| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive equation from area/geometry |
| Difficulty | Standard +0.3 This is a standard sector geometry problem requiring formula recall (area of sector = ½r²θ, area of triangle = ½r²sinθ) and basic algebraic manipulation. Part (i) involves setting up an equation from equal areas, leading to a simple rearrangement. Part (ii) is straightforward application of arc length and chord length formulas. Both parts are routine applications of AS-level circle geometry with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.05c Area of triangle: using 1/2 ab sin(C)1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\frac{1}{2}r^2\theta = \frac{1}{2}r^2\theta - \frac{1}{2}r^2\sin\theta\) → \(2\sin\theta = \theta\) → \(p = 2.\) | B1 B1 [2] | Correct equation. All ok – answer given. |
| (ii) Chord length = \(8\sin1.2 \times 2\) (14.9) (or from cosine rule) Arc length = 2.4 × 8 (19.2) Perimeter = sum of these = 34.1 | M1 B1 A1 [3] | Needs ×2. Any method ok. co |
**(i)** $\frac{1}{2}r^2\theta = \frac{1}{2}r^2\theta - \frac{1}{2}r^2\sin\theta$ → $2\sin\theta = \theta$ → $p = 2.$ | B1 B1 [2] | Correct equation. All ok – answer given.
**(ii)** Chord length = $8\sin1.2 \times 2$ (14.9) (or from cosine rule) Arc length = 2.4 × 8 (19.2) Perimeter = sum of these = 34.1 | M1 B1 A1 [3] | Needs ×2. Any method ok. co4\\
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The diagram shows a sector of a circle with radius $r \mathrm {~cm}$ and centre $O$. The chord $A B$ divides the sector into a triangle $A O B$ and a segment $A X B$. Angle $A O B$ is $\theta$ radians.\\
(i) In the case where the areas of the triangle $A O B$ and the segment $A X B$ are equal, find the value of the constant $p$ for which $\theta = p \sin \theta$.\\
(ii) In the case where $r = 8$ and $\theta = 2.4$, find the perimeter of the segment $A X B$.
\hfill \mbox{\textit{CAIE P1 2014 Q4 [5]}}