Standard +0.3 This is a standard Further Maths binomial expansion question requiring (a) routine application of the generalized binomial theorem with a negative fractional power, (b) multiplying two expansions and collecting terms (straightforward algebra), and (c) numerical substitution. While it involves multiple parts and careful algebraic manipulation, it follows a well-practiced template with no novel insight required, making it slightly easier than average.
11. a. Find the binomial expansion of \(( 4 - x ) ^ { - \frac { 1 } { 2 } }\), up to and including the term in \(x ^ { 2 }\).
Given that the binomial expansion of \(\mathrm { f } ( x ) = \sqrt { \frac { 1 + 2 x } { 4 - x } } , | x | < \frac { 1 } { 4 }\), is
$$\frac { 1 } { 2 } + \frac { 9 } { 16 } x - A x ^ { 2 } + \cdots$$
b. Show that the value of the constant \(A\) is \(\frac { 45 } { 256 }\)
c. By substituting \(x = \frac { 1 } { 4 }\) into the answer for (b) find an approximate for \(\sqrt { 10 }\), giving your answer to 3 decimal places.
## Question 11(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f\left(\frac{1}{4}\right) = \sqrt{\frac{1+2(0.25)}{4-0.25}} = \frac{\sqrt{10}}{5}$ | M1 | Substitutes $x = \frac{1}{4}$ into both sides and attempts to find at least one side. Multiplying by 5 first is acceptable |
| $\frac{\sqrt{10}}{5} = \frac{2579}{4096}$ | A1 | Finds both sides leading to a correct equation |
| $\sqrt{10} = \frac{12695}{4096} = 3.148193 \approx 3.148$ | A1 | States $\sqrt{10} \approx 3.148$ |
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11. a. Find the binomial expansion of $( 4 - x ) ^ { - \frac { 1 } { 2 } }$, up to and including the term in $x ^ { 2 }$.
Given that the binomial expansion of $\mathrm { f } ( x ) = \sqrt { \frac { 1 + 2 x } { 4 - x } } , | x | < \frac { 1 } { 4 }$, is
$$\frac { 1 } { 2 } + \frac { 9 } { 16 } x - A x ^ { 2 } + \cdots$$
b. Show that the value of the constant $A$ is $\frac { 45 } { 256 }$\\
c. By substituting $x = \frac { 1 } { 4 }$ into the answer for (b) find an approximate for $\sqrt { 10 }$, giving your answer to 3 decimal places.\\
\hfill \mbox{\textit{Edexcel PMT Mocks Q11 [10]}}