4MA0 Higher June 2015 Paper 4 Q22
22 The diagram shows two triangles, A and B.
Diagram NOT accurately drawn
The area of triangle B is 3 times the area of triangle A.
Given that \(b \gt 4\), find an expression for \(a\) in terms of \(b\).
(5)
| Scheme | Marks |
|---|---|
\(\dfrac{1}{2}ab \sin 150\) oe or \(\dfrac{1}{2}(a + 1)(b + 2)\) oe (must be sin150, not sin C) | M1 |
| \(3 \times \dfrac{1}{2}ab \sin 150 = \dfrac{1}{2}(a + 1)(b + 2)\) oe | M1 |
| \(3 \times \dfrac{1}{2}ab \sin 150 = \dfrac{1}{2}(ab + b + 2a + 2)\) oe | M1 |
| \(3ab - 2ab - 4a = 2b + 4\) oe or \(3ab \sin 150 - ab - 2a = b + 2\) oe | M1 |
| \(\dfrac{2b + 4}{b - 4}\) oe | A1 |
| (5) |
Notes
M1: Or \(\dfrac{1}{2}(a + 1)(b + 2) \sin 90\)
M1: correct equation, eg may see \(\tfrac{3}{4}ab = \ldots\)
M1: expansion of brackets in a correct equation
M1: isolation of terms in \(a\) in a correct equation (may be on either side of equation & can still have sin150)
A1: eg \(\dfrac{-2b - 4}{4 - b}\) ; \(\dfrac{\frac{1}{2}b + 1}{\frac{1}{4}b - 1}\), \(\dfrac{b + 2}{3b \sin 150 - b - 2}\), etc