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

To find the value of k, we can use point P(-4, 12). Substituting the values into the equation:

$12 = (-4)^2 + k(-4) + 8$

This yields:

$12 = 16 - 4k + 8$

Simplifying:

$12 = 24 - 4k$

Rearranging gives:

$4k = 24 - 12$

So, we find:

$4k = 12 \Rightarrow k = 3$

Thus, the updated equation of the curve becomes:

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

To determine if triangle ABQ is isosceles, we need to find the lengths of the sides AB, AQ, and BQ using the distance formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

1. Calculate AB:

   • A(0, 2) and B(4, 5) $AB = \sqrt{(4 - 0)^2 + (5 - 2)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

2. Calculate AQ:

   • A(0, 2) and Q(0, y) where y is calculated from the curve equation for x=0: $AQ = |y - 2| = |8 - 2| = 6$

3. Calculate BQ:

   • B(4, 5) and Q(0, y) $BQ = \sqrt{(4 - 0)^2 + (5 - y)^2}$

Using the points through which the line passes and the previous calculations, we determine that:

• If AQ ≠ BQ, then triangle ABQ is not isosceles.

The conclusion is that Diann is incorrect.