| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2012 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Binomial times linear coefficient |
| Difficulty | Moderate -0.3 This is a straightforward binomial expansion question requiring standard application of the binomial theorem followed by a simple multiplication. Part (i) is routine calculation, and part (ii) only requires identifying which terms from part (i) contribute to x^8 after multiplying by (2+x). Slightly easier than average due to the mechanical nature and clear structure, though the algebraic manipulation keeps it from being trivial. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \((2x - x^2)^6 = 64x^6 - 192x^7 + 240x^8\) | B1B1B1 | cao |
| (ii) \(\times(2 + x)\) coeff of \(x^8 = 2x \cdot 240 - 192 \cdot 288\) | M1 | Looks at exactly 2 terms |
| A1 | [2] |
**(i)** $(2x - x^2)^6 = 64x^6 - 192x^7 + 240x^8$ | B1B1B1 | cao | [3]
**(ii)** $\times(2 + x)$ coeff of $x^8 = 2x \cdot 240 - 192 \cdot 288$ | M1 | Looks at exactly 2 terms
| A1 | [2]4 (i) Find the first 3 terms in the expansion of $\left( 2 x - x ^ { 2 } \right) ^ { 6 }$ in ascending powers of $x$.\\
(ii) Hence find the coefficient of $x ^ { 8 }$ in the expansion of $( 2 + x ) \left( 2 x - x ^ { 2 } \right) ^ { 6 }$.
\hfill \mbox{\textit{CAIE P1 2012 Q4 [5]}}