| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2010 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Transformations of quadratic graphs |
| Difficulty | Moderate -0.8 This is a straightforward C1 question testing routine completion of the square, basic graph sketching, and standard transformations. All parts follow textbook procedures with no problem-solving required—students simply apply learned techniques mechanically. The multi-part structure adds marks but not conceptual difficulty. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02n Sketch curves: simple equations including polynomials1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| \(V = \pi\int_0^1 \cos(x^2)\,dx\) | M1, A1 | \(\int \cos(x^2)\,dx\); Correct limits (condone \(k\) or missing \(\pi\) until final mark) |
| Answer | Marks | Guidance |
|---|---|---|
| \(x\): \(0\), \(0.25\), \(0.5\), \(0.75\), \(1\) | B1 | PI |
| \(Y = y^2\): \(1\), \(0.9980(47)\), \(0.9689(12)\), \(0.8459(24)\), \(0.5403(02)\) | B1 | PI |
| \(\frac{0.25}{3} \times \{Y(0) + Y(1) + 4[Y(0.25) + Y(0.75)] + 2Y(0.5)\}\) | M1 | Use of Simpson's rule |
| \(V = \pi \times \frac{10.8539\ldots}{12}\), so \(V = 2.8416\) (to 4 d.p.) | A1 | 6 |
# Question 5 [XMCA2]:
$V = \pi\int_0^1 \cos(x^2)\,dx$ | M1, A1 | $\int \cos(x^2)\,dx$; Correct limits (condone $k$ or missing $\pi$ until final mark)
Applying Simpson's rule to $\int_0^1 \cos(x^2)\,dx$:
$x$: $0$, $0.25$, $0.5$, $0.75$, $1$ | B1 | PI
$Y = y^2$: $1$, $0.9980(47)$, $0.9689(12)$, $0.8459(24)$, $0.5403(02)$ | B1 | PI
$\frac{0.25}{3} \times \{Y(0) + Y(1) + 4[Y(0.25) + Y(0.75)] + 2Y(0.5)\}$ | M1 | Use of Simpson's rule
$V = \pi \times \frac{10.8539\ldots}{12}$, so $V = 2.8416$ (to 4 d.p.) | A1 | **6** | CAO
---5
\begin{enumerate}[label=(\alph*)]
\item Express $( x - 5 ) ( x - 3 ) + 2$ in the form $( x - p ) ^ { 2 } + q$, where $p$ and $q$ are integers.\\
(3 marks)
\item \begin{enumerate}[label=(\roman*)]
\item Sketch the graph of $y = ( x - 5 ) ( x - 3 ) + 2$, stating the coordinates of the minimum point and the point where the graph crosses the $y$-axis.
\item Write down an equation of the tangent to the graph of $y = ( x - 5 ) ( x - 3 ) + 2$ at its vertex.
\end{enumerate}\item Describe the geometrical transformation that maps the graph of $y = x ^ { 2 }$ onto the graph of $y = ( x - 5 ) ( x - 3 ) + 2$.
\end{enumerate}
\hfill \mbox{\textit{AQA C1 2010 Q5 [11]}}