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TMUA 2023 · Paper 1 · Question 7 of 20

TMUA 2023 Paper 1 Question 7

Exponentials and logarithms — Exponential equations · hidden quadratic. Try it first; the answer and a full worked solution are below.

TMUA 2023 · Paper 1Exponentials and logarithmsExponential equations · hidden quadratic7 options
P(x) and Q(x) are defined as follows: P(x) = 2x + 4 Q(x) = 2(2x−2) − 2(x+2) + 16 Find the largest value of x such that P(x) and Q(x) are in the ratio 4 : 1, respectively.
  1. A5
  2. B12
  3. C32
  4. Dlog2 3
  5. Elog2 5
  6. Flog2 12
  7. Glog2 20
Show the answer and worked solution
answer · F
  1. A5
  2. B12
  3. C32
  4. Dlog2 3
  5. Elog2 5
  6. Flog2 12
  7. Glog2 20
Put t = 2x. Then P = t + 4 and Q = t24 − 4t + 16 = (t−8)24. The ratio 4:1 means P = 4Q, so t + 4 = (t−8)2. Expanding gives t2 − 17t + 60 = 0, so (t−5)(t−12) = 0 and t = 5 or t = 12. Since x = log2 t, the largest value is log2 12.