CAT 2021 Slot 1 QA Question 15: If $r$ is a constant such that $|x^2 - 4x - 13| = r$ has exactly three distinct real roots, then the value of $r$ is

Question

If $r$ is a constant such that $|x^2 - 4x - 13| = r$ has exactly three distinct real roots, then the value of $r$ is

Accepted Answer

Rewrite the quadratic inside the modulus.
$x^2 - 4x - 13 = (x-2)^2 - 17$
For $|x^2 - 4x - 13| = r$, we need to solve two equations.
$x^2 - 4x - 13 = r$
$x^2 - 4x - 13 = -r$
The first equation gives two real roots for every non-negative $r$. For exactly three distinct roots, the second equation must

If rrr is a constant such that ∣x2−4x−13∣=r|x^2 - 4x - 13| = r∣x2−4x−13∣=r has exactly three distinct real roots, then the value of rrr is

If $r$ is a constant such that $|x^2 - 4x - 13| = r$ has exactly three distinct real roots, then the value of $r$ is