Topical Revision Worksheet 7 Linear Law Solution (LC)

Topical Revision Worksheet 7 Topic: Linear Law Question 1 [2008 EOY, Q4] marks Two variables x and y are such that when is drawn against , a straight line of gradient is obtained, passing through the point . (i) Express y in terms of x. Solution: Hence [4] (ii) Find the value of x when . Solution: When , [2] Question 2 [2009 EOY, Q15] marks Answer the whole of this question on a sheet of graph paper. The table shows experimental values of two variables, x and y, which are known to be related by the equation . x x 1 2 3 4 5 y 2.4 1.1 0.2 – 0.5 – 1.1 (i) Choosing a suitable scale, plot (xy – x) against x2 and draw a straight line. Solution: xy − x 1.4 0.2 − 2.4 − 6 − 10.5 1 4 9 16 25 [5]

(ii) Use your graph to estimate the values of p and q. Solution: Fr graph, q = 1.89, [2] (iii) Explain how you would use your graph to determine if the set of experimental values x = 10 and y = − 4 are related by the equation . Solution: xy − x = − 50 , = 100. Extend the line. If it passes through (100, −50), then the values are related by the equation . [1] xy − x

Question 3 [2010 EOY, Q8] marks The table below shows experimental readings of y for given values of x such that where a and b are real constants. 0 1 2 3 4 5 6 By drawing a straight line graph of against , identify the incorrect reading, Solution: so let and 2.21 1.39 2.21 2.77 From graph, the incorrect reading is 1.96 [4]

(ii) estimate the values of a and b. Solution: From the graph, [2] Question 4 [2012 EOY, Q12] marks The table shows experimental values of two variables, x and y, which are known to be related by the equation , where a and b are constants. x 1.0 1.5 2.0 2.5 3.0 y 7.3 3.5 2.0 1.3 0.9 (i) Express this equation in a form suitable for drawing a straight line graph. Solution: [2] (ii) Using a suitable scale, plot a straight line graph of lg y against lg x. Use your graph to estimate the value of a and of b. Solution: lg x 0 0.18 0.30 0.40 0.48 lg y 0.86 0.54 0.30 0.11 −0.05 [6]

Table of values Label axes Plot points correctly Best fit line From graph, Question 5 [2013 EOY, Q15] marks The table shows the experimental values of two variables x and y . x 1 2 3 4 5 y 3.8 11.2 20.5 33.2 50.6 3.8 5.6 8.3 It is known that x and y are related by the equation Plot against x and draw a suitable straight line. Solution: x 1 2 3 4 5 y 3.8 11.2 20.5 33.2 50.6 3.8 5.6 6.8 8.3 10.1 lg x lg y

(a) Use the graph to estimate the value of a and of b, Solution: (a) Plot against x. From the diagram, The gradient b = 1.6 and the vertical intercept is a = 2.3. [4] (ii) the value of x when y = 8x. Solution: When y = 8x . From the diagram, the value of x is 3.6 [1] (b) The diagram shows the straight line graph of xy against passing through the points A(0, 6) and B(8, 4). Express y in terms of x. [3]

Solution: xy – intercept is 6 gradient equation of straight line is of the form “ y = mx + c “ i.e. Hence Question 6 [2014 EOY, Q10] marks Answer the whole of this question on a sheet of graph paper. The table shows experimental values of two variables x and y. It is known that x and y are related by an equation , and that one of these values of y given in the table is subject to an abnormally large error. x 1.5 2.5 3.5 4.5 5.0 6.0 y 9.9 6.0 3.6 2.2 1.7 1.4 (i) Draw a straight line graph of against x, using a scale of 2 cm to 1 unit on the x-axis and 1 cm to 0.2 units on the -axis.

Solution: x 1.5 2.5 3.5 4.5 5.0 6.0 y 9.9 6.0 3.6 2.2 1.7 1.4 2.29 1.79 1.28 0.79 0.53 0.34 [4] (ii) Use your graph to estimate the value of a and of b. Solution: [3] (iii) Identify the abnormal reading of y and estimate its correct value. Solution: The abnormal reading is 1.4. [B1] When x = 6.0, is estimated to be 0.04. Hence, the correct value of . (accept 1.00 to 1.04) [2]

Question 7 [2015 EOY, Q8] marks Answer the whole of this question on a sheet of graph paper. Two variables x and y are related such that , where a and b are unknown constants. In an experiment, values of y are found for several values of x and the values are tabulated as shown below. It is also found that one particular value of y is incorrectly recorded. x 1 2 3 4 5 6 y 2.07 3.77 5.45 6.81 8.81 10.48 (i) Using a scale of 2 cm to 1 unit for the horizontal axis and 1 cm to 1 unit for the vertical axes, plot against x and draw a straight line graph, and use it to identify the incorrect value of y recorded. Solution: [4]

Incorrect value of y recorded is 6.81.