| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2003 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Quadratic trigonometric equations |
| Type | Quartic in sin or cos substitution |
| Difficulty | Moderate -0.3 This is a standard A-level trigonometric equation requiring routine substitution using the Pythagorean identity (cos²θ = 1 - sin²θ), solving a quadratic equation, then finding angles. Part (i) is guided algebraic manipulation, and part (ii) involves straightforward application of inverse trig functions. Slightly easier than average due to the scaffolding in part (i) and use of standard techniques throughout. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(4s^4+s=7(1-s^2) \rightarrow 4x^2+7x-2=0\) | B1 | Use of \(s^2+c^2=1\). Answer given. |
| Answer | Marks | Guidance |
|---|---|---|
| \(\rightarrow \theta = 30°\) and \(150°\) and \(\theta = 210°\) and \(330°\) | M1, A1, A1, A1 [4] | Recognition of quadratic in \(s^2\). Co. For \(180° -\) "his value". For other 2 answers from "his value", providing no extra answers in the range or answers from \(s^2= -1\). |
**(i)** $4s^4+s=7(1-s^2) \rightarrow 4x^2+7x-2=0$ | B1 | Use of $s^2+c^2=1$. Answer given.
**(ii)** $4s^4+7s^2-2=0$
$\rightarrow s^2 = \frac{1}{4}$ or $s^2 = -2$
$\rightarrow \sin\theta = \pm\frac{1}{2}$
$\rightarrow \theta = 30°$ and $150°$ and $\theta = 210°$ and $330°$ | M1, A1, A1, A1 [4] | Recognition of quadratic in $s^2$. Co. For $180° -$ "his value". For other 2 answers from "his value", providing no extra answers in the range or answers from $s^2= -1$.2 (i) Show that the equation $4 \sin ^ { 4 } \theta + 5 = 7 \cos ^ { 2 } \theta$ may be written in the form $4 x ^ { 2 } + 7 x - 2 = 0$, where $x = \sin ^ { 2 } \theta$.\\
(ii) Hence solve the equation $4 \sin ^ { 4 } \theta + 5 = 7 \cos ^ { 2 } \theta$, for $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P1 2003 Q2 [5]}}