4MA0 Foundation June 2014 Paper 1R Q15
15 The diagram shows a rectangle \(PQRS\).
\(PQ = 14\) cm and \(QR = 9\) cm.
The point \(A\) lies on \(PS\) so that \(PA = 5\) cm.
The point \(B\) lies on \(SR\) so that \(BR = 8\) cm.
Diagram NOT accurately drawn
Give your answer correct to 3 significant figures. (3)
| Scheme | Marks |
|---|---|
| \(\tfrac{1}{2} \times 8 \times 9\) or \(\tfrac{1}{2} \times 5 \times 14\) or 36 or 35 | M1 |
| \(\tfrac{1}{2} \times 4 \times 6\) or 12 | M1 |
| \(9 \times 14 - \tfrac{1}{2} \times 4 \times 6 - \tfrac{1}{2} \times 8 \times 9 - \tfrac{1}{2} \times 5 \times 14\) or 126 − 12 − 36 − 35 | M1 |
| 43 | A1 |
| (4) |
Notes
M1: Correct expression for area of \(RQB\) or \(PQA\).
M1: Correct expression for area of \(ABS\).
M1: Area of rectangle – their three triangles
Alternative
| Scheme | Marks |
|---|---|
| \(AB = \sqrt{52}\), \(BQ = \sqrt{145}\), \(AQ = \sqrt{221}\) | M1 |
| \(ABQ = 97.9434\ldots\) or \(BQA = 28.7126\ldots\) or \(BAQ = 53.3438\ldots\) | M1 |
\(\tfrac{1}{2}(\sqrt{52})(\sqrt{145})\sin(97.9..)\) or \(\tfrac{1}{2}(\sqrt{145})(\sqrt{221})\sin(28.7..)\) or \(\tfrac{1}{2}(\sqrt{52})(\sqrt{221})\sin(53.3..)\) oe | M1 |
| 43 | A1 |
Notes
M1: A correct method to find all 3 sides of triangle \(ABQ\)
M1: A correct method to find an angle in \(ABQ\) (cosine rule or 180 — use of trig in 2 smaller triangles)
M1: Correct use of formula \(\tfrac{1}{2}ab\sin C\) to find area of \(ABQ\)
A1: Must be exact answer – not from rounding.
Alternative
| Scheme | Marks |
|---|---|
| M2 | |
| M1 | |
| 43 | A1 |
Notes
M2: For a correct method to find 2 sides and the correct included angle (by use of trig and angles on a straight line).
M1: Correct use of formula \(\tfrac{1}{2}ab\sin C\) to find area of \(ABQ\) (see above)
A1: Must be exact answer – not from rounding.
| Scheme | Marks |
|---|---|
| \(5^2 + 14^2\) or 25 + 196 or 221 | M1 |
| \(\sqrt{5^2 + 14^2}\) or \(\sqrt{25 + 196}\) or \(\sqrt{221}\) | M1 |
| 14.9 | A1 |
| (3) | |
| (Total 7 marks) |
Notes
M1: For squaring and adding
M1: dep for square root
A1: For answer rounding to 14.9