| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2012 |
| Session | June |
| Marks | 9 |
| Topic | Tangents, normals and gradients |
| Type | Optimization with constraints |
| Difficulty | Moderate -0.3 This is a straightforward optimization problem using standard circle sector formulas. Part (i) requires basic recall of sector area and arc length formulas. Part (ii) involves simple algebraic manipulation to eliminate r using the constraint P=20. Part (iii) is routine differentiation and finding a maximum. While it requires multiple steps, each step uses standard A-level techniques with no novel insight needed, making it slightly easier than average. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(P = 2r + 2rx\); \(A = r^2x\) | B1, B1 [2] | |
| (ii) \(P = 20\) implies \(r = \frac{10}{1+x}\) | M1 | \(r = f(x)\) |
| so \(A = \left(\frac{10}{1+x}\right)^2 x = \frac{100x}{(1+x)^2}\) AG | A1 [2] | |
| (iii) Use quotient rule: \(\frac{dA}{dx} = \frac{100(1+x)^2 - 200x(1+x)}{(1+x)^4} = \left[\frac{100(1-x)}{(1+x)^3}\right]\) | M1, A1, A1 | |
| Set equal to zero and find \(x = 1\); Attempt to show with first differential test that it is max; Completely correct solution | M1, A1 [5] | [9] |
**(i)** $P = 2r + 2rx$; $A = r^2x$ | B1, B1 [2] |
**(ii)** $P = 20$ implies $r = \frac{10}{1+x}$ | M1 | $r = f(x)$
so $A = \left(\frac{10}{1+x}\right)^2 x = \frac{100x}{(1+x)^2}$ AG | A1 [2] |
**(iii)** Use quotient rule: $\frac{dA}{dx} = \frac{100(1+x)^2 - 200x(1+x)}{(1+x)^4} = \left[\frac{100(1-x)}{(1+x)^3}\right]$ | M1, A1, A1 |
Set equal to zero and find $x = 1$; Attempt to show with first differential test that it is max; Completely correct solution | M1, A1 [5] | [9] | Allow $\pm 1$; Or equivalent\includegraphics{figure_9}
The diagram shows a sector of a circle, $OMN$. The angle $MON$ is $2x$ radians, the radius of the circle is $r$ and $O$ is the centre.
\begin{enumerate}[label=(\roman*)]
\item Find expressions, in terms of $r$ and $x$, for the area, $A$, and perimeter, $P$, of the sector. [2]
\item Given that $P = 20$, show that $A = \frac{100x}{(1 + x)^2}$. [2]
\item Find $\frac{dA}{dx}$, and hence find the value of $x$ for which the area of the sector is a maximum. [5]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2012 Q9 [9]}}