Moderate -0.8 This is a straightforward application of the binomial theorem requiring students to expand (1+px)^9 to find the first three terms, then solve simple equations to find p and q. The algebra is routine: the coefficient of x gives 9p=36 so p=4, then the coefficient of x² gives q directly. This is easier than average as it's a standard textbook exercise with no problem-solving insight required.
2. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of
$$( 1 + p x ) ^ { 9 }$$
where \(p\) is a constant.
These first 3 terms are \(1,36 x\) and \(q x ^ { 2 }\), where \(q\) is a constant.
(b) Find the value of \(p\) and the value of \(q\).
(a) $(1+px)^9 = 1 + 9px + \binom{9}{2}(px)^2$ | B1 B1 | (2 marks)
(b) $9p = 36$, so $p=4$ | M1 A1
$q = \frac{9 \times 8}{2}p^2$ or $36p^2$ or $36p$ if that follows from their (a) | M1
So $q = 576$ | A1 cao | (4 marks)
**Guidance:**
- (a) 2nd B1 for $\binom{9}{2}(px)^2$ or better. Condone "." not "+".
- (b) 1st M1 for a linear equation for $p$. 2nd M1 for either printed expression, follow through their $p$.
**Note:** $1 + 9px + 36p^2$ leading to $p = 4, q = 144$ scores B1 B0 M1 A1 M1 A0 i.e 4/6
**Total: 6 marks**
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2. (a) Find the first 3 terms, in ascending powers of $x$, of the binomial expansion of
$$( 1 + p x ) ^ { 9 }$$
where $p$ is a constant.
These first 3 terms are $1,36 x$ and $q x ^ { 2 }$, where $q$ is a constant.\\
(b) Find the value of $p$ and the value of $q$.\\
\hfill \mbox{\textit{Edexcel C2 2006 Q2 [6]}}