The diagram shows the line $y = 24$ and part of the graph of $y = |x^2 - 16x + 39|$.
The graph crosses the line $y = 24$ at the points $P$ and $R$ and meets the $x$-axis at $Q$.

1. Find the coordinates of the turning point of the quadratic curve to determine the height of the 'hump'.

Let $f(x) = x^2 - 16x + 39$.
Completing the square:
$f(x) = (x - 8)^2 - 64 + 39 = (x - 8)^2 - 25$
The minimum point of $y = f(x)$ is $(8, -25)$.
Therefore, the maximum point of the reflected part of $y = |x^2 - 16x + 39|$ is $(8, 25)$.

2. Determine the number of intersections between the curve and the horizontal line $y = k$.

The number of solutions to $|x^2 - 16x + 39| = k$ corresponds to the number of intersection points between the graph of $y = |x^2 - 16x + 39|$ and the horizontal line $y = k$.

• For $k < 0$: 0 intersections.
• For $k = 0$: 2 intersections (at the $x$-intercepts).
• For $0 < k < 25$: 4 intersections.
• For $k = 25$: 3 intersections.
• For $k > 25$: 2 intersections.

3. State the set of values of $k$ for which there are exactly 2 solutions.

From the intersection analysis, the line $y = k$ intersects the curve exactly twice when $k = 0$ and when $k > 25$.