| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2007 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Solve equation with tan(θ ± α) |
| Difficulty | Moderate -0.3 This is a straightforward application of the tan addition formula followed by solving a quadratic equation in tan x. Part (i) is algebraic manipulation with the formula provided, and part (ii) requires using the quadratic formula and finding angles from calculator values. While it requires multiple steps, each step follows standard procedures with no novel insight needed, making it slightly easier than average. |
| Spec | 1.09a Sign change methods: locate roots1.09b Sign change methods: understand failure cases |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use correct \(\tan(A+B)\) formula to obtain an equation in \(\tan x\) | M1* | |
| Use \(\tan 45° = 1\) | M1(dep) | |
| Obtain the given answer | A1 | [3] |
| (ii) Make reasonable attempt to solve the given quadratic for one value of \(\tan x\) | M1 | |
| Obtain \(\tan x = -1 \pm \sqrt{2}\), or equivalent in the form \((\pm \sqrt{b})/c\) (accept \(0.4, -2.4\)) | A1 | |
| Obtain answer \(x = 22.5°\) | A1 | |
| Obtain second answer \(x = 112.5\) and no others in the range | A1 | |
| [Ignore answers outside the range.] | [4] |
**(i)** Use correct $\tan(A+B)$ formula to obtain an equation in $\tan x$ | M1* |
Use $\tan 45° = 1$ | M1(dep) |
Obtain the given answer | A1 | [3]
**(ii)** Make reasonable attempt to solve the given quadratic for one value of $\tan x$ | M1 |
Obtain $\tan x = -1 \pm \sqrt{2}$, or equivalent in the form $(\pm \sqrt{b})/c$ (accept $0.4, -2.4$) | A1 |
Obtain answer $x = 22.5°$ | A1 |
Obtain second answer $x = 112.5$ and no others in the range | A1 |
[Ignore answers outside the range.] | | [4]
[Treat answers in radians as a MR and deduct one mark from the marks for the angles.]5 (i) Show that the equation
$$\tan \left( 45 ^ { \circ } + x \right) - \tan x = 2$$
can be written in the form
$$\tan ^ { 2 } x + 2 \tan x - 1 = 0$$
(ii) Hence solve the equation
$$\tan \left( 45 ^ { \circ } + x \right) - \tan x = 2$$
giving all solutions in the interval $0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P3 2007 Q5 [7]}}