| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Coefficient zero after multiplying binomial |
| Difficulty | Moderate -0.8 This is a straightforward binomial expansion question requiring only routine application of the binomial theorem and coefficient matching. Part (i) is direct formula application, part (ii) involves simple algebraic manipulation to find 'a' by setting the x-coefficient to zero, and part (iii) requires collecting x² terms. All steps are standard textbook exercises with no problem-solving insight needed, making it easier than average but not trivial due to the multi-part structure. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(1 + 5ax + 10a^2x^2\) | B2,1 [2] | Loses 1 mark for each incorrect term. |
| (ii) \(× (1 - 2x) \rightarrow 5ax - 2x \rightarrow a = \frac{5}{8}\) | M1, A1 [2] | Needs to consider exactly 2 terms. co. |
| (iii) Coeff of \(x^2\) is \(-10a + 10a^2 \rightarrow -4 + 1.6 = -2.4\) | M1 A1√, A1 [3] | Needs to consider exactly 2 terms. co. |
(i) $1 + 5ax + 10a^2x^2$ | B2,1 [2] | Loses 1 mark for each incorrect term.
(ii) $× (1 - 2x) \rightarrow 5ax - 2x \rightarrow a = \frac{5}{8}$ | M1, A1 [2] | Needs to consider exactly 2 terms. co.
(iii) Coeff of $x^2$ is $-10a + 10a^2 \rightarrow -4 + 1.6 = -2.4$ | M1 A1√, A1 [3] | Needs to consider exactly 2 terms. co.\begin{enumerate}[label=(\roman*)]
\item Find the first 3 terms in the expansion of $(1 + ax)^4$ in ascending powers of $x$. [2]
\item Given that there is no term in $x$ in the expansion of $(1 - 2x)(1 + ax)^5$, find the value of the constant $a$. [2]
\item For this value of $a$, find the coefficient of $x^2$ in the expansion of $(1 - 2x)(1 + ax)^5$. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2010 Q6 [7]}}