| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Express in terms of one trig function |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question testing standard trigonometric identities and equation solving. Part (i) requires using sin²x + cos²x = 1 to rewrite the expression (routine manipulation), part (ii) involves finding range from the rewritten form (direct observation), and part (iii) is a basic quadratic-type equation in cos²x. All techniques are standard P1 material with no novel insight required, making it slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(2(1 - \cos^2 x) - 3\cos^2 x\) | M1 | Uses \(s^2 + c^2 = 1\) |
| \(\rightarrow 2 - 5\cos^2 x\ (a=2,\ b=-5)\) | A1 [2] | co |
| Answer | Marks |
|---|---|
| Values are \(-3\) and \(2\) | B1\(\sqrt{}\) B1\(\sqrt{}\) [2] |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = 0.685,\ 2.46\) (accept 0.684) | B1\(\sqrt{}\), B1 B1\(\sqrt{}\) [3] | co \(\sqrt{}\) for \(\pi -\) (first answer). SC B1 for both 39.2 and 140.8 |
## Question 5:
$x \mapsto 2\sin^2 x - 3\cos^2 x$
**Part (i):**
$2(1 - \cos^2 x) - 3\cos^2 x$ | M1 | Uses $s^2 + c^2 = 1$
$\rightarrow 2 - 5\cos^2 x\ (a=2,\ b=-5)$ | A1 [2] | co
**Part (ii):**
Values are $-3$ and $2$ | B1$\sqrt{}$ B1$\sqrt{}$ [2] |
**Part (iii):**
$2 - 5\cos^2 x = -1 \rightarrow \cos^2 x = 0.6$
$x = 0.685,\ 2.46$ (accept 0.684) | B1$\sqrt{}$, B1 B1$\sqrt{}$ [3] | co $\sqrt{}$ for $\pi -$ (first answer). SC B1 for both 39.2 and 140.8
---5 The function f is such that $\mathrm { f } ( x ) = 2 \sin ^ { 2 } x - 3 \cos ^ { 2 } x$ for $0 \leqslant x \leqslant \pi$.\\
(i) Express $\mathrm { f } ( x )$ in the form $a + b \cos ^ { 2 } x$, stating the values of $a$ and $b$.\\
(ii) State the greatest and least values of $\mathrm { f } ( x )$.\\
(iii) Solve the equation $\mathrm { f } ( x ) + 1 = 0$.
\hfill \mbox{\textit{CAIE P1 2010 Q5 [7]}}