Higher November 2023 Paper 3 Q16
16 \(PQ\) is a diameter of a circle.
\(QR\) is a tangent to the circle.
\(PQ = QR\)
\(PR = 10\) cm
Not drawn accurately
Work out the radius of the circle.
Give your answer as a decimal. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(d^2 + d^2 = 10^2\) or \(2d^2 = 100\) or \((2r)^2 + (2r)^2 = 10^2\) or \(8r^2 = 100\) | M1 | oe must use same letter for \(PQ\) and \(QR\) |
| \((d =)\ \sqrt{\dfrac{100}{2}}\) or \((d =)\ 5\sqrt{2}\) or \((d =)\ 7.07(1…)\) or \((d =)\ 7.1\) or \((r^2 =)\ \dfrac{100}{8}\) or \((r =)\ \dfrac{5}{2}\sqrt{2}\) | M1dep | oe eg \((d =)\ \sqrt{50}\) |
| 3.5(3…) or 3.54 or 3.55 | A1 | |
| Alternative method 2 | ||
| \(\sin 45 = \dfrac{d}{10}\) or \(\cos 45 = \dfrac{d}{10}\) | M1 | oe eg \(\sin 45 = \dfrac{2r}{10}\) or \(\sin 45 = \dfrac{r}{5}\) |
| \((d =)\ 10 \times \sin 45\) or \((d =)\ 10 \times \cos 45\) or \((d =)\ 5\sqrt{2}\) or \((d =)\ 7.07(1…)\) or \((d =)\ 7.1\) | M1dep | oe eg \((2r =)\ 10 \times \sin 45\) or \((r =)\ 5 \times \sin 45\) or \((r =)\ \dfrac{5\sqrt{2}}{2}\) |
| 3.5(3…) or 3.54 or 3.55 | A1 | |
Additional guidance
Alt method 1
If working with diameter, square root is required for 2nd M1
If working with radius, square root is not required for 2nd M1
Alt method 1 \(2r^2 + 2r^2 = 10^2\) is M0M0A0 unless recovered
Use of sine rule follows Alt method 2