Higher June 2024 Paper 2 Q21
21 The diagram shows a square \(ABCD\) and a circle.
Diagram NOT accurately drawn
The sides of the square are tangents to the circle.
The total area of the shaded regions is 80 cm\(^2\)
Work out the length of \(AC\)
Give your answer correct to 3 significant figures.
(5)
| Scheme | Marks |
|---|---|
| \((2r)^2 - \pi r^2\) oe or \(x^2 - \pi \times (0.5x)^2\) | M1 |
| \(4r^2 - \pi r^2 = 80\) oe eg \(r^2 - 0.25\pi r^2 = 20\) or \(x^2 - 0.25\pi x^2 = 80\) or \(4x^2 - \pi x^2 = 320\) oe | M1 |
\(r^2 = \dfrac{80}{4 - \pi}\) ( = 93.19…) or \(r = \sqrt{\dfrac{80}{4 - \pi}}\) ( = 9.65…) \(x^2 = \dfrac{80}{1 - 0.25\pi}\) (= 372.78..) or \(x = \sqrt{\dfrac{80}{1 - 0.25\pi}}\) (19.307...) oe eg \(\sqrt{\dfrac{320}{4 - \pi}}\) | M1 |
\((AC =)\sqrt{(2 \times \text{``}{9.65}\text{''})^2 + (2 \times \text{``}{9.65}\text{''})^2}\) oe or \((AC =)\; 2 \times \sqrt{\text{``}{9.65}\text{''}^2 + \text{``}{9.65}\text{''}^2}\) \((AC =)\sqrt{\text{``}{19.307}\text{''}^2 + \text{``}{19.307...}\text{''}^2}\) oe eg \(\sqrt{8 \times \dfrac{80}{4 - \pi}}\) oe or \((AC =)\dfrac{2 \times \text{``}{9.65}\text{''}}{\sin 45}\) or \(\dfrac{2 \times \text{``}{9.65}\text{''}}{\cos 45}\) | M1 |
| Correct answer scores full marks (unless from obvious incorrect working) Answer: 27.3 | A1 |
| (5) | |
| (5 marks) |
Notes
M1: A correct expression for the area of the shaded parts in one variable only
for this mark only, accept without brackets
(eg \(2r^2 - \pi r^2\) or \(x^2 - \pi \times \dfrac{1}{2}x^2\))
(any letter can be used eg \(AB\), \(x\), \(y\) etc, here, \(r\) = radius, \(x\) = side of square)
M1: A correct equation in one variable with brackets expanded (may be seen later in working)
M1: A correct expression for the radius squared or radius or for the side of the square squared or for the side of the square
M1: For a correct calculation to find the length of \(AC\)
A1: 27.3 – 27.5