Intermediate June 2023 Unit 2 Q15
15. In the diagram below, AD is a straight line.
\(B\widehat{A}C\) = 90°, \(B\widehat{D}E\) = 90° and \(C\widehat{B}E\) = 90°.
AC = 7·7 cm, BC = 11·3 cm and BD = 13·1 cm.
Diagram not drawn to scale
(a) Calculate the value of \(x\). [3]
(b) Hence find the length DE. [4]
| Scheme | Marks | Comments | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| (\(x\) =) \(\sin^{-1}\dfrac{7{\cdot}7}{11{\cdot}3}\) or \(\sin^{-1}\dfrac{7{\cdot}7 \times \sin 90}{11{\cdot}3}\) or equivalent | M2 | Check diagram for answers Award M1 for one of the following:
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| Allow an answer between 42·8 and 43(°) ISW | A1 | Allow correct angles given in radians or gradians:
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Alternative method
| Scheme | Marks | Comments |
|---|---|---|
| Correct use of a ‘two-step’ method. | M2 | A partial trigonometric method is M0. |
| Allow an answer between 42·8 and 43(°) ISW | A1 | Allow 42·8(…°) Allow correct angles given in radians or gradians. |
| Scheme | Marks | Comments | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| DBE = (90 – 43) = 47(°) OR BED = 43(°) | B1 | Check diagram for answers. Strict FT for DBE = 90 – ‘their \(x\)’ or BED = ‘their \(x\)’, provided ‘their \(x\)’ ≠ 45°. Note: DBE must be acute for B1. May be implied in further work. | ||||||||||||
| Valid method to find the length DE DE = 13·1 × tan 47 \(DE = \dfrac{13{\cdot}1}{\tan 43}\) \(DE = \dfrac{13{\cdot}1 \times \sin 47}{\sin 43}\) | M2 | If B1 already awarded for ‘their angle DBE’ but then ‘their angle BED’ is incorrect and ‘their BED’ is then used (or vice versa) for either M2 or M1, then award B0 previously. Or award M2 for correct use of a ‘two-step’ method (e.g. ‘Pythagoras and similar triangles’ or ‘Pythagoras and correct trigonometric relationship’). FT ‘their angle DBE’ or ‘their angle BED’ provided not 0°, 45°, 90° or 180°. Award M1 for one of the following:
For \(\dfrac{13{\cdot}1 \times \sin 47}{\sin 43}\) FT their clearly stated or shown angles BED and DBE only if BED + DBE = 90°. | ||||||||||||
| DE in the range 14·04 to 14·1 (cm) ISW | A1 | Allow 14 from correct workings. FT from M2 only and provided that angle is acute and leads to a positive answer. Award B1M2A0 for any of the following unsupported answers:
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