June 2025 Paper 2 Q7
7
(a)
\(2^{5a + 1} = 2^a\)
Work out the value of \(a\). [2 marks]
(b)
\(3^{-7} \div 3^c = \left(3^{-3}\right)^{2c}\)
Work out the value of \(c\). [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(5a + 1 = a\) or \(2^{5a - a + 1} = 1\) or \(2^{5a - a + 1} = 2^0\) | M1 | oe linear equation eg \(4a = -1\) or oe index equation in the form \(2^{\mathrm{f}(a)} = 1\) or \(2^{\mathrm{f}(a)} = 2^0\) |
| \(-\dfrac{1}{4}\) or \(-0.25\) | A1 | oe value |
Additional guidance
| Only \(\dfrac{2^{5a + 1}}{2^a} = 1\) | M0 |
| Answer | Mark | Comments |
|---|---|---|
| \(3^{-7 - c}\) or \(3^{-6c}\) or \(-7 - c\) and \(-6c\) | M1 | oe eg \(\dfrac{1}{3^{7 + c}}\) or \(\dfrac{1}{3^{6c}}\) or \(3^{-3 \times 2c}\) implied by a correct equation equating powers of 3 eg \(3^{-7} = 3^{-5c}\) |
| \(-7 - c = -6c\) | M1dep | oe linear equation eg \(5c = 7\) |
| \(\dfrac{7}{5}\) or 1.4 | A1 | oe value condone \(\dfrac{-7}{-5}\) |
Additional guidance
| \(\dfrac{1}{3^7} \div 3^c\) needs further processing for M1 | |
| \(\dfrac{1}{(3^3)^{2c}}\) needs further processing for M1 |