| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2011 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Optimization with constraint |
| Difficulty | Standard +0.3 This is a straightforward constrained optimization problem requiring substitution from xy=600 into the first equation, then differentiating to find the minimum. The algebra is simple, and the method is standard textbook fare for P1 calculus, making it slightly easier than average. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(z = 3x + 2\left(\frac{600}{x}\right)\) or \(z = \frac{z - 3x}{2} = 600\) OE | B1 [1] | |
| \(\rightarrow\) AG | ||
| (ii) \(\frac{dz}{dx} = 3 - \frac{1200}{x^2}\) or \(\frac{dz}{dy} = 2 - \frac{1800}{y^2}\) | B1 | |
| \(= 0 \Rightarrow x = 20\) or \(= 0 \Rightarrow y = 30\) | M1 A1 | Set to 0 & attempt to solve. Allow ±20 Ft from their x provided positive Or other valid method |
| \(z = 60 + \frac{120}{20} = 120\) | A1 ✓ | |
| \(\frac{d^2z}{dx^2} = \frac{2400}{x^3}\) | B1 ✓ | Dep. on \(\frac{d^2z}{dx^2} = \frac{k}{x^3}\) (k > 0) or other valid method. |
| \(> 0 \Rightarrow\) minimum | B1 [6] |
**(i)** $z = 3x + 2\left(\frac{600}{x}\right)$ or $z = \frac{z - 3x}{2} = 600$ OE | B1 [1] |
$\rightarrow$ AG |
**(ii)** $\frac{dz}{dx} = 3 - \frac{1200}{x^2}$ or $\frac{dz}{dy} = 2 - \frac{1800}{y^2}$ | B1 |
$= 0 \Rightarrow x = 20$ or $= 0 \Rightarrow y = 30$ | M1 A1 | Set to 0 & attempt to solve. Allow ±20 Ft from their x provided positive Or other valid method
$z = 60 + \frac{120}{20} = 120$ | A1 ✓ |
$\frac{d^2z}{dx^2} = \frac{2400}{x^3}$ | B1 ✓ | Dep. on $\frac{d^2z}{dx^2} = \frac{k}{x^3}$ (k > 0) or other valid method.
$> 0 \Rightarrow$ minimum | B1 [6] |
---6 The variables $x , y$ and $z$ can take only positive values and are such that
$$z = 3 x + 2 y \quad \text { and } \quad x y = 600 .$$
(i) Show that $z = 3 x + \frac { 1200 } { x }$.\\
(ii) Find the stationary value of $z$ and determine its nature.
\hfill \mbox{\textit{CAIE P1 2011 Q6 [7]}}