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 ${{\left( 2k \right)}^{\text{th}}}$ move, $P$ has made $k$ horizontal moves and $k$ vertical moves. For example, at the end of ${{5}^{\text{th}}}$ move, $P$ has made $3$ horizontal moves and $2$ vertical moves and at the end of the ${{6}^{\text{th}}}$ 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.
(i)
Find the exact coordinates of $P$ at the end of the ${{9}^{\text{th}}}$ move, showing your working clearly.
[4]
$P$ moves from $A$ to $B$ in the ${{\left( 2k-1 \right)}^{\text{th}}}$ and ${{\left( 2k \right)}^{\text{th}}}$ moves. (see Fig. 2)
(ii)
Find, in terms of $k$, the distance travelled by $P$ in the ${{\left( 2k-1 \right)}^{\text{th}}}$ and ${{\left( 2k \right)}^{\text{th}}}$ moves respectively.
[2]
(iii)
Hence, find the least value of $k$ such that the gradient of $AB$ is at least $-0.003$.
[3]
Suggested Handwritten and Video Solutions
Share with your friends!