IGCSE/O LEVEL

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LINEAR LAW

Past Papers
(2011-2021)
Including Variants
1 The figure shows the graph of a straight line with 1g y plotted against x. The straight line passes For
through the points A (5,3) and B (15,5). Examiner’s
Use
lg y
B (15,5)

A (5,3)

O x

(i) Express lg y in terms of x. [3]

(ii) Show that y = a (10bx) where a and b are to be found. [3]

© UCLES 2011 4037/12/M/J/11

2 The variables x and y are related so that when lg y is plotted against lg x a straight line graph For
passing through the points (4, 12) and (6, 17) is obtained. Examiner’s
Use

lg y

(6,17)

(4,12)

O lg x

(i) Express y in terms of x, giving your answer in the form y = axb. [6]

(ii) Find the value of x when y = 300. [2]

© UCLES 2011 4037/21/M/J/11

3 Variables t and N are such that when 1g N is plotted against 1g t, a straight line graph passing For
through the points (0.45, 1.2) and (1, 3.4) is obtained. Examiner’s
Use
lg N
(1, 3.4)

(0.45, 1.2)

O lg t

(i) Express the equation of the straight line graph in the form 1g N = m 1g t + 1g c, where m and c
are constants to be found. [4]

(ii) Hence express N in terms of t. [1]

© UCLES 2011 4037/11/O/N/11 [Turn over

4 Variables x and y are such that, when y2 is plotted against 2x, a straight line graph is obtained. For
This line has a gradient of 5 and passes through the point (16,81). Examiner’s
Use

y2
(16,81)

O 2x

(i) Express y2 in terms of 2x. [3]

(ii) Find the value of x when y = 6. [3]

© UCLES 2011 4037/12/O/N/11 [Turn over

5 The table shows experimental values of two variables x and y. For
Examiner’s
Use
x 1 2 3 4 5
y 3.40 2.92 2.93 3.10 3.34

It is known that x and y are related by the equation y = a + bx, where a and b are constants.
x

(i) Complete the following table.

x x

y x
[1]

(ii) On the grid on page 11 plot y x against x x and draw a straight line graph. [2]

(iii) Use your graph to estimate the value of a and of b. [3]

(iv) Estimate the value of y when x is 1.5. [1]

© UCLES 2011 4037/21/O/N/11

 For
y√x Examiner’s
Use

O 2 4 6 8 10 12 x√x

© UCLES 2011 4037/21/O/N/11 [Turn over

6 The table shows values of variables x and y. For
Examiner’s
Use
x 1 3 6 10 14
y 2.5 4.5 0 –20 –56

(i) By plotting a suitable straight line graph, show that y and x are related by the equation
y = Ax + Bx2, where A and B are constants. [4]

© UCLES 2012 4037/11/M/J/12

 (ii) Use your graph to find the value of A and of B. [4] For
Examiner’s
Use

© UCLES 2012 4037/11/M/J/12 [Turn over

7 The table shows experimental values of variables x and y. For
Examiner’s
Use
x 5 30 150 400
y 8.9 21.9 48.9 80.6

(i) By plotting a suitable straight line graph, show that y and x are related by the equation
y = axb, where a and b are constants. [4]

© UCLES 2012 4037/22/M/J/12

 (ii) Use your graph to estimate the value of a and of b. [4] For
Examiner’s
Use

(iii) Estimate the value of y when x = 100. [2]

© UCLES 2012 4037/22/M/J/12 [Turn over

8 The variables s and t are related by the equation t = ks n , where k and n are constants. The table For
below shows values of variables s and t. Examiner’s
Use

s 2 4 6 8
t 25.00 6.25 2.78 1.56

(i) A straight line graph is to be drawn for this information with lg t plotted on the vertical axis.
State the variable which must be plotted on the horizontal axis. [1]

(ii) Draw this straight line graph on the grid below. [3]

1g t

O
© UCLES 2013 4037/13/O/N/13
 (iii) Use your graph to find the value of k and of n. [4] For
Examiner’s
Use

(iv) Estimate the value of s when t = 4. [2]

© UCLES 2013 4037/13/O/N/13

9 The table shows experimental values of two variables x and y. For
Examiner’s
Use
x 2 4 6 8
y 9.6 38.4 105 232

It is known that x and y are related by the equation y = ax 3 + bx , where a and b are constants.
y
(i) A straight line graph is to be drawn for this information with on the vertical axis. State the
x
variable which must be plotted on the horizontal axis. [1]

(ii) Draw this straight line graph on the grid below. [2]

y
x
30

20

10

© UCLES 2013 4037/22/O/N/13

 (iii) Use your graph to estimate the value of a and of b. [3] For
Examiner’s
Use

(iv) Estimate the value of x for which 2y = 25x . [2]

