| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2017 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Optimization with constraint |
| Difficulty | Standard +0.3 This is a standard optimization problem requiring volume-to-constraint substitution, surface area formula derivation, differentiation, and second derivative test. While multi-step, each component uses routine P1 techniques (differentiation of powers, solving equations) with no novel insight required. Slightly easier than average due to guided structure and standard methodology. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx1.07p Points of inflection: using second derivative |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Volume \(= \left(\frac{1}{2}\right)x^2\frac{\sqrt{3}}{2}h = 2000 \rightarrow h = \frac{8000}{\sqrt{3}x^2}\) | M1 | Use of (area of triangle, with attempt at height) \(\times h = 2000\), \(h = f(x)\) |
| \(A = 3xh + (2)\times\left(\frac{1}{2}\right)\times x^2 \times \frac{\sqrt{3}}{2}\) | M1 | Uses 3 rectangles and at least one triangle |
| Sub for \(h \rightarrow A = \frac{\sqrt{3}}{2}x^2 + \frac{24000}{\sqrt{3}}x^{-1}\) | A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{dA}{dx} = \frac{\sqrt{3}}{2}\cdot 2x - \frac{24000}{\sqrt{3}}x^{-2}\) | B1 | CAO, allow decimal equivalent |
| \(= 0\) when \(x^3 = 8000 \rightarrow x = 20\) | M1 A1 | Sets their \(\frac{dA}{dx}\) to 0 and attempts to solve for \(x\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{d^2A}{dx^2} = \frac{\sqrt{3}}{2}\cdot 2 + \frac{48000}{\sqrt{3}}x^{-3} > 0\) | M1 | Any valid method, ignore value of \(\frac{d^2A}{dx^2}\) providing it is positive |
| \(\rightarrow\) Minimum | A1 FT | FT on their \(x\) providing it is positive |
## Question 6:
**Part 6(i):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Volume $= \left(\frac{1}{2}\right)x^2\frac{\sqrt{3}}{2}h = 2000 \rightarrow h = \frac{8000}{\sqrt{3}x^2}$ | M1 | Use of (area of triangle, with attempt at height) $\times h = 2000$, $h = f(x)$ |
| $A = 3xh + (2)\times\left(\frac{1}{2}\right)\times x^2 \times \frac{\sqrt{3}}{2}$ | M1 | Uses 3 rectangles and at least one triangle |
| Sub for $h \rightarrow A = \frac{\sqrt{3}}{2}x^2 + \frac{24000}{\sqrt{3}}x^{-1}$ | A1 | AG |
**Part 6(ii):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dA}{dx} = \frac{\sqrt{3}}{2}\cdot 2x - \frac{24000}{\sqrt{3}}x^{-2}$ | B1 | CAO, allow decimal equivalent |
| $= 0$ when $x^3 = 8000 \rightarrow x = 20$ | M1 A1 | Sets their $\frac{dA}{dx}$ to 0 and attempts to solve for $x$ |
**Part 6(iii):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{d^2A}{dx^2} = \frac{\sqrt{3}}{2}\cdot 2 + \frac{48000}{\sqrt{3}}x^{-3} > 0$ | M1 | Any valid method, ignore value of $\frac{d^2A}{dx^2}$ providing it is positive |
| $\rightarrow$ Minimum | A1 FT | FT on their $x$ providing it is positive |
---6 The horizontal base of a solid prism is an equilateral triangle of side $x \mathrm {~cm}$. The sides of the prism are vertical. The height of the prism is $h \mathrm {~cm}$ and the volume of the prism is $2000 \mathrm {~cm} ^ { 3 }$.\\
(i) Express $h$ in terms of $x$ and show that the total surface area of the prism, $A \mathrm {~cm} ^ { 2 }$, is given by
$$A = \frac { \sqrt { } 3 } { 2 } x ^ { 2 } + \frac { 24000 } { \sqrt { } 3 } x ^ { - 1 }$$
(ii) Given that $x$ can vary, find the value of $x$ for which $A$ has a stationary value.\\
(iii) Determine, showing all necessary working, the nature of this stationary value.\\
\hfill \mbox{\textit{CAIE P1 2017 Q6 [8]}}