| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | March |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sine and Cosine Rules |
| Type | Algebraic side lengths |
| Difficulty | Challenging +1.2 Part (a) is a straightforward application of the cosine rule with given angle and sides, requiring only substitution and calculation. Part (b) requires algebraic manipulation of the cosine rule to express k in terms of θ, then using the given inequality to determine bounds—this involves some problem-solving beyond routine application, but the steps are guided and the techniques are standard for A-level. |
| Spec | 1.05b Sine and cosine rules: including ambiguous case1.05e Small angle approximations: sin x ~ x, cos x ~ 1-x^2/2, tan x ~ x |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\Delta ADE = \frac{1}{2}(ka)^2 \sin\frac{\pi}{6}\) | M1 | Attempt to find the area of \(\Delta ADE\) |
| \(\frac{1}{4}k^2a^2\) | A1 | OE |
| Sector \(ABC = \frac{1}{2}a^2\frac{\pi}{6}\) | B1 | |
| \(2 \times \frac{1}{4}k^2a^2 = \frac{1}{2}a^2\frac{\pi}{6}\) | M1 | OE. For \(2\times\Delta ADE = \text{sector}ABC\) with at least one correct area |
| \(k = \sqrt{\frac{\pi}{6}} = 0.7236\) | A1 | |
| Total: 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(2\times\frac{1}{2}(ka)^2\sin\theta = \frac{1}{2}a^2\theta\) | M1 | Condone omission of '2' or '1/2' on LHS for M1 only |
| \(k^2 = \frac{\theta}{2\sin\theta}\) | A1 | |
| \(k^2 > \frac{1}{2}\) leading to \(\frac{1}{\sqrt{2}} < k < 1\) | A1 | OE. Accept \(k > \frac{1}{\sqrt{2}}\) or \(k > 0.707\) (AWRT) or \(0.707\text{(AWRT)} < k < 1\) or \(k > \sqrt{\frac{1}{2}}\) OE |
| Total: 3 |
## Question 10(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\Delta ADE = \frac{1}{2}(ka)^2 \sin\frac{\pi}{6}$ | M1 | Attempt to find the area of $\Delta ADE$ |
| $\frac{1}{4}k^2a^2$ | A1 | OE |
| Sector $ABC = \frac{1}{2}a^2\frac{\pi}{6}$ | B1 | |
| $2 \times \frac{1}{4}k^2a^2 = \frac{1}{2}a^2\frac{\pi}{6}$ | M1 | OE. For $2\times\Delta ADE = \text{sector}ABC$ with at least one correct area |
| $k = \sqrt{\frac{\pi}{6}} = 0.7236$ | A1 | |
| **Total: 5** | | |
## Question 10(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $2\times\frac{1}{2}(ka)^2\sin\theta = \frac{1}{2}a^2\theta$ | M1 | Condone omission of '2' or '1/2' on LHS for M1 only |
| $k^2 = \frac{\theta}{2\sin\theta}$ | A1 | |
| $k^2 > \frac{1}{2}$ leading to $\frac{1}{\sqrt{2}} < k < 1$ | A1 | OE. Accept $k > \frac{1}{\sqrt{2}}$ or $k > 0.707$ (AWRT) or $0.707\text{(AWRT)} < k < 1$ or $k > \sqrt{\frac{1}{2}}$ OE |
| **Total: 3** | | |\begin{enumerate}[label=(\alph*)]
\item For the case where angle $B A C = \frac { 1 } { 6 } \pi$ radians, find $k$ correct to 4 significant figures.
\item For the general case in which angle $B A C = \theta$ radians, where $0 < \theta < \frac { 1 } { 2 } \pi$, it is given that $\frac { \theta } { \sin \theta } > 1$. Find the set of possible values of $k$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2021 Q10 [8]}}