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. 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 \(f\).

      [3]

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]