FHSS 2025 P1 Q13

The diagram shows part of the curve \(y = \sqrt{2x + 3} - x\) , and two vertical lines \(x = -1\) and \(x = 6\) .

1. Explain why the curve intersects the \(x\)-axis at only one point, \(A\).

   Find the \(x\)-coordinate of point \(A\).

   [3]
2. Point \(B\) is moving along the curve such that its \(y\)-coordinate is decreasing at a constant rate of \(0.6\) units per second. Find the rate of change of point \(B\)'s \(x\)-coordinate at where the curve intersects the \(x\)-axis.[3]

3. Jenny claimed that \(\int_{-1}^{6} (\sqrt{2x + 3} - x) \,\mathrm{d}x\) can be used to find the area bounded by the curve, the two vertical lines and \(x\)-axis.

   Do you agree with her? Explain your answer clearly.

   [2]