© UCLES 2013 4037/22/O/N/13

10 The table shows values of variables V and p.

V 10 50 100 200
p 95.0 8.5 3.0 1.1

(i) By plotting a suitable straight line graph, show that V and p are related by the equation p = kV n ,
where k and n are constants. [4]

© UCLES 2014 4037/11/M/J/14

 Use your graph to find

(ii) the value of n, [2]

(iii) the value of p when V = 35. [2]

© UCLES 2014 4037/11/M/J/14 [Turn over

11 Two variables x and y are connected by the relationship y = Ab x , where A and b are constants.

(i) Transform the relationship y = Ab x into a straight line form. [2]

An experiment was carried out measuring values of y for certain values of x. The values of ln y and x
were plotted and a line of best fit was drawn. The graph is shown on the grid below.

ln y

0 1 2 3 4 5 6 x
–1

(ii) Use the graph to determine the value of A and the value of b, giving each to 1 significant figure.
[4]

(iii) Find x when y = 220. [2]

© UCLES 2014 4037/22/M/J/14 [Turn over

12 The relationship between experimental values of two variables, x and y, is given by y = Ab x , where A
and b are constants.

(i) By transforming the relationship y = Ab x , show that plotting ln y against x should produce a
straight line graph. [2]

(ii) The diagram below shows the results of plotting ln y against x for 7 different pairs of values of
variables, x and y. A line of best fit has been drawn.

ln y
12

11

10

0 1 2 3 4 5 6 x

By taking readings from the diagram, find the value of A and of b, giving each value correct to
1 significant figure. [4]

(iii) Estimate the value of y when x = 2.5. [2]

© UCLES 2015 4037/21/M/J/15

 9

13 Two variables, x and y, are such that y = Ax b , where A and b are constants. When ln y is plotted
against ln x , a straight line graph is obtained which passes through the points ^1.4, 5.8h and ^2.2, 6.0h.

(i) Find the value of A and of b. [4]

(ii) Calculate the value of y when x = 5. [2]

© UCLES 2015 4037/12/O/N/15 [Turn over

 14

14 The trees in a certain forest are dying because of an unknown virus.

The number of trees, N, surviving t years after the onset of the virus is shown in the table below.

t 1 2 3 4 5 6
N 2000 1300 890 590 395 260

The relationship between N and t is thought to be of the form N = Ab -t .

(i) Transform this relationship into straight line form. [1]

(ii) Using the given data, draw this straight line on the grid below. [3]

© UCLES 2015 4037/23/O/N/15

 (iii) Use your graph to estimate the value of A and of b. [3]

If the trees continue to die in the same way, find

(iv) the number of trees surviving after 10 years, [1]

(v) the number of years taken until there are only 10 trees surviving. [2]

© UCLES 2015 4037/23/O/N/15

15
lg y
1.0

0.9

0.8
(1, 0.73)
0.7

0.6

0.5

0.4

0.3

0.2

0.1 (4, 0.10)

0.0
0 1 2 3 4 x2

Variables x and y are such that when lg y is plotted against x 2 , the straight line graph shown above is
obtained.
2
(i) Given that y = Ab x , find the value of A and of b. [4]

(ii) Find the value of y when x = 1.5. [2]

(iii) Find the positive value of x when y = 2. [2]

© UCLES 2016 4037/12/M/J/16

16 The variables x and y are such that when ln y is plotted against x, a straight line graph is obtained. This
line passes through the points x = 4, ln y = 0.20 and x = 12, ln y = 0.08 .

(i) Given that y = Ab x , find the value of A and of b. [5]

(ii) Find the value of y when x = 6 . [2]

(iii) Find the value of x when y = 1.1 . [2]

© UCLES 2016 4037/12/O/N/16

17
ln y

(1.5, 3.5)

(4.0, 1.5)

0 1 2 3 4 5 1
x

1
The variables x and y are such that when ln y is plotted against the straight line graph shown above is
x
obtained.
b
(i) Given that y = Ae x , find the value of A and of b. [4]

© UCLES 2016 4037/13/O/N/16

 (ii) Find the value of y when x = 0.32 . [2]

(iii) Find the value of x when y = 20 . [2]

© UCLES 2016 4037/13/O/N/16

18 It is given that y = A (10 bx) , where A and b are constants. The straight line graph obtained when lg y is
plotted against x passes through the points (0.5, 2.2) and (1.0, 3.7) .

(i) Find the value of A and of b. [5]

Using your values of A and b, find

(ii) the value of y when x = 0.6 , [2]

(iii) the value of x when y = 600 . [2]

© UCLES 2017 4037/12/M/J/17 [Turn over

19 When lg y is plotted against x, a straight line is obtained which passes through the points (0.6, 0.3) and
(1.1, 0.2).

