| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2020 |
| Session | February |
| Marks | 10 |
| Topic | Areas by integration |
| Type | Area with trigonometric functions |
| Difficulty | Challenging +1.2 This is a multi-part question requiring integration of trigonometric functions, finding intersection points, and using small angle approximations. Part (a) is standard integration with double angle formulas. Part (b) requires Taylor series approximations (cos²x ≈ 1-x², sin 2x ≈ 2x) which is A-level content but straightforward application. Part (c) is routine quadratic solving. The question is slightly above average due to the combination of techniques and the small angle approximation step, but each individual component is standard A-level material with no novel insight required. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.05e Small angle approximations: sin x ~ x, cos x ~ 1-x^2/2, tan x ~ x1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.08e Area between curve and x-axis: using definite integrals |
10 The diagram shows part of the graphs of $y = \cos ^ { 2 } x$ and $y = 5 \sin 2 x$ for small positive values of $x$. The graphs meet at the point $A$ with $x$-coordinate $\alpha$.\\
\includegraphics[max width=\textwidth, alt={}, center]{022274c9-7ed2-4436-ae97-d410d7d566fc-14_652_561_402_735}
\begin{enumerate}[label=(\alph*)]
\item Find the exact area contained between the two graphs (between $x = 0$ and $x = \alpha$ ) and the $y$-axis. Give your answer in terms of $\alpha , \cos 2 \alpha$ and/or $\sin 2 \alpha$.
\item Using the fact that $\alpha$ is a small positive solution to the equation $\cos ^ { 2 } x = 5 \sin 2 x$, show that $\alpha$ satisfies approximately the equation $\alpha ^ { 2 } + 10 \alpha - 1 = 0$, if terms in $\alpha ^ { 3 }$ and higher are ignored.
\item Use the equation in part (b) to find an approximate value of $\alpha$, correct to 3 significant figures.
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2020 Q10 [10]}}