| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Second derivative test justification |
| Difficulty | Standard +0.3 This is a standard multi-part calculus question covering routine techniques: finding stationary points using first/second derivatives, verifying a gradient calculation, finding a point with given gradient, and computing a triangle area. All steps are straightforward applications of C2 methods with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.07i Differentiate x^n: for rational n and sums1.07m 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 |
|---|---|
| B1 | condone one error |
| Answer | Marks |
|---|---|
| A2 | A1 for one correct or \(x = 0, 2\) |
| Answer | Marks |
|---|---|
| T1 | SC B1 for \((0,1)\) from their \(y'\) |
| Answer | Marks |
|---|---|
| B1 | Dep't on M1 or \(y\) either side or clear cubic sketch |
| Answer | Marks |
|---|---|
| M1 | ft for their \(y'\) |
| Answer | Marks |
|---|---|
| A1 | implies the M1 |
| Answer | Marks |
|---|---|
| B1 | ft their \((3, 1)\) and their grad, not 9 |
| Answer | Marks |
|---|---|
| B1 | ft their normal (linear) |
**Question 3:**
**i**
B1 | condone one error
**ii**
$y' = 3x^2 - 6x$
M1
$y' = 0$
A2 | A1 for one correct or $x = 0, 2$
$(0, 1)$ or $(2, -3)$
T1 | SC B1 for $(0,1)$ from their $y'$
sign of $y''$ used to test or $y'$ either side
B1 | Dep't on M1 or $y$ either side or clear cubic sketch
$y'(-1) = 3 + 6 = 9$
M1 | ft for their $y'$
$3x^2 - 6x = 9$
$x = 3$
A1 | implies the M1
At P $y = 1$
B1
grad normal $= -\frac{1}{9}$ cao
B1 | ft their $(3, 1)$ and their grad, not 9
$y - 1 = -\frac{1}{9}(x - 3)$
M1
intercepts $12$ and $\frac{4}{3}$ or use of $\int_0^{12} \frac{4}{3} - \frac{1}{9}x \, dx$ (their normal)
B1 | ft their normal (linear)
$\frac{1}{2} \times 12 \times \frac{4}{3}$ cao
A1
**Total: 13 marks (5 marks for i, 8 marks for ii)**3 A cubic curve has equation $y = x ^ { 3 } - 3 x ^ { 2 } + 1$.\\
(i) Use calculus to find the coordinates of the turning points on this curve. Determine the nature of these turning points.\\
(ii) 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.
\hfill \mbox{\textit{OCR MEI C2 Q3 [13]}}