Worksheet Chapter 6: Linear Law

1. 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.

2. 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 __________.

3. 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.

4. 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.

5. 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.

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. 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.

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

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. 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.