2022 SAJC Promo Q10

Starting from the coordinates \(\left( 0,8 \right)\), a particle \(P\) moves in the positive \(x\)-direction and then in the negative \(y\)-direction alternatively (see Fig. 1). Hence, at the end of the \({{\left( 2k \right)}^{\mathrm{th}}}\) move, \(P\) has made \(k\) horizontal moves and \(k\) vertical moves. For example, at the end of \({{5}^{\mathrm{th}}}\) move, \(P\) has made \(3\) horizontal moves and \(2\) vertical moves and at the end of the \({{6}^{\mathrm{th}}}\) move, \(P\) has made \(3\) horizontal moves and \(3\) vertical moves.

The distance in the first horizontal move is \(1\) unit, second horizontal move is \(1.8\) units, and each successive horizontal move is \(80\%\) more than the previous horizontal move.

The distance in the first vertical move is \(2\) units, second vertical move is \(5\) units, and each successive vertical move is increased by \(3\) units.

1. Find the exact coordinates of \(P\) at the end of the \({{9}^{\mathrm{th}}}\) move, showing your working clearly.[4]

\(P\) moves from \(A\) to \(B\) in the \({{\left( 2k-1 \right)}^{\mathrm{th}}}\) and \({{\left( 2k \right)}^{\mathrm{th}}}\) moves. (see Fig. 2)

2. Find, in terms of \(k\), the distance travelled by \(P\) in the \({{\left( 2k-1 \right)}^{\mathrm{th}}}\) and \({{\left( 2k \right)}^{\mathrm{th}}}\) moves respectively.[2]
3. Hence, find the least value of \(k\) such that the gradient of \(AB\) is at least \(-0.003\).[3]