A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 8: The binomial expansion · Lesson 3

Binomial estimation

70 marks

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Quick questions to start.

A1
Last lesson
Find the coefficient of $x^{2}$ in $(1 - 4 x)^{7}$ .
[2 marks]
Answer
A2
Last lesson
Find the constant term in $(x - \frac{3}{x})^{4}$ .
[2 marks]
Answer
A3
Needed today
Find $x$ when $1 - 3 x = 0.988$ .
[2 marks]
Answer
A4
Needed today
Find the percentage error when $12.4$ is used for $12.5$ .
[2 marks]
Answer

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

Find the first four terms of $(1 - \frac{1}{4} x)^{8}$ . Use them to estimate $0.9 7^{8}$ to $4$ d.p.

Hint Choose

x

so that

1 - \frac{1}{4} x = 0.97

B1
Find the first four terms of $(1 + \frac{1}{5} x)^{6}$ . Use them to estimate $1.0 4^{6}$ to $4$ d.p.
[4 marks]
Answer

Example 2

Find the first three terms of $(2 - x)^{7}$ . Use them to estimate $1.9 9^{7}$ to $2$ d.p.

Hint

2 - x = 1.99

gives

x = 0.01

B2
Find the first three terms of $(3 + \frac{1}{2} x)^{6}$ . Use them to estimate $3.0 1^{6}$ to $2$ d.p.
[3 marks]
Answer

Example 3

$x$ is small, so terms in $x^{3}$ and higher can be ignored. Show that $(1 + 2 x) (1 - x)^{6} \approx 1 - 4 x + 3 x^{2}$ .

Hint Expand

(1 - x)^{6}

only up to

x^{2}

, then multiply out.

B3
$x$ is small, so terms in $x^{3}$ and higher can be ignored. $(3 - x) (1 + 2 x)^{5} \approx a + b x + c x^{2}$ . Find $a$ , $b$ and $c$ .
[3 marks]
Answer

Example 4

Use the first three terms of $(1 - 2 x)^{9}$ to estimate $0.99 6^{9}$ . Find the percentage error, to $2$ s.f.

Hint Percentage error

= \frac{estimate - true}{true} \times 100

B4
Use the first three terms of $(1 + 3 x)^{8}$ to estimate $1.00 3^{8}$ . Find the percentage error, to $2$ s.f.
[4 marks]
Answer

A-Level

C1
$x$ is small, so terms in $x^{3}$ and higher can be ignored. $(1 - x) (1 + k x)^{6} \approx 1 + 11 x + a x^{2}$ . Find the constants $k$ and $a$ .
[4 marks]
Answer

C2
Without a calculator, use the expansion of $(1 + x)^{5}$ to find the exact value of $1. 1^{5}$ .
[3 marks]
Answer
C3
The first three terms of $(1 + \frac{1}{2} x)^{10}$ are used to estimate $1.0 5^{10}$ . Work out the estimate. Is it too large or too small? Give a reason.
[4 marks]

Edexcel exam style

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

Colour one circle on each line.

	Not yet	Nearly	Yes
I can estimate a power from the first terms of an expansion.
I can approximate a product, ignoring x cubed and higher.
I can find a percentage error and say why x must be small.
I can say if an estimate is too large or too small.

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