Higher June 2024 Paper 2 Q23
23 Simplify \(\quad \dfrac{30 \times 25^{2x + 7}}{\sqrt{180} \times \left(\sqrt{5}\right)^{4x + 9}}\)
Give your answer in the form \(5^w\) where \(w\) is an expression in terms of \(x\)
Show each stage of your working clearly.
(3)
| Scheme | Marks |
|---|---|
For 2 of \(30 = 5 \times 6\) or \(30 = 5 \times 2 \times 3\) oe (for numerator) or \(\sqrt{180} = 6\sqrt{5}\) oe or \(\sqrt{180} = 2 \times 3 \times \sqrt{5}\) (for denominator) or \(25^{2x + 7} = \left(5^2\right)^{2x + 7}\) or \(5^{2(2x + 7)}\) oe or \(\left(\sqrt{5}\right)^{4x + 9} = \left(5^{\frac{1}{2}}\right)^{4x + 9}\) or \(5^{\frac{1}{2}(4x + 9)}\) oe or \(\dfrac{30}{\sqrt{180}} = \sqrt{5}\) oe or \(5^{4x + 15}\) as the numerator or \(5^{2x + 5}\) as the denominator | M1 |
\(\dfrac{6 \times 5 \times 5^{4x + 14}}{6 \times 5^{0.5} \times 5^{2x + 4.5}}\) oe eg \(\dfrac{5^{4x + 15}}{5^{2x + 5}}\) or \(\dfrac{\sqrt{5}^{\,8x + 30}}{\sqrt{5}^{\,4x + 10}}\) oe or \(\dfrac{25^{2x + 7.5}}{25^{x + 2.5}}\) oe | M1 |
| working required Answer: \(5^{2x + 10}\) | A1 |
| (3) | |
| (3 marks) |
Notes
M1: or \(5^{4x + 15}\) as the numerator or
\(5^{2x + 5}\) as the denominator or
\(30 = 5 \times 6\) or \(30 = 5 \times 2 \times 3\) oe or
\(\sqrt{180} = 6\sqrt{5}\) oe or \(\sqrt{180} = 2 \times 3 \times \sqrt{5}\) or
\(25^{2x + 7} = \left(5^2\right)^{2x + 7}\) or \(5^{2(2x + 7)}\) oe or
\(\left(\sqrt{5}\right)^{4x + 9} = \left(5^{\frac{1}{2}}\right)^{4x + 9}\) or \(5^{\frac{1}{2}(4x + 9)}\) oe or
M1: Correct expression in terms of 6 (or 2 and 3) and 5 with indices
Some cancellation could have taken place
or fully correct with \(\sqrt{5}\) and powers or 25 and powers
A1: dep on M2
allow \(w = 2x + 10\)