Higher June 2025 Paper 2 Q20
20
In the diagram
\(A\) is the point \((0, 8)\)
\(B\) is the point \((16, 0)\)
The point \(D\) divides the line segment \(AB\) in the ratio 1 : 3
The line L passes through \(D\).
The gradient of L is \(\sqrt{3}\)
L passes through the point with coordinates \((-2, f)\)
Show that \(f \lt -4\) (5)
| Answer | Mark | Mark scheme |
|---|---|---|
| Shown | M1 | for the method to find a coordinate of the point \(D\) eg \(\dfrac{1}{1 + 3} \times 16\ (= 4)\) or \(\dfrac{3}{1 + 3} \times 8\ (= 6)\) or (4, 6) labelled |
| M1 | for a correct form for L, eg \(y = \sqrt{3}x + c\) OR a correct equation for the gradient of L, eg \(\dfrac{[6] - f}{[4] - -2} = \sqrt{3}\) | |
| M1 | for correct substitution to find \(c\) eg \([6] = \sqrt{3} \times [4] + c\) or \(y - [6] = \sqrt{3}(x - [4])\) or \(c = [6] - [4]\sqrt{3}\ (= -0.928\ldots)\) OR starts to rearrange equation for gradient, eg \([6] - f = ([4] - -2)\sqrt{3}\) | |
| M1 | (dep on previous M1) for the method to substitute in \(-2\), eg \(\sqrt{3} \times (-2) + [6] - \sqrt{3} \times [4]\) or \(\sqrt{3} \times (-2) + [c]\) OR a correct unevaluated expression for \(f\), eg \(f = [6] - ([4] - -2)\sqrt{3}\) | |
| C1 | accurate figure eg \(f = -4.39\ldots\) or \(-4.39\ldots \lt -4\) |
Additional guidance
First two marks may be seen in either order
Accept 4 or 6 stated or (6, 4)
Condone incorrect value for \(c\) when awarding this mark
[4] must be clearly identified as the \(x\)-coordinate of \(D\) if incorrect
Award of this mark implies the previous mark
[6] must be clearly identified as the \(y\)-coordinate of \(D\) if incorrect
[\(c\)] must be clearly what they have found to be the \(y\)-intercept of L and must have come from correct processes to evaluate
\(-4.39\ldots\) must come from correct working
Accept \(-4.4\) or better