SPS SPS SM Pure 2021 September — Question 6 8 marks

Exam BoardSPS
ModuleSPS SM Pure (SPS SM Pure)
Year2021
SessionSeptember
Marks8
TopicSine and Cosine Rules
TypeShaded region with arc
DifficultyStandard +0.3 This is a straightforward multi-part question on sine/cosine rules and sector areas. Part (a) is routine cosine rule application, part (b) is standard area formula, and part (c) requires setting up an equation where sector area equals half the triangle area, then finding the perimeter. All steps are standard techniques with no novel insight required, making it slightly easier than average.
Spec1.05b Sine and cosine rules: including ambiguous case1.05c Area of triangle: using 1/2 ab sin(C)1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta
6. The diagram shows a triangle \(A B C\). \includegraphics[max width=\textwidth, alt={}, center]{cee51b6b-40d2-4abb-acf7-47c73a919bf9-14_499_718_219_703} The lengths of \(A B , B C\) and \(A C\) are \(8 \mathrm {~cm} , 5 \mathrm {~cm}\) and 9 cm respectively.
Angle \(B A C\) is \(\theta\) radians.
  1. Show that \(\theta = 0.586\), correct to three significant figures.
    [0pt] [2 marks]
  2. Find the area of triangle \(A B C\), giving your answer, in \(\mathrm { cm } ^ { 2 }\), to three significant figures.
    [0pt] [2 marks]
  3. A circular sector, centre \(A\) and radius \(r \mathrm {~cm}\), is removed from triangle \(A B C\). The remaining shape is shown shaded in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{cee51b6b-40d2-4abb-acf7-47c73a919bf9-14_488_700_1409_685} Given that the area of the sector removed is equal to the area of the shaded shape, find the perimeter of the shaded shape. Give your answer in cm to three significant figures.
    [0pt] [4 marks]
6. The diagram shows a triangle $A B C$.\\
\includegraphics[max width=\textwidth, alt={}, center]{cee51b6b-40d2-4abb-acf7-47c73a919bf9-14_499_718_219_703}

The lengths of $A B , B C$ and $A C$ are $8 \mathrm {~cm} , 5 \mathrm {~cm}$ and 9 cm respectively.\\
Angle $B A C$ is $\theta$ radians.
\begin{enumerate}[label=(\alph*)]
\item Show that $\theta = 0.586$, correct to three significant figures.\\[0pt]
[2 marks]
\item Find the area of triangle $A B C$, giving your answer, in $\mathrm { cm } ^ { 2 }$, to three significant figures.\\[0pt]
[2 marks]
\item A circular sector, centre $A$ and radius $r \mathrm {~cm}$, is removed from triangle $A B C$. The remaining shape is shown shaded in the diagram below.\\
\includegraphics[max width=\textwidth, alt={}, center]{cee51b6b-40d2-4abb-acf7-47c73a919bf9-14_488_700_1409_685}

Given that the area of the sector removed is equal to the area of the shaded shape, find the perimeter of the shaded shape. Give your answer in cm to three significant figures.\\[0pt]
[4 marks]
\end{enumerate}

\hfill \mbox{\textit{SPS SPS SM Pure 2021 Q6 [8]}}
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