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TMUA 2020 · Paper 2 · Question 18 of 20

TMUA 2020 Paper 2 Question 18

Algebra and functions — Cubics · a difference that is always positive. Try it first; the answer and a full worked solution are below.

TMUA 2020 · Paper 2Algebra and functionsCubics · a difference that is always positive8 options
In this question, f(x) = ax3 + bx2 + cx + d and g(x) = px3 + qx2 + rx + s are cubic polynomials.

If f(x) − g(x) > 0 for every real x, which of the following is/are necessarily true?

I    a > p
II   if b = q then c = r
III  d > s

  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
Show the answer and worked solution
answer · G
  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
The difference f(x) − g(x) = (a−p)x3 + (b−q)x2 + (c−r)x + (d−s) has to be positive everywhere. A cubic with non-zero leading coefficient runs to −∞ at one end, so a − p must be 0, that is a = p — statement I is false rather than true. Statement III follows by putting x = 0: the difference there is d − s > 0. For II, if also b = q the difference is the linear function (c−r)x + (d−s), and a linear function is positive for all x only if its gradient is zero, so c = r. Hence II and III.