LINEAR LAW
PAPER 1
1
x and y are related by the equation
y  px 2  qx , where p and q are constants.
A straight line is obtained by plotting
y
against x , as shown in Diagram 1.
x
Calculate the value of p and q.
[4 marks]
DIAGRAM 1
Answer:
p  ……………………………
q  ……………………………
2.
Diagram 2 shows a straight line graph of
y
against x . Given that y  6 x  x 2 ,
x
calculate the value of k and of h .
[3 marks]
y
x
• (2, k )
• ( h , 1)
x
O
DIAGRAM 2
Answer: k  ……………………………
h  ……………………………
3.
The variables x and y are related by the
equation y  kx4 , where k is a constant.
(a) Convert the equation to linear form.
(b) Diagram 3 shows the straight line
obtained by plotting log 10 y against
log 10 x .
Find the value of
(i) log 10 k
(ii) h .
[4 marks]
log 10 y
(2, h)
• ( 0 , 3)
O
DIAGRAM 3
log 10 x
Answer: (a) ……………………………….
(b) (i) ……………………………
(ii) …………………………...
4
Diagram 4 shows the graph of
1
y
against .
x
x
y
x
Express y in terms of x .
• (4, 13)
[3 marks]
(  1 , 3)
•
1
x
O
O
DIAGRAM 4
Answer: ………………………………….
2
5
DIAGRAM 5(a)
DIAGRAM 5(b)
Diagram 5(a) shows the curve y = –3 x 2 + 5. Diagram 5(b) shows the straight line graph
obtained when y = –3 x 2 + 5 is expressed in the linear form Y = 5X + c.
Express X and Y in terms of x and /or y
[3 marks]
Answer: ………………………………….
3
PAPER 2
6.
Table 1 shows the values of two variables x and y related by the equation y  ab x 1 ,
where a and b are constants.
x
y
3
12.1
4
6.46
5
3.47
6
1.89
8
0.52
TABLE 1
(a) Draw the graph of log 10 y against ( x  1) .
[4 marks]
(b) From your graph, find
(i) the value of y when x  7
(ii) the value of a
(iii) the value of b .
[6 marks]
7. Table 2 shows the values of two variables, x and y, obtained from an experiment.
Variables x and y are related by the equation y = p k x , where p and k are constants.
x
y
2
3.16
4
5.50
6
9.12
8
16.22
10
28.84
12
46.77
TABLE 2
(a) Plot log 10 y against x by using a scale of 2 cm to 2 units on the x-axis and 2 cm to 0.2
unit on the log 10 y -axis.
Hence, draw the line of best fit.
[4 marks]
(b) Use your graph from (a) to find the value of
(i)
p
(ii)
k
[6 marks]
8. Table 3 shows the values of two variables, x and y, obtained from an experiment.
2
It is known that x and y are related by the equation y  pk x , where p and k are constants.
x
y
1.5
1.59
2.0
1.86
2.5
2.40
3.0
3.17
3.5
4.36
4.0
6.76
TABLE 3
2
(a) Plot log y against x .
Hence, draw the line of best fit.
[5 marks]
(b) Use the graph in (a) to find the values of
(i) p
(ii) k .
[5 marks]
4
9. Table 4 shows the values of two variables, x and y, obtained from an experiment. The
r
variables x and y are related by the equation y  px 
, where p and r are constants
px
x
y
1.0
5.5
2.0
4.7
3.0
5.0
4.0
6.5
5.0
7.7
5.5
8.4
TABLE 4
(a) Plot xy against x 2 , by using a scale of 2 cm to 5 units on both axes.
Hence, draw the line of best fit.
(b) Use your graph from (a) to find the value of
(i) p
(ii) r .
[5 marks]
[5 marks]
10. Table 5 shows the values of two variables, x and y, obtained from an experiment.
The variables x and y are related by the equation y  pk x 1 , where p and k are constants
x
y
1
4.0
2
5.7
3
8.7
4
13.2
5
20.0
6
28.8
TABLE 5
(a) Plot log y against (x+1), using a scale of 2 cm to 1 unit on the (x+1)-axis and 2 cm
to 0.2 unit on the log y-axis. Hence, draw the line of best fit.
[5 marks]
(b) Use your graph from (a) to find the value of
(i) p
(ii) k.
[5 marks]
5
Linear Law
Answer
1. p  2, q  13
2. h  5, k  4
3. (a) log 10 y  4 log 10 x  log 10 k
(b) (i) 3
(ii) 11
y  5x  2
4
1
5. X = 2
x
y
Y= 2
x
6.
x -1
2
3
4
5
7
Log10y 1.08 0.81 0.54 0.28 -0.28
(b) (i) y = 1
(ii) a = 42.66
(iii) b = 0.5337
6
7 (a)
x
2
4
6
8
10
12
Log10y 0.50 0.74 0.96 1.21 1.46 1.67
(b) (i) p = 1.920
(ii) k = 1.309
8 (a)
x2
2.25
4.0
6.25
9.0
12.25
16.0
log 10 y
0.20
0.27
0.38
0.50
0.64
0.83
(b) (i) p  1.259
(ii) k  1.109
7
9(a)
1
4
9 16 25 30.25
x2
xy 5.5 9.4 15 26 38.5 46.2
(b) (i) p  1.37
(ii) r  5.48
10. (a)
x+1
Log y
2
0.60
3
0.76
4
0.94
5
1.12
6
1.30
7
1.46
(b) p = 1.778
k = 1.483
8