| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2002 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Graphs & Exact Values |
| Type | Find function constants from given conditions |
| Difficulty | Easy -1.2 This is a straightforward question requiring substitution of exact trig values (sin(π/2)=1, sin(3π/2)=-1) to form two simultaneous equations, then solving for a and b. Part (ii) involves basic equation solving and calculator work, while part (iii) is a routine sketch. All steps are standard procedures with no problem-solving insight required, making it easier than average. |
| Spec | 1.02m Graphs of functions: difference between plotting and sketching1.02u Functions: definition and vocabulary (domain, range, mapping)1.05f Trigonometric function graphs: symmetries and periodicities1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Solution: \(a = -5\), \(b = -3\) | B1, B1, B1 | Correct only; Correct only; Correct only |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = 0.64\) or \(x = 2.50\) | B1\*, B1\* | For \(\sin^{-1}(-b/a)\); For \(\pi\) - his answer |
| (iii) Graph | B2\*, 2 | Just one cycle. Starts on negative y-axis. Max about correct. Min about correct |
$f(x) = a\sin x + b$
**(i)** $f(\pi/2) = 2$
$f(3\pi/2) = -8$
Solution: $a = -5$, $b = -3$ | B1, B1, B1 | Correct only; Correct only; Correct only
**(ii)** $5\sin x - 3 = 0$, $\sin x = \frac{3}{5}$
$x = 0.64$ or $x = 2.50$ | B1\*, B1\* | For $\sin^{-1}(-b/a)$; For $\pi$ - his answer
**(iii)** Graph | B2\*, 2 | Just one cycle. Starts on negative y-axis. Max about correct. Min about correct6 The function f , where $\mathrm { f } ( x ) = a \sin x + b$, is defined for the domain $0 \leqslant x \leqslant 2 \pi$. Given that $\mathrm { f } \left( \frac { 1 } { 2 } \pi \right) = 2$ and that $\mathrm { f } \left( \frac { 3 } { 2 } \pi \right) = - 8$,\\
(i) find the values of $a$ and $b$,\\
(ii) find the values of $x$ for which $\mathrm { f } ( x ) = 0$, giving your answers in radians correct to 2 decimal places,\\
(iii) sketch the graph of $y = \mathrm { f } ( x )$.
\hfill \mbox{\textit{CAIE P1 2002 Q6 [7]}}