| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2020 |
| Session | March |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Single coefficient given directly |
| Difficulty | Standard +0.3 This is a straightforward binomial expansion problem requiring identification of the correct term for a given power of x, solving a simple equation (likely quadratic), then finding another coefficient. The mechanics are routine for A-level, though the negative powers require careful bookkeeping. Slightly above average due to the two-part structure and algebraic manipulation, but still a standard textbook exercise. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| \(5C2\left[2(x)\right]^3\left[\frac{a}{(x^2)}\right]^2\) | B1 | SOI. Can include correct \(x\)'s |
| \(10\times8\times a^2\left(\frac{x^3}{x^4}\right) = 720\left(\frac{1}{x}\right)\) | B1 | SOI. Can include correct \(x\)'s |
| \(a = \pm3\) | B1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| \(5C4\left[2(x)\right]\left[\frac{\text{their}\,a}{(x^2)}\right]^4\) | B1 | SOI. *Their a* can be just one of their values (e.g. just 3). Can gain mark from within an expansion but must use *their* value of \(a\) |
| 810 identified | B1 | Allow with \(x^{-7}\) |
| Total: 2 |
## Question 6(a):
| $5C2\left[2(x)\right]^3\left[\frac{a}{(x^2)}\right]^2$ | B1 | SOI. Can include correct $x$'s |
|---|---|---|
| $10\times8\times a^2\left(\frac{x^3}{x^4}\right) = 720\left(\frac{1}{x}\right)$ | B1 | SOI. Can include correct $x$'s |
| $a = \pm3$ | B1 | |
| **Total: 3** | | |
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## Question 6(b):
| $5C4\left[2(x)\right]\left[\frac{\text{their}\,a}{(x^2)}\right]^4$ | B1 | SOI. *Their a* can be just one of their values (e.g. just 3). Can gain mark from within an expansion but must use *their* value of $a$ |
|---|---|---|
| 810 identified | B1 | Allow with $x^{-7}$ |
| **Total: 2** | | |
---6 The coefficient of $\frac { 1 } { x }$ in the expansion of $\left( 2 x + \frac { a } { x ^ { 2 } } \right) ^ { 5 }$ is 720 .
\begin{enumerate}[label=(\alph*)]
\item Find the possible values of the constant $a$.
\item Hence find the coefficient of $\frac { 1 } { x ^ { 7 } }$ in the expansion.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2020 Q6 [5]}}