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TMUA 2016 · Paper 1 · Question 1 of 20

TMUA 2016 Paper 1 Question 1

Algebra and functions — Binomial expansion of a cube · matching surd coefficients. Try it first; the answer and a full worked solution are below.

TMUA 2016 · Paper 1Algebra and functionsBinomial expansion of a cube · matching surd coefficients8 options
It is given that the expansion of (ax + b)3 is 8x3 − px2 + 18x − 3√3, where a, b and p are real constants.

What is the value of p?

  1. A−12√3
  2. B−6√3
  3. C−4√3
  4. D−√3
  5. E√3
  6. F4√3
  7. G6√3
  8. H12√3
Show the answer and worked solution
answer · H
  1. A−12√3
  2. B−6√3
  3. C−4√3
  4. D−√3
  5. E√3
  6. F4√3
  7. G6√3
  8. H12√3
Expand: (ax+b)3 = a3x3 + 3a2b x2 + 3ab2 x + b3. The two end terms pin down a and b on their own: a3 = 8 gives a = 2, and b3 = −3√3 gives b = −√3, since (−√3)3 = −3√3. Check against the x term: 3ab2 = 3⋅ 2⋅ 3 = 18, as required. The x2 coefficient is 3a2b = 3⋅ 4⋅(−√3) = −12√3, and the expansion writes that coefficient as −p, so p = 12√3. The sign trap is dropping the minus in −px2 and answering −12√3.