Moderate -0.3 This question requires applying the binomial theorem twice and equating coefficients, but the steps are straightforward: find the coefficient of x^4 in (3+x)^5 using C(5,4)·3^1·x^4, find the general term in (2x+a/x)^6, identify which term gives x^2, then solve a simple equation. It's slightly easier than average because it's a direct application of standard binomial expansion techniques with no conceptual surprises.
1 The coefficient of \(x ^ { 4 }\) in the expansion of \(( 3 + x ) ^ { 5 }\) is equal to the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 2 x + \frac { a } { x } \right) ^ { 6 }\).
Find the value of the positive constant \(a\).
Condone inclusion of \(x^4\). Can be seen as part of an expansion.
Coefficient of \(x^2 = 240a^2\)
B1
Condone inclusion of \(x^2\). Can be seen as part of an expansion.
'Their \(240\)'\(a^2\) – 'their \(15\)'
M1
Forming an equation of the form \(pa^2 = q\), where \(p\) and \(q\) are constants. Condone inclusion of powers of \(x\) as long as they then disappear.
\(a = \frac{1}{4}\) or \(0.25\)
A1
OE. Do not condone extra 'answer' of \(-\frac{1}{4}\), or allow \(\sqrt{\frac{1}{16}}\) or similar.
Total: 4
## Question 1:
| Answer | Mark | Guidance |
|--------|------|----------|
| Coefficient of $x^4 = 15$ | B1 | Condone inclusion of $x^4$. Can be seen as part of an expansion. |
| Coefficient of $x^2 = 240a^2$ | B1 | Condone inclusion of $x^2$. Can be seen as part of an expansion. |
| 'Their $240$'$a^2$ – 'their $15$' | M1 | Forming an equation of the form $pa^2 = q$, where $p$ and $q$ are constants. Condone inclusion of powers of $x$ as long as they then disappear. |
| $a = \frac{1}{4}$ or $0.25$ | A1 | OE. Do not condone extra 'answer' of $-\frac{1}{4}$, or allow $\sqrt{\frac{1}{16}}$ or similar. |
| **Total: 4** | | |
---
1 The coefficient of $x ^ { 4 }$ in the expansion of $( 3 + x ) ^ { 5 }$ is equal to the coefficient of $x ^ { 2 }$ in the expansion of $\left( 2 x + \frac { a } { x } \right) ^ { 6 }$.
Find the value of the positive constant $a$.\\
\hfill \mbox{\textit{CAIE P1 2022 Q1 [4]}}