| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2016 |
| Session | Specimen |
| Marks | 9 |
| Topic | Tangents, normals and gradients |
| Type | Optimization with constraints |
| Difficulty | Standard +0.3 This is a standard optimization problem with a constraint. Part (i) requires basic circle sector formulas (arc length and area), part (ii) involves algebraic manipulation to eliminate r using the constraint P=20, and part (iii) is routine differentiation using the quotient rule followed by setting the derivative to zero. All steps are textbook procedures with no novel insight required, 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 |
**(i)** $P = 2r + 2rx$ **B1**
$A = r^2x$ **B1**
**(ii)** $P = 20$ implies $r = \frac{10}{1+x}$ **M1**
so $A = \left(\frac{10}{1+x}\right)^2 x = \frac{100x}{(1+x)^2}$ AG **A1**
**(iii)** Use quotient rule **M1**
$\frac{dA}{dx} = \frac{100(1+x)^2 - 200x(1+x)}{(1+x)^4} = \frac{100(1-x)}{(1+x)^3}$ **A1**
Set equal to zero and find $x = 1$ **A1**
Show with first differential test that it is maximum. o.e. **M1 A1**5\\
\includegraphics[max width=\textwidth, alt={}, center]{1c957cfe-bead-41d9-8985-479e876e1616-3_577_743_287_662}
The diagram shows a sector of a circle, $O M N$. The angle $M O N$ is $2 x$ radians, the radius of the circle is $r$ and $O$ is the centre.\\
(i) Find expressions, in terms of $r$ and $x$, for the area, $A$, and the perimeter, $P$, of the sector.\\
(ii) Given that $P = 20$, show that $A = \frac { 100 x } { ( 1 + x ) ^ { 2 } }$.\\
(iii) Find $\frac { \mathrm { d } A } { \mathrm {~d} x }$, and hence find the value of $x$ for which the area of the sector is a maximum.
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2016 Q5 [9]}}