Higher June 2025 Paper 2 Q10
10 \(ABD\) and \(ABC\) are right-angled triangles.
Diagram NOT accurately drawn
\(AB = 16\) cm \(BC = 12\) cm \(AD = 1.5 \times AC\)
Find the length of \(CD\)
Give your answer correct to 3 significant figures.
(5)
| Scheme | Marks |
|---|---|
\(\left(AC^2 =\right)12^2 + 16^2 \;(= 144 + 256 = 400)\) or \((BAC =)\tan^{-1}\left(\dfrac{12}{16}\right)\left(= 36.8(698\ldots)\right)\) or 36.9 or \((BCA =)\tan^{-1}\left(\dfrac{16}{12}\right)\left(= 53.1(301\ldots)\right)\) | M1 |
\((AC =)\sqrt{12^2 + 16^2}\left(= \sqrt{144 + 256} = \sqrt{400} = 20\right)\) or \((AC =)\frac{16}{\cos\text{``}{36.8}\text{''}}(= 20)\) or \((AC =)\frac{12}{\sin\text{``}{36.8}\text{''}}(= 20)\) or \((AC =)\frac{16}{\sin\text{``}{53.1}\text{''}}(= 20)\) or \((AC =)\frac{12}{\cos\text{``}{53.1}\text{''}}(= 20)\) | M1 |
\(\left(BD^2 =\right)\left(1.5 \times \text{``}{20}\text{''}\right)^2 - 16^2 \;(= 644)\) or \(\left(BD^2 =\right)30^2 - 16^2 \;(= 900 - 256 = 644)\) or \((BAD =)\cos^{-1}\left(\dfrac{16}{\text{``}{30}\text{''}}\right)\left(= 57.7(690\ldots)\right)\) or 57.8 or \((BDA =)\sin^{-1}\left(\dfrac{16}{\text{``}{30}\text{''}}\right)\left(= 32.2(309\ldots)\right)\) or \((BCA =)\sin^{-1}\left(\dfrac{16}{\text{``}{20}\text{''}}\right)\left(= 53.1(301\ldots)\right)\) and \(\dfrac{(CD)}{\sin(180 - 126.9 - 32.2)} = \dfrac{\text{``}{30}\text{''}}{\sin(180 - 53.1)}\) oe | M1 |
\((BD =)\sqrt{\left(1.5 \times \text{``}{20}\text{''}\right)^2 - 16^2}\) (= 25.3(771…)) or \((BD =)\sqrt{30^2 - 16^2}\left(= \sqrt{900 - 256} = \sqrt{644} = 2\sqrt{161} = 25.3(771\ldots)\right)\) or \((BD =)16 \times \tan\text{``}{57.7}\text{''}\left(= 25.3(771\ldots)\right)\) or \((BD =)\text{``}{30}\text{''} \times \sin\text{``}{57.7}\text{''}\left(= 25.3(771\ldots)\right)\) or \((BD =)\sqrt{16^2 + \text{``}{30}\text{''}^2 - 2 \times 16 \times \text{``}{30}\text{''} \times \cos\text{``}{57.7}\text{''}}\left(= 25.3(771\ldots)\right)\) or \((BD =)30 \times \cos\text{``}{32.2}\text{''}\left(= 25.3(771\ldots)\right)\) or \((BD =)\dfrac{16}{\tan\text{``}{32.2}\text{''}}\left(= 25.3(771\ldots)\right)\) or \((BD =)\dfrac{16}{\sin\text{``}{32.2}\text{''}} \times \sin\text{``}{57.7}\text{''}\left(= 25.3(771\ldots)\right)\) or \((CD =)\dfrac{\text{``}{30}\text{''}}{\sin\text{``}{126.9}\text{''}} \times \sin\text{``}{20.9}\text{''}\) oe | M1 |
| Working not required, so correct answer scores full marks (unless from obvious incorrect working) Answer: 13.4 | A1 |
| (5) | |
| (5 marks) |
Notes
M1: for a correct method using triangle \(ABC\)
M1: for a correct method to find \(AC\)
M1: for a correct method using triangle to find \(BD^2\) or angle \(BAD\) or angle \(BDA\) or for a correct equation for side \(CD\)
M1: for a correct method to find \(BD\) or \(CD\)
A1: awrt 13.4