| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2008 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Standard product of two binomials |
| Difficulty | Moderate -0.8 This is a straightforward binomial expansion question requiring routine application of the binomial theorem for part (i), followed by a simple multiplication in part (ii). The 'hence' structure guides students directly to the method, and the calculations involve only small positive integers with no algebraic manipulation challenges. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| \(\rightarrow 32 + 80x^3 + 80x^6\) | 3 × B1 | If coeffs ok but x and \(x^2\), allow B1 special case. Allow 80, 80 if in (ii). [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Product has 3 terms in \(x^4\) | B1 | Anywhere. |
| M1 | Must be attempt at more than 1 term. | |
| \(\rightarrow 80 + 160 + 32 = 272\) | A1√ | For follow-through on both expansions, providing there are 3 terms added. [3] |
**(i)** $(2 + x^3)^5 = 2^5 + 5.2^4.x^3 + 10.2^3.x^4$
$\rightarrow 32 + 80x^3 + 80x^6$ | 3 × B1 | If coeffs ok but x and $x^2$, allow B1 special case. Allow 80, 80 if in (ii). [3]
(allow $2^5$ for 32)
**(ii)** $(1 + x^2)^2 = 1 + 2x^2 + x^4$
Product has 3 terms in $x^4$ | B1 | Anywhere.
| M1 | Must be attempt at more than 1 term.
$\rightarrow 80 + 160 + 32 = 272$ | A1√ | For follow-through on both expansions, providing there are 3 terms added. [3]3 (i) Find the first 3 terms in the expansion, in ascending powers of $x$, of $\left( 2 + x ^ { 2 } \right) ^ { 5 }$.\\
(ii) Hence find the coefficient of $x ^ { 4 }$ in the expansion of $\left( 1 + x ^ { 2 } \right) ^ { 2 } \left( 2 + x ^ { 2 } \right) ^ { 5 }$.
\hfill \mbox{\textit{CAIE P1 2008 Q3 [6]}}