Edexcel IAL P3 October 2023 question paper and mark scheme (WMA13/01)

Try 3 P3 questions like these

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1. Further Differentiation · 5 marks

   A curve has the equation $y = e^{-x}\sin x$. Find the exact x-coordinate of the first positive turning point of the curve.

   Worked answer

   1. M1 for using the product rule to differentiate: $\frac{dy}{dx} = -e^{-x}\sin x + e^{-x}\cos x$.
   2. M1 for setting the derivative equal to 0 to find turning points: $e^{-x}(\cos x - \sin x) = 0$.
   3. M1 for dividing by $e^{-x}$ (since $e^{-x} \ne 0$) to get $\cos x = \sin x$.
   4. M1 for dividing by $\cos x$ to obtain $\tan x = 1$.
   5. A1 for identifying the first positive solution as $x = \frac{\pi}{4}$.

2. 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 $330^\circ$.

3. Exponentials and Logarithms · 4 marks

   Solve the equation $2\ln x = \ln(3x + 4)$.

   Worked answer

   1. M1 for applying the power law to the LHS: $\ln(x^2) = \ln(3x + 4)$.
   2. M1 for equating the arguments: $x^2 = 3x + 4$.
   3. M1 for rearranging to $x^2 - 3x - 4 = 0$ and finding roots $x = 4$ and $x = -1$.
   4. A1 for rejecting $x = -1$ (since $\ln x$ must have $x > 0$) and giving $x = 4$.

More on these topics: Further Differentiation · Trigonometric Functions · Exponentials and Logarithms

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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Formula booklet

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