4MA0 Higher June 2013 Paper 3 Q20
20
Diagram NOT accurately drawn
The diagram shows a rectangle.
The length of the rectangle is \(x\) cm.
The length of a diagonal of the rectangle is 8 cm.
The perimeter of the rectangle is 20 cm.
(a) Show that \(x^2 - 10x + 18 = 0\) (4)
(b) Solve \(x^2 - 10x + 18 = 0\)
Give your solutions correct to 3 significant figures.
Show your working clearly. (3)
Give your solutions correct to 3 significant figures.
Show your working clearly. (3)
| Scheme | Marks |
|---|---|
\(\dfrac{20 - 2x}{2}\) or \(10 - x\) seen as the width or \(\sqrt{8^2 - x^2}\) oe | B1 |
\(x^2 + (10 - x)^2 = 8^2\) or \(x^2 + (10 - x)^2 = 64\) or \(2x + 2\sqrt{8^2 - x^2} = 20\) or \(x + \sqrt{8^2 - x^2} = 10\) | M1 |
\(x^2 + 100 - 10x - 10x + x^2 = 64\) or \((2\sqrt{8^2 - x^2})^2 = 400 - 40x - 40x + 4x^2\) or \((\sqrt{8^2 - x^2})^2 = 100 - 10x - 10x + x^2\) | M1 |
| \(x^2 - 10x + 18 = 0\) | A1 |
| (4) |
Notes
M1: accept \(\dfrac{20 - 2x}{2}\) in place of \(10 - x\) for all method marks
M1: (dep on previous M1)
for correct expansion of \((10 - x)^2\) or correct expansion of \((20 - 2x)^2\)
A1: for correct manipulation resulting in given equation
| Scheme | Marks |
|---|---|
| \(\dfrac{--10 \pm \sqrt{(-10)^2 - 4 \times 1 \times 18}}{2 \times 1}\) or for this expression with one or more of \((--10)\), \((-10)^2\), \(10^2\), \(-4 \times 1 \times 18\), \(2 \times 1\), \((-10)^2 - 4 \times 1 \times 18\) evaluated e.g. \(\dfrac{10 \pm \sqrt{28}}{2}\) | M1 |
| \(\sqrt{28}\) or \(2\sqrt{7}\) or \(\sqrt{100 - 72}\) or 5.29... | M1 |
| 2.35 7.65 | A1 |
| (3) |
Notes
M1: correct substitution brackets not necessary (accept \(10^2\))
or
\((x - 5)^2 - 25 + 18 = 0\) oe
M1: (independent) for correct simplification of discriminant (if evaluated at least 3sf rounded or truncated)
or \(x - 5 = \pm\sqrt{7}\) oe
A1: for values rounding to 2.35 and 7.65 (2.35424... 7.64575...)
Award full marks for correct solutions if at least 1 method mark scored.