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AS Pure Mathematics · Edexcel 8MA0 · free A-level maths worksheet

Binomial expansion

(a + bx)^n for positive integer n, nCr, finding coefficients and unknowns, approximations.

12 original questions, 50 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

Expand (2 + x)4 fully, simplifying each term.

3 marks
Show answer
16 + 32x + 24x2 + 8x3 + x4
2

Find the first four terms, in ascending powers of x, of the binomial expansion of (1 − 3x)6. Give each term in its simplest form.

3 marks
Show answer
1 - 18x + 135x2 - 540x3
3

Find the coefficient of x3 in the binomial expansion of (3 + 2x)7

2 marks
Show answer
22680
4

In the binomial expansion of (1 + kx)9, where k is a positive constant, the coefficient of x2 is 252

Find the exact value of k.

3 marks
Show answer
k = √7
5
  1. Find the first four terms, in ascending powers of x, of the binomial expansion of (2 − x4)8. Give each term in its simplest form.
  2. Use your expansion, with a suitable value of x, to find an estimate for 1.9758. Give your answer to 3 decimal places.
6 marks
Show answer
a) 256 - 256x + 112x2 - 28x3 b) 231.492 (x = 0.1)
6

In the binomial expansion of (a + 3x)6, where a is a non-zero constant, the coefficient of x is equal to the coefficient of x2

Find the value of a.

4 marks
Show answer
a = 15/2
7

In the binomial expansion of (1 + 2x)n, where n is a positive integer, the coefficient of x2 is 180

Find the value of n.

4 marks
Show answer
n = 10
8
  1. Find the first four terms, in ascending powers of x, of the binomial expansion of (2 − x)6
  2. Hence find the coefficient of x3 in the expansion of (1 + 4x)(2 − x)6
5 marks
Show answer
a) 64 - 192x + 240x2 - 160x3 b) 800
9

Find the term independent of x in the binomial expansion of (2x − 3x)6

4 marks
Show answer
-4320
10

The first three terms, in ascending powers of x, of the binomial expansion of (1 + px)n are

1 + 12x + 63x2

where n is a positive integer and p is a constant.

Find the value of n and the value of p.

5 marks
Show answer
n = 8, p = 3/2
11

Priya invests £5000 in an account that pays r% interest per year, compounded annually. After 10 years the account holds £5000(1 + 0.01r)10

  1. Use the first three terms of the binomial expansion of (1 + y)10, in ascending powers of y, to estimate the amount in the account after 10 years when r = 2
  2. Explain why your answer to part (a) is less than the true amount.
  3. Find the percentage error in your estimate. Give your answer to 2 significant figures.
5 marks
Show answer
a) £6090 b) The terms left out (120y3 and higher) are all positive, so the estimate is too small c) 0.082%
12
  1. Find the first three terms, in ascending powers of x, of the binomial expansion of (2 + kx)5, where k is a non-zero constant. Give each term in terms of k.
  2. Given that the coefficient of x2 in the expansion of (3 − x)(2 + kx)5 is 160, find the possible values of k.
6 marks
Show answer
a) 32 + 80kx + 80k2x2 b) k = 1 or k = -2/3

Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.

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