chp 2 linear law (addmaths form 5)

7/29/2019 Chp 2 Linear Law (AddMaths Form 5)

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cikgubid/AMF5/W4/Mon-2013

Chp 2: Linear Law

Addmaths Form 5[cikgubid/AMF5/W4-2013/Mon/KYRHG]

IMPORTANT NOTES:1. Line of Best Fit

* A straight line drawn that passes through as many points aspossible.

* The number of points that do not lie on the straight line drawn should be more or less the same on both

sides of the straight line.

y

++

+

+

+

+

0 x2. Non-linear Function

* A function that has one or more variables,x ory, which are not in the first degree.

* A non-linear function that consists of variables x andy (not in a straight-line graph) can be reduced or

converted to the linear form, Y =mX + c,where X and Y represent the functions ofx ory or both (with

a straight-line graph).

3. Steps to Find Values of Constants in a Non-linear Function* Reduce or convert the non-linear function with variables x and y to the linear form, Y = mX + c, where

X and Y represent the functions ofx ory or both.

* Prepare a table for the values ofX and Y.* Choose a suitable scale to draw the graph as large as possibleand label both axes.

* Plot the graph ofY against X and draw the line of best fit.

* Construct a right-angled triangle on the drawn line of best fit, to calculate the gradient of the straight

line.

y

+

+ (x2, y2)

+

++

+ (x1, y1)

0 x* Determine the Y-intercept, which is represented by c, from the straight-line graph.

4. To Determine Variables ofx ory* The values of certain variables, eitherx ory, can be determined;

(i) from the graph of the line of best fit, or

(ii) from the equation of the line of best fit that is formed.

Gradient, m =12

12

xx

yy

Name:............................................


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Exercise 1Line of Best Fit

1. (a) Draw the line of best fit fory againstx on a graph paper from the data shown on the following table.(b) From the line of the best fit you have drawn;

(i) find the value ofy whenx = 18,

(ii) find the value ofx, wheny = 40,

(iii) form a straight-line equation.

Answer:

(i)

(ii)

(iii)

2. (a) Draw the line of best fit fory againstx on a graph paper from the data shown on the following table.(b) From the line of the best fit you have drawn;

(i) find the value ofywhenx = 0.4,



x -2 -1 0 1 2 3

y 1 4 6 8 11 13

Answer:

(i)

(ii)

(iii)

3. (a) Draw the line of best fit fory againstx on a graph paper from the data shown on the following table.

(b) From the line of the best fit you have drawn;

(i) find the value ofy whenx = 0.3,



x 0.2 0.4 0.6 0.8 1.0 1.2

y 66 60 54 49 43 36

Answer:

(i)

(ii)

(iii)

x 5 10 15 20 25

y 16 28 36 50 62


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Exercise 2Application of Linear Law to Non-Linear Functions

1. Express the following non-linear equation to the linear form Y = mX + c.

Hence, state the Y, m,Xand c.

No. Non-linear Equation Linear Form Y m X c

1. y2

= + 4

2. y = 2x25x

3. y x = 10

4. y = a x +x

b

5. y =qx

p

6. ax2 + by 2 = x

7. y = abx

8. ay = bx +x2

9. y = axn

10. y = ax +b

x 2

x

3


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2. The following straight-line graph drawn to represent the equationy = ax2

+ bx, where a and b are constant.

Find the value ofa and ofb.

x

y

(1, 4)

0 x

(5, 0)

3. The following straight-line graph drawn to represent the equationy = ax+x

b, where a and b are constant.

Find the value ofa and ofb.xy

5

(4, 3)

0 x2

4. The following straight-line graph drawn to represent the equationy =2x

a+

x

b, where a and b are constant.

Find the value ofa

and ofb

.

xy

(4, 7)

(2, 3)

0 1/x

5. The following straight-line graph drawn to represent the equationy = abx, wherea andb are constant. Findthe value ofa and ofb.

log y

(9, 7)

(1, 3)

0 x


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Exercise 3Problem Solving I

1. The following table shows the experimental values of two variables, x and y. It is known that x and y are

related by an equationax + by = x2, wherea andb are constants.

