| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2006 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Determine nature of stationary points |
| Difficulty | Moderate -0.3 This is a standard C2 calculus question covering routine techniques: finding turning points via differentiation, determining their nature with the second derivative, finding tangent/normal equations, and basic coordinate geometry. Part (i) is textbook-standard (5 marks). Part (ii) involves more steps but uses only familiar methods—verifying a gradient, solving a quadratic for another point with the same gradient, finding a normal equation, and calculating a triangle area. While the multi-step nature adds some challenge, all techniques are core C2 material with no novel insight required, making it slightly easier than the average A-level question. |
| Spec | 1.07m Tangents and normals: gradient and equations1.07n Stationary points: find maxima, minima using derivatives1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| i \(y = 3x^2 - 6x\) | B1 | condone one error |
| use of \(y' = 0\) | M1 | |
| \((0, 1)\) or \((2, -3)\) | A2 | A1 for one correct or \(x = 0, 2\) |
| sign of \(y'\) used to test or \(y'\) either side | T1 | |
| Dep't on M1 or \(y\) either side or clear cubic sketch | \(5\) | |
| ii \(y'(-1) = 3 + 6 = 9\) | B1 | ft for their \(y'\) |
| \(3x^2 - 6x = 9\) | M1 | implies the M1 |
| \(x = 3\) | A1 | |
| At P \(y = 1\) | B1 | |
| grad normal \(= -1/9\) cao | B1 | |
| \(y - 1 = -1/9(x - 3)\) | M1 | |
| intercepts \(12\) and \(4\)/3or use of \(\int_0^{12} \frac{4}{3}x \, dx\) (their normal) | B1 | ft their \((3, 1)\) and their grad, not 9; ft their normal (linear) |
| \(\frac{1}{2} \times 12 \times 4/3\) cao | A1 |
i $y = 3x^2 - 6x$ | B1 | condone one error |
use of $y' = 0$ | M1 | |
$(0, 1)$ or $(2, -3)$ | A2 | A1 for one correct or $x = 0, 2$ |
sign of $y'$ used to test or $y'$ either side | T1 | |
| | Dep't on M1 or $y$ either side or clear cubic sketch | $5$ |
ii $y'(-1) = 3 + 6 = 9$ | B1 | ft for their $y'$ |
$3x^2 - 6x = 9$ | M1 | implies the M1 |
$x = 3$ | A1 | |
At P $y = 1$ | B1 | |
grad normal $= -1/9$ cao | B1 | |
$y - 1 = -1/9(x - 3)$ | M1 | |
intercepts $12$ and $4$/3or use of $\int_0^{12} \frac{4}{3}x \, dx$ (their normal) | B1 | ft their $(3, 1)$ and their grad, not 9; ft their normal (linear) |
$\frac{1}{2} \times 12 \times 4/3$ cao | A1 | | $8$ | $13$ |A cubic curve has equation $y = x^3 - 3x^2 + 1$.
\begin{enumerate}[label=(\roman*)]
\item Use calculus to find the coordinates of the turning points on this curve. Determine the nature of these turning points. [5]
\item Show that the tangent to the curve at the point where $x = -1$ has gradient 9.
Find the coordinates of the other point, P, on the curve at which the tangent has gradient 9 and find the equation of the normal to the curve at P.
Show that the area of the triangle bounded by the normal at P and the $x$- and $y$-axes is 8 square units. [8]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C2 2006 Q11 [13]}}