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TMUA 2018 · Paper 1 · Question 7 of 20

TMUA 2018 Paper 1 Question 7

Sequences and series — Binomial expansion · matching coefficients. Try it first; the answer and a full worked solution are below.

TMUA 2018 · Paper 1Sequences and seriesBinomial expansion · matching coefficients6 options
The non-zero constant k is chosen so that the coefficients of x6 in the expansions of (1 + kx2)7 and (k + x)10 are equal.

What is the value of k?

  1. A16
  2. B6
  3. C√66
  4. D√6
  5. E√3030
  6. F√30
Show the answer and worked solution
answer · A
  1. A16
  2. B6
  3. C√66
  4. D√6
  5. E√3030
  6. F√30
In (1 + kx2)7 the general term is (7r)(kx2)r, so x6 comes from r = 3 and its coefficient is (73)k3 = 35k3. In (k + x)10 the x6 term is (106)k4x6, with coefficient 210k4. Setting 35k3 = 210k4 and cancelling k3 (legitimate since k ≠ 0) gives 1 = 6k, so k = 16.