| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | October |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Applied context modeling |
| Difficulty | Standard +0.3 This is a standard A-level harmonic form question with routine application. Part (a) uses the textbook method for expressing a cos x + b sin x in R cos(x - α) form. Parts (b) and (c) apply this result to find maximum/minimum values and solve equations—all straightforward techniques covered in P3 with no novel problem-solving required. Slightly easier than average due to its predictable structure. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
(a) $R = \sqrt{17}$ B1
$\tan \alpha = 4 \Rightarrow \alpha = $ awrt $1.326$ M1 A1
(3 marks)
(b) Minimum height $= \frac{24}{3+R} = 3.37$ (metres) M1 A1
(2 marks)
(c) Uses part (a): $10 = \frac{24}{3 + \sqrt{17}\cos\left(t-1.326\right)} \Rightarrow \cos\left(t - 1.326\right) = -\frac{1}{2}$ M1 A1
$t = $ awrt $6.09$ M1 A1
(4 marks)
(9 marks)7. (a) Express $\cos x + 4 \sin x$ in the form $R \cos ( x - \alpha )$ where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$ Give the exact value of $R$ and give the value of $\alpha$, in radians, to 3 decimal places.
A scientist is studying the behaviour of seabirds in a colony.
She models the height above sea level, $H$ metres, of one of the birds in the colony by the equation
$$H = \frac { 24 } { 3 + \cos \left( \frac { 1 } { 2 } t \right) + 4 \sin \left( \frac { 1 } { 2 } t \right) } \quad 0 \leqslant t \leqslant 6.5$$
where $t$ seconds is the time after it leaves the nest.
Find, according to the model,\\
(b) the minimum height of the seabird above sea level, giving your answer to the nearest cm,\\
(c) the value of $t$, to 2 decimal places, when $H = 10$
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\hfill \mbox{\textit{Edexcel P3 2020 Q7 [9]}}