4. (a) Find the first three terms, in ascending powers of x, of the binomial expansion of $$ \frac{1}{\sqrt{4 - x}} $$ giving each coefficient in its simplest form. The expansion can be used to find an approximation to \sqrt{2}. Possible values of x that could be substituted into this expansion are: (i) x = -14 because $$ \frac{1}{\sqrt{4 - x}} = \frac{1}{\sqrt{18}} = \frac{1}{6}\sqrt{2} $$ (ii) x = 2 because $$ \frac{1}{\sqrt{4 - x}} = \frac{1}{\sqrt{2}} = \frac{1}{2}\sqrt{2} $$ (iii) x = \frac{-1}{2} because $$ \frac{1}{\sqrt{4 - x}} = \frac{1}{\sqrt{\frac{9}{2}}} = \frac{\sqrt{2}}{3} $$ (b) Without evaluating your expansion, (i) state, giving a reason, which of the three values of x should not be used (ii) state, giving a reason, which of the three values of x would lead to the most accurate approximation to \sqrt{2}

A-Level Edexcel Maths Pure Trigonometric Equations: 4. (a) Find the first three term

4. (a) Find the first three terms, in ascending powers of x, of the binomial expansion of $$ \frac{1}{\sqrt{4 - x}} $$ giving each coefficient in its simplest form - Edexcel - A-Level Maths Pure - Question 6 - 2019 - Paper 1

Question 6

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4. (a) Find the first three terms, in ascending powers of x, of the binomial expansion of $$ \frac{1}{\sqrt{4 - x}} $$ giving each coefficient in its simplest form.... show full transcript

Worked Solution & Example Answer:4. (a) Find the first three terms, in ascending powers of x, of the binomial expansion of $$ \frac{1}{\sqrt{4 - x}} $$ giving each coefficient in its simplest form - Edexcel - A-Level Maths Pure - Question 6 - 2019 - Paper 1

Step 1

Find the first three terms, in ascending powers of x, of the binomial expansion of \frac{1}{\sqrt{4 - x}}

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Answer

To find the first three terms of the binomial expansion for \frac{1}{\sqrt{4 - x}}$, we can rewrite it as:

\frac{1}{\sqrt{4(1 - \frac{x}{4})}} = \frac{1}{2}\cdot (1 - \frac{x}{4})^{-\frac{1}{2}}.

Using the binomial expansion formula:

$(1 + u)^n = 1 + nu + \frac{n(n-1)}{2}u^2 + \cdots$ where (n = -\frac{1}{2}) and (u = -\frac{x}{4}), we find:

First term: ( \frac{1}{2} )
Second term: ( \frac{1}{2} \cdot \left(-\frac{1}{2}\right)(-\frac{x}{4}) = \frac{x}{16} )
Third term: ( \frac{1}{2} \cdot \left(-\frac{1}{2}\right)\left(-\frac{3}{2}\right) \frac{x^2}{16} = \frac{3x^2}{512} )

Thus, the first three terms are:

\frac{1}{2} + \frac{x}{16} + \frac{3x^2}{512}.

Step 2

Without evaluating your expansion, state, giving a reason, which of the three values of x should not be used

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Answer

The value x = -14 should not be used because substituting it into the expression leads to:

\sqrt{4 - (-14)} = \sqrt{18},

which is outside the radius of convergence of the binomial expansion for \left| \frac{x}{4} \right| < 1. Because ( |-14| > 4 ), the expansion is invalid.

Step 3

Without evaluating your expansion, state, giving a reason, which of the three values of x would lead to the most accurate approximation to \sqrt{2}

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Answer

The value x = 2 would lead to the most accurate approximation because:

\frac{1}{\sqrt{4 - 2}} = \frac{1}{\sqrt{2}},

which directly computes the value of \sqrt{2} and does not require further expansion or approximation.

4. (a) Find the first three terms, in ascending powers of x, of the binomial expansion of $$ \frac{1}{\sqrt{4 - x}} $$ giving each coefficient in its simplest form - Edexcel - A-Level Maths Pure - Question 6 - 2019 - Paper 1

Worked Solution & Example Answer:4. (a) Find the first three terms, in ascending powers of x, of the binomial expansion of $$ \frac{1}{\sqrt{4 - x}} $$ giving each coefficient in its simplest form - Edexcel - A-Level Maths Pure - Question 6 - 2019 - Paper 1

Find the first three terms, in ascending powers of x, of the binomial expansion of \frac{1}{\sqrt{4 - x}}

Without evaluating your expansion, state, giving a reason, which of the three values of x should not be used

Without evaluating your expansion, state, giving a reason, which of the three values of x would lead to the most accurate approximation to \sqrt{2}

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