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]
Mark scheme
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
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]
Mark scheme
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 \(48ax^2\)
oe the terms may not have been collected could do \((3 + 2x)^2 \times (3 + 2x)^2\). If they use this method (doesn’t refer to \((3 + 2x)^3\)) then award this mark for answer expanded correctly but with one numerical error. Terms must be collected
terms must be collected could do \((3 + 2x)^2 \times (3 + 2x)^3\). If they use this method (doesn’t refer to \((3 + 2x)^4\)) then award this mark for answer expanded correctly but with one numerical error. Terms must be collected would imply first 2 M marks if done correctly
Working could be seen as a list or a grid. This can be awarded full marks if done correctly Candidates could use a combination of methods. Use whichever alt method works best (probably alt 2) Missing brackets must be recovered
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]
Mark scheme
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