(a) Draw the graph ofx

yagainstx.

(b) From the graph, find

(i) the values ofa,

(ii) the value ofb,(iii) the value ofywhenx = 3.5.

x 1 2 3 4 5 6

y 0.50 0.33 0.50 1.99 4.17 7.01

2. The following table shows the experimental values of two variables, xand y. It is known that x and y are

related by an equationy = px +x

q, wherep and q are constants.

(a) Draw the graph ofxy againstx2.

(b) From the graph, find(i) the values ofp,

(ii) the value ofq,

(iii) the value ofy whenx = 5.7.

x 1 2 3 4 5 6

y 7.2 8.4 10.9 13.8 16.8 19.9


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3. The following table shows the experimental values of two variables, x and y. It is known that x and y are

related by an equationy =bx

a, where a and b are constants.

(a) Draw the graph ofy

1againstx.

(b) From the graph, find

(i) the values ofa,

(ii) the value ofb,(iii) the value ofx wheny = 1.8

x 2 4 6 8 10 12

y 3.20 2.44 1.96 1.64 1.41 1.23

4. The following table shows the experimental values of two variables, x and y. It is known that x and yare

related by an equationy = axb, wherea andb are constants.

(a) Convert the equation into linear form, hence draw the linear graph.(b) From the graph, find

(i) the values ofa,

(ii) the value ofb,

x 2 3 4 5 6

y 11.3 20.8 32.0 44.7 58.8


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Exercise 4Problem Solving II

1. The following straight-line graph is obtained by plotting log3y againstx.

log3 y (a) Express log3y in term ofx.

(3, 10) (b) Expressy in term ofx.

4 (c) Find the value ofy whenx = -1

0 x

2. The following straight-line graph is obtained by plottingy

1against

x

1.

y

1

(a) Express y

1

in term ofx.

(b) Find the value ofy whenx = 3.6

0 4x

1


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Exercise 5Past Years SPM Papers

1. The variablesx andy are related by the equationy = kx4, wherek is a constant.

(a) Convert the equationy = kx4 to linear form.

(b) The following diagram shows the straight line obtained by plotting log10 y against log10x.

Find the value of;

log10y (i) log10k,

(2, h) (ii) h. (4 marks)

SPM 2005/Paper 1)

(0, 3)

0 log10 x

Answer: (a) ....

(b) (i) ...

(ii) ..

2. The following diagram shows a straight line graph ofx

yagainstx. Given thaty = 6xx

2, calculate the

value ofk and ofh. (3 marks)

x

y (SPM 2004/Paper 1)

(2, k)

(h, 3)

0 x1

Answer: k = .....

h = ....

3. The variablesx andy are related by the equationy = px2

+ qx, wherep and q are constants. A straight line is

obtained by plottingx

yagainstx, as shown in the diagram below. Calculate the values ofp and q.

(4 marks)

(SPM 2003/Paper 1)

x

y

(2, 9)

(6, 1) Answer: p= ..0 x

q = ....


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4. Diagram 4(a) shows the curvey = 3x2

+ 5.

Diagram 4(b) shows the straight line graph obtained wheny = 3x2

+ 5 is expressed in the linear form

Y = 5X + c. ExpressXand Yin terms ofx and /ory. (3 marks)

(SPM 2006/Paper 1)

y Y

y = -3x2

+ 5 X

x 00

-3

DIAGRAM 4(a) DIAGRAM 4(b)

Answer: X= .

Y = ....

5. The variables x and y are related by the equation

4

my , where m is a constant. The following diagram

shows the straight line graph obtained by plotting log10 y againstx. (3 marks)

log10y (SPM2008/Paper1)

(a) Express the equation4

my in its linear

form used to obtain the straight line graph. x

0

(b) Find the value ofm. (0, -4)

Answer: (a) ...................................................

