| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Product with unknown constant to determine |
| Difficulty | Standard +0.3 This is a straightforward binomial expansion question requiring students to write out terms using the binomial theorem, then multiply by a simple factor and equate coefficients. The algebra is routine and the method is standard textbook fare, making it slightly easier than average but still requiring careful manipulation of the unknown constant a. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((a-x)^6 = a^6 - 6a^5x + 15a^4x^2 - 20a^3x^3 + \ldots\) | B2, 1, 0 | Allow extra terms. Terms may be listed. Allow \(a^6x^0\). |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\left(1 + \dfrac{2}{ax}\right)(\ldots 15a^4x^2 - 20a^3x^3 + \ldots)\) leading to \([x^2](15a^4 - 40a^2)\) | M1 | Attempting to find 2 terms in \(x^2\) |
| \(15a^4 - 40a^2 = -20\) leading to \(15a^4 - 40a^2 + 20[=0]\) | A1 | Terms on one side of the equation |
| \((5a^2 - 10)(3a^2 - 2)[=0]\) | M1 | OE. M1 for attempted factorisation or solving for \(a^2\) or \(u(=a^2)\) using e.g. formula or completing the square |
| \(a = \pm\sqrt{2},\ \pm\sqrt{\dfrac{2}{3}}\) | B1 B1 | OE exact form only. If B0B0 scored then SC B1 for \(\sqrt{2}, \sqrt{\frac{2}{3}}\) WWW or \(\pm1.41, \pm0.816\) WWW |
## Question 7:
### Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(a-x)^6 = a^6 - 6a^5x + 15a^4x^2 - 20a^3x^3 + \ldots$ | B2, 1, 0 | Allow extra terms. Terms may be listed. Allow $a^6x^0$. |
### Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\left(1 + \dfrac{2}{ax}\right)(\ldots 15a^4x^2 - 20a^3x^3 + \ldots)$ leading to $[x^2](15a^4 - 40a^2)$ | M1 | Attempting to find 2 terms in $x^2$ |
| $15a^4 - 40a^2 = -20$ leading to $15a^4 - 40a^2 + 20[=0]$ | A1 | Terms on one side of the equation |
| $(5a^2 - 10)(3a^2 - 2)[=0]$ | M1 | OE. M1 for attempted factorisation or solving for $a^2$ or $u(=a^2)$ using e.g. formula or completing the square |
| $a = \pm\sqrt{2},\ \pm\sqrt{\dfrac{2}{3}}$ | B1 B1 | OE exact form only. If B0B0 scored then **SC B1** for $\sqrt{2}, \sqrt{\frac{2}{3}}$ WWW or $\pm1.41, \pm0.816$ WWW |7
\begin{enumerate}[label=(\alph*)]
\item Write down the first four terms of the expansion, in ascending powers of $x$, of $( a - x ) ^ { 6 }$.
\item Given that the coefficient of $x ^ { 2 }$ in the expansion of $\left( 1 + \frac { 2 } { a x } \right) ( a - x ) ^ { 6 }$ is - 20 , find in exact form the possible values of the constant $a$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2021 Q7 [7]}}