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4PM1/02R Jan 2020, Question 9: Binomial Series for Any Rational Power

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 10 marks

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The question, in short

Rewrite 1/(2 − x)³ in the form p(1 − qx)^(−3), expand it, then use the given first terms of f(x) = (a + bx)/(2 − x)³ to find a, b and c.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (a) · 2 marks

Find the constants p and q so that 1/(2 − x)³ equals p(1 − qx)^(−3).

  1. 1(2 − x)^(−3) = 2^(−3)(1 − x/2)^(−3)
  2. 2p = 1/8, q = 1/2

Answerp = 1/8, q = 1/2

Part (b) · 3 marks

Expand 1/(2 − x)³ in ascending powers of x through the x³ term, giving each coefficient as an exact fraction in lowest terms.

  1. 1(1 − x/2)^(−3) = 1 + (−3)(−x/2) + (−3)(−4)/2! (−x/2)² + (−3)(−4)(−5)/3! (−x/2)³
  2. 2= 1 + (3/2)x + (3/2)x² + (5/4)x³
  3. 3Multiply by 1/8: 1/8 + (3/16)x + (3/16)x² + (5/32)x³

Answer1/8 + (3/16)x + (3/16)x² + (5/32)x³

Part (c) · 3 marks

Given f(x) = (a + bx)/(2 − x)³ with integer a, b and first terms 3/8 − (43/16)x + cx², find a and b.

  1. 1(a + bx)(1/8 + (3/16)x + (3/16)x² + …)
  2. 2Constant: a/8 = 3/8, so a = 3
  3. 3x term: 3a/16 + b/8 = −43/16
  4. 49/16 + 2b/16 = −43/16, so 2b = −52, b = −26

Answera = 3, b = −26

Part (d) · 2 marks

Find the exact value of the coefficient c.

  1. 1x² coefficient: 3a/16 + 3b/16
  2. 2= 9/16 − 78/16 = −69/16

Answerc = −69/16

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