| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2012 |
| Session | Specimen |
| Marks | 10 |
| Topic | Areas by integration |
| Type | Region bounded by two curves |
| Difficulty | Moderate -0.3 This is a straightforward area-between-curves question requiring finding intersection points by solving a quadratic, sketching two simple curves, and evaluating a definite integral. All steps are routine A-level techniques with no novel insight required, making it slightly easier than average but not trivial since it requires correct setup and integration of a cubic. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.08f Area between two curves: using integration |
**(i)** Parabola correct **B1**
Line correct **B1**
**(ii)** Equating and attempting to solve equation **M1**
Obtain $x = -1$ and $x = 2$ **A1**
*EITHER:* Attempt subtraction $f(x) - g(x)$ in the correct order **M1**
Obtain $2 + x - x^2$ **A1**
Attempt integration of their difference **M1**
Obtain $2x - \dfrac{1}{2}x^2 - \dfrac{1}{3}x^3$ **A1**
Use limits correctly **M1**
Obtain $4\dfrac{1}{2}$ **A1**
*OR:* Attempt $\displaystyle\int(3 + 2x - x^2)\,\mathrm{d}x$ **M1**
Obtain $3x + x^2 - \dfrac{1}{3}x^3$ **A1**
Use limits correctly **M1**
Obtain $9$ **A1**
Calculate area of triangle as $\dfrac{1}{2} \times 3 \times 3 = 4\dfrac{1}{2}$ **M1**
Subtract to obtain area between curve and line as $4\dfrac{1}{2}$ **A1**
**Total: 10 marks**9 (i) On the same axes, sketch the curves $y = 3 + 2 x - x ^ { 2 }$ and $y = x + 1$.\\
(ii) Find the exact area of the region contained between the curves $y = 3 + 2 x - x ^ { 2 }$ and $y = x + 1$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2012 Q9 [10]}}