GATE CS 2014 Set 2 QGA-8 — General Aptitude

Given the equation:

Since $x$ is a real number, we analyze the two possible cases for the absolute value:

Case 1: $x^2 - 2x + 3 = 11$

This gives the real roots:

Case 2: $x^2 - 2x + 3 = -11$

The discriminant of this quadratic equation is:

Since the discriminant is negative, there are no real roots in this case.

Thus, the only possible real values for $x$ are $x = 4$ and $x = -2$.

Now, we evaluate the expression $E = |-x^3 + x^2 - x|$ for these values of $x$:

• For $x = 4$: $$E = |-(4)^3 + (4)^2 - 4| = |-64 + 16 - 4| = |-52| = 52$$
• For $x = -2$: $$E = |-(-2)^3 + (-2)^2 - (-2)| = |8 + 4 + 2| = |14| = 14$$

Therefore, the possible values of $|-x^3 + x^2 - x|$ are $14$ and $52$.