Edexcel IAL P4 June 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. Proof · 4 marks

   Prove by contradiction that there are infinitely many prime numbers.

   Worked solution

   1. M1 for assuming that there is a finite number of prime numbers, listing them as $p_1,p_2,\dots ,p_n$.
   2. M1 for constructing a new number $N=(p_1\times p_2\times \cdots \times p_n)+1$.
   3. M1 for arguing that dividing $N$ by any of the primes $p_i$ leaves a remainder of $1$, meaning $N$ has no prime factors from the finite list.
   4. A1 for concluding that $N$ must either be a new prime itself or have a prime factor not in the list, contradicting the assumption that the list contains all primes.

3. The Binomial Expansion · 5 marks

   The binomial expansion of $(a+bx{)}^{-2}$, where $a>0$, begins $\frac{1}{9}-\frac{2}{27}x+\dots$. Find the values of the constants $a$ and $b$.

   Worked solution

   1. M1 for writing $(a+bx{)}^{-2}=a^{-2}(1+\frac{b}{a}x{)}^{-2}$.
   2. M1 for equating the constant term: $a^{-2}=\frac{1}{9}$.
   3. A1 for finding $a=3$ (since $a>0$).
   4. M1 for equating the $x$ coefficient: $a^{-2}(-2)(\frac{b}{a})=-\frac{2}{27}$.
   5. A1 for substituting $a=3$ to get $\frac{1}{9}(\frac{-2b}{3})=-\frac{2}{27}⟹-\frac{2b}{27}=-\frac{2}{27}⟹b=1$.

More on these topics: Vectors · Proof · The Binomial Expansion

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 4 to 12 marks.
• Biggest question: Q7, 12 marks: parametric curve: tangent and area.
• Marks by main topic: Integration 18, Differentiation 16, Parametric Equations 12, Partial Fractions 9, Vectors 9, The Binomial Expansion 7, Proof 4.
• More Parametric Equations in other sittings: October 2022 Q6 (8 marks), January 2022 Q5 (10 marks), January 2023 Q8 (11 marks).
• Against the other 14 sittings we mapped: fewer on Differential Equations (0 marks here, 7 on average).
• Not on this paper: Differential Equations, 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 with a constant k (7 marks) · P4 The Binomial Expansion
2. Q2Partial fractions and a differential equation (9 marks) · P4 Partial Fractions, P4 Differential Equations
3. Q3Rates of change for a dissolving tablet (8 marks) · P4 Differentiation
4. Q4Implicit curve with a constant k (8 marks) · P4 Differentiation
5. Q5Substitution x = 2 sin u (8 marks) · P4 Integration
6. Q6Position vectors and a line (9 marks) · P4 Vectors
7. Q7Parametric curve: tangent and area (12 marks) · P4 Parametric Equations, P4 Integration
8. Q8Volume of revolution for a solid (10 marks) · P4 Integration
9. Q9Proof by contradiction with divisibility (4 marks) · P4 Proof

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