CHAPTER 2 : LINEAR LAW Contents Page 3.0 CONCEPT MAP UNDERSTAND AND USE THE CONCEPT OF LINES OF BEST FIT Draw lines of best fit b inspecti

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1 ADDITIONAL ATHEMATICS FORM 5 MODULE 3 LINEAR LAW

2 CHAPTER 2 : LINEAR LAW Contents Page 3.0 CONCEPT MAP UNDERSTAND AND USE THE CONCEPT OF LINES OF BEST FIT Draw lines of best fit b inspection of given data Eercises Write equations for lines of best fit Eercises Determine values of variables from a) lines of best fit b) equations of lines of best fit Eercises APPLY LINEAR LAW TO NON- LINEAR RELATIONS Reduce non-linear relations to linear form Eercises Determine values of constants of non-lineas relations given Eercises Steps to plot a straight line Eercises SPM Questions Assessment test Answers 30 1

3 CONCEPT MAP CHAPTER 2 : LINEAR LAW LINEAR LAW Linear Relation Non Linear Relation Reducing non-linear relation to linear form Lines of best fit Determining values of constants Formation of equation Obtaining information 2

4 3.1 UNDERSTAND AND USE THE CONCEPT OF LINES OF BEST FIT THE CONCEPT OF LINES OF BEST FIT. The properties of the line of best fit. The straight line is drawn such awa that it passes through as man points as possible. The number of points that do not lie on the straight line drawn should be more or less the same both sides of the straight line. Eample. Draw the line of best fit for the given graph below. EXERCISE 1 1. Draw the line of best fit for the following diagrams. Eercises 1 (a) (b) 3

5 3.1.2 Write equations for lines of best fit Eercises 2 Eample 1 (3, 8) Eample 2 (1, 3) (3, 7) The equation that can be formed is: Y = mx + c 8 2 Gradient, m = = Y intercept = 2 Therefore, = Linear form 0 The equation that can be formed is: Y = mx + c 7 3 Gradient, m = = The straight line also passes through the Point (1, 3) Y Another equation can be formed is, 3 = 2(1) + c.c = 1 Therefore, = Linear form a) 9 b) (3, 6). (8, 1)

6 c) ( 5, 2) d) (8, 8) ( 6,4) Determine values of variables from: a) lines of best fit b) Equations of lines of best fit Eample 1 X (7, k) Eample 1 (3, k) 2 Gradient = Therefore, (2, 4) 4 2 = k 4 = k 2 = 5.k = = The straight line also passes through the point (3, k) Another equation that can be formed is.k = 3(3) + 4.k = = 13 5

7 Eercises 3 a) lines of best fit b) Equations of lines of best fit i) X (6, k) (3, 3) 1 0 i) (k, 3) X. = ii) ii) X (k, 6) = 6 - (5, 2) 0 7 (3, k) 0 iii) X (8, 8) (6, 4) (k, 2) 0 iii) X (4, k). =

8 3.2 APPLY LINEAR LAW TO NON- LINEAR RELATIONS Reduce non- linear relations to linear form Y = mx + c Eample 1 = a + b 2 Non- linear Eample 1 a) = p q Non- linear 2 a = + b 1 = a + b Y = c + Xm Divided b Linear form log 10 = log 10 (p q ) log 10 = log 10 p + log 10 q log 10 = log 10 p + qlog 10 Y = c m X Take log of Both siders Linear form Eercises 4 Equation Linear form Y X m c a) 2 = a + b b) = a 2 + b c) a b = + 1 d) 2 = e) = f) = ab 4 g) = 2 a ( + b)2 7

9 3.2.4 Determine values of constants of non linear relations given Eercises 5 Eample 1 X (11, 6) a) (1, 2) The above figure shows part of a straight line graph drawn to represent the equation. = a 2 + b Find the value of a and b 6 2 Gradient, a = A = (4, 2) 0 The above figure shows part of a straight line graph drawn to represent the equation. = a 2 + b Find the value of a and b, Therefore, = b Another equation that can be formed is.(1)(2) = 5 2 (1) 2 + b b, = = 5 5 Hence a = 5 2, b = 5 8 8

10 b) Log c) (3, 6) (1, 9) 1.lg 0 The above figure shows part of a straight line graph drawn to represent the equation The above figure shows part of a straight line graph drawn to represent the equation.of = a + b Find the value of a and b,.of = a b Find the value of a and b, 9

11 3.3 STEPS TO PLOT A STRAIGHT LINE Using a graph paper. Change the non-linear function with variables and to a linear form Y = mx + c Construct a table Choose a suitable scale Label both aes Line of best fit Determine : gradient m Y-intercept c QUESTIONS X Y Table The above table shows the eperimental values of two variables, and. It is know that and are related b the equation = p 2 + q Non- linear a) Draw the line of best fit for against b) From our graph, find, i) the initial velocit ii) the acceleration 10

