Question 9:
Table 2.1 shows the values of two variables, x and y, obtained from an experiment. The variables x and y are related by the equation y = a/ b–x, such that a and b are constants.
(a) Based on Table 2.1, fill in the Table 2.2.[1 mark]
(b) Plot 1/y against x, using a scale of 2 cm to 0.5 unit on the 1/y-axis and 2 cm to 1 unit on the x-axis.
Hence, draw the line of best fit.
[3 marks]
(c) Using the graph in (b),
(i) find the value of y when x = 4.5,
(ii) reduce y = a/ b–x to linear form, hence find the value of a and of b.
[6 marks]
Answer:
(a)
(b)
(c)(i)
$$ \begin{aligned} & \text { When } x=4.5, \\ & \frac{1}{y}=1.825 \\ & y=\frac{1}{1.825} \\ & y=0.5479 \end{aligned} $$
(c)(ii)
$$ \begin{aligned} & \frac{y}{1}=\frac{a}{b-x} \\ & \frac{1}{y}=\frac{b-x}{a} \\ & \frac{1}{y}=-\frac{1}{a} x+\frac{b}{a} \\ & Y=m X+C \end{aligned} $$
$$ \begin{aligned} -\frac{1}{a} & =m \\ -\frac{1}{a} & =\frac{3.5-3.85}{1.2-0.5} \\ -\frac{1}{a} & =-\frac{1}{2} \\ a & =2 \end{aligned} $$
$$ \begin{aligned} \frac{b}{a} & =C \\ \frac{b}{2} & =4.10 \\ b & =2(4.10) \\ b & =8.20 \end{aligned} $$
Table 2.1 shows the values of two variables, x and y, obtained from an experiment. The variables x and y are related by the equation y = a/ b–x, such that a and b are constants.
(a) Based on Table 2.1, fill in the Table 2.2.[1 mark]
(b) Plot 1/y against x, using a scale of 2 cm to 0.5 unit on the 1/y-axis and 2 cm to 1 unit on the x-axis.
Hence, draw the line of best fit.
[3 marks]
(c) Using the graph in (b),
(i) find the value of y when x = 4.5,
(ii) reduce y = a/ b–x to linear form, hence find the value of a and of b.
[6 marks]
Answer:
(a)
(b)
(c)(i)
$$ \begin{aligned} & \text { When } x=4.5, \\ & \frac{1}{y}=1.825 \\ & y=\frac{1}{1.825} \\ & y=0.5479 \end{aligned} $$
(c)(ii)
$$ \begin{aligned} & \frac{y}{1}=\frac{a}{b-x} \\ & \frac{1}{y}=\frac{b-x}{a} \\ & \frac{1}{y}=-\frac{1}{a} x+\frac{b}{a} \\ & Y=m X+C \end{aligned} $$
$$ \begin{aligned} -\frac{1}{a} & =m \\ -\frac{1}{a} & =\frac{3.5-3.85}{1.2-0.5} \\ -\frac{1}{a} & =-\frac{1}{2} \\ a & =2 \end{aligned} $$
$$ \begin{aligned} \frac{b}{a} & =C \\ \frac{b}{2} & =4.10 \\ b & =2(4.10) \\ b & =8.20 \end{aligned} $$