Moderate -0.3 Part (a) is routine application of binomial expansion formula for the first four terms. Part (b) requires setting up and solving a simple equation relating two coefficients, which is a standard C2 exercise requiring minimal problem-solving beyond the formula application.
3. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 + a x ) ^ { 10 }\), where \(a\) is a non-zero constant. Give each term in its simplest form.
Given that, in this expansion, the coefficient of \(x ^ { 3 }\) is double the coefficient of \(x ^ { 2 }\),
(b) find the value of \(a\).
Evidence from one of these terms is sufficient. Requires correct structure: binomial coefficient and correct power of \(x\)
\(+45a^2x^2,\ +120a^3x^3\)
A1, A1
1st A1: correct \(x^2\) term. 2nd A1: correct \(x^3\) term (must be simplified). \(\binom{10}{2}\) and \(\binom{10}{3}\) or equivalent such as \({}^{10}C_2\) and \({}^{10}C_3\) are acceptable
Part (b)
Answer
Marks
Guidance
Answer/Working
Marks
Guidance
\(120a^3 = 2\times 45a^2\)
M1
Equating coefficient of \(x^3\) to twice coefficient of \(x^2\), or vice versa
\(a = \frac{3}{4}\) or equiv. \(\left(e.g.\ \frac{90}{120},\ 0.75\right)\), ignore \(a=0\) if seen
A1
Beware: \(a=\frac{3}{4}\) following \(120a = 90a\) is A0
## Question 3:
### Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(1+ax)^{10} = 1 + 10ax + \ldots$ (not unsimplified versions) | B1 | |
| $+ \frac{10\times 9}{2}(ax)^2 + \frac{10\times 9\times 8}{6}(ax)^3$ | M1 | Evidence from one of these terms is sufficient. Requires correct structure: binomial coefficient and correct power of $x$ |
| $+45a^2x^2,\ +120a^3x^3$ | A1, A1 | 1st A1: correct $x^2$ term. 2nd A1: correct $x^3$ term (must be simplified). $\binom{10}{2}$ and $\binom{10}{3}$ or equivalent such as ${}^{10}C_2$ and ${}^{10}C_3$ are acceptable |
### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $120a^3 = 2\times 45a^2$ | M1 | Equating coefficient of $x^3$ to twice coefficient of $x^2$, or vice versa |
| $a = \frac{3}{4}$ or equiv. $\left(e.g.\ \frac{90}{120},\ 0.75\right)$, ignore $a=0$ if seen | A1 | Beware: $a=\frac{3}{4}$ following $120a = 90a$ is A0 |
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3. (a) Find the first 4 terms, in ascending powers of $x$, of the binomial expansion of $( 1 + a x ) ^ { 10 }$, where $a$ is a non-zero constant. Give each term in its simplest form.
Given that, in this expansion, the coefficient of $x ^ { 3 }$ is double the coefficient of $x ^ { 2 }$,\\
(b) find the value of $a$.\\
\hfill \mbox{\textit{Edexcel C2 2008 Q3 [6]}}