TMUA Logic Drill · Sufficient Conditions

Question 8 · Hard · Paper 2 Style

Which one of the following is a sufficient condition for the equation

x³ − 3x² + a = 0,

where a is a constant, to have exactly one real root?

A. a > 0 B. a ≤ 0 C. a ≥ 4 D. a < 4 E. |a| > 4 F. |a| ≤ 4 G. a = 9/4 H. |a| = 3/2

Correct answer: E

Solution:

Let y = x³ − 3x² + a. Changing a simply translates the cubic vertically.

Differentiate:
dy/dx = 3x² − 6x = 3x(x − 2).

So the stationary points occur at x = 0 and x = 2. Their y-values are:

at x = 0: a,
at x = 2: a − 4.

The cubic has exactly one real root if the local maximum is below the x-axis or the local minimum is above the x-axis. This happens when:

a < 0 or a > 4.

So the full necessary and sufficient condition is a < 0 or a > 4. But the question asks only for a sufficient condition from the options.

Option E says |a| > 4, meaning a > 4 or a < −4. Both parts lie inside the valid range a > 4 or a < 0. Therefore option E is sufficient.