| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2010 |
| Session | November |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Binomial times linear coefficient |
| Difficulty | Moderate -0.8 This is a straightforward binomial expansion question requiring routine application of the binomial theorem for positive integer n=8, followed by simple multiplication to find a specific coefficient. The techniques are standard and mechanical with no problem-solving insight needed, making it easier than average but not trivial since it requires careful algebraic manipulation across two parts. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(1 + 8(-2x^2) + {}^8C_2(-2x^2)^2\) | B2,1 | Loses 1 for each error |
| \(1 - 16x^2 + 112x^4\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \((2-x^2) \times \text{their } (1 - 16x^2 + 112x^4)\) | M1 | Must consider exactly 2 terms |
| \((2 \times \text{their } 112) - \text{their}(-16)\) | ||
| \(240\) | A1\(\sqrt{}\) |
## Question 1:
**Part (i)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $1 + 8(-2x^2) + {}^8C_2(-2x^2)^2$ | B2,1 | Loses 1 for each error |
| $1 - 16x^2 + 112x^4$ | | |
**Part (ii)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(2-x^2) \times \text{their } (1 - 16x^2 + 112x^4)$ | M1 | Must consider exactly 2 terms |
| $(2 \times \text{their } 112) - \text{their}(-16)$ | | |
| $240$ | A1$\sqrt{}$ | |
---1 (i) Find the first 3 terms in the expansion, in ascending powers of $x$, of $\left( 1 - 2 x ^ { 2 } \right) ^ { 8 }$.\\
(ii) Find the coefficient of $x ^ { 4 }$ in the expansion of $\left( 2 - x ^ { 2 } \right) \left( 1 - 2 x ^ { 2 } \right) ^ { 8 }$.
\hfill \mbox{\textit{CAIE P1 2010 Q1 [4]}}