(i) Find lg y in terms of x. [4]

(ii) Find y in terms of x, giving your answer in the form y = A ^10 bxh, where A and b are constants.
[3]

© UCLES 2017 4037/12/O/N/17 [Turn over

20 When ln y is plotted against x 2 a straight line is obtained which passes through the points (0.2, 2.4) and
(0.8, 0.9).

(i) Express ln y in the form px 2 + q , where p and q are constants. [3]

2
(ii) Hence express y in terms of z, where z = e x . [3]

© UCLES 2017 4037/13/O/N/17

21 The variables x and y are such that when e y is plotted against x 2, a straight line graph passing through
the points (5, 3) and (3, 1) is obtained. Find y in terms of x. [5]

© UCLES 2018 4037/12/M/J/18

22 An experiment was carried out recording values of y for certain values of x. The variables x and y are
thought to be connected by the relationship y = ax n , where a and n are constants.

(i) Transform the relationship y = ax n into straight line form. [2]

The values of ln y and ln x were plotted and a line of best fit drawn. This is shown in the diagram
below.

ln y
5

0
0 2 4 6 8 ln x

(ii) Use the graph to find the value of a and of n, stating the coordinates of the points that you use. [3]

(iii) Find the value of x when y = 50. [2]

© UCLES 2018 4037/21/M/J/18

23 Variables x and y are such that when y2 is plotted against e2x a straight line is obtained which passes through
the points (1.5, 5.5) and (3.7, 12.1). Find

(i) y in terms of e2x, [3]

(ii) the value of y when x = 3, [1]

(iii) the value of x when y = 50 . [3]

© UCLES 2018 4037/23/O/N/18 [Turn over

24 When lg y is plotted against x2 a straight line graph is obtained which passes through the points (2, 4) and
(6, 16).
2
(i) Show that y = 10 A + Bx , where A and B are constants. [4]

1
(ii) Find y when x = . [2]
3

(iii) Find the positive value of x when y = 2. [3]

© UCLES 2019 4037/11/M/J/19 [Turn over

 1
25 When e y is plotted against , a straight line graph passing through the points (2, 20) and (4, 8) is obtained.
x
(i) Find y in terms of x. [5]

(ii) Hence find the positive values of x for which y is defined. [1]

(iii) Find the exact value of y when x = 3. [1]

(iv) Find the exact value of x when y = 2. [2]

© UCLES 2019 4037/12/M/J/19 [Turn over

26 When lg y 2 is plotted against x, a straight line is obtained passing through the points (5, 12) and (3, 20).
Find y in terms of x, giving your answer in the form y = 10 ax + b , where a and b are integers. [5]

© UCLES 2019 4037/12/O/N/19

27
x 1 1.5 2 2.5 3
y 6 14.3 48 228 1536

2
The table shows values of the variables x and y such that y = Ab x , where A and b are constants.
2
(i) Draw a straight line graph to show that y = Ab x . [4]

© UCLES 2019 4037/13/O/N/19

 (ii) Use your graph to find the value of A and of b. [4]

(iii) Estimate the value of x when y = 100. [2]

© UCLES 2019 4037/13/O/N/19

28 Variables x and y are such that, when 4 1
y is plotted against , a straight line graph passing through the
x
points (0.5, 9) and (3, 34) is obtained. Find y as a function of x. [4]

© UCLES 2020 4037/21/M/J/20

29 It is known that y = A # 10 bx , where A and b are constants. When lg y is plotted against x 2 , a
2

straight line passing through the points (3.63, 5.25) and (4.83, 6.88) is obtained.

(a) Find the value of A and of b. [4]

Using your values of A and b, find

(b) the value of y when x = 2 , [2]

(c) the positive value of x when y = 4 . [2]

© UCLES 2020 4037/13/O/N/20

30 Variables x and y are such that y = Ax b , where A and b are constants. When lg y is plotted against lg x,
a straight line graph passing through the points (0.61, 0.57) and (5.36, 4.37) is obtained.

(a) Find the value of A and of b. [5]

Using your values of A and b, find

(b) the value of y when x = 3, [2]

(c) the value of x when y = 3. [2]

© UCLES 2021 4037/12/M/J/21

31 When e 2y is plotted against x 2 , a straight line graph passing through the points (4, 7.96) and (2, 3.76) is
obtained.

(a) Find y in terms of x. [5]

(b) Find y when x = 1. [2]

(c) Using your equation from part (a), find the positive values of x for which the straight line exists.
[3]

© UCLES 2021 4037/13/O/N/21

32 Variables x and y are such that when y is plotted against log 2 (x + 1), where x 2- 1, a straight line is
obtained which passes through (2, 10.4) and (4, 15.4) .

