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

Binomial expansion for any rational n

(1 + x)^n and (a + bx)^n for negative and fractional n, validity, approximations, expansions after partial fractions.

12 original questions, 76 marks. Type your answers below for an instant mark and a score (algebra in any equivalent form, e.g. 2sqrt(3) or 2√3), or print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

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The questions

1
  1. Find the first four terms, in ascending powers of x, of the binomial expansion of (1 + 3x)−2, giving each term in its simplest form.
  2. State the range of values of x for which the expansion is valid.
4 marks
Show answer
a) 1 - 6x + 27x2 - 108x3 b) |x| < 13
2

In the binomial expansion of (1 + kx)−3, where k is a negative constant, the coefficient of x2 is 54

  1. Find the value of k.
  2. Find the coefficient of x3 in the expansion.
4 marks
Show answer
a) k = -3 b) 270
3
  1. Find the first four terms, in ascending powers of x, of the binomial expansion of 1√(4 + x), giving each coefficient as a simplified fraction.
  2. State the range of values of x for which the expansion is valid.
5 marks
Show answer
a) 12 - 116x + 3256x2 - 52048x3 b) |x| < 4
4
  1. Find the first four terms, in ascending powers of x, of the binomial expansion of √(1 + 8x), giving each term in its simplest form.
  2. Explain why substituting x = 18 into this expansion would not give a valid approximation for √2
  3. By substituting x = 132 into your expansion, find an approximation for √5, giving your answer to 4 decimal places.
7 marks
Show answer
a) 1 + 4x - 8x2 + 32x3 b) the expansion is only valid for |x| < 18, and x = 18 is not inside that interval c) 2.2363
5
  1. Show that the binomial expansion of (8 − 3x)13, in ascending powers of x up to and including the term in x2, is

2 − 14x − 132x2

  1. State the range of values of x for which the expansion is valid.
  2. Use the expansion with a suitable value of x to find an estimate for ∛7.97, giving your answer to 6 decimal places.
6 marks
Show answer
a) shown b) |x| < 83 c) 1.997497
6

The binomial expansion of (a + bx)−2, where a and b are positive constants, in ascending powers of x, begins

19 − 427x + cx2 + ...

  1. Find the values of a and b.
  2. Find the value of c.
  3. State the range of values of x for which the expansion is valid.
6 marks
Show answer
a) a = 3, b = 2 b) c = 427 c) |x| < 32
7
  1. Find the binomial expansion of (2 + x)−2, in ascending powers of x up to and including the term in x2, giving each coefficient as a simplified fraction.
  2. Hence find the first three terms, in ascending powers of x, of the expansion of 3 − x(2 + x)2
  3. State the range of values of x for which this expansion is valid.
6 marks
Show answer
a) 14 - 14x + 316x2 b) 34 - x + 1316x2 c) |x| < 2
8
  1. Use binomial expansions to show that

√(1 + x1 − x) ≈ 1 + x + 12x2

  1. By substituting x = 19 into both sides, find an approximation for √5, giving your answer as a fraction in its simplest form.
  2. Find the percentage error in this approximation, giving your answer to 2 significant figures.
8 marks
Show answer
a) shown b) 18181 c) 0.067%
9

The binomial expansion of (1 + ax)n, in ascending powers of x, begins

1 − 6x + 24x2 + ...

where a and n are constants and n is not a positive integer.

  1. Find the value of n and the value of a.
  2. Find the coefficient of x3
  3. State the range of values of x for which the expansion is valid.
6 marks
Show answer
a) n = -3, a = 2 b) -80 c) |x| < 12
10

f(x) = 4 + 5x(1 + 2x)(1 − x)

  1. Express f(x) in partial fractions.
  2. Hence find the series expansion of f(x), in ascending powers of x up to and including the term in x3, giving each coefficient in its simplest form.
  3. State the range of values of x for which the expansion is valid, giving a reason.
9 marks
Show answer
a) 11 + 2x + 31 - x b) 4 + x + 7x2 - 5x3 c) |x| < 12: both expansions must be valid, (1 + 2x)-1 needs |x| < 1/2 and (1 - x)-1 needs |x| < 1
11
  1. Find the binomial expansion of 1√(1 + x2), in ascending powers of x up to and including the term in x4
  2. Hence find an approximation for

∫00.2 1√(1 + x2) dx

giving your answer to 6 significant figures.

  1. Explain why this approximation is likely to be very accurate.
7 marks
Show answer
a) 1 - 12x2 + 38x4 b) 0.198691 c) on 0 ≤ x ≤ 0.2, x2 ≤ 0.04, well inside |x| < 1, so the missing terms are very small
12

In physics, the total energy E of a particle of mass m moving at speed v is

E = mc2(1 − v2c2)−12

where c is the speed of light and 0 ≤ v < c. The kinetic energy is K = E − mc2

  1. Use a binomial expansion to show that, when v is small compared with c,

K ≈ 12mv2 + 38mv4c2

  1. Explain why the expansion is valid for all speeds allowed in the model.
  2. A particle moves at v = 0.1c. Find the percentage error when its kinetic energy is approximated by 12mv2, giving your answer to 2 significant figures.
8 marks
Show answer
a) shown b) the expansion needs |v2/c2| < 1, which is true because 0 ≤ v < c c) 0.75%

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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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