Moderate -0.3 This is a standard sector/segment problem requiring straightforward application of sector area formula and rectangle area, with basic trigonometry to find rectangle dimensions. The algebraic manipulation to reach the required form is routine. Slightly easier than average due to being a well-practiced question type with clear geometric setup.
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\includegraphics[max width=\textwidth, alt={}, center]{dd2cb0ec-5df9-4d99-9e15-5ae1f1c07b96-2_536_606_735_772}
In the diagram, \(A O B\) is a sector of a circle with centre \(O\) and radius 12 cm . The point \(A\) lies on the side \(C D\) of the rectangle \(O C D B\). Angle \(A O B = \frac { 1 } { 3 } \pi\) radians. Express the area of the shaded region in the form \(a ( \sqrt { } 3 ) - b \pi\), stating the values of the integers \(a\) and \(b\).
\(OC = 6\sqrt{3}\) and \(AC = 6\) Sector area = \(\frac{1}{2}r^2\theta = 24\pi\) Area of rectangle = \(12 \times 6\sqrt{3}\) Area of triangle = \(\frac{1}{2} \times 6 \times 6\sqrt{3}\)
B1 B1 M1 M1 M1 [6]
Wherever these come (must have \(\sqrt{3}\)). Use of correct formula with radians. Use of base \(\times\) height (not for 12 × 12). Use of \(\frac{1}{2}\) base \(\times\) height (needs trig)
\(\rightarrow 54\sqrt{3} - 24\pi\)
A1
co. Ok without stating \(a=54, b=24\).
$OC = 6\sqrt{3}$ and $AC = 6$ Sector area = $\frac{1}{2}r^2\theta = 24\pi$ Area of rectangle = $12 \times 6\sqrt{3}$ Area of triangle = $\frac{1}{2} \times 6 \times 6\sqrt{3}$ | B1 B1 M1 M1 M1 [6] | Wherever these come (must have $\sqrt{3}$). Use of correct formula with radians. Use of base $\times$ height (not for 12 × 12). Use of $\frac{1}{2}$ base $\times$ height (needs trig)
$\rightarrow 54\sqrt{3} - 24\pi$ | A1 | co. Ok without stating $a=54, b=24$.
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\includegraphics[max width=\textwidth, alt={}, center]{dd2cb0ec-5df9-4d99-9e15-5ae1f1c07b96-2_536_606_735_772}
In the diagram, $A O B$ is a sector of a circle with centre $O$ and radius 12 cm . The point $A$ lies on the side $C D$ of the rectangle $O C D B$. Angle $A O B = \frac { 1 } { 3 } \pi$ radians. Express the area of the shaded region in the form $a ( \sqrt { } 3 ) - b \pi$, stating the values of the integers $a$ and $b$.
\hfill \mbox{\textit{CAIE P1 2006 Q3 [6]}}