| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2012 |
| Session | Specimen |
| Marks | 7 |
| Topic | Radians, Arc Length and Sector Area |
| Type | Simultaneous equations with arc/area |
| Difficulty | Moderate -0.3 This is a straightforward application of standard arc length and sector area formulas (s=rθ, A=½r²θ) leading to simultaneous equations. The algebra is routine: substitution gives a perfect square quadratic (r-10)²=0, yielding r=10 immediately, then θ=2. While it requires multiple steps, each is standard bookwork with no conceptual challenges or problem-solving insight needed—slightly easier than a typical A-level question. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
**(i)** Form the equation $2r + r\theta = 40$ **B1**
and $\dfrac{1}{2}r^2\theta = 100$ **B1**
**(ii)** Use $\theta = \dfrac{200}{r^2}$, or equivalent, to eliminate $\theta$ **M1**
Obtain $r^2 - 20r + 100 = 0$ Answer given **A1**
Solve quadratic for $r$ **M1**
Obtain correct value $r = 10$ **A1**
Substitute and obtain correct value $\theta = 2$ **A1**
**Total: 7 marks**4 A sector $A O B$ of a circle has radius $r \mathrm {~cm}$ and the angle $A O B$ is $\theta$ radians. The perimeter of the sector is 40 cm and its area is $100 \mathrm {~cm} ^ { 2 }$.\\
(i) Write down equations for the perimeter and area of the sector in terms of $r$ and $\theta$.\\
(ii) Use your equations to show that $r ^ { 2 } - 20 r + 100 = 0$ and hence find the value of $r$ and of $\theta$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2012 Q4 [7]}}