Curve motion: find x-coordinate

The equation of a curve is \(y = (5-2x)^{\frac{1}{2}} + 5\) for \(x < \frac{5}{2}\).

1. A point \(P\) is moving along the curve in such a way that the \(y\)-coordinate of point \(P\) is decreasing at 5 units per second. Find the rate at which the \(x\)-coordinate of point \(P\) is increasing when \(y = 32\). [4]
2. Point \(A\) on the curve has \(y\)-coordinate 32. Point \(B\) on the curve is such that the gradient of the curve at \(B\) is \(-3\). Find the equation of the perpendicular bisector of \(AB\). Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [6]

A curve has equation \(y = \frac{12}{5 - 2x}\).

1. Find \(\frac{dy}{dx}\). [2]

A point moves along this curve. As the point passes through \(A\), the \(x\)-coordinate is increasing at a rate of 0.15 units per second and the \(y\)-coordinate is increasing at a rate of 0.4 units per second.

1. Find the possible \(x\)-coordinates of \(A\). [4]

A curve is such that \(\frac{\mathrm{d}y}{\mathrm{d}x} = \sqrt{(4x + 1)}\) and \((2, 5)\) is a point on the curve.

1. Find the equation of the curve. [4]
2. A point \(P\) moves along the curve in such a way that the \(y\)-coordinate is increasing at a constant rate of 0.06 units per second. Find the rate of change of the \(x\)-coordinate when \(P\) passes through \((2, 5)\). [2]
3. Show that \(\frac{\mathrm{d}^2y}{\mathrm{d}x^2} \times \frac{\mathrm{d}y}{\mathrm{d}x}\) is constant. [2]