Higher June 2023 Paper 2 Q23
23 The diagram shows a cuboid with a square cross section.
Diagram NOT accurately drawn
The volume of the cuboid is \(\left(13 + 6\sqrt{5}\right)\) cm³
Without using a calculator, find the value of \(x\)
Give your answer in the form \(\;a + \sqrt{b}\;\) where \(a\) and \(b\) are integers.
Show your working clearly.
(4)
| Scheme | Marks |
|---|---|
| \((x^2 =)\; \dfrac{13 + 6\sqrt{5}}{2\sqrt{5} - 3}\) | M1 |
| \(\dfrac{13 + 6\sqrt{5}}{2\sqrt{5} - 3} \times \dfrac{2\sqrt{5} + 3}{2\sqrt{5} + 3}\) or \(\dfrac{13 + 6\sqrt{5}}{2\sqrt{5} - 3} \times \dfrac{-2\sqrt{5} - 3}{-2\sqrt{5} - 3}\) | M1 |
eg \(\dfrac{13 + 6\sqrt{5}}{2\sqrt{5} - 3} \times \dfrac{2\sqrt{5} + 3}{2\sqrt{5} + 3} = \dfrac{99 + 44\sqrt{5}}{11}\) or eg \(\dfrac{13 + 6\sqrt{5}}{2\sqrt{5} - 3} \times \dfrac{2\sqrt{5} + 3}{2\sqrt{5} + 3} = \dfrac{26\sqrt{5} + 39 + 60 + 18\sqrt{5}}{20 - 9}\) or eg \(\dfrac{13 + 6\sqrt{5}}{2\sqrt{5} - 3} \times \dfrac{2\sqrt{5} + 3}{2\sqrt{5} + 3} = \dfrac{26\sqrt{5} + 39 + 12(\sqrt{5})^2 + 18\sqrt{5}}{(2\sqrt{5})^2 - 3^2}\) | M1 |
Working required Answer: \(2 + \sqrt{5}\) | A1 |
| (4) | |
| (4 marks) |
Notes
M1: expression for \(x^2\)
M1: dep on previous M1 showing a correct product to rationalise the denominator (must be correct \(x^2\))
M1: dep on previous M1
continuing the expansion of the product on the numerator and denominator – maybe one of these forms or a combination of forms
A1: dep on M3 accept \(a = 2\), \(b = 5\)