Edexcel IAL P4 October 2021 question paper and mark scheme (WMA14/01)

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1. Parametric Equations · 8 marks

   A curve $C$ has parametric equations $x = t^2$, $y = t^3 - 3t$.
   (a) Find $\dfrac{dy}{dx}$ in terms of $t$. (3)
   (b) Find the coordinates of the points on $C$ where the tangent is horizontal. (3)
   (c) Find the Cartesian equation of the tangent at $t = 2$. (2)

   Worked answer

   (a) $\tfrac{dx}{dt} = 2t$, $\tfrac{dy}{dt} = 3t^2 - 3$   M1 A1. $\tfrac{dy}{dx} = \tfrac{3t^2 - 3}{2t} = \tfrac{3(t^2 - 1)}{2t}$   A1.
   (b) Horizontal iff $\tfrac{dy}{dt} = 0$: $3t^2 = 3 \Rightarrow t = \pm 1$   M1. $t = 1$: $(1, -2)$; $t = -1$: $(1, 2)$   A1 A1.
   (c) At $t = 2$: $(x, y) = (4, 2)$; gradient $= \tfrac{3 \cdot 3}{4} = \tfrac{9}{4}$; equation: $y - 2 = \tfrac{9}{4}(x - 4)$, i.e. $y = \tfrac{9}{4}x - 7$   M1 A1.

2. Differential Equations · 4 marks

   Use the method of separation of variables to find the general solution to the differential equation: \[\frac{dy}{dx} = \frac{x}{y}\] Express your answer in the form \(y^2 = f(x)\).

   Worked answer

   Separate the variables by multiplying both sides by \(y\) and \(dx\): M1
   \(y \, dy = x \, dx\).
   Integrate both sides: \(\int y \, dy = \int x \, dx\). M1
   \(\frac{y^2}{2} = \frac{x^2}{2} + C\). A1
   Multiply entirely by 2 to isolate \(y^2\):
   \(y^2 = x^2 + 2C\). (Note: \(2C\) can just be written as a new constant \(K\)). A1

3. Proof · 4 marks

   Prove by contradiction that there are infinitely many prime numbers.

   Worked answer

   1. M1 for assuming that there is a finite number of prime numbers, listing them as $p_1, p_2, \dots, p_n$.
   2. M1 for constructing a new number $N = (p_1 \times p_2 \times \dots \times p_n) + 1$.
   3. M1 for arguing that dividing $N$ by any of the primes $p_i$ leaves a remainder of $1$, meaning $N$ has no prime factors from the finite list.
   4. A1 for concluding that $N$ must either be a new prime itself or have a prime factor not in the list, contradicting the assumption that the list contains all primes.

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)