Higher June 2023 Paper 1R Q25
25 Here is a triangle \(ABC\)
Diagram NOT accurately drawn
The area of the triangle is \(\;(x^2 + x - 3.75)\) cm2
Find the size of the largest angle in triangle \(ABC\)
Give your answer correct to the nearest degree.
(6)
| Scheme | Marks |
|---|---|
eg \(\dfrac{1}{2}(2x - 1)(2x + 1)\sin 30 = x^2 + x - 3.75\) oe | M1 |
| 3.5 | A1 |
\((BC^2 =)\;\text{``}6\text{''}^2 + \text{``}8\text{''}^2 - (2 \times \text{``}6\text{''} \times \text{``}8\text{''} \times \cos 30)\;(= 16.8(615\ldots))\) oe or \((BC =)\;\sqrt{\text{``}16.8\ldots\text{''}}\) (= 4.10(628…)) | M1 |
\(\dfrac{\sin(ABC)}{\text{``}8\text{''}} = \dfrac{\sin 30}{\sqrt{\text{``}16.8\text{''}}}\) oe or \(\dfrac{\sin(BCA)}{\text{``}6\text{''}} = \dfrac{\sin 30}{\sqrt{\text{``}16.8\text{''}}}\) oe or \(\text{``}6\text{''}^2 = \text{``}8\text{''}^2 + \left(\sqrt{\text{``}16.8\text{''}}\right)^2 - \left(2 \times \text{``}8\text{''} \times \sqrt{\text{``}16.8\text{''}} \times \cos(BCA)\right)\) oe or \(\text{``}8\text{''}^2 = \text{``}6\text{''}^2 + \left(\sqrt{\text{``}16.8\text{''}}\right)^2 - \left(2 \times \text{``}6\text{''} \times \sqrt{\text{``}16.8\text{''}} \times \cos(ABC)\right)\) oe | M1 |
\((\sin ABC =)\;\dfrac{\sin 30 \times \text{``}8\text{''}}{\sqrt{\text{``}16.8\text{''}}}\;(= 0.974\ldots)\) oe or \(ABC = 76.9\ldots\) or \((\sin BCA =)\;\dfrac{\sin 30 \times \text{``}6\text{''}}{\sqrt{\text{``}16.8\text{''}}}\;(= 0.730\ldots)\) oe or \(BCA = 46.9\ldots\) or \((\cos BCA =)\;\dfrac{\text{``}8\text{''}^2 + \left(\sqrt{\text{``}16.8\text{''}}\right)^2 - \text{``}6\text{''}^2}{2 \times \text{``}8\text{''} \times \left(\sqrt{\text{``}16.8\text{''}}\right)}\;(= 0.682\ldots)\) oe or \(BCA = 46.9\ldots\) or \((\cos ABC =)\;\dfrac{\text{``}6\text{''}^2 + \left(\sqrt{\text{``}16.8\text{''}}\right)^2 - \text{``}8\text{''}^2}{2 \times \text{``}6\text{''} \times \left(\sqrt{\text{``}16.8\text{''}}\right)}\;(= -0.226\ldots)\) oe or \(ABC = 103.0\ldots\) | M1 |
| Correct answer scores full marks (unless from obvious incorrect working) Answer: 103 | A1 |
| (6) | |
| (6 marks) |
Notes
M1: for equating area of triangle with the given area
A1: for the value of \(x\)
M1: ft dep on M1 for a correct method to find \(BC^2\) or \(BC\)
(\(AB = 6\) and \(AC = 8\))
M1: ft dep on previous M1 for a correct method to find angle \(ABC\) or angle \(BCA\)
M1: ft dep on previous M1 for a correct rearrangement for \(\sin ABC\) or \(\sin BCA\) or \(\cos BCA\) or \(\cos ABC\)
A1: accept awrt 103