Edexcel IAL P4 January 2022 question paper and mark scheme (WMA14/01)

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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. 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 $a^2 - 4b \ne 2$.

More on these topics: Parametric Equations · Differential Equations · Proof

P4 topics this paper can test

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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)