Standard +0.3 Part (a) is a straightforward binomial expansion requiring routine application of the formula with fractional coefficients. Part (b) adds a mild twist by requiring students to expand a quadratic factor and collect terms, but this is a standard 'hence' extension that follows directly from part (a) with careful algebraic manipulation. The question tests technique rather than insight.
6. (a) Find, in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), the binomial expansion of
$$\left( 1 + \frac { 1 } { 4 } x \right) ^ { 12 }$$
giving each term in its simplest form.
(b) Hence find the coefficient of \(x\) in the expansion of
$$\left( 3 + \frac { 2 } { x } \right) ^ { 2 } \left( 1 + \frac { 1 } { 4 } x \right) ^ { 12 }$$
Attempt at binomial expansion to get third and/or fourth term; correct binomial coefficient combined with correct power of \(x\); accept \(^{12}C_2\), \(^{12}C_3\), \(\binom{12}{2}\), \(\binom{12}{3}\), 66, 220, \(\frac{12\times11}{2!}\), \(\frac{12\times11\times10}{3!}\)
B1 for \(1+3x\) only (accept \(1+3x^1\), \(1+\frac{3x^1}{1}\), \(1,3x\); NOT \(1+\binom{12}{1}\frac{1}{4}x^1\)); A1 for \(+\frac{33}{8}x^2\) (accept \(4.125x^2\)); A1 for \(+\frac{55}{16}x^3\) (accept \(3.4375x^3\)); fractions must be simplified
Or unsimplified equivalent; two \(\frac{6}{x}\) terms do not need combining; no additional terms
At least two of: \(9\times3x\), \(\frac{12}{x}\times\frac{33}{8}x^2\), \(\frac{4}{x^2}\times\frac{55}{16}x^3\)
M1
Attempt to multiply expansion of \(\left(3+\frac{2}{x}\right)^2\) by answer to (a) to correctly find at least two \(x\) terms; expansions must be of form \(\left(a+\frac{b}{x}+\frac{c}{x^2}\right)(P+Qx+Rx^2+Sx^3)\) where \(a,c\neq 0\)
Attempts to add all necessary \(x\) terms to get single term in \(x\); from \(aQx + bRx + cSx = \ldots x\) with all non-zero
\(= \frac{361}{4}\)
A1
Or exact equivalent including 90.25; do not accept \(\frac{361}{4}x\). Note: expansion of \(\left(9+\frac{4}{x^2}\right)\) in part (b) scores maximum B0M1M1A0
## Question 6:
### Part (a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\left(1+\frac{1}{4}x\right)^{12} = 1 + \binom{12}{1}\left(\frac{1}{4}x\right) + \binom{12}{2}\left(\frac{1}{4}x\right)^2 + \binom{12}{3}\left(\frac{1}{4}x\right)^3 + \ldots$ | M1 | Attempt at binomial expansion to get third and/or fourth term; correct binomial coefficient combined with correct power of $x$; accept $^{12}C_2$, $^{12}C_3$, $\binom{12}{2}$, $\binom{12}{3}$, 66, 220, $\frac{12\times11}{2!}$, $\frac{12\times11\times10}{3!}$ |
| $= 1 + 3x + \frac{33}{8}x^2 + \frac{55}{16}x^3 + \ldots$ | B1A1A1 | B1 for $1+3x$ only (accept $1+3x^1$, $1+\frac{3x^1}{1}$, $1,3x$; NOT $1+\binom{12}{1}\frac{1}{4}x^1$); A1 for $+\frac{33}{8}x^2$ (accept $4.125x^2$); A1 for $+\frac{55}{16}x^3$ (accept $3.4375x^3$); fractions must be simplified |
### Part (b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\left(3+\frac{2}{x}\right)^2 = 9 + \frac{12}{x} + \frac{4}{x^2}$ | B1 | Or unsimplified equivalent; two $\frac{6}{x}$ terms do not need combining; no additional terms |
| At least two of: $9\times3x$, $\frac{12}{x}\times\frac{33}{8}x^2$, $\frac{4}{x^2}\times\frac{55}{16}x^3$ | M1 | Attempt to multiply expansion of $\left(3+\frac{2}{x}\right)^2$ by answer to (a) to correctly find at least two $x$ terms; expansions must be of form $\left(a+\frac{b}{x}+\frac{c}{x^2}\right)(P+Qx+Rx^2+Sx^3)$ where $a,c\neq 0$ |
| $9\times3x + \frac{12}{x}\times\frac{33}{8}x^2 + \frac{4}{x^2}\times\frac{55}{16}x^3 = \ldots x$ | dM1 | Attempts to add all necessary $x$ terms to get single term in $x$; from $aQx + bRx + cSx = \ldots x$ with all non-zero |
| $= \frac{361}{4}$ | A1 | Or exact equivalent including 90.25; do not accept $\frac{361}{4}x$. Note: expansion of $\left(9+\frac{4}{x^2}\right)$ in part (b) scores maximum B0M1M1A0 |
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6. (a) Find, in ascending powers of $x$, up to and including the term in $x ^ { 3 }$, the binomial expansion of
$$\left( 1 + \frac { 1 } { 4 } x \right) ^ { 12 }$$
giving each term in its simplest form.\\
(b) Hence find the coefficient of $x$ in the expansion of
$$\left( 3 + \frac { 2 } { x } \right) ^ { 2 } \left( 1 + \frac { 1 } { 4 } x \right) ^ { 12 }$$
\hfill \mbox{\textit{Edexcel C12 2019 Q6 [8]}}