2025 Presbyterian High School Prelims A Math Paper 2
The time taken, \(t\) in minutes, to download a particular gaming software is related to internet speed, \(x\) Mbps. The variables \(x\) and \(t\) are related by the formula \(t=\frac{a}{b+x}\), where \(a\) and \(b\) are positive constants. The data below shows some measured values of \(x\) and \(t\).
\(x\) (Mbps)
50
100
200
500
1000
\(t\) (minutes)
23
13
7.3
3.2
1.6
Plot \(\frac{1}{t}\) against \(x\) and draw a straight line graph.[2]
Use the graph to estimate the value of \(a\) and of \(b\), correct to 2 significant figures.[4]
Use the graph to estimate the time taken for John to download the gaming software if the speed of his internet is 240 Mbps.[2]
Suggest an alternative set of variables to be plotted on each axis to draw a straight line to represent the formula.[1]
The diagram shows part of a straight line graph drawn to represent the equation \(y=px^q\). Given that the line passes through the point \((2.5,8.1)\) and has a gradient of 3, find the value of \(p\) correct to the nearest integer.[3]
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Answer:(a)(i) plot \((50,0.0435),\ (100,0.0769),\ (200,0.137),\ (500,0.313),\ (1000,0.625)\) (ii) \(a=1600,\ b=22\) (iii) \(6.26\) minutes (iv) plot \(xt\) against \(t\) (b) \(p=4\)