4MA0 Higher January 2016 Paper 3 Q22
22 The diagram shows a rectangle.
Diagram NOT accurately drawn
The width of the rectangle is \(x\) cm.
The length of a diagonal of the rectangle is 12 cm.
The perimeter of the rectangle is 28 cm.
Find the possible values of \(x\).
Give your values correct to 3 significant figures.
Show your working clearly.
(7)
| Scheme | Marks |
|---|---|
(length = ) \(\dfrac{28 - 2x}{2}\) or \(14 - x\) or \(\sqrt{12^2 - x^2}\) | M1 |
| \(12^2 = x^2 + (14 - x)^2\) oe | M1 |
\(144 = x^2 + 196 - 28x + x^2\) or \(144 = x^2 + \dfrac{784 - 112x + 4x^2}{4}\) | M1 |
| \(2x^2 - 28x + 52 = 0\) or \(x^2 - 14x + 26 = 0\) | A1 |
| eg. \(x = \dfrac{--14 \pm \sqrt{(-14)^2 - 4 \times 1 \times 26}}{2 \times 1}\) | M1 |
| eg. \(\dfrac{14 \pm \sqrt{92}}{2}\) or \(7 \pm \sqrt{23}\) | M1 |
| 11.8, 2.20 | A1 |
| (7) |
Notes
M1: correct expression for length of rectangle OR
a pair of correct simultaneous equations
eg. \(x^2 + y^2 = 12^2\) and \(2x + 2y = 28\)
A1: for a correct simplified quadratic equation
M1: ft for correct substitution into quadratic formula for their quadratic (condone one sign error; condone missing brackets and \(14^2\))
M1: (indep) correct simplification of discriminant for correct quadratic equation
A1: answers in the ranges 11.7 - 11.8 and 2.2 - 2.21
dep on a correct quadratic equation and at least M4