7. (a) Find the first four terms, in ascending powers of x, of the binomial expansion of \[\sqrt{4 - 9x}\] writing each term in simplest form. A student uses this expansion with $ x = \frac{1}{9} $ to find an approximation for $ \sqrt{3} $ Using the answer to part (a) and without doing any calculations, (b) state whether this approximation will be an overestimate or an underestimate of $ \sqrt{3} $ giving a brief reason for your answer.

A-Level Edexcel Maths Pure Modelling with Trigonometric Functions: 7. (a) Find the first four terms

7. (a) Find the first four terms, in ascending powers of x, of the binomial expansion of \[\sqrt{4 - 9x}\] writing each term in simplest form - Edexcel - A-Level Maths Pure - Question 9 - 2022 - Paper 2

Question 9

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7. (a) Find the first four terms, in ascending powers of x, of the binomial expansion of \[\sqrt{4 - 9x}\] writing each term in simplest form. A student uses this e... show full transcript

Worked Solution & Example Answer:7. (a) Find the first four terms, in ascending powers of x, of the binomial expansion of \[\sqrt{4 - 9x}\] writing each term in simplest form - Edexcel - A-Level Maths Pure - Question 9 - 2022 - Paper 2

Step 1

Find the first four terms, in ascending powers of x, of the binomial expansion of $\sqrt{4 - 9x}$

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Answer

To expand (\sqrt{4 - 9x}), we can rewrite it as (\sqrt{4(1 - \frac{9x}{4})} = 2\sqrt{1 - \frac{9x}{4}}).

Using the binomial expansion formula ((1 + u)^n \approx 1 + nu + \frac{n(n-1)}{2!}u^2 + \frac{n(n-1)(n-2)}{3!}u^3 + ...), where (u = -\frac{9x}{4}) and (n = \frac{1}{2}), we can write:

The first term: [ 2(1) = 2 ]
The second term: [ 2 \cdot \frac{1}{2}\left(-\frac{9x}{4}\right) = -\frac{9x}{4} ]
The third term: [ 2 \cdot \frac{1}{2}\cdot \frac{-\frac{1}{2}(-\frac{9}{4})}{2!}\left(-\frac{9x}{4}\right)^2 = \frac{81x^2}{32} ]
The fourth term: [ 2 \cdot \frac{1}{2}\cdot \frac{\frac{-1}{2}(\frac{-\frac{9}{4}}{2})}{3!}\left(-\frac{9x}{4}\right)^3 = -\frac{729x^3}{512} ]

Therefore, the first four terms in ascending powers of ( x ) are:
[ 2 - \frac{9x}{4} + \frac{81x^2}{32} - \frac{729x^3}{512} ]

Step 2

state whether this approximation will be an overestimate or an underestimate of $\sqrt{3}$ giving a brief reason for your answer.

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Answer

The approximation will be an overestimate of (\sqrt{3}). This is because in the binomial expansion, all terms after the first one are negative when evaluated around (x = 0) (as shown in part (a)). Hence as more terms from the series are added, the value decreases from the initial estimate, indicating that the approximation at (x = \frac{1}{9}) would lead to an overestimate.

7. (a) Find the first four terms, in ascending powers of x, of the binomial expansion of \[\sqrt{4 - 9x}\] writing each term in simplest form - Edexcel - A-Level Maths Pure - Question 9 - 2022 - Paper 2

Worked Solution & Example Answer:7. (a) Find the first four terms, in ascending powers of x, of the binomial expansion of \[\sqrt{4 - 9x}\] writing each term in simplest form - Edexcel - A-Level Maths Pure - Question 9 - 2022 - Paper 2

Find the first four terms, in ascending powers of x, of the binomial expansion of \(\sqrt{4 - 9x}\)

state whether this approximation will be an overestimate or an underestimate of \(\sqrt{3}\) giving a brief reason for your answer.

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