| 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 (area = ½r²θ, arc length = rθ). Part (ii) involves algebraic manipulation to eliminate r using the constraint P=20. Part (iii) is routine differentiation using the quotient rule and solving dA/dx=0. All steps are textbook-standard 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 sums |
$P = 2r + 2rx$ [B1]
$A = r^2x$ [B1]
$P = 20$ implies $r = \frac{10}{1+x}$ [M1]
so $A = \left(\frac{10}{1+x}\right)^2 x = \frac{100x}{(1+x)^2}$ AG [A1]
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]{ac5bf967-8b97-4bf3-991f-28c3ec7a25da-3_570_736_292_667}
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]}}