6 (a) Find the first three terms, in ascending powers of x, of the binomial expansion of $$rac{1}{rac{1}{4} + x}$$ 6 (b) Hence, find the first three terms of the binomial expansion of $$rac{1}{rac{1}{4} - x^3}$$ 6 (c) Using your answer to part (b), find an approximation for $$\int_0^1 \frac{1}{\frac{1}{4} - x^3} \,dx$$, giving your answer to seven decimal places. 6 (d) (i) Edward, a student, decides to use this method to find a more accurate value for the integral by increasing the number of terms of the binomial expansion used. Explain clearly whether Edward's approximation will be an overestimate, an underestimate, or if it is impossible to tell. 6 (d) (ii) Edward goes on to use the expansion from part (b) to find an approximation for $$\int_0^{\frac{1}{2}} \frac{1}{\frac{1}{4} - x^3} \,dx$$ Explain why Edward's approximation is invalid.

A-Level AQA Maths Pure Integration: 6 (a) Find the first three terms

6 (a) Find the first three terms, in ascending powers of x, of the binomial expansion of $$rac{1}{rac{1}{4} + x}$$ 6 (b) Hence, find the first three terms of the binomial expansion of $$rac{1}{rac{1}{4} - x^3}$$ 6 (c) Using your answer to part (b), find an approximation for $$\int_0^1 \frac{1}{\frac{1}{4} - x^3} \,dx$$, giving your answer to seven decimal places - AQA - A-Level Maths Pure - Question 6 - 2018 - Paper 1

Question 6

$6-(a)-Find-the-first-three-terms,-in-ascending-powers-of-x,-of-the-binomial-expansion-of--$$-rac{1}{-rac{1}{4}-+-x}$$--6-(b)-Hence,-find-the-first-three-terms-of-the-binomial-expansion-of--$$-rac{1}{-rac{1}{4}---x^3}$$--6-(c)-Using-your-answer-to-part-(b),-find-an-approximation-for--$$\int_0^1-\frac{1}{\frac{1}{4}---x^3}-\,dx$$,-giving-your-answer-to-seven-decimal-places-AQA-A-Level Maths Pure-Question 6-2018-Paper 1.png$

6 (a) Find the first three terms, in ascending powers of x, of the binomial expansion of $$rac{1}{rac{1}{4} + x}$$ 6 (b) Hence, find the first three terms of the... show full transcript

Worked Solution & Example Answer:6 (a) Find the first three terms, in ascending powers of x, of the binomial expansion of $$rac{1}{rac{1}{4} + x}$$ 6 (b) Hence, find the first three terms of the binomial expansion of $$rac{1}{rac{1}{4} - x^3}$$ 6 (c) Using your answer to part (b), find an approximation for $$\int_0^1 \frac{1}{\frac{1}{4} - x^3} \,dx$$, giving your answer to seven decimal places - AQA - A-Level Maths Pure - Question 6 - 2018 - Paper 1

Step 1

Find the first three terms, in ascending powers of x, of the binomial expansion of $rac{1}{rac{1}{4} + x}$

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Answer

To find the binomial expansion, rewrite it in a suitable form:

$rac{1}{ rac{1}{4} + x} = 4(1 + 4x)^{-1}$

Using the binomial expansion formula $(1 + kx)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \ldots$ for n = -1:

The first term is: [ 1 ]
The second term is: [ -4x ]
The third term is: [ \frac{(-1)(-2)}{2} \cdot 16x^2 = 8x^2 ]

So the first three terms are: $4 - 16x + 32x^2$ .

Step 2

Hence, find the first three terms of the binomial expansion of $rac{1}{rac{1}{4} - x^3}$

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Answer

Rewrite the expression:

$rac{1}{ rac{1}{4} - x^3} = 4 (1 - 4x^3)^{-1}$

Using the same binomial expansion:

The first term is: [ 1 ]
The second term is: [ 4x^3 ]
The third term, since there are no terms higher than $x^3$ , is: [ 0 ]

Thus, the first three terms are: $4 + 16x^3 + 0$ .

Step 3

Using your answer to part (b), find an approximation for $ \int_0^1 \frac{1}{\frac{1}{4} - x^3} \,dx$

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Answer

Use the expansion from part (b) and integrate:

$\int_0^1 (4 + 16x^3) \,dx$ .

Calculating the terms:

For $\,4 \,dx$ : [ 4x \bigg|_0^1 = 4 ]
For $\,(16x^3) \,dx$ : [ 16 \cdot \frac{x^4}{4} \bigg|_0^1 = 4 ]

Total approximation: $4 + 4 = 8$ .

So, the approximation is 8.

Step 4

Explain clearly whether Edward's approximation will be an overestimate, an underestimate, or if it is impossible to tell.

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Answer

Each term in the expansion is positive. Since increasing the number of terms will provide a more accurate estimate, Edward's approximation will therefore be an underestimate.

Step 5

Explain why Edward's approximation is invalid.

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Answer

The binomial expansion is valid for $|x| < \frac{1}{4}$ . However, when evaluating $\,\int_0^{\frac{1}{2}} \frac{1}{\frac{1}{4} - x^3} \,dx$ , it exceeds this limit since $\frac{1}{2} > \frac{1}{4}$ . Thus, the approximation derived is invalid.

Find the first three terms, in ascending powers of x, of the binomial expansion of $ rac{1}{ rac{1}{4} + x}$

Hence, find the first three terms of the binomial expansion of $ rac{1}{ rac{1}{4} - x^3}$

Using your answer to part (b), find an approximation for $ \int_0^1 \frac{1}{\frac{1}{4} - x^3} \,dx$

Explain clearly whether Edward's approximation will be an overestimate, an underestimate, or if it is impossible to tell.

Explain why Edward's approximation is invalid.

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Find the first three terms, in ascending powers of x, of the binomial expansion of $rac{1}{rac{1}{4} + x}$

Hence, find the first three terms of the binomial expansion of $rac{1}{rac{1}{4} - x^3}$