SPM Practice 3 (Linear Law) – Question 1

Question 1 (10 marks):
Use a graph to answer this question.
Table 1 shows the values of two variables, x and y, obtained from an experiment.
The variables x and y are related by the equation $y-\sqrt{h}=\frac{hk}{x}$ , where h and k are constants.


(a) Plot xy against x, using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the xy-axis.
Hence, draw the line of best fit.

(b) Using the graph in 9(a), find
(i) the value of h and of k,
(ii) the correct value of y if one of the values of y has been wrongly recorded during the experiment.

Solution: 
(a)




(b)
$\begin{array}{l}y-\sqrt{h}=\frac{hk}{x}\\ xy-\sqrt{h}x=hk\\ xy=\sqrt{h}x+hk\\ Y=mX+C\\ Y=xy,m=\sqrt{h},C=hk\end{array}$


(b)(i)
$\begin{array}{l}m=\frac{36.5}{5.1}\\ \sqrt{h}=\frac{36.5}{5.1}\\ \sqrt{h}=7.157\\ h=51.22\\ \\ C=-4\\ hk=-4\\ k=\frac{-4}{h}\\ k=-\frac{4}{51.22}\\ k=-0.0781\end{array}$


(b)(ii)
$\begin{array}{l}xy=21\\ 3.5y=21\\ y=\frac{21}{3.5}=6.0\\ \text{Correct value of }y\text{ is 6}\text{.0}\text{.}\end{array}$