2025 School of Science and Technology Prelims A Math Paper 2
The mass, \(y\) mg, of a radioactive substance decreases with time, \(x\) hours, after the start of the experiment. The research analyst claims that the data can be modelled by an equation of the form \(x^2y=a+bx^2\), where \(a\) and \(b\) are constants. Values of \(y\) for different values of \(x\) have been collected. Explain how a straight line graph can be drawn to represent the formula, and state how the values of \(a\) and \(b\) could be obtained from the line.[4]
The table shows experimental values of two variables \(R\) and \(t\).
\(t\)
10
20
30
40
50
\(R\)
2.32
2.79
3.31
4.74
4.90
It is known that \(R\) and \(t\) are related by the equation \(R=R_0(3^{-kt})\), where \(R_0\) and \(k\) are constants. An error was made in recording one of the values of \(R\).
Plot \(\ln R\) against \(t\) and draw a straight line graph to illustrate the information.[2]
From your straight line graph, identify the incorrect reading and suggest a corrected value for \(R\).[2]
Use your graph to estimate the values of \(R_0\) and \(k\).[3]
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Answer:(a) \(x^2y=bx^2+a\). Plot \(x^2y\) against \(x^2\). Vertical intercept \(=a\), gradient \(=b\). (b)(i) \(\ln R=(-k\ln3)t+\ln R_0\). Plot \(\ln R\) against \(t\). (b)(ii) Incorrect reading: \(R=4.74\) when \(t=40\). \(\ln R\approx1.40\), so \(R\approx4.06\). (b)(iii) \(\ln R_0\approx0.660\), so \(R_0\approx1.93\). \(-k\ln3\approx0.0187\), so \(k\approx-0.0170\).