Moderate -0.8 This is a straightforward binomial expansion question requiring recall of the formula and basic algebraic manipulation. Part (a) is routine application of (a+bx)^n expansion, and part (b) involves setting up and solving a simple linear equation from the coefficient relationship. The question is easier than average as it requires no problem-solving insight, just methodical application of a standard technique.
2. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of
$$( 3 + b x ) ^ { 5 }$$
where \(b\) is a non-zero constant. Give each term in its simplest form.
Given that, in this expansion, the coefficient of \(x ^ { 2 }\) is twice the coefficient of \(x\),
(b) find the value of \(b\).
Establishes equation from coefficients. Condone 2 on wrong side. An equation in \(b\) alone is required
\(b = \frac{810}{270} \Rightarrow b = 3\)
A1
\(b=3\) (ignore \(b=0\) if seen). Answer of 3 from no working scores M1A0
# Question 2:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| 243 as constant term | B1 | Just writing $(3)^5$ is B0 |
| $405bx$ | B1 | Term must be simplified to $405bx$; note $405+bx$ is B0 |
| $(3+bx)^5 = (3)^5 + {}^5C_1(3)^4(bx) + {}^5C_2(3)^3(bx)^2 + \ldots$ | M1 | For either the $x$ term or $x^2$ term; requires correct binomial coefficient in any form with correct power of $x$ |
| $= 243 + 405bx + 270b^2x^2 + \ldots$ | A1 | For $270b^2x^2$ or $270(bx)^2$. If $270bx^2$ follows $270(bx)^2$, isw and allow A1 |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2(\text{coeff } x) = \text{coeff } x^2 \Rightarrow 2(405b) = 270b^2$ | M1 | Establishes equation from coefficients. Condone 2 on wrong side. An equation in $b$ alone is required |
| $b = \frac{810}{270} \Rightarrow b = 3$ | A1 | $b=3$ (ignore $b=0$ if seen). Answer of 3 from no working scores M1A0 |
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2. (a) Find the first 3 terms, in ascending powers of $x$, of the binomial expansion of
$$( 3 + b x ) ^ { 5 }$$
where $b$ is a non-zero constant. Give each term in its simplest form.
Given that, in this expansion, the coefficient of $x ^ { 2 }$ is twice the coefficient of $x$,\\
(b) find the value of $b$.\\
\hfill \mbox{\textit{Edexcel C2 2011 Q2 [6]}}