Higher November 2022 Paper 2 Q13
13 Trapezium DEFG is formed by joining
triangle DEH
to
rectangle EFGH.
ABC is similar to DEH.
Work out the area of DEFG. [5 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 – Works out \(BC\) using Pythagoras then works out \(EH\) | ||
| \(7^2\) or 49 and \(4.2^2\) or 17.64 | M1 | oe |
| \(\sqrt{7^2 - 4.2^2}\) or \(\sqrt{49 - 17.64}\) or \(\sqrt{31.36}\) or 5.6 | M1dep | oe implied by 11.76 as the area of the smaller triangle may be on diagram |
| \(6 \div 4.2 \times\) their 5.6 or 8 | M1dep | oe full method to work out \(EH\) may be on diagram as \(EH\) or \(FG\) implied by 24 as the area of the larger triangle or 60 as the area of the rectangle |
| \(0.5 \times\) their \(8 \times 6\) or 24 and their \(8 \times 7.5\) or 60 | M1dep | oe eg \(0.5 \times\) their \(5.6 \times 4.2 \times (6 \div 4.2)^2\) and their \(8 \times 7.5\) or \(0.5 \times\) their \(8 \times (7.5 + 13.5)\) |
| 84 | A1 | |
| Alternative method 2 – Works out \(ED\) using similar triangles then works out \(EH\) | ||
| \(6 \div 4.2 \times 7\) or 10 | M1 | oe may be on diagram |
| (their \(10)^2\) or 100 and \(6^2\) or 36 | M1dep | oe |
| \(\sqrt{(\text{their } 10)^2 - 6^2}\) or \(\sqrt{100 - 36}\) or \(\sqrt{64}\) or 8 | M1dep | oe full method to work out \(EH\) may be on diagram as \(EH\) or \(FG\) implied by 24 as the area of the larger triangle or 60 as the area of the rectangle |
| \(0.5 \times\) their \(8 \times 6\) or 24 and their \(8 \times 7.5\) or 60 | M1dep | oe eg \(0.5 \times\) their \(5.6 \times 4.2 \times (6 \div 4.2)^2\) and their \(8 \times 7.5\) or \(0.5 \times\) their \(8 \times (7.5 + 13.5)\) |
| 84 | A1 | |
| Alternative method 3 – Uses trigonometry to work out \(BC\) then works out \(EH\) or uses trigonometry to work out \(EH\) | ||
| (angle \(ABC =\)) \(\sin^{-1}\left(\dfrac{4.2}{7}\right)\) or (angle \(ABC =\)) [36.8, 36.9] or (angle \(BAC =\)) \(\cos^{-1}\left(\dfrac{4.2}{7}\right)\) or (angle \(BAC =\)) [53.1, 53.2] | M1 | oe full method to work out \(ABC\) or \(BAC\) |
| \(7 \times \cos\) (their [36.8, 36.9]) or \(7 \times \sin\) (their [53.1, 53.2]) or 5.6 or \(\tan\) (their [36.8, 36.9]) \(= \dfrac{6}{EH}\) or \(\tan\) (their [53.1, 53.2]) \(= \dfrac{EH}{6}\) | M1dep | oe full method to work out \(BC\) or partial method to work out \(EH\) |
| \(6 \div 4.2 \times\) their 5.6 or 8 or \(6 \div \tan\) (their [36.8, 36.9]) or \(6 \times \tan\) (their [53.1, 53.2]) | M1dep | oe full method to work out \(EH\) may be on diagram as \(EH\) or \(FG\) implied by 24 as the area of the larger triangle or 60 as the area of the rectangle |
| \(0.5 \times\) their \(8 \times 6\) or 24 and their \(8 \times 7.5\) or 60 | M1dep | oe eg \(0.5 \times\) their \(5.6 \times 4.2 \times (6 \div 4.2)^2\) and their \(8 \times 7.5\) or \(0.5 \times\) their \(8 \times (7.5 + 13.5)\) |
| 84 | A1 | |
Additional guidance
Up to M3 may be awarded for correct work with no answer, or incorrect answer, even if this is seen amongst multiple attempts