Edexcel IAL P1 October 2021 question paper and mark scheme (WMA11/01)

Try 3 P1 questions like these

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1. Trigonometry · 5 marks

   Two identical circles of radius $8$ cm intersect such that the circumference of each passes through the centre of the other. Find the exact area of the overlapping region.

   Worked answer

   1. M1 for recognising that lines drawn from the intersection points to the centres form two equilateral triangles, hence the central angle in each circle covering the overlap is $\frac{2\pi}{3}$.
   2. M1 for establishing the overlapping region consists of two identical segments, so Area $= 2 \times \frac{1}{2}(8^2)(\frac{2\pi}{3} - \sin\frac{2\pi}{3})$.
   3. M1 for substituting $\sin\frac{2\pi}{3} = \frac{\sqrt{3}}{2}$.
   4. M1 for expanding the expression $64(\frac{2\pi}{3} - \frac{\sqrt{3}}{2})$.
   5. A1 for the exact simplified area $\frac{128\pi}{3} - 32\sqrt{3} \text{ cm}^2$.

2. Integration · 6 marks

   Given that $f''(x) = 12x - 4$, $f'(1) = 2$ and $f(-1) = 5$, find $f(x)$.

   Worked answer

   1. M1 for integrating $f''(x)$ to find $f'(x) = 6x^2 - 4x + c_1$.
   2. M1 for substituting $f'(1) = 2$ to find $c_1$: $2 = 6(1)^2 - 4(1) + c_1 \implies c_1 = 0$.
   3. M1 for substituting $c_1 = 0$ and integrating again to find $f(x) = 2x^3 - 2x^2 + c_2$.
   4. M1 for substituting $f(-1) = 5$: $5 = 2(-1)^3 - 2(-1)^2 + c_2$.
   5. M1 for solving for $c_2$: $5 = -2 - 2 + c_2 \implies c_2 = 9$.
   6. A1 for the final correct function $f(x) = 2x^3 - 2x^2 + 9$.

3. Graphs and Transformations · 4 marks

   A cubic curve crosses the x-axis at $x = -2$, $x = 1$, and $x = 4$. It crosses the y-axis at the point $(0, 16)$. Find the equation of the curve in the form $y = ax^3 + bx^2 + cx + d$.

   Worked answer

   1. M1 for writing the equation in the factored form $y = k(x+2)(x-1)(x-4)$.
   2. M1 for substituting $(0, 16)$ to find $k$: $16 = k(2)(-1)(-4) \implies 16 = 8k$.
   3. A1 for finding $k = 2$.
   4. A1 for expanding to get $y = 2(x^3 - 3x^2 - 6x + 8) = 2x^3 - 6x^2 - 12x + 16$.

More on these topics: Trigonometry · Integration · Graphs and Transformations

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• Algebra
• Coordinate Geometry
• Trigonometry
• Differentiation
• Integration
• Graphs and Transformations

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Formula booklet

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