| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | March |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Product of expansions |
| Difficulty | Standard +0.3 Part (a) is a standard binomial expansion requiring identification of the term where powers of x cancel (r=2 gives x^0). Part (b) adds one extra step of multiplying two terms from different expansions. This is a routine textbook exercise with clear methodology, slightly above average due to the x^(-2) term and the product in part (b), but still straightforward application of the binomial theorem. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(^6C_2 \times (3x)^4 \left(\frac{2}{x^2}\right)^2\) | B1 | Can be seen within an expansion |
| \(15 \times 3^4 \times 2^2\) | B1 | Identified. Powers must be correct |
| \(4860\) | B1 | Without any power of \(x\) |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Their \(4860\) and one other relevant term | M1 | Using their \(4860\) and an attempt to find a term in \(x^{-3}\) |
| Other term \(= 6C3(3x)^3\left(\frac{2}{x^2}\right)^3\) or \(6C3 \times 3^3 \times 2^3\) or \(4320\) | A1 | Must be identified. If M0 scored then SC B1 for \(4320\) as the only answer |
| \([4860 - 4320 =]\ 540\) | A1 | |
| 3 |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $^6C_2 \times (3x)^4 \left(\frac{2}{x^2}\right)^2$ | B1 | Can be seen within an expansion |
| $15 \times 3^4 \times 2^2$ | B1 | Identified. Powers must be correct |
| $4860$ | B1 | Without any power of $x$ |
| | **3** | |
## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Their $4860$ and one other relevant term | M1 | Using their $4860$ and an attempt to find a term in $x^{-3}$ |
| Other term $= 6C3(3x)^3\left(\frac{2}{x^2}\right)^3$ or $6C3 \times 3^3 \times 2^3$ or $4320$ | A1 | Must be identified. If M0 scored then SC B1 for $4320$ as the only answer |
| $[4860 - 4320 =]\ 540$ | A1 | |
| | **3** | |3 Find the term independent of $x$ in each of the following expansions.
\begin{enumerate}[label=(\alph*)]
\item $\left( 3 x + \frac { 2 } { x ^ { 2 } } \right) ^ { 6 }$
\item $\left( 3 x + \frac { 2 } { x ^ { 2 } } \right) ^ { 6 } \left( 1 - x ^ { 3 } \right)$
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2022 Q3 [6]}}