| Exam Board | OCR |
|---|---|
| Module | PURE |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Product with unknown constant to determine |
| Difficulty | Moderate -0.3 Part (a) is straightforward binomial expansion requiring routine application of the formula. Part (b) involves multiplying two expansions and equating coefficients, which adds a small algebraic step but remains a standard textbook exercise with no conceptual difficulty beyond the basic technique. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| \((2-5x)^5 = 2^5 + {}^5C_1 2^4(-5x) + {}^5C_2 2^3(-5x)^2 + \ldots\) | M1 | Attempt at least 2 terms – products of binomial coefficients and correct powers of 2 and \(-5x\); allow \(\pm 5x\) |
| \(32 - 400x\) | A1 | |
| \(+2000x^2\) | A1 | Do not allow from \(+5x\) |
| Answer | Marks | Guidance |
|---|---|---|
| \((1 + 2ax + a^2x^2)(32 - 400x + 2000x^2 + \ldots)\) | M1\* | Expand first bracket, multiply by part (a) to obtain the two relevant terms in \(x\) |
| \(64a - 400 = 48 \Rightarrow a = \ldots\) | Dep\*M1 | Equate sum of the two relevant terms to 48 and attempt to solve for \(a\); M1 only for \(2a - 400 = 48\) |
| \(a = 7\) | A1 | Obtain \(a = 7\) only |
# Question 4:
## Part (a):
$(2-5x)^5 = 2^5 + {}^5C_1 2^4(-5x) + {}^5C_2 2^3(-5x)^2 + \ldots$ | **M1** | Attempt at least 2 terms – products of binomial coefficients and correct powers of 2 and $-5x$; allow $\pm 5x$
$32 - 400x$ | **A1** |
$+2000x^2$ | **A1** | Do not allow from $+5x$
## Part (b):
$(1 + 2ax + a^2x^2)(32 - 400x + 2000x^2 + \ldots)$ | **M1\*** | Expand first bracket, multiply by part (a) to obtain the two relevant terms in $x$
$64a - 400 = 48 \Rightarrow a = \ldots$ | **Dep\*M1** | Equate sum of the two relevant terms to 48 and attempt to solve for $a$; M1 only for $2a - 400 = 48$
$a = 7$ | **A1** | Obtain $a = 7$ only
---4
\begin{enumerate}[label=(\alph*)]
\item Find and simplify the first three terms in the expansion of $( 2 - 5 x ) ^ { 5 }$ in ascending powers of $x$.
\item In the expansion of $( 1 + a x ) ^ { 2 } ( 2 - 5 x ) ^ { 5 }$, the coefficient of $x$ is 48 .
Find the value of $a$.
\end{enumerate}
\hfill \mbox{\textit{OCR PURE Q4 [6]}}