| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2004 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Quadratic trigonometric equations |
| Type | Show then solve: sin²/cos² substitution |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring standard techniques: using the identity cos²x = 1 - sin²x to simplify, solving a quadratic equation in sin x, and finding range from the simplified form. All steps are routine for P1 level with no novel insight required, making it slightly easier than average. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(5s^2 + 3c^2 = 5s^2 + 3(1-s^2) \rightarrow 3 + 2\sin^2 x\), \(a=3\), \(b=2\) | M1, A1 [2] | Use of \(s^2 + c^2 = 1\); \(3 + 2\sin^2 x\) gets both marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(3 + 2s^2 = 7s\); sets to 0 and solves; \(s = \frac{1}{2}\) or \(s = 3\) | M1 | Sets to \(0\) + correct method of solution |
| Only values are \(\frac{\pi}{6}\) and \(\frac{5\pi}{6}\) | A1A1\(\sqrt{}\) [3] | Co for one value; other \(\pi = \) "1st"; if degrees give A0,A1\(\sqrt{}\) for \(180-\) |
| Answer | Marks | Guidance |
|---|---|---|
| Minimum value \(= "a" = 3\); Maximum value \(= "a+b" = 5\) | ||
| Range \(3 \leq f(x) \leq 5\) | B1\(\sqrt{}\)B1\(\sqrt{}\) [2] | For his "\(a\)" and "\(a+b\)"; condone \(<\); allow 3 and 5 on their own |
# Question 6:
## Part (i)
| $5s^2 + 3c^2 = 5s^2 + 3(1-s^2) \rightarrow 3 + 2\sin^2 x$, $a=3$, $b=2$ | M1, A1 [2] | Use of $s^2 + c^2 = 1$; $3 + 2\sin^2 x$ gets both marks |
## Part (ii)
| $3 + 2s^2 = 7s$; sets to 0 and solves; $s = \frac{1}{2}$ or $s = 3$ | M1 | Sets to $0$ + correct method of solution |
| Only values are $\frac{\pi}{6}$ and $\frac{5\pi}{6}$ | A1A1$\sqrt{}$ [3] | Co for one value; other $\pi = $ "1st"; if degrees give A0,A1$\sqrt{}$ for $180-$ |
## Part (iii)
| Minimum value $= "a" = 3$; Maximum value $= "a+b" = 5$ | | |
| Range $3 \leq f(x) \leq 5$ | B1$\sqrt{}$B1$\sqrt{}$ [2] | For his "$a$" and "$a+b$"; condone $<$; allow 3 and 5 on their own |
---6 The function $\mathrm { f } : x \mapsto 5 \sin ^ { 2 } x + 3 \cos ^ { 2 } x$ is defined for the domain $0 \leqslant x \leqslant \pi$.\\
(i) Express $\mathrm { f } ( x )$ in the form $a + b \sin ^ { 2 } x$, stating the values of $a$ and $b$.\\
(ii) Hence find the values of $x$ for which $\mathrm { f } ( x ) = 7 \sin x$.\\
(iii) State the range of f .
\hfill \mbox{\textit{CAIE P1 2004 Q6 [7]}}