Standard +0.3 This is a standard C3/C4 binomial expansion question requiring routine application of (1+x)^n for fractional n, followed by straightforward algebraic manipulation and substitution. Part (a) involves expanding two binomial expressions and multiplying them—a textbook exercise. Parts (b) and (c) are simple arithmetic substitutions requiring no problem-solving insight. Slightly above average difficulty only due to the multi-part nature and careful algebraic manipulation needed.
6. (a) Use binomial expansions to show that, for \(| x | < \frac { 1 } { 2 }\)
(b) Find the exact value of \(\sqrt { \frac { 1 + 2 x } { 1 - x } }\) when \(x = \frac { 1 } { 10 }\)
Give your answer in the form \(k \sqrt { 3 }\), where \(k\) is a constant to be determined.
(c) Substitute \(x = \frac { 1 } { 10 }\) into the expansion given in part (a) and hence find an approximate value for \(\sqrt { 3 }\)
Give your answer in the form \(\frac { a } { b }\) where \(a\) and \(b\) are integers.
$$\sqrt { \frac { 1 + 2 x } { 1 - x } } \approx 1 + \frac { 3 } { 2 } x + \frac { 3 } { 8 } x ^ { 2 }$$
M1 for binomial expansion with index \(\frac{1}{2}\) or \(-\frac{1}{2}\); A1 correct unsimplified form for one expression
Both expansions simplified correctly: \((1+x-\frac{1}{2}x^2+\ldots)\times(1+\frac{1}{2}x+\frac{3}{8}x^2+\ldots)\)
A1
Correct simplified form for both expressions
Multiplying terms to find six required terms
M1
Allow terms in \(x^3\) and greater; follow through on their expansions
\(= 1+\frac{3}{2}x+\frac{3}{8}x^2\)
A1*
Correct solution only (6)
Part (b):
Answer
Marks
Guidance
\(\sqrt{\frac{1+\frac{2}{10}}{1-\frac{1}{10}}} = \frac{2}{3}\sqrt{3}\) or statement \(k=\frac{2}{3}\)
B1
(1)
Part (c):
Answer
Marks
Guidance
Sub \(x=\frac{1}{10}\) into both sides of (a) \(\Rightarrow \frac{2}{3}\sqrt{3} = 1+\frac{3}{2}\left(\frac{1}{10}\right)+\frac{3}{8}\left(\frac{1}{10}\right)^2\)
M1
Do not allow \(k=1\)
\(\sqrt{3} \approx \frac{2769}{1600}\) or exact equivalent
A1
Condone \(\sqrt{3}\approx\frac{1600}{923}\) which follows from (b) \(=\frac{2}{\sqrt{3}}\) (2)
# Question 6:
## Part (a):
| $\sqrt{\frac{1+2x}{1-x}} = (1+2x)^{\frac{1}{2}}(1-x)^{-\frac{1}{2}}$ | B1 | May be implied by working; must be form that can lead to answer |
| $=\left(1+\frac{1}{2}(2x)+\frac{(\frac{1}{2})(-\frac{1}{2})}{2!}(2x)^2+\ldots\right)\times\left(1+\left(-\frac{1}{2}\right)(-x)+\frac{(-\frac{1}{2})(-\frac{3}{2})}{2!}(-x)^2+\ldots\right)$ | M1 A1 | M1 for binomial expansion with index $\frac{1}{2}$ or $-\frac{1}{2}$; A1 correct unsimplified form for one expression |
| Both expansions simplified correctly: $(1+x-\frac{1}{2}x^2+\ldots)\times(1+\frac{1}{2}x+\frac{3}{8}x^2+\ldots)$ | A1 | Correct simplified form for both expressions |
| Multiplying terms to find six required terms | M1 | Allow terms in $x^3$ and greater; follow through on their expansions |
| $= 1+\frac{3}{2}x+\frac{3}{8}x^2$ | A1* | Correct solution only **(6)** |
## Part (b):
| $\sqrt{\frac{1+\frac{2}{10}}{1-\frac{1}{10}}} = \frac{2}{3}\sqrt{3}$ or statement $k=\frac{2}{3}$ | B1 | **(1)** |
## Part (c):
| Sub $x=\frac{1}{10}$ into both sides of (a) $\Rightarrow \frac{2}{3}\sqrt{3} = 1+\frac{3}{2}\left(\frac{1}{10}\right)+\frac{3}{8}\left(\frac{1}{10}\right)^2$ | M1 | Do not allow $k=1$ |
| $\sqrt{3} \approx \frac{2769}{1600}$ or exact equivalent | A1 | Condone $\sqrt{3}\approx\frac{1600}{923}$ which follows from (b) $=\frac{2}{\sqrt{3}}$ **(2)** |
6. (a) Use binomial expansions to show that, for $| x | < \frac { 1 } { 2 }$\\
(b) Find the exact value of $\sqrt { \frac { 1 + 2 x } { 1 - x } }$ when $x = \frac { 1 } { 10 }$
Give your answer in the form $k \sqrt { 3 }$, where $k$ is a constant to be determined.\\
(c) Substitute $x = \frac { 1 } { 10 }$ into the expansion given in part (a) and hence find an approximate value for $\sqrt { 3 }$
Give your answer in the form $\frac { a } { b }$ where $a$ and $b$ are integers.
$$\sqrt { \frac { 1 + 2 x } { 1 - x } } \approx 1 + \frac { 3 } { 2 } x + \frac { 3 } { 8 } x ^ { 2 }$$
\hfill \mbox{\textit{Edexcel C34 2018 Q6 [9]}}