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

Algebraic methods and partial fractions

Simplifying algebraic fractions, algebraic division with a remainder, partial fractions with distinct, repeated and improper denominators.

12 original questions, 72 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

Simplify fully

  1. 3x2 − 12x2 + x − 6
  2. x2 + 5xx2 − 1 ÷ x2 + 3x − 10x2 − x
5 marks
Show answer
a) 3(x + 2)x + 3 b) x2(x + 1)(x - 2)
2

Express 2x2 + 6x − 2x(x − 1)(x + 2) in partial fractions.

4 marks
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1x + 2x - 1 - 1x + 2
3

Given that

2x3 + 3x2 − 5x + 4x + 2 ≡ Ax2 + Bx + C + Dx + 2,   x ≠ −2

find the values of the constants A, B, C and D.

4 marks
Show answer
A = 2, B = -1, C = -3, D = 10
4

f(x) = 2 − 4x + 1 − 16x2 − 2x − 3,   x > 3

  1. Show that f(x) = 2x − 10x − 3
  2. Hence solve the equation f(x) = x − 6
7 marks
Show answer
a) f(x) = 2x - 10x - 3 (shown) b) x = 4 or x = 7
5

Express 3x2 − x + 8(x − 3)(x + 1)2 in partial fractions.

4 marks
Show answer
2x - 3 + 1x + 1 - 3(x + 1)2
6

Express 2x2 + 1(x − 2)(x + 1) in the form A + Bx − 2 + Cx + 1, where A, B and C are constants to be found.

4 marks
Show answer
2 + 3x - 2 - 1x + 1
7

Given that

x4 + 2x3 − 3x2 + x − 5x2 + x − 1 ≡ ax2 + bx + c + dx + ex2 + x − 1

find the values of the constants a, b, c, d and e.

4 marks
Show answer
a = 1, b = 1, c = -3, d = 5, e = -8
8
  1. Express 4x + 3(2x − 1)(x + 2) in partial fractions.
  2. Hence find the exact value of ∫13 4x + 3(2x − 1)(x + 2) dx, giving your answer in the form ln k, where k is a rational number.
6 marks
Show answer
a) 22x - 1 + 1x + 2 b) ln 253
9

f(x) = kx + 6(x − 1)(x + 2),   x > 1, where k is a positive constant.

  1. Express f(x) in partial fractions in terms of k.
  2. Given that ∫210 f(x) dx = 10 ln 3, find the value of k.
7 marks
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a) k + 63(x - 1) + 2k - 63(x + 2) b) k = 6
10

The function f is defined by

f(x) = 3x2 + 13x − 8(x − 1)(x + 3),   x > 1

  1. Express f(x) in the form A + Bx − 1 + Cx + 3, where A, B and C are constants.
  2. Hence find f′(x) and explain why f is a decreasing function.
  3. State the range of f.
9 marks
Show answer
a) f(x) = 3 + 2x - 1 + 5x + 3 b) f'(x) = -2(x - 1)2 - 5(x + 3)2, which is negative for all x > 1 c) f(x) > 3
11

f(x) = 18x2 − 9x + 8(3x − 1)2(x + 2),   x > 13

Given that f(x) can be written in the form

A3x − 1 + B(3x − 1)2 + Cx + 2

where A, B and C are constants,

  1. (i) find the value of B and the value of C,
    (ii) show that A = 0
  2. Hence find the exact value of ∫12 f(x) dx, giving your answer in the form p + q ln r, where p, q and r are rational.
8 marks
Show answer
a) B = 3, C = 2, A = 0 (shown) b) 310 + 2 ln 43
12

Rainwater flows into a garden water butt. The rate of flow, R litres per hour, t hours after the rain starts is modelled by

R = 30t(t + 1)(t + 6),   0 ≤ t ≤ 10

  1. Express 30t(t + 1)(t + 6) in partial fractions.
  2. Hence find, according to the model, the exact volume of water that flows into the water butt in the first 4 hours. Give your answer in the form a ln 5 − b ln 3, where a and b are integers.
  3. Using your answer to part (a), find the greatest rate of flow according to the model, giving your answer to 3 significant figures.
10 marks
Show answer
a) 36t + 6 - 6t + 1 b) 30 ln 5 - 36 ln 3 litres (about 8.73 litres) c) 2.52 litres per hour (when t = √6)

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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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