| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2013 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Coefficient zero after multiplying binomial |
| Difficulty | Moderate -0.8 Part (i) is straightforward binomial expansion requiring direct application of the formula for three terms. Part (ii) requires multiplying the expansion by (1+ax) and setting the x² coefficient to zero—a standard exercise with minimal problem-solving demand, making this easier than average but not trivial. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(64 + 576x + 2160x^2\) | B1B1B1 | Can score in (ii) |
| (ii) \(576a(x^2) + 2160(x^2) = 0\) | M1 | |
| \(a = -\frac{2160}{576}\) oe (eg \(-\frac{15}{4}\)) or \(-3.75\) | A1 | [2] |
(i) $64 + 576x + 2160x^2$ | B1B1B1 | Can score in (ii)
(ii) $576a(x^2) + 2160(x^2) = 0$ | M1 |
$a = -\frac{2160}{576}$ oe (eg $-\frac{15}{4}$) or $-3.75$ | A1 | [2]1 (i) Find the first three terms when $( 2 + 3 x ) ^ { 6 }$ is expanded in ascending powers of $x$.\\
(ii) In the expansion of $( 1 + a x ) ( 2 + 3 x ) ^ { 6 }$, the coefficient of $x ^ { 2 }$ is zero. Find the value of $a$.
\hfill \mbox{\textit{CAIE P1 2013 Q1 [5]}}