The diagram shows a straight line that passes through points A and B, and a curve that passes through points P and Q - OCR - GCSE Maths - Question 5 - 2018 - Paper 1

From the given information, we see that the curve defined by the equation passes through points P(-4, 12) and Q(4, 5).

Plugging in point P(-4, 12):

$12 = (-4)^2 + k(-4) + 8\n12 = 16 - 4k + 8\n12 = 24 - 4k\n4k = 24 - 12\n4k = 12\nk = 3$

Now substituting 'k' back, the equation becomes:

$y = x^2 + 3x + 8$

To determine if triangle ABQ is isosceles, we need to find and compare the lengths of sides AB, AQ, and BQ.

1. Distance AB:

Using the distance formula:

$AB = \sqrt{(4 - 0)^2 + (5 - 2)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

2. Distance AQ:

Using point Q(4, 5):

$AQ = \sqrt{(4 - 0)^2 + (5 - 2)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

3. Distance BQ:

$BQ = \sqrt{(4 - 4)^2 + (5 - 12)^2} = \sqrt{0 + 49} = 7$

Since AB = AQ = 5 and BQ = 7, triangle ABQ has two equal sides. Therefore, Diann is correct; triangle ABQ is indeed isosceles.