| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Show formula then optimise: cylinder/prism (single variable) |
| Difficulty | Moderate -0.3 This is a standard constrained optimization problem requiring substitution of a constraint into a formula, differentiation of a polynomial, and finding a maximum using first/second derivative tests. All steps are routine C2 techniques with no novel insight required, making it slightly easier than average but still requiring multiple connected steps. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(200 - 2\pi r^2 = 2\pi r h\) | M1 | \(100 = \pi r^2 + \pi r h\). Alternative (sc3 for complete argument working backwards): \(V = 100r - \pi r^3 \Rightarrow \pi r^2 h = 100r - \pi r^3 \Rightarrow \pi r h = 100 - \pi r^2 \Rightarrow 100 = \pi r h + \pi r^2 \Rightarrow 200 = A = 2\pi r h + 2\pi r^2\); sc0 if argument is incomplete |
| \(h = \frac{200 - 2\pi r^2}{2\pi r}\) o.e. | M1 | \(100r = \pi r^3 + \pi r^2 h\). Or: M1 for \(h = \frac{V}{\pi r^2}\) |
| Substitution of correct \(h\) into \(V = \pi r^2 h\) | M1 | \(100r = \pi r^3 + V\). Or: M1 for \(200 = 2\pi r^2 + 2\pi r \times \frac{V}{\pi r^2}\); M1 for \(200 = 2\pi r^2 + \frac{2V}{r}\) |
| \(V = 100r - \pi r^3\) convincingly obtained | A1 | A1 for \(V = 100r - \pi r^3\) convincingly obtained |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{dV}{dr} = 100 - 3\pi r^2\) | B2 | B1 for each term; allow \(9.42(\ldots)\,r^2\) or better if decimalised |
| \(\frac{d^2V}{dr^2} = -6\pi r\) | B1 | \(-18.8(\ldots)\,r\) or better if decimalised |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Their \(\frac{dV}{dr} = 0\) s.o.i. | M1 | Must contain \(r\) as the only variable |
| \(r = 3.26\) c.a.o. | A2 | A1 for \(r = (\pm)\sqrt{\frac{100}{3\pi}}\); may be implied by \(3.25\ldots\) |
| \(V = 217\) c.a.o. | A1 | Deduct 1 mark only in this part if answers not given to 3 s.f.; there must be evidence of use of calculus |
## Question 3:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $200 - 2\pi r^2 = 2\pi r h$ | **M1** | $100 = \pi r^2 + \pi r h$. Alternative (sc3 for complete argument working backwards): $V = 100r - \pi r^3 \Rightarrow \pi r^2 h = 100r - \pi r^3 \Rightarrow \pi r h = 100 - \pi r^2 \Rightarrow 100 = \pi r h + \pi r^2 \Rightarrow 200 = A = 2\pi r h + 2\pi r^2$; **sc0** if argument is incomplete |
| $h = \frac{200 - 2\pi r^2}{2\pi r}$ o.e. | **M1** | $100r = \pi r^3 + \pi r^2 h$. Or: **M1** for $h = \frac{V}{\pi r^2}$ |
| Substitution of correct $h$ into $V = \pi r^2 h$ | **M1** | $100r = \pi r^3 + V$. Or: **M1** for $200 = 2\pi r^2 + 2\pi r \times \frac{V}{\pi r^2}$; **M1** for $200 = 2\pi r^2 + \frac{2V}{r}$ |
| $V = 100r - \pi r^3$ convincingly obtained | **A1** | **A1** for $V = 100r - \pi r^3$ convincingly obtained |
---
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dV}{dr} = 100 - 3\pi r^2$ | **B2** | **B1** for each term; allow $9.42(\ldots)\,r^2$ or better if decimalised |
| $\frac{d^2V}{dr^2} = -6\pi r$ | **B1** | $-18.8(\ldots)\,r$ or better if decimalised |
---
### Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Their $\frac{dV}{dr} = 0$ s.o.i. | **M1** | Must contain $r$ as the only variable |
| $r = 3.26$ c.a.o. | **A2** | **A1** for $r = (\pm)\sqrt{\frac{100}{3\pi}}$; may be implied by $3.25\ldots$ |
| $V = 217$ c.a.o. | **A1** | Deduct 1 mark only in this part if answers not given to 3 s.f.; there must be evidence of use of calculus |
---3 (i) The standard formulae for the volume $V$ and total surface area $A$ of a solid cylinder of radius $r$ and height $h$ are
$$V = \pi r ^ { 2 } h \quad \text { and } \quad A = 2 \pi r ^ { 2 } + 2 \pi r h .$$
Use these to show that, for a cylinder with $A = 200$,
$$V = 100 r - \pi r ^ { 3 }$$
(ii) Find $\frac { \mathrm { d } V } { \mathrm {~d} r }$ and $\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} r ^ { 2 } }$.\\
(iii) Use calculus to find the value of $r$ that gives a maximum value for $V$ and hence find this maximum value, giving your answers correct to 3 significant figures.
\hfill \mbox{\textit{OCR MEI C2 Q3 [11]}}