Worksheet Chapter 6: Linear Law

1. An experiment measures the decay of a substance where mass M (in grams) at time t (in hours) follows M = M₀ e^(-kt). Applying natural logarithms yields ln M = -kt + ln M₀.
A straight-line plot of ln M against t has a slope of -0.15 and an vertical intercept of 4.605.
(a) Determine the initial mass M₀ (given e⁴.605 ≈ 100).
(b) Find the decay constant k.

2. The variables x and y are related by y = kx + h. A straight-line graph of 1y against x has a gradient of 0.5 and a vertical intercept of 2.
(a) Express 1y in terms of x, h, and k.
(b) Find the values of h and k.

3. True or False: A line of best fit must pass through every single plotted data point on the graph.

4. Variables x and y are related by the non-linear equation y = a b^x. Experimental values converted to linear form give a straight line passing through (1, 1) and (3, 5) on a graph of lg y against x.
(a) Calculate the gradient and vertical intercept of the straight line.
(b) Find the values of constants a and b.

5. True or False: When converting y = a x^b into linear form using base-10 logarithms, plotting lg y against lg x yields a straight line with gradient b and vertical intercept lg a.

6. Fill in the blank: When plotting a graph to convert non-linear data into a straight line, estimating a value within the range of given data points is called __________.

7. A straight line of best fit obtained by plotting xy against x² passes through the points (2, 7) and (5, 16).
(a) Find the equation of the line of best fit in terms of x and y.
(b) Calculate the value of y when x = 2.

8. The variables x and y are related by the equation y = px² + qx.
(a) Reduce the equation to the linear form Y = mX + c such that the horizontal axis represents x.
(b) State what Y, m, and c represent.

9. The table below shows corresponding values of x and y obtained from an experiment:
x | 1 | 2 | 3 | 4
y | 5 | 11 | 19 | 29
It is known that x and y are related by y = ax² + b.
(a) Construct a table of values for x² and y.
(b) Determine the values of constants a and b by finding the gradient and Y-intercept.

10. Fill in the blank: To reduce a non-linear equation to linear form Y = mX + c, the non-linear relationship must be expressed such that m and c are __________.