| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Find stationary point then sketch curve |
| Difficulty | Moderate -0.8 This is a straightforward C2 question testing routine differentiation and stationary point analysis. Part (i) requires basic polynomial differentiation and solving a quadratic, part (ii) involves factorising a cubic, and part (iii) is a standard sketch. All techniques are textbook exercises with no problem-solving insight required, making it easier than average. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| \(y'=6x^2-18x+12\) | M1 | condone one error |
| \(=12\) | M1 | subst of \(x=3\) in their \(y'\) |
| \(y=7\) when \(x=3\) | B1 | |
| tgt is \(y-7=12(x-3)\) | M1 | f.t. their \(y\) and \(y'\) |
| verifying \((-1,-41)\) on tgt | A1 | or B2 for showing line joining \((3,7)\) and \((-1,-41)\) has gradient 12 |
| Answer | Marks | Guidance |
|---|---|---|
| \(y'=0\) s.o.i. | M1 | Their \(y'\) |
| quadratic with 3 terms | M1 | Any valid attempt at solution |
| \(x=1\) or \(2\) | A1 | or A1 for \((1,3)\) and A1 for \((2,2)\) marking to benefit of candidate |
| \(y=3\) or \(2\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| cubic curve correct orientation touching \(x\)-axis only at \((0.2,0)\) | G1 | |
| max and min correct | G1 | f. |
| curve crossing \(y\)-axis only at \(-2\) | G1 |
## Question 4:
**Part i:**
$y'=6x^2-18x+12$ | M1 | condone one error
$=12$ | M1 | subst of $x=3$ in their $y'$
$y=7$ when $x=3$ | B1 |
tgt is $y-7=12(x-3)$ | M1 | f.t. their $y$ and $y'$
verifying $(-1,-41)$ on tgt | A1 | or B2 for showing line joining $(3,7)$ and $(-1,-41)$ has gradient 12 | 5
**Part ii:**
$y'=0$ s.o.i. | M1 | Their $y'$
quadratic with 3 terms | M1 | Any valid attempt at solution
$x=1$ or $2$ | A1 | or A1 for $(1,3)$ and A1 for $(2,2)$ marking to benefit of candidate
$y=3$ or $2$ | A1 | | 4
**Part iii:**
cubic curve correct orientation touching $x$-axis only at $(0.2,0)$ | G1 |
max and min correct | G1 | f.
curve crossing $y$-axis only at $-2$ | G1 | | 3
---4 (i) Differentiate $x ^ { 3 } - 3 x ^ { 2 } - 9 x$. Hence find the $x$-coordinates of the stationary points on the curve $y = x ^ { 3 } - 3 x ^ { 2 } - 9 x$, showing which is the maximum and which the minimum.\\
(ii) Find, in exact form, the coordinates of the points at which the curve crosses the $x$-axis.\\
(iii) Sketch the curve.
\hfill \mbox{\textit{OCR MEI C2 Q4 [11]}}