Higher November 2018 Paper 1 Q26
26 The area of this triangle is \(25\sqrt{3}\) cm2
Not drawn accurately
Work out the value of \(w\).
Give your answer in the form \(\ a\sqrt{b}\ \) where \(a\) and \(b\) are integers greater than 1 [5 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(0.5 \times 20 \times x \times \sin 60\) or \(10x \sin 60\) or \(5\sqrt{3}\,x\) | M1 | oe |
| \(0.5 \times 20 \times x \times \sin 60 = 25\sqrt{3}\) or \(x = 5\) | M1dep | oe equation |
| (their 5)\(^2 + 20^2\) \(-\ 2 \times\) their \(5 \times 20 \times \cos 60\) or \(25 + 400 - 200\cos 60\) or 325 | M1 | oe their 5 must be their value of \(x\) |
| \(\sqrt{\text{their } 325}\) | M1dep | dep on 3rd M1 their 325 can be unsimplified |
| \(5\sqrt{13}\) | A1 | |
| Alternative method 2 | ||
| \(0.5 \times 20 \times h = 25\sqrt{3}\) or \(h = \dfrac{5\sqrt{3}}{2}\) | M1 | oe any letter \(h\) is perpendicular height for 20 cm base |
| \(\sin 60 = \dfrac{\text{their } \frac{5\sqrt{3}}{2}}{x}\) or \(x = 5\) | M1dep | oe |
| (their 5)\(^2 + 20^2\) \(-\ 2 \times\) their \(5 \times 20 \times \cos 60\) or \(25 + 400 - 200\cos 60\) or 325 | M1 | oe their 5 must be their value of \(x\) |
| \(\sqrt{\text{their } 325}\) | M1dep | dep on 3rd M1 their 325 can be unsimplified |
| \(5\sqrt{13}\) | A1 | |
| Alternative method 3 | ||
| \(0.5 \times 20 \times h = 25\sqrt{3}\) or \(h = \dfrac{5\sqrt{3}}{2}\) | M1 | oe any letter \(h\) is perpendicular height for 20 cm base |
| \(\tan 60 = \dfrac{\text{their } h}{c}\) or \(c = \dfrac{5}{2}\) | M1dep | oe any letter \(c\) is part of 20 cm base |
| \(\left(\text{their } \dfrac{5\sqrt{3}}{2}\right)^2 + \left(20 - \text{their } \dfrac{5}{2}\right)^2\) or \(\left(\text{their } \dfrac{5\sqrt{3}}{2}\right)^2 + \left(\dfrac{35}{2}\right)^2\) or 325 | M1dep | |
| \(\sqrt{\left(\text{their } \frac{5\sqrt{3}}{2}\right)^2 + \left(20 - \text{their } \frac{5}{2}\right)^2}\) or \(\sqrt{\text{their } 325}\) | M1dep | |
| \(5\sqrt{13}\) | A1 | |
Additional guidance
| Omitting 0.5 in area formula can score a maximum of M0M0M1M1A0 | |
| \(\sqrt{(\text{their } 5)^2 + 20^2 - 2 \times \text{their } 5 \times 20 \times \cos 60}\) | M0M0M1M1A0 |