| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2020 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Ratio of coefficients condition |
| Difficulty | Standard +0.3 This is a straightforward binomial expansion question requiring students to find specific term coefficients, set them equal, and solve for a constant. Part (a) involves standard binomial coefficient calculation and simple algebra. Part (b) adds one extra step of multiplying by (1-x³) but remains routine. The question tests procedural fluency rather than problem-solving insight, making it slightly easier than average. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| \(6C2 \times \left[2(x^2)\right]^4 \times \left[\frac{a}{x}\right]^2\), \(6C3 \times \left[2(x^2)\right]^3 \times \left[\frac{a}{x}\right]^3\) | B1 B1 | SOI Can be seen in an expansion |
| \(15 \times 2^4 \times a^2 = 20 \times 2^3 \times a^3\) | M1 | SOI Terms must be from a correct series |
| \(a = \frac{15 \times 2^4}{20 \times 2^3} = \frac{3}{2}\) | A1 | OE |
| Answer | Marks |
|---|---|
| \(0\) | B1 |
## Question 5(a):
| $6C2 \times \left[2(x^2)\right]^4 \times \left[\frac{a}{x}\right]^2$, $6C3 \times \left[2(x^2)\right]^3 \times \left[\frac{a}{x}\right]^3$ | B1 B1 | SOI Can be seen in an expansion |
|---|---|---|
| $15 \times 2^4 \times a^2 = 20 \times 2^3 \times a^3$ | M1 | SOI Terms must be from a correct series |
| $a = \frac{15 \times 2^4}{20 \times 2^3} = \frac{3}{2}$ | A1 | OE |
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## Question 5(b):
| $0$ | B1 | |
|---|---|---|
---5 In the expansion of $\left( 2 x ^ { 2 } + \frac { a } { x } \right) ^ { 6 }$, the coefficients of $x ^ { 6 }$ and $x ^ { 3 }$ are equal.
\begin{enumerate}[label=(\alph*)]
\item Find the value of the non-zero constant $a$.
\item Find the coefficient of $x ^ { 6 }$ in the expansion of $\left( 1 - x ^ { 3 } \right) \left( 2 x ^ { 2 } + \frac { a } { x } \right) ^ { 6 }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2020 Q5 [5]}}