Edexcel IAL P3 January 2025 question paper and mark scheme (WMA13/01)

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1. Functions · 4 marks

   Given that $f(x) = \frac{x+1}{x-2}$, $x \ne 2$, find an expression for $ff(x)$ in its simplest form.

   Worked answer

   1. M1 for substituting $f(x)$ into itself: $ff(x) = \frac{\frac{x+1}{x-2} + 1}{\frac{x+1}{x-2} - 2}$.
   2. M1 for multiplying the numerator and the denominator by $(x-2)$ to clear the nested fractions.
   3. M1 for obtaining the expanded fraction $\frac{(x+1) + (x-2)}{(x+1) - 2(x-2)}$.
   4. A1 for fully simplifying to $\frac{2x-1}{5-x}$ (or $\frac{1-2x}{x-5}$).

2. Numerical Methods · 6 marks

   The equation $\cos x = x$ has a unique root $\alpha$ in $[0, 1]$. Using the iteration $x_{n+1} = \cos x_n$ with $x_0 = 0.5$, find $x_1$, $x_2$, $x_3$, and $x_4$ each to $4$ d.p. (Angle in radians.)

   Worked answer

   $x_1 = \cos 0.5 = 0.8776$   M1 A1. $x_2 = \cos 0.8776 = 0.6390$   A1. $x_3 = \cos 0.6390 = 0.8027$   A1. $x_4 = \cos 0.8027 = 0.6948$   A1 A1.

3. Trigonometric Functions · 6 marks

   Solve $3 \cot^2 \theta + 5 \text{cosec} \theta + 1 = 0$ for $0^\circ \le \theta \le 360^\circ$.

   Worked answer

   1. M1 for substituting $\cot^2 \theta = \text{cosec}^2 \theta - 1$ to form $3(\text{cosec}^2 \theta - 1) + 5\text{cosec} \theta + 1 = 0$.
   2. M1 for simplifying to the quadratic $3\text{cosec}^2 \theta + 5\text{cosec} \theta - 2 = 0$.
   3. M1 for factorising to $(3\text{cosec} \theta - 1)(\text{cosec} \theta + 2) = 0$.
   4. M1 for showing $\text{cosec} \theta = \frac{1}{3} \implies \sin \theta = 3$ yields no real solutions.
   5. M1 for solving $\text{cosec} \theta = -2 \implies \sin \theta = -0.5$.
   6. A1 for finding the two correct angles: $210^\circ$ and $330^\circ$.

More on these topics: Functions · Numerical Methods · Trigonometric Functions

P3 topics this paper can test

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• Functions
• Numerical Methods
• Further Differentiation
• Further Integration
• Algebraic Methods
• Trigonometric Functions
• Trigonometric Addition Formulae
• Exponentials and Logarithms

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What the examiners said about this paper

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

• WMA13/01: Functions (Q6); Modulus equations and inequalities (Q7); Calculator in the wrong mode (and used instead of calculus) (Q1, Q9). 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)