12 SOLUTION STEP 1 = p 2 + q = p 2 q + Reduce the non-linear To the linear form The equation is divided throughout d To create a constant that is free from On the right-hand side i.e, q = p + q Linear form Y = mx + c Y = mx + c STEP 2 construct table STEP 3 Using graph paper, - Choose a suitable scale so that the graph drawn is as big as possible. - Label both ais - Plot the graph of Y against X and draw the line of best fit 11

13 STEP 4 From the graph, find m and c Gradient, m = = 2 Construct a right-angled triangle, So that two vertices are on the line of best fit, calculate the gradient, m Y- intercept = c.= -3 Determine the Y-intercept, c.from the straight line graph 12

14 Eercises 6 1. The table below shows some eperimental data of two related variable and It is know that and are related b an equation in the form = a + b 2, where. a and b are constants a) Draw the straight line graph of against b) Hence, use the graph to find the values of a and b 13

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16 2. The table below shows some eperimental data of two related variable and It is known that and are related b an equation in the form a. = + b, where a and b are constants. a) Draw the straight line graph 2 against b) Hence, use the graph to find the values of a and b 15

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18 4.0 SPM QUESTIONS 1. SPM 2003( paper 1, question no 10) and are related b equation = p 2 + q, where p and q are constants. A straight line is obtained b plotting against, as show in Diagram below (2, 9) (6, 1) 0 Calculate the values af p and q. 17

19 2. SPM 2004( paper 1, question no 13) Diagram below shows a straight line graph of against (2, k) a. (h, 3) Gigen that = 6 2, calculate the value of k and of h 18

20 3. SPM 2005( paper 1, question no 13) The variable and are related b the equation = k 4 where k is a constant. a) Convert the equation = k 4 to linear form b) Diagram below shows the straight line obtained b plotting log 10 against log 10 log 10 (0, 3). (2, h) 0 Find the value of i) log 10 k ii) h. log 10 19

21 4. SPM 2003( paper 2, question no 7) Table 1 shows the values of two variables, and, obtained from on eperiment. It is known that and are related b the equation = pk 2, where p and k are constants (a) Plot log against 2. Hence, draw the line of best fit (b) Use the graph in (a) to find value of ( i ) p ( ii ) k 20

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23 5. SPM 2004( paper 2, question no 7) Table 1 shows the values of two variables, and, obtained from on eperiment. Variables and are related b the equation = pk, where p and k are constants (a) Plot log 10 against b using scala of 2 cm to 2 units on the -ais and 2 cm to 0.2 unit on the log 10 -ais. Hence, draw the line of best fit (b) Use the graph in (a) to find value of ( i ) p ( ii ) k 22

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25 6. SPM 2005( paper 2, question no 7) Table 1 shows the values of two variables, and, obtained from on eperiment. r The variables and are related b the equation = p +, p where p and r are constants (b) Plot against 2 b using scala of 2 cm to 5 units on both aes. Hence, draw the line of best fit (b) Use the graph in (a) to find value of ( i ) p ( ii ) r 24

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27 5.0 ASSESSMENT TEST 1. (0, h). (8, k) 0 The above figure shows part of a straight-line graph drawn to represent the equation 6 1 = - Find the value of h and of k 2 26

28 3. The table below shows some eperimental data of two related variable and Y It is know that and are related b an equation in the from. = a 3 - b, where a and b are constants. a) Change the equation to the linear form and hence draw the straight line graph for values of and b) From our graph, I).determine the values of a and b II).find the value of when =

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30 6.0 ANSWERS Eercises 2 a) = b) = 3 3 c) = d) = 2-8 Eercises 4 Y X m c a). 2 a b b) a b c) 1 1 b 1 a a d) e) 3 5 f) lg lg b lg a 2 2b g) a a Eercises 3 a) i) k = 5 b) i) k = 3 ii) k = 1 ii) k = 3 iii) k = 5 iii) k = 6 Eercises 5 a) a = 1, b = 2 b) a = 11, b = 8 c) a = 3, b = 3 Eercises 5 1) a = - 1, b = 10 2) a = 0.5, b = 2.8 SPM Questions 1) p = -2, q = 13 2) h = 3, k = 4 3) a) lg = 4lg + lg k b) k = 100, h = 11 4) p = 1.259, k = ) p = 1.82, k = ) P = 1.37, r = 5.48 Assessment test 1) h = 6, k = 2 2) a) = a 2 b b) i. a = 2, b = 3.6 ii. =

CHAPTER 2 LINEAR LAW FORM 5 PAPER 1. Diagram 1 Diagram 1 shows part of a straight line graph drawn to represent

CHAPTER 2 LINEAR LAW FORM 5 PAPER 1. Diagram 1 Diagram 1 shows part of a straight line graph drawn to represent PAPER. n ( 8, k ) Diagram Diagram shows part of a straight line graph drawn to represent and n.. Find the values of k [4 marks] 2. log ( 3,9 ) ( 7,) log Diagram 2 Diagram 2 shows part of a straight line