(a) Find the binomial expansion of $(4 + 5x)^{\frac{1}{2}}$, $ |x| < \frac{4}{5} $ give each coefficient in its simplest form. (b) Find the exact value of $(4 + 5x)^{\frac{1}{2}}$ when $x = \frac{1}{10}$. Give your answer in the form $k\sqrt{2}$, where $k$ is a constant to be determined. (c) Substitute $x = \frac{1}{10}$ into your binomial expansion from part (a) and hence find an approximate value for $\sqrt{2}$. Give your answer in the form $\frac{p}{q}$ where $p$ and $q$ are integers.

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(a) Find the binomial expansion of $(4 + 5x)^{\frac{1}{2}}$, $ |x| < \frac{4}{5} $ give each coefficient in its simplest form - Edexcel - A-Level Maths Pure - Question 2 - 2015 - Paper 4

Question 2

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(a) Find the binomial expansion of $(4 + 5x)^{\frac{1}{2}}$, $ |x| < \frac{4}{5} $ give each coefficient in its simplest form. (b) Find the exact value of $(4... show full transcript

Worked Solution & Example Answer:(a) Find the binomial expansion of $(4 + 5x)^{\frac{1}{2}}$, $ |x| < \frac{4}{5} $ give each coefficient in its simplest form - Edexcel - A-Level Maths Pure - Question 2 - 2015 - Paper 4

Step 1

Find the binomial expansion of $(4 + 5x)^{\frac{1}{2}}$

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Answer

To find the binomial expansion of $(4 + 5x)^{\frac{1}{2}}$ , we can use the binomial theorem, which states:

$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$

In this case, let:

$a = 4$
$b = 5x$
$n = \frac{1}{2}$

Thus, we get:

$(4 + 5x)^{\frac{1}{2}} = \sum_{k=0}^{n} \binom{\frac{1}{2}}{k} 4^{\frac{1}{2}-k} (5x)^k$

Calculating the first few terms:

For $k=0$ : (inom{\frac{1}{2}}{0} 4^{\frac{1}{2}} (5x)^0 = 2)
For $k=1$ : (inom{\frac{1}{2}}{1} 4^{-\frac{1}{2}} (5x)^1 = \frac{1}{2} \cdot \frac{1}{2} \cdot 5x = \frac{5}{4}x)
For $k=2$ : (inom{\frac{1}{2}}{2} 4^{-\frac{3}{2}} (5x)^2 = \frac{1}{2} \cdot -\frac{1}{2} \cdot \frac{1}{12} \cdot (25x^2) = -\frac{25}{48}x^2)

Thus, the expansion up to $x^2$ is:

$2 + \frac{5}{4} x - \frac{25}{48} x^2$ .

Step 2

Find the exact value of $(4 + 5x)^{\frac{1}{2}}$ when $x = \frac{1}{10}$

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Answer

Substituting $x = \frac{1}{10}$ into the expression derived in part (a):

$(4 + 5 \cdot \frac{1}{10})^{\frac{1}{2}} = (4 + \frac{5}{2})^{\frac{1}{2}} = (\frac{13}{2})^{\frac{1}{2}} = \frac{\sqrt{13}}{\sqrt{2}} = k\sqrt{2}$

We can express $\sqrt{13} = k\sqrt{2}$ where $k = \frac{\sqrt{13}}{\sqrt{2}}$ .

Step 3

Substitute $x = \frac{1}{10}$ into your binomial expansion and find an approximate value for $\sqrt{2}$

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Answer

Substituting $x = \frac{1}{10}$ into the binomial expansion:

$2 + \frac{5}{4} \cdot \frac{1}{10} - \frac{25}{48} \cdot \left(\frac{1}{10}\right)^2$

Calculating each part:

First term: $2$
Second term: $\frac{5}{4} \cdot \frac{1}{10} = \frac{5}{40} = \frac{1}{8}$
Third term: $- \frac{25}{48} \cdot \frac{1}{100} = -\frac{25}{4800}$

Adding these together: $\text{Approximation } = 2 + \frac{1}{8} - \frac{25}{4800}$

Converting $2$ to a common denominator gives: $\frac{9600}{4800} + \frac{600}{4800} - \frac{25}{4800} = \frac{9600 + 600 - 25}{4800} = \frac{10175}{4800}$

Thus, the approximate value of $\sqrt{2}$ can be expressed as: $\frac{10175}{4800}$

(a) Find the binomial expansion of $(4 + 5x)^{\frac{1}{2}}$, $ |x| < \frac{4}{5} $ give each coefficient in its simplest form - Edexcel - A-Level Maths Pure - Question 2 - 2015 - Paper 4

Worked Solution & Example Answer:(a) Find the binomial expansion of $(4 + 5x)^{\frac{1}{2}}$, $ |x| < \frac{4}{5} $ give each coefficient in its simplest form - Edexcel - A-Level Maths Pure - Question 2 - 2015 - Paper 4

Find the binomial expansion of $(4 + 5x)^{\frac{1}{2}}$

Find the exact value of $(4 + 5x)^{\frac{1}{2}}$ when $x = \frac{1}{10}$

Substitute $x = \frac{1}{10}$ into your binomial expansion and find an approximate value for $\sqrt{2}$

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