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

Try 3 P4 questions like these

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1. Parametric Equations · 7 marks

   A curve has parametric equations $x = t + 1$, $y = t^3 - 12t$.
   (a) Find the values of $t$ at the stationary points on the curve. (3)
   (b) Find the coordinates of the stationary points. (2)
   (c) Determine the nature of each stationary point. (2)

   Worked answer

   (a) $\tfrac{dy}{dx} = \tfrac{3t^2 - 12}{1} = 3t^2 - 12$   M1 A1. Set $= 0$: $t^2 = 4 \Rightarrow t = \pm 2$   A1.
   (b) $t = 2$: $(3, -16)$; $t = -2$: $(-1, 16)$   A1 A1.
   (c) $\tfrac{d^2y}{dx^2} = 6t \cdot \tfrac{dt}{dx} = 6t$ (since $\tfrac{dt}{dx} = 1$)   M1. $t = 2$: $12 > 0$ ⇒ minimum; $t = -2$: $-12 < 0$ ⇒ maximum   A1.

2. Differential Equations · 6 marks

   Find the exact particular solution to the differential equation: \[\frac{dy}{dx} = y \cos x\] given the boundary condition that \(y(0) = e^2\).

   Worked answer

   Separate variables: \(\frac{1}{y} dy = \cos x \, dx\). M1
   Integrate both sides: \(\int \frac{1}{y} dy = \int \cos x \, dx\). M1
   \(\ln |y| = \sin x + C\). A1A1
   Convert to exponential form: \(y = e^{\sin x + C} = A e^{\sin x}\) (where \(A = e^C\)). M1
   Substitute initial condition \(y(0) = e^2\):
   \(e^2 = A e^{\sin(0)} \implies e^2 = A e^0 \implies A = e^2\).
   Exact particular solution: \(y = e^2 e^{\sin x}\) or \(y = e^{\sin x + 2}\). A1

3. 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: $(\frac{1}{2}, \frac{1}{2})$.

More on these topics: Parametric Equations · Differential Equations · Coordinate Geometry (parametric)

P4 topics this paper can test

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

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

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Formula booklet

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