2021 H2 Mathematics Paper 2 Question 4

Arithmetic and Geometric Progressions (APs, GPs)

Answers

(a)

325 seconds{325 \textrm{ seconds}}

(b)

1.17 m/s{1.17 \textrm{ m/s}}

(c)

Away from her starting point

Full solutions

(a)

u10=25a+9d=2540+9d=25d=−53\begin{gather*} u_{10} = 25 \\ a+9d = 25 \\ 40+9d = 25 \\ d = - \frac{5}{3} \end{gather*}
Total time taken to swim 10 lengths
S10=n2(2a+(n−1)d)=102(2(40)+(10−1)(−53))=325 s  ■\begin{align*} S_{10} &= \frac{n}{2} \Big( 2a + \left( n - 1 \right) d \Big) \\ &= \frac{10}{2} \Big( 2(40) + (10-1)(- \frac{5}{3}) \Big) \\ &= 325 \textrm{ s}\; \blacksquare \end{align*}

(b)

For Alfie's programme,
u10=40ar9=4025r9=40r9=85r=1.0536\begin{gather*} u_{10} = 40 \\ ar^9 = 40 \\ 25r^9 = 40 \\ r^9 = \frac{8}{5} \\ r = 1.0536 \end{gather*}
Total time taken to swim 10 lengths
S10=a(1−rn)1−r=25(1.053610−1)1.0536−1=319.80\begin{align*} S_{10} &= \frac{a\left(1-r^{n}\right)}{1-r} \\ &= \frac{25(1.0536^{10}-1)}{1.0536-1} \\ &= 319.80 \end{align*}
Total time for all 30 lengths
=325+(25×10)+319.80=894.80\begin{align*} &= 325 + (25\times 10) + 319.80 \\ &= 894.80 \end{align*}
Average speed
=35×30894.80=1.17 m/s (3 sf)  ■\begin{align*} &= \frac{35 \times 30}{894.80} \\ &= 1.17 \textrm{ m/s (3 sf)} \; \blacksquare \end{align*}

(c)

8 minutes = 480 seconds
Hence she is 480−325=155{480-325=155} seconds into swimming 10 lengths at 25 seconds each
155=6(25)+5155 = 6(25)+5
Hence she is 5 seconds seconds into her 7th lap of 10 lengths at 25 seconds each
Since 5 is less than half of 25, she is swimming away from her starting point ■{\blacksquare}