| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2020 |
| Session | Specimen |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Product with unknown constant to determine |
| Difficulty | Moderate -0.5 Part (a) is straightforward binomial expansion requiring direct application of the formula to find two coefficients. Part (b) requires multiplying the expansion by (3x+1) and collecting terms, which adds one modest problem-solving step but remains a standard textbook exercise with clear methodology. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Coefficient of \(x^2\) is 240 | 1 B1 | |
| Coefficient of \(x^3\) is \(20 \times 8 \times (-1) = -160\) | 2 B2 | B1 for \(+160\) |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Product needs exactly 2 terms | 1 M1 | \(3 \times\) their \(240 +\) their \(-160\) |
| \(720 - 160 = 560\) | 1 A1FT | FT for candidate's answers |
| Total | 2 |
## Question 6(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Coefficient of $x^2$ is 240 | 1 B1 | |
| Coefficient of $x^3$ is $20 \times 8 \times (-1) = -160$ | 2 B2 | B1 for $+160$ |
| **Total** | **3** | |
## Question 6(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Product needs exactly 2 terms | 1 M1 | $3 \times$ their $240 +$ their $-160$ |
| $720 - 160 = 560$ | 1 A1FT | FT for candidate's answers |
| **Total** | **2** | |6 (a) Find the coefficients of $x ^ { 2 }$ and $x ^ { 3 }$ in the expansion of $( 2 - x ) ^ { 6 }$.\\
(b) Hence find the coefficient of $x ^ { 3 }$ in the expansion of $( 3 x + 1 ) ( 2 - x ) ^ { 6 }$.\\
\hfill \mbox{\textit{CAIE P1 2020 Q6 [5]}}