2025 Nov TZ3 P1 Q8

Consider the function defined by \(f(x) = \frac{1}{2}x^2 + kx + 13\), where \(x \in \mathbb{R}\) and \(k \in \mathbb{Z}^+\).

1. Given that the equation \(f(x) = 0\) has no real roots, show that the greatest possible value of \(k\) is \(5\).

   [2]

For the remainder of this question, consider the case \(k = 5\).

2. [3]

   1. Write down the equation of the axis of symmetry of the graph of \(f\).

   2. Hence, or otherwise, determine the coordinates of the minimum point on the graph of \(f\).

The following diagram shows the graph of \(f\) and a line \(L\) which is normal to the curve at \(x = -3\). The shaded area shown is bounded by the curve, the line \(L\) and the \(y\)-axis.

3. Show that the equation of \(L\) is given by \(y = -\frac{1}{2}x + 1\).

   [5]

4. Hence, find the shaded area.

   [5]