Edexcel IAL P2 June 2019 question paper (WMA12/01)

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1. Circles · 6 marks

   The circle $C$ has centre $(a, b)$ and passes through the points $P(0, 0)$, $Q(6, 0)$, and $R(0, 8)$. Find the values of $a$, $b$, and the radius of $C$.

   Worked answer

   $P$ and $Q$ both on circle ⇒ centre lies on the perpendicular bisector of $PQ$, so $a = 3$   M1 A1. Similarly $P$ and $R$ give $b = 4$   M1 A1. Radius $= |CP| = \sqrt{9 + 16} = 5$   M1 A1.

2. Differentiation (stationary points) · 4 marks

   Find the exact coordinates of the stationary point on the curve $y = \sqrt{x} - 2x$, for $x > 0$.

   Worked answer

   1. M1 for rewriting as $y = x^{0.5} - 2x$ and differentiating: $\frac{dy}{dx} = 0.5x^{-0.5} - 2$.
   2. M1 for setting to zero and rearranging: $\frac{1}{2\sqrt{x}} = 2 \implies \sqrt{x} = 0.25$.
   3. A1 for solving to get $x = 0.0625$ or $\frac{1}{16}$.
   4. A1 for finding the y-coordinate: $\sqrt{\frac{1}{16}} - 2(\frac{1}{16}) = \frac{1}{4} - \frac{1}{8} = \frac{1}{8}$, giving $(\frac{1}{16}, \frac{1}{8})$.

3. Integration (definite integrals and area) · 7 marks

   Find the exact total area of the finite regions bounded by the curves $y = x^3 + x^2$ and $y = 2x^2 + 2x$.

   Worked answer

   1. M1 for setting $x^3 + x^2 = 2x^2 + 2x \implies x^3 - x^2 - 2x = 0$.
   2. M1 for factoring $x(x-2)(x+1)=0$ to find the intersection points $x=-1, 0, 2$.
   3. M1 for recognising two distinct regions and setting up integrals: $\int_{-1}^{0} (x^3 - x^2 - 2x) \, dx$ and $\int_{0}^{2} (2x^2 + 2x - x^3 - x^2) \, dx$.
   4. M1 for integrating the generic polynomial to $\pm[\frac{x^4}{4} - \frac{x^3}{3} - x^2]$.
   5. M1 for evaluating the area of the first region to $\frac{5}{12}$.
   6. M1 for evaluating the area of the second region to $\frac{8}{3}$.
   7. A1 for calculating the combined total area: $\frac{5}{12} + \frac{8}{3} = \frac{37}{12}$.

More on these topics: Circles · Differentiation (stationary points) · Integration (definite integrals and area)

P2 topics this paper can test

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• Sequences and Series
• Logarithms and Exponentials
• Circles
• Further Trigonometry
• Algebraic Methods
• Proof
• The Binomial Expansion
• Differentiation (stationary points)
• Integration (definite integrals and area)

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Formula booklet

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