Standard +0.3 This question requires applying the binomial theorem to find specific coefficients in two expansions, then solving a linear equation relating them. While it involves multiple steps (expanding two binomials, identifying correct terms, setting up and solving an equation), each step uses standard techniques with no novel insight required. The algebra is straightforward and the question is more computational than conceptual, making it slightly easier than average.
4 The coefficient of \(x\) in the expansion of \(\left( 4 x + \frac { 10 } { x } \right) ^ { 3 }\) is \(p\). The coefficient of \(\frac { 1 } { x }\) in the expansion of \(\left( 2 x + \frac { k } { x ^ { 2 } } \right) ^ { 5 }\) is \(q\).
Given that \(p = 6 q\), find the possible values of \(k\).
Term in \(\frac{1}{x}\) or \(q=\) \([10\times](2x)^3\left(\frac{k}{x^2}\right)^2\)
M1
Appropriate term identified and selected
\([10\times 2^3k^2=]\ 80k^2\)
A1
Allow \(\frac{80k^2}{x}\)
\(p=6q\) used \((480=6\times 80k^2\) or \(80=80k^2)\)
M1
Correct link used for *their* coefficient of \(x\) and \(\frac{1}{x}\) (\(p\) and \(q\)) with no \(x\)'s
\([k^2=1\Rightarrow]\ k=\pm 1\)
A1
A0 if a range of values given. Do not allow \(\pm\sqrt{1}\)
5
## Question 4:
| Answer | Marks | Guidance |
|--------|-------|----------|
| [Coefficient of $x$ or $p=$] $480$ | B1 | SOI. Allow $480x$ even in an expansion |
| Term in $\frac{1}{x}$ or $q=$ $[10\times](2x)^3\left(\frac{k}{x^2}\right)^2$ | M1 | Appropriate term identified and selected |
| $[10\times 2^3k^2=]\ 80k^2$ | A1 | Allow $\frac{80k^2}{x}$ |
| $p=6q$ used $(480=6\times 80k^2$ or $80=80k^2)$ | M1 | Correct link used for *their* coefficient of $x$ and $\frac{1}{x}$ ($p$ and $q$) with no $x$'s |
| $[k^2=1\Rightarrow]\ k=\pm 1$ | A1 | A0 if a range of values given. Do not allow $\pm\sqrt{1}$ |
| | **5** | |
4 The coefficient of $x$ in the expansion of $\left( 4 x + \frac { 10 } { x } \right) ^ { 3 }$ is $p$. The coefficient of $\frac { 1 } { x }$ in the expansion of $\left( 2 x + \frac { k } { x ^ { 2 } } \right) ^ { 5 }$ is $q$.
Given that $p = 6 q$, find the possible values of $k$.\\
\hfill \mbox{\textit{CAIE P1 2021 Q4 [5]}}