November 2021 Paper 2 Q16
16 The coefficient of \(x^4\) in the expansion of \(\quad (a + 2x)^6 \quad\) is 1500
Work out the two possible values of \(a\). [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(15 \times 2^4\) or \(15 \times 16\) or 240 | M1 | oe eg \(\dbinom{6}{4}2^4\) or \(2^6 \times \dfrac{6 \times 5}{2} \times \left(\dfrac{1}{2}\right)^2\) may include \(a^2\) and/or \(x^4\) allow embedded eg \({}^6\mathrm{C}_4\,a^2(2x)^4\) |
| \(240a^2 = 1500\) or \(a^2 = \dfrac{1500}{240}\) or \((\pm)\sqrt{\dfrac{1500}{240}}\) or \(\dfrac{5}{2}\) or \(-\dfrac{5}{2}\) | M1dep | must evaluate \(\dbinom{6}{4}\) oe eg \(15 \times 2^4a^2 = 1500\) or \((\pm)\sqrt{\dfrac{1500}{15 \times 2^4}}\) may include \(x^4\) on both sides of an equation |
| \(\dfrac{5}{2}\) and \(-\dfrac{5}{2}\) with no other values | A1 | oe eg 2.5 and \(-2.5\) SC2 [2.236, 2.24] and [\(-2.24\), \(-2.236\)] SC1 [2.236, 2.24] or [\(-2.24\), \(-2.236\)] |
Additional guidance
| The relevant term must be selected from a full expansion but the other terms can be ignored | |
| Allow \(\dbinom{6}{4}\) to be \(\dbinom{6}{2}\) | |
| \(240a^2x^4 = 1500x^4\) | M1M1 |
| \(240a^2x^4 = 1500\) recovered to \((\pm)\sqrt{\dfrac{1500}{240}}\) oe | M1M1 |
| \(240a^2x^4 = 1500\) not recovered to \((\pm)\sqrt{\dfrac{1500}{240}}\) oe | M1M0 |