Higher June 2023 Paper 1 Q26
26 A circle, centre \(O\), has circumference \(20\pi\) cm
\(Q\) is a point on the circle.
\(OPQR\) is a square.
Not drawn accurately
perimeter of the square : circumference of the circle \(= \sqrt{a} : \pi \quad\) where \(a\) is an integer.
Work out the value of \(a\).
You must show your working. [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(20\pi \div 2\pi\) or 10 | M1 | oe may be seen on diagram implied by diameter = 20 |
| \(x^2 + x^2 = (\text{their } 10)^2\) or \(2x^2 = 100\) or \(x^2 = 50\) or their \(10 \times \sin 45\) or their \(10 \times \cos 45\) or their \(10 \times \dfrac{1}{\sqrt{2}}\) | M1 | oe any letter (condone \(a\)) their 10 is their length \(OQ\) (the radius of the circle) |
| \(\sqrt{\text{their } 10^2 \div 2}\) or \(\sqrt{50}\) or \(5\sqrt{2}\) or \(4 \times \sqrt{50}\) or \(4 \times\) their \(10 \times \sin 45\) or \(4 \times\) their \(10 \times \cos 45\) or \(40 \times \dfrac{1}{\sqrt{2}}\) or \(\dfrac{40\sqrt{2}}{2}\) or \(20\sqrt{2}\) | M1dep | oe value for the length of one side of the square or the perimeter of the square eg \(\dfrac{10}{\sqrt{2}}\) dep on previous mark |
| 2 with full working seen for M3 | A1 | |
| Alternative method 2 | ||
| \(20\pi \div 2\pi\) or 10 or side length of square \(= 5\sqrt{a}\) | M1 | oe may be seen on diagram implied by diameter = 20 |
| (Perimeter of square \(= 20\sqrt{a}\) and) side length of square \(= 5\sqrt{a}\) and \(\left(5\sqrt{a}\right)^2 + \left(5\sqrt{a}\right)^2 = (\text{their } 10)^2\) | M1 | oe their 10 is their length \(OQ\) (the radius of the circle) condone missing brackets if recovered |
| \(25a + 25a = (\text{their } 10)^2\) or \(50a = 100\) | M1dep | dep on M1M1 |
| 2 with full working seen for M3 | A1 | |
Additional guidance
| 2 with no working | M0M0M0A0 |
| \(\sqrt{2}\) on answer line (may score method marks) | A0 |