Edexcel IAL P3 October 2024 question paper and mark scheme (WMA13/01)

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1. Functions · 5 marks

   Solve the inequality $|2x + 1| \le |x - 5|$.

   Worked answer

   1. M1 for squaring both sides to remove the modulus signs: $(2x+1)^2 \le (x-5)^2$.
   2. M1 for expanding the brackets: $4x^2 + 4x + 1 \le x^2 - 10x + 25$.
   3. M1 for rearranging to form a quadratic inequality: $3x^2 + 14x - 24 \le 0$.
   4. M1 for factorising to find the critical values: $(3x-4)(x+6) \le 0$, yielding critical values $-6$ and $\frac{4}{3}$.
   5. A1 for stating the correct closed interval $-6 \le x \le \frac{4}{3}$.

2. Trigonometric Addition Formulae · 6 marks

   Solve the equation \(\cos 2x = \sin x\) for \(-\pi \le x \le \pi\). Give your answers as exact radian values.

   Worked answer

   Substitute the double angle identity \(\cos 2x = 1 - 2\sin^2 x\). M1
   \(1 - 2\sin^2 x = \sin x \implies 2\sin^2 x + \sin x - 1 = 0\). A1
   Factorise the quadratic: \((2\sin x - 1)(\sin x + 1) = 0\). M1A1
   Case 1: \(2\sin x - 1 = 0 \implies \sin x = \frac{1}{2}\).
   Within the domain \(-\pi \le x \le \pi\), solutions are \(x = \frac{\pi}{6}\) and \(x = \frac{5\pi}{6}\). A1
   Case 2: \(\sin x + 1 = 0 \implies \sin x = -1\).
   Within the domain \(-\pi \le x \le \pi\), the solution is \(x = -\frac{\pi}{2}\). A1

3. Exponentials and Logarithms · 5 marks

   The variables $x$ and $y$ satisfy the equation $y = a b^x$. A graph of $\ln y$ against $x$ is plotted and forms a straight line passing through $(0, 2)$ and $(4, 10)$. Find the exact values of the constants $a$ and $b$.

   Worked answer

   1. M1 for taking natural logarithms of both sides to get $\ln y = \ln a + x \ln b$.
   2. M1 for calculating the gradient of the line: $\ln b = \frac{10 - 2}{4 - 0} = 2$.
   3. A1 for solving to find exactly $b = e^2$.
   4. M1 for identifying the y-intercept from the coordinate $(0, 2)$, so $\ln a = 2$.
   5. A1 for solving to find exactly $a = e^2$.

More on these topics: Functions · Trigonometric Addition Formulae · Exponentials and Logarithms

P3 topics this paper can test

Every P3 paper can test any of the 8 P3 topics in the specification.

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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 (Q3); Exponential models (Q8). 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)