2025 Crescent Girls' School Prelims A Math Paper 2
For the equation \(py = xy + 1\) where \(p\) is an unknown constant, may be represented by a straight line, expressed in the form \(Y = mX + c\), where \(X\) and \(Y\) are functions of \(x\) and/or \(y\) and \(m\) and \(c\) are constants. Using the following table, insert in it an expression for \(Y, X, m\) and \(c\).[2]
\(Y\)
\(X\)
\(m\)
\(c\)
\(py = xy + 1\)
The variables \(x\) and \(y\) are related by the equation \(x^a y = b\), where \(a\) and \(b\) are constants. The table below shows some corresponding values of \(x\) and \(y\).
\(x\)
1.31
1.63
1.97
2.54
4.03
\(y\)
1270
398
131
35.1
2.63
Plot \(\lg y\) against \(\lg x\) and draw a straight line graph.[3]
Use your graph to estimate the value of \(a\) and of \(b\).[4]
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Answer:(a) \(Y=\dfrac{1}{y}\), \(X=x\), \(m=-1\), \(c=p\). (b)(i) Plot \((0.117,3.10)\), \((0.212,2.60)\), \((0.294,2.12)\), \((0.405,1.55)\) and \((0.605,0.420)\), then draw a straight line of best fit. (b)(ii) \(a\approx5.50\), \(b\approx5.62\times10^3\).