Moderate -0.3 This is a straightforward binomial expansion problem requiring identification of x² terms in two expansions and solving a linear equation. While it involves multiple steps, the techniques are standard and the algebra is routine for A-level students who have learned the binomial theorem.
1 The coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 2 + \frac { x } { 2 } \right) ^ { 6 } + ( a + x ) ^ { 5 }\) is 330 . Find the value of the constant \(a\).
Coefficient of \(x^2\) in \(\left(2+\frac{x}{2}\right)^6\) is \(_{6}C_{2}\times2^4\times(\frac{1}{2})^2\ (x^2)\ (=60)\)
B2,1,0
3 things wanted – 1 each incorrect component, must be multiplied together. Allow \(_{6}C_{4}\), \(\binom{6}{4}\) and factorial equivalents. Marks can be awarded for correct term in an expansion.
Coefficient of \(x^2\) in \((a+x)^5\) is \(_{5}C_{2}\times a^3\ (x^2)\ (=10a^3)\)
B1
Marks can be awarded for correct term in an expansion.
\(\rightarrow 60 + 10a^3 = 330\)
M1
Forms an equation '*their* \(60\)' + '*their* \(10a^3\)' \(= 330\), OK with \(x^2\) in all three terms initially. This can be recovered by a correct answer.
\(a = 3\)
A1
Condone \(\pm3\) as long as \(+3\) is selected.
5
**Question 1:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Coefficient of $x^2$ in $\left(2+\frac{x}{2}\right)^6$ is $_{6}C_{2}\times2^4\times(\frac{1}{2})^2\ (x^2)\ (=60)$ | **B2,1,0** | 3 things wanted – 1 each incorrect component, must be multiplied together. Allow $_{6}C_{4}$, $\binom{6}{4}$ and factorial equivalents. Marks can be awarded for correct term in an expansion. |
| Coefficient of $x^2$ in $(a+x)^5$ is $_{5}C_{2}\times a^3\ (x^2)\ (=10a^3)$ | **B1** | Marks can be awarded for correct term in an expansion. |
| $\rightarrow 60 + 10a^3 = 330$ | **M1** | Forms an equation '*their* $60$' + '*their* $10a^3$' $= 330$, OK with $x^2$ in all three terms initially. This can be recovered by a correct answer. |
| $a = 3$ | **A1** | Condone $\pm3$ as long as $+3$ is selected. |
| | **5** | |
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1 The coefficient of $x ^ { 2 }$ in the expansion of $\left( 2 + \frac { x } { 2 } \right) ^ { 6 } + ( a + x ) ^ { 5 }$ is 330 . Find the value of the constant $a$.\\
\hfill \mbox{\textit{CAIE P1 2018 Q1 [5]}}