| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Coefficient zero after multiplying binomial |
| Difficulty | Standard +0.3 Part (a) is routine binomial expansion with fractional terms. Part (b) requires identifying specific terms from the expansion, setting up two simultaneous equations from coefficient conditions, and solving—a standard multi-step application but straightforward once the method is recognized. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x^5 + 10x^3 + 40x + \frac{80}{x} + \frac{80}{x^3} + \frac{32}{x^5}\) or \(x^5 + 10x^3 + 40x + 80x^{-1} + 80x^{-3} + 32x^{-5}\) | B2, 1, 0 | B2 all terms correct, B1 5 terms correct. Terms must be simplified. Lists of terms allowed. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| their \(40 \times a + (\text{their coefficient of } x^{-1}) \times b = 0\) | M1 | Coefficients of \(a\) and \(b\) must be non-zero, allow \(x\)'s so long as they are dealt with correctly |
| \((\text{their coefficient of } x^{-1}) \times a + (\text{their coefficient of } x^{-3}) \times b = 80\) | M1 | Coefficients of \(a\) and \(b\) must be non-zero, allow \(x\)'s as long as they are dealt with correctly |
| \(a = 2 \quad b = -1\) | A1 A1 | Dependent on both M marks, may be seen without working |
## Question 3:
**Part (a):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x^5 + 10x^3 + 40x + \frac{80}{x} + \frac{80}{x^3} + \frac{32}{x^5}$ or $x^5 + 10x^3 + 40x + 80x^{-1} + 80x^{-3} + 32x^{-5}$ | B2, 1, 0 | B2 all terms correct, B1 5 terms correct. Terms must be simplified. Lists of terms allowed. |
**Part (b):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| their $40 \times a + (\text{their coefficient of } x^{-1}) \times b = 0$ | M1 | Coefficients of $a$ and $b$ must be non-zero, allow $x$'s so long as they are dealt with correctly |
| $(\text{their coefficient of } x^{-1}) \times a + (\text{their coefficient of } x^{-3}) \times b = 80$ | M1 | Coefficients of $a$ and $b$ must be non-zero, allow $x$'s as long as they are dealt with correctly |
| $a = 2 \quad b = -1$ | A1 A1 | Dependent on both M marks, may be seen without working |
---3
\begin{enumerate}[label=(\alph*)]
\item Give the complete expansion of $\left( x + \frac { 2 } { x } \right) ^ { 5 }$.
\item In the expansion of $\left( a + b x ^ { 2 } \right) \left( x + \frac { 2 } { x } \right) ^ { 5 }$, the coefficient of $x$ is zero and the coefficient of $\frac { 1 } { x }$ is 80 . Find the values of the constants $a$ and $b$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2023 Q3 [6]}}