. Q5Integration by parts and by substitution (10 marks) · P4 Integration
6. Q6Car engine cooling model (9 marks) · P4 Differential Equations
7. Q7Proof that a root is irrational (4 marks) · P4 Proof
8. Q8Parametric curve: tangent and area (12 marks) · P4 Parametric Equations, P4 Integration

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