LINEARIZATION OF GRAPHS
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1 LINEARIZATION OF GRAPHS
2 Question 1 (**) The table below shows eperimental data connecting two variables and y y It is assumed that and y are related by an equation of the form y = ab, where a and b are non zero constants. a) Find an equation of a straight line, in terms of well defined constants, in order to investigate the validity of this assumption. b) Plot a suitable graph to show that the assumption of part (a) is valid. c) Use the graph of part (b) to estimate, correct to 1 decimal place, the value of a and the value of b. d) Estimate the value of y when = 2.5. MP1-M, log y = log b + log a, a 9.5, b 1.2, y 15.0
3 Question 2 (**) The table below shows eperimental data connecting two variables t and W. t W It is assumed that t and W are related by an equation of the form W t = ab, where a and b are non zero constants. a) Find an equation of a straight line, in terms of well defined constants, in order to investigate the validity of this assumption. b) Plot a suitable graph to show that the assumption of part (a) is valid. c) Use the information from the graph to estimate, correct to 2 decimal places, the value of a and the value of b. d) Estimate the value of W when t = 20. SYN-E, logw = tlogb + log a, a 1.26, b 1.48, W 3200
4 Question 3 (**) The following table shows some eperimental data y It is assumed that the two variables and y are related by the formula y = ab, where a and b are non zero constants. a) Use a graphical method to show that the data is consistent with this assumption. b) Find estimates for the values of a and b, correct to one decimal place. c) Use the estimated values of a and b, to find an estimate for the value of y when 60 =. MP1-O, a 0.7, b 1.2, y 39000
5 Question 4 (**) The following table shows some eperimental data. t P It is assumed that the two variables t and P are related by the formula b P = at, where a and b are non zero constants. a) Use a graphical method to show that the data is consistent with this assumption. b) Determine estimates for the value of a and the value of b, correct to one decimal place. c) Use the estimated values of a and b, to find an estimate for the value of P when 20 t =. MP1-B, a 6.3, b 1.6, P 760
6 Question 5 (**) The table below shows eperimental data connecting two variables t and H. t H It is assumed that t and H are related by an equation of the form H n = kt, where k and n are non zero constants. a) Find an equation of a straight line, in terms of well defined constants, in order to investigate the validity of this assumption. b) Plot a suitable graph to show that the assumption of part (a) is valid. c) Use the graph to estimate, correct to 2 significant figures, the value of k and the value of n. SYN-F, log H = nlogt + log k, k 0.66, n 1.1
7 Question 6 (**+) The table below shows eperimental data connecting two variables and y y It is assumed that and y are related by an equation of the form 2 y = A + B, where A and B are non zero constants. By plotting accurately the equation of a suitable straight line, estimate correct to 1 significant figure the value of A and the value of B. A 0.2, B 50
8 Question 7 (**+) The table below shows eperimental data connecting two variables and y y It is assumed that and y are related by an equation of the form 2 y = A + B, where A and B are non zero constants. By plotting accurately the equation of a suitable straight line, estimate correct to 1 decimal place the value of A and the value of B. A 0.6, B 1.9
9 Question 8 (**+) The table below shows eperimental data connecting two variables and y y It is assumed that and y are related by an equation of the form A y = + B, 2 where A and B are non zero constants. By plotting accurately the equation of a suitable straight line, estimate correct to 2 significant figures the value of A and the value of B. SYNF-C, A 800, B 13
10 Question 9 (**+) The variables and y are thought to obey a law of the form n y = a, where a and n are non zero constants. Let X = log10 and Y = log10 y. a) Show there is a linear relationship between X and Y. The figure below shows the graph of Y against X. Y A( 0,2) O B( 6,0) X b) Determine the value of a and the value of n. SYN-I, n = 1, a = 100 3
11 Question 10 (**+) The variables and y are thought to obey a law of the form y = a k, where a and k are positive constants. Let Y = log10 y. a) Show there is a linear relationship between and Y. The figure below shows the graph of Y against. Y B( 0,3) 3 (,0 ) A 2 O b) Determine the value of a and the value of k. MP1-Q, a = 1000, k = 100
12 Question 11 (***) log10 y B( 8,21) O A( 2,3) log10 The figure above shows a set of aes where log10 y is plotted against log10. A straight line passes through the points A ( 2,3) and ( 8,21) B. Epress y in terms of. MP1-V, y =
13 Question 12 (***) ln y B( 2,4ln 5) A( 0,2ln5) O t The figure above shows a set of aes where ln y is plotted against t. A straight line passes through the points A ( 0,2ln5) and ( 2,4ln 5) B. Epress y in terms of t. MP1-K, y = 2 5 t +
14 Question 13 (***) In each of the following equations and y are variables, and A, B and k are non zero constants. a) 2 y = A + B. b) y = A B +. c) y = Ae k d) ( ) y A = B. Epress each of these equations in straight line form and state the variables to be plotted in the and y ais. the gradient and the y intercept of the straight line. X = y Y = m = A c = B, X = 1 Y = y 1 m = A B c = A, X = Y = ln( y ) m = k c = ln A, 1 X = Y = y 2 2 m = B c = A
15 Question 14 (***) The following table shows some eperimental data y It is assumed that and y are related by an equation of the form where A and B are non zero constants. B y = A +, By plotting accurately the equation of a suitable straight line estimate the value of A and the value of B. A 2, B 120
16 Question 15 (***) The following table shows some eperimental data y It is assumed that and y are related by an equation of the form 2 B y = A +, where A and B are non zero constants. By plotting accurately the equation of a suitable straight line estimate the value of A and the value of B. A 2, B 420
17 Question 16 (***) The following table shows some eperimental data y It is assumed that and y are related by an equation of the form y = A + Bln, where A and B are non zero constants. By plotting accurately the equation of a suitable straight line estimate the value of A and the value of B. A 0.10, B 2.0
18 Question 17 (***+) log2 y A( 2,8) B( 5,2) O log2 The figure above shows a set of aes where log2 y is plotted against log2. A straight line passes through the points A ( 2,3) and ( 8,21) B. Determine the value of y at the point where y =. MP1-X, y = 16
19 Question 18 (***+) The following table shows some eperimental data y It is assumed that and y are related by an equation of the form y = A B + C 2, where A, B and C are non zero constants. By plotting accurately the equation of a suitable straight line, estimate correct to 2 significant figures the value of A, B and C. A 2500, B 5000, C 21
20 Question 19 (*****) The table below shows eperimental data connecting two variables and y. t P It is assumed that t and P are related by an equation of the form k P = A t, where A and k are non zero constants. By linearizing the above equation, and using partial differentiation to obtain a line of least squares determine the value of A and the value of k. A 250, k 0.2
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