June 2024 Paper 2 Q19
19 In the expansion of \(\quad (3 + ax)^5 \quad\) where \(a\) is a non-zero constant
\[8 \times \text{coefficient of } x^2 = \text{coefficient of } x^4\]Work out the possible values of \(a\). [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(10 \times 3^3 \times a^2\) or \(10 \times 27 \times a^2\) or \(270a^2\) | M1 | oe eg \(\dbinom{5}{2} \times 3^3 \times a^2\) or \({}^5\mathrm{C}_2\, 3^{5-2} a^2\) \(a\) may be \((ax)\) may be implied eg \(2160a^2\) or \(2160a^2x^2\) |
| \(5 \times 3^{(1)} \times a^4\) or \(15a^4\) | M1 | oe eg \(\dbinom{5}{4} \times 3 \times a^4\) or \({}^5\mathrm{C}_4\, 3^{5-4} a^4\) \(a\) may be \((ax)\) may be implied eg \(15a^4x^4\) |
| \(8 \times\) their \(270a^2 =\) their \(15a^4\) or \(2160a^2 = 15a^4\) | M1dep | oe eg \(a^2 = 144\) or \((a =)\ 12\) or \((a =)\ -12\) their \(270a^2\) must be \(ka^2\) where \(k\) is a non-zero constant their \(15a^4\) must be \(ma^4\) where \(m\) is a non-zero constant dep on M2 |
| \((a =)\ 12\) and \((a =)\ -12\) with no other values | A1 | oe eg \((a =)\ \pm 12\) with no other values |
Additional guidance
| After M1M1 an equation in terms of \(x\) and \(a\) cannot score M1dep even if recovered | |
| Ignore other terms in the expansions | |
| \((a =)\ 12\) and \((a =)\ -12\) and \((a =)\ 0\) | M3A0 |