4MA0 Higher January 2018 Paper 4 Q20
20
(a) Simplify fully \(\left(\dfrac{125e^{12}}{27f^3}\right)^{-\frac{2}{3}}\) (3)
(b) Given that \(2^{\frac{1}{2}} \times 2^{\frac{n}{3}} = \dfrac{8^x}{4^n}\)
express \(x\) in terms of \(n\). (4)
express \(x\) in terms of \(n\). (4)
| Scheme | Marks |
|---|---|
eg \(\left(\dfrac{27f^3}{125e^{12}}\right)^{\frac{2}{3}}\) or \(\dfrac{1}{\left(\dfrac{125e^{12}}{27f^3}\right)^{\frac{2}{3}}}\) or \(\left(\dfrac{15625e^{24}}{729f^6}\right)^{-\frac{1}{3}}\) or \(\left(\dfrac{5e^4}{3f}\right)^{-2}\) | M1 |
eg \(\left(\dfrac{3f}{5e^4}\right)^2\) or \(\dfrac{1}{\left(\dfrac{15625e^{24}}{729f^6}\right)^{\frac{1}{3}}}\) or \(\left(\dfrac{25e^8}{9f^2}\right)^{-1}\) or \(\dfrac{1}{\left(\dfrac{5e^4}{3f}\right)^2}\) or \(\left(\dfrac{729f^6}{15625e^{24}}\right)^{\frac{1}{3}}\) | M1 |
| \(\dfrac{9f^2}{25e^8}\) | A1 |
| (3) |
Notes
M1: For dealing with either negative index or cube root or square
or an expression including
\(\dfrac{9}{25}\) or \(\left(\dfrac{25}{9}\right)^{-1}\) or \(\dfrac{f^2}{e^8}\) or \(\dfrac{e^{-8}}{f^{-2}}\) (oe)
M1: For dealing with two elements
or a correct expression not simplified eg \(\dfrac{\frac{1}{25}e^{-8}}{\frac{1}{9}f^{-2}}\), \(\dfrac{0.04e^{-8}}{0.\dot{1}f^{-2}}\)
A1: allow \(\dfrac{9e^{-8}}{25f^{-2}}\), \(0.36\dfrac{f^2}{e^8}\), \(\dfrac{9}{25}e^{-8}f^2\)
| Scheme | Marks |
|---|---|
| \(8^x = (2^3)^x\) or \(2^{3x}\) \(4^n = (2^2)^n\) or \(2^{2n}\) \(2^{\frac{1}{2}} \times 2^{\frac{n}{3}} = 2^{\frac{1}{2} + \frac{n}{3}}\) | M2 |
| eg \(\dfrac{3 + 2n}{6} = 3x - 2n\) or \(\dfrac{1}{2} + \dfrac{n}{3} + 2n = 3x\) oe | M1 |
Eg \(x = \dfrac{1}{6} + \dfrac{n}{9} + \dfrac{2n}{3}\), \(x = \dfrac{1}{6} + \dfrac{7n}{9}\) oe Answer: \(x = \dfrac{14n + 3}{18}\) | A1 |
| (4) |
Notes
M2: For all of:
\(8^x\) written as a power or 2
\(4^n\) written as a power of 2
LHS written as a single 2 with a power
M1 for 2 of these 3
M1: A correct equation using only the powers
A1: oe