2024 TJC H1 Q3

A ball is rolling in a straight line such that its distance away from the starting point, \(s\) cm, can be modelled using the equation

\[s = at + \frac{b}{\sqrt{t + 4}} + c,\]

where \(t\) is the time taken in seconds, and \(a\), \(b\) and \(c\) are real constants.

The ball is at the starting point when \(t = 0\), and moved \(10\) cm in the first \(5\) seconds. It moved another \(9\) cm in the next \(16\) seconds. Find the ball’s distance away from the starting point when \(t = 50\).[4]

1. Differentiate \(\ln \sqrt{1-3x}\) with respect to \(x\).[3]
2. A curve \(C\) has equation \(y = 2 - \mathrm{e}^{1-\frac{x}{3}}\). Using an algebraic method, find the range of values of \(x\) for which the gradient of \(C\) is less than \(\frac{1}{6}\).[3]