| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Compound angle with reciprocal functions |
| Difficulty | Standard +0.8 This question requires converting reciprocal trig functions to standard form, then applying compound angle formulas in reverse (harmonic form), followed by solving a double-angle equation. Part (a) involves non-trivial algebraic manipulation (multiplying by cos x, rearranging to harmonic form) and recognizing the R cos(x+α) structure. Part (b) requires careful handling of the double angle substitution and finding all solutions in the given range. While the techniques are A-level standard, the combination of reciprocal functions, harmonic form, and double angles with exact/decimal answers makes this moderately challenging, above average difficulty. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Rearrange and obtain \(4\cos x - \sin x = \sqrt{5}\) | B1 | |
| State \(R = \sqrt{17}\) | B1 | |
| Use trig formulae to find \(\alpha\) | M1 | |
| Obtain \(\alpha = 14.04°\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Evaluate \(\cos^{-1}\!\left(\dfrac{\sqrt{5}}{\sqrt{17}}\right)\) | B1 FT | FT *their* \(R\) |
| Carry out a correct method to find a value of \(x\) in the given interval | M1 | |
| Obtain answer, e.g. \(21.6°\) | A1 | |
| Obtain a second answer, e.g. \(144.4°\) and no other in the interval | A1 | Treat answers in radians as a misread. Ignore answers outside the given interval. |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Rearrange and obtain $4\cos x - \sin x = \sqrt{5}$ | B1 | |
| State $R = \sqrt{17}$ | B1 | |
| Use trig formulae to find $\alpha$ | M1 | |
| Obtain $\alpha = 14.04°$ | A1 | |
## Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Evaluate $\cos^{-1}\!\left(\dfrac{\sqrt{5}}{\sqrt{17}}\right)$ | B1 FT | FT *their* $R$ |
| Carry out a correct method to find a value of $x$ in the given interval | M1 | |
| Obtain answer, e.g. $21.6°$ | A1 | |
| Obtain a second answer, e.g. $144.4°$ and no other in the interval | A1 | Treat answers in radians as a misread. Ignore answers outside the given interval. |
---7
\begin{enumerate}[label=(\alph*)]
\item Show that the equation $\sqrt { 5 } \sec x + \tan x = 4$ can be expressed as $R \cos ( x + \alpha ) = \sqrt { 5 }$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$. Give the exact value of $R$ and the value of $\alpha$ correct to 2 decimal places. [4]
\item Hence solve the equation $\sqrt { 5 } \sec 2 x + \tan 2 x = 4$, for $0 ^ { \circ } < x < 180 ^ { \circ }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2022 Q7 [8]}}