| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2020 |
| Session | March |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard trigonometric equations |
| Type | Convert to quadratic in tan |
| Difficulty | Standard +0.3 Part (a) is a straightforward quadratic in tan x requiring factorization and inverse tan. Part (b) involves discriminant analysis but is standard. Part (c) requires recognizing that tan x has a discontinuity at 90°, making it slightly more conceptually demanding but still routine for P1 level with clear structure and standard techniques throughout. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02g Inequalities: linear and quadratic in single variable1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((\tan x - 2)(3\tan x + 1) = 0\), or formula or completing the square | M1 | Allow reversal of signs in the factors. Must see a method |
| \(\tan x = 2\) or \(-\frac{1}{3}\) | A1 | |
| \(x = 63.4°\) (only value in range) | B1FT | |
| \(x = 161.6°\) (only value in range) | B1FT | |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Apply \(b^2 - 4ac < 0\) | M1 | SOI. Expect \(25 - 4(3)(k) < 0\), \(\tan x\) must not be in coefficients |
| \(k > \frac{25}{12}\) | A1 | Allow \(b^2 - 4ac = 0\) leading to correct \(k > \frac{25}{12}\) for M1A1 |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(k = 0\) | M1 | SOI |
| \(\tan x = 0\) or \(\frac{5}{3}\) | A1 | |
| \(x = 0°\) or \(180°\) or \(59.0°\) | A1 | All three required |
| Total: 3 |
## Question 11(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(\tan x - 2)(3\tan x + 1) = 0$, or formula or completing the square | M1 | Allow reversal of signs in the factors. Must see a method |
| $\tan x = 2$ or $-\frac{1}{3}$ | A1 | |
| $x = 63.4°$ (only value in range) | B1FT | |
| $x = 161.6°$ (only value in range) | B1FT | |
| **Total: 4** | | |
## Question 11(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Apply $b^2 - 4ac < 0$ | M1 | SOI. Expect $25 - 4(3)(k) < 0$, $\tan x$ must not be in coefficients |
| $k > \frac{25}{12}$ | A1 | Allow $b^2 - 4ac = 0$ leading to correct $k > \frac{25}{12}$ for M1A1 |
| **Total: 2** | | |
## Question 11(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $k = 0$ | M1 | SOI |
| $\tan x = 0$ or $\frac{5}{3}$ | A1 | |
| $x = 0°$ or $180°$ or $59.0°$ | A1 | All three required |
| **Total: 3** | | |11
\begin{enumerate}[label=(\alph*)]
\item Solve the equation $3 \tan ^ { 2 } x - 5 \tan x - 2 = 0$ for $0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }$.
\item Find the set of values of $k$ for which the equation $3 \tan ^ { 2 } x - 5 \tan x + k = 0$ has no solutions.
\item For the equation $3 \tan ^ { 2 } x - 5 \tan x + k = 0$, state the value of $k$ for which there are three solutions in the interval $0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }$, and find these solutions.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2020 Q11 [9]}}