June 2023 Paper 1 Q16
16 The expansions of \(\quad (1 + 12x)^4 \quad\) and \(\quad (a + 4x)^3 \quad\) have the same coefficient of \(x^2\)
Work out the value of \(a\). [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| (1 4) 6 (4 1) or (1) 3 (3 1) seen and used correctly (in correct place in expansion) as part of Pascal’s triangle or \(\dbinom{4}{2}\) or \({}^4\mathrm{C}_2\) or \(\dbinom{3}{1}\) or \({}^3\mathrm{C}_1\) or \(\dbinom{3}{2}\) or \({}^3\mathrm{C}_2\) seen and used | B1 | \(6(1^2)12x^2\) would be B1 \(6(1^2)12^2x^2\) would be B1 leading onto M1dep \(6(1^2) + 12x^2\) would be B0 or equivalent in other expansion |
| \(6 \times 12 \times 12\) \((\times x^2)\) or \(864(\times x^2)\) | M1dep | oe could be within an expansion dependent on B mark only |
| \(3 \times a \times 4 \times 4\) \((\times x^2)\) or \(48a(\times x^2)\) | M1dep | oe could be within an expansion dependent on B mark only |
| 18 | A1 |
Additional guidance
| (1) 3 (3 1) used in \((1 + 12x)^4\) would gain no marks | B0 |
| Either M mark implies the B mark | |
| Candidates attempting to expand the brackets will need to get either M mark to get the B mark. They could have made errors in their expansions as long as they get \(864x^2\) or \(48ax^2\) |