| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2009 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Product with unknown constant to determine |
| Difficulty | Moderate -0.3 Part (i) is straightforward binomial expansion requiring recall of the formula and basic arithmetic. Part (ii) requires collecting terms to find the x² coefficient and solving for a constant, which is a standard textbook exercise with minimal problem-solving demand. The question is slightly easier than average due to small powers and straightforward algebra. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
(i) $(2 – x)^6$
M1: $64 − 192x + 240x^2$
M1: Allow $2^6$
A1: Considers at least 2 terms in $x^2$
[3]
(ii) $(1 + 2x + ax^2)(2 – x)^6$
M1: Coeff of $x^2 = 240 − 384 + 64a$
M1: Considers exactly 3 terms + solution
A1: Equates to 48 $\to a = 3$
[3]3 (i) Find the first 3 terms in the expansion of $( 2 - x ) ^ { 6 }$ in ascending powers of $x$.\\
(ii) Given that the coefficient of $x ^ { 2 }$ in the expansion of $\left( 1 + 2 x + a x ^ { 2 } \right) ( 2 - x ) ^ { 6 }$ is 48 , find the value of the constant $a$.
\hfill \mbox{\textit{CAIE P1 2009 Q3 [6]}}