SPM Additional Mathematics 2019, Paper 2 (Question 11)

Question 11:
Use a graph paper to answer this question.
Table 2 shows the values of two variables, x and y, obtained from an experiment. The variables x and y are related by the equation

y = \frac{a}{{}^{b}{\sqrt{x}}},

where a and b are constants.

Table 2

(a) Based on Table 2, construct a table for the values of log₁₀x and log₁₀y. [2 marks]

(b) Plot log₁₀y against log₁₀x, using a scale of 2 cm to 0.1 unit on the X-axis and 2 cm to 0.2 unit on the Y-axis.
Hence, draw the line of best fit. [3 marks]

(c) Using the graph in 11(b), find the value of [5 marks]
(i) a,
(ii) b.

Solution:
(a)

(b)

(c)

\begin{array}{l} y = \frac{a}{{}^{b}{\sqrt{x}}} \\ \log_{10} y = \log_{10} (\frac{a}{{}^{b}{\sqrt{x}}}) \\ \log_{10} y = \log_{10} a - \log_{10} {}^{b}{\sqrt{x}} \\ \log_{10} y = \log_{10} a - \log_{10} x^{\frac{1}{b}} \\ \log_{10} y = - \frac{1}{b} \log_{10} x + \log_{10} a \\ Y = \log_{10} y, m = - \frac{1}{b}, X = \log_{10} x, c = \log_{10} a \end{array}

(c)(i)

\begin{array}{l} c = \log_{10} a \\ 0.44 = \log_{10} a \\ a = 10^{0.44} \\ = 2.754 \end{array}

(c)(ii)

\begin{array}{l} m = - \frac{1.16}{0.44} \\ = - 2.636 \\ - \frac{1}{b} = - 2.636 \\ b = 0.3794 \end{array}

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