Edexcel IAL P4 October 2024 question paper and mark scheme (WMA14/01)

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1. Proof · 4 marks

   Prove by contradiction that if $a$ and $b$ are integers, then $a^2 - 4b \ne 2$.

   Worked answer

   1. M1 for assuming that there exist integers $a$ and $b$ such that $a^2 - 4b = 2$, which rearranges to $a^2 = 4b + 2 = 2(2b + 1)$.
   2. M1 for deducing that since $a^2$ is an even number, $a$ must also be an even number.
   3. M1 for letting $a = 2k$, substituting to get $(2k)^2 = 4b + 2 \implies 4k^2 = 4b + 2$, and dividing by $2$ to get $2k^2 = 2b + 1 \implies 2(k^2 - b) = 1$.
   4. A1 for observing that the left-hand side $2(k^2 - b)$ is even while the right-hand side $1$ is odd, which is a contradiction, so $a^2 - 4b \ne 2$.

2. Coordinate Geometry (parametric) · 5 marks

   A curve $C$ has parametric equations $x = \sin \theta$ and $y = \cos 2\theta$ for $0 \le \theta \le \pi$. Find the exact coordinates of the points where $C$ intersects the line $y = x$.

   Worked answer

   1. M1 for setting $x = y$, resulting in $\sin \theta = \cos 2\theta$.
   2. M1 for substituting the double angle formula: $\sin \theta = 1 - 2\sin^2 \theta$.
   3. M1 for rearranging into a quadratic $2\sin^2 \theta + \sin \theta - 1 = 0$ and factorising $(2\sin \theta - 1)(\sin \theta + 1) = 0$.
   4. A1 for finding valid solutions for $\sin \theta$: $\sin \theta = \frac{1}{2}$ (rejecting $\sin \theta = -1$ since $0 \le \theta \le \pi$).
   5. A1 for calculating the corresponding $(x, y)$ coordinates: $(\frac{1}{2}, \frac{1}{2})$.

3. Integration · 5 marks

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

   Worked answer

   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: Proof · Coordinate Geometry (parametric) · Integration

P4 topics this paper can test

Every P4 paper can test any of the 9 P4 topics in the specification.

We haven't mapped this paper question by question yet. Each topic below opens our P4 page at that topic: what it covers, a worked example with its mark scheme, then practice questions.

• Vectors
• Partial Fractions
• Parametric Equations
• Differential Equations
• Proof
• The Binomial Expansion
• Coordinate Geometry (parametric)
• Differentiation
• Integration

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What the examiners said about this paper

Summarised in our own words from the board's examiner report (Examiner Insights).

• WMA14/01: Binomial (any index) (Q1); Proof by contradiction (Q2); Connected rates of change (Q5). Official examiner report ↗

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Formula booklet

Mathematical Formulae and Statistical Tables (International AS/A Level Mathematics) · Pearson Edexcel · For the 2018 IAL specification (current)