(a) Find y in terms of log 2 (x + 1) . [4]

(b) Find the value of y when x = 15. [1]

© UCLES 2021 4037/22/O/N/21

 (c) Find the value of x when y = 25. [3]

© UCLES 2021 4037/22/O/N/21

 Answers
1 (ii)𝑦 = 348 (iii) 𝑥 = 0.876
1.(i)𝑙𝑔𝑦 = 5 (𝑥 − 5) + 3
14. (i)𝑙𝑜𝑔𝑁 = 𝑙𝑜𝑔𝐴 − 𝑡𝑙𝑜𝑔𝑏 6
1
(ii)𝑏 = 5 , 𝑎 = 100 25.(i)𝑦 = ln (32 − 𝑥)
(iii)𝐴 = 2950, 𝑏 = 1.5 3
2.(i)𝑦 = 100𝑥 2.5 (ii)𝑥 > 16
(iv)51
(ii)1.55 (iii)𝑦 = 𝑙𝑛30
(v)14
3.(i)𝑚 = 4, 𝑐 = 0.251 6
15a.(i) 𝐴 = 8.71 𝑏 = 0.617 (iv)𝑥 = 32−𝑒 2
(ii)𝑁 = 0.251𝑡 4
(ii)𝑦 = 2.93 26.𝑦 = 10−2𝑥+16
4.(i) 𝑦 2 = 5(2𝑥 ) + 1
(iii) 𝑥 = 1.74 27.(ii)𝐴 = 3 𝑏 = 2
(ii)𝑥 = 2.81
16.(i)𝐴 = 1.30 𝑏 = 0.985 (iii)2.25
5.(i) 10
(ii)1.19 28. 𝑦 = ( 𝑥 + 4)4
𝑥 √𝑥 1 2.83 5.20 8 11.18
(iii)11
y√ 𝑥 3.40 4.13 5.07 6.20 7.47 29a. 𝐴 = 2.07 𝑏 = 1.36
(iii)𝑎 = 3 ± 0.1 𝑏 = 0.4 ± 0.001 17.(i)𝐴 = 110, 𝑏 = −0.8 b. (533 − 576) × 103
(iv)3.05 (ii)𝑦 = 9 c. 0.46
6.(ii)𝐴 = 3 𝐵 = −0.5 (iii)𝑥 = 0.47 30a. 𝐴 = 1.14, 𝑏 = 0.84
7.(ii)𝑏 = 0.5 ± 0.03 𝑎 = 4 ± 0.3 18.(i) 𝐴 = 5.01 𝑏 = 3 b. 𝑦 = 2.87
(iii)32 − 49 (ii)316 c. 𝑥 = 3.16
8. (i)𝑙𝑔𝑠 (iii)𝑥 = 0.693 1
31a. 𝑦 = 2 ln (2.1𝑥 2 − 0.44)
21 𝑥
(iii)𝑛 = −2 𝑘 = 100 18.(i)𝑙𝑔𝑦 = 50 − 5
b. 0.253
(iv)𝑠 = 4.9 (ii)𝐴 = 2.63 𝑏 = −0.2 c. 𝑥 > 0.458
9.(i)𝑥 2 5
20.(i)𝑙𝑛𝑦 = − 2 𝑥2 + 2.9 32a. √𝑦 = 2.5 log 2 (𝑥 + 1) + 5.4
(iii)𝑎 = 0.4 ± 0.02 𝑏 = 3.2 ± 0.4 5
(ii)𝑦 = 18.2𝑧 −2 b. 𝑦 = 237.16
(iv)4.8
21.(ii)𝑦 = ln (𝑥 2 − 2) c. 𝑥 = −0.105
10.(i)𝑙𝑔𝑝 = 𝑛𝑙𝑔𝑘 + 𝑙𝑔𝑣
22.(i) 𝑙𝑔𝑦 = 𝑙𝑔𝑎 + 𝑛𝑙𝑔𝑥
(ii)𝑛 = −1.5
(ii) 𝑛 = −0.2 𝑎 = 𝑒 4.7
(iii)13 𝑡𝑜 16
(iii)22
11.(i)log 𝑦 = 𝑙𝑜𝑔𝐴 + 𝑥𝑙𝑜𝑔𝑏
23.(i) 𝑦 = ±√3𝑒 2𝑥 + 1
(ii)𝐴 =0.5, B=3
(ii)±34.8
(iii)4.4
(iii)3.36
12.(ii)𝐴 = 90000 𝑏 = 0.4
24.(i) 𝐴 = −2, 𝐵 = 3
(iii)𝑒 9
(ii) 0.1
13.(i)𝐴 = 233 𝑏 = 0.25