(b) ...................................................

6. The variablesx andy are related by equation y2= 4x(10 2x). A straight line graph is obtained by plotting

x

y2againstx, as shown in the diagram below. Find the values ofp and q. (3 marks)

x

y2 (SPM2007/Paper 1)

(3, q)

0 x

(p, 0) Answer: (a) ....

(b) ......


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7. Use the graph paper provided to answer this question.

The following table shows the values of two variables,x and y, obtained from an experiment. The variablesx

andy are related by the equation y = px +px

r, wherep and r are constants.

x 1.0 2.0 3.0 4.0 5.0 5.5

y 5.5 4.7 5.0 6.5 7.7 8.4

(a) Plotxy againstx2, by using a scale of 2 cm to 5 units on both axes. Hence, draw the line of best fit.(5 marks)

(b) Use the graph from (a) to find the value of(i) p,

(ii) r, (5marks)

(SPM 2005/Paper 2)


The following table shows the values of two variables, x and y, obtained from an experiment. It is known

thatx andy are related by the equationy = pk

2x

, wherep andk are constants.

x 1.5 2.0 2.5 3.0 3.5 4.0

y 1.59 1.86 2.40 3.17 4.36 6.76

(a) Plot log10 y againstx,2. Hence draw the line of best fit. (5 marks)

(b) Use the graph in (a) to find the value of

(i) p,

(ii) k, (5 marks)

(SPM 2003/Paper 2)


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The following table shows the values of two variables, x and y, obtained from an experiment. Variables x

andy are related by the equationy = pkx, wherep andk are constants.

x 2 4 6 8 10 12

y 3.16 5.50 9.12 16.22 28.84 46.77

(a) Plot log10y againstx, by using a scale of 2 cm to 2 units on thex-axis and 2 cm to 0.2 unit on the log10y-axis. Hence, draw the line of best fit. (4 marks)

(b) Use the graph from (a) to find the value of(i) p,

(ii)k, (6 marks)

(SPM 2004/Paper 2)


Table 2 shows the values of two variables, x and y, obtained from an experiment. Variables x and y are

related by the equation 1xpky wherep andk are constants.

x 1 2 3 4 5 6

y 4.0 5.7 8.7 13.2 20.0 28.8

TABLE 2

(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 7(a) to find the value of

(i) p,

(ii)k, (5 marks)(SPM 2006/Paper 2)


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ik bid/AMF5/W4/M 2013

11. Use the graph paper to answer this question.

Table 8 shows the values of two variables,x andy, obtained from an experiment. The variablesxandy are

related by the equation 1x

p

y

k, where kandp are constants.

x 1.5 2.0 3.0 4.0 5.0 6.0

y 2.502 0.770 0.465 0.385 0.351 0.328

Table 8

(a) Based on Table 8, construct a table for the values of .11y

andx

(2 marks)

(b) Plot ,11x

againsty

using a scale of 2 cm to 0.1 unit on thex

1-axis and 2 cm to 0.5 unit on the

.1

axisy Hence, draw the line of best fit. (3 marks)

(c) Use the graph in 11(b) to find the value of(i) k,

(ii) p. (5 marks)(SPM2009/Paper 2)

12. Use graph paper to answer this question.The table below shows the values of two variables s andy, obtained from an experiment. Variablesx andy

are related by the equationy = hk2x

, wherehandkare constants.

(a) Based on the table, construct a table for the values oflog10y. (1 mark)

(b) Plot log10y againstx, using a scale of 2 cm to 1 unit on thex-axis and 2 cm to 0.1 unit on the log10y-

axis. Hence, draw the line of best fit. (4 marks)

(c) Use the graph in (b) to find the value of

(i) x wheny = 4.8,(ii) h,(iii) k. (5 marks)

(SPM2008/Paper 2)

x 1.5 3.0 4.5 6.0 7.5 9.0

y 2.51 3.24 4.37 5.75 7.76 10.00

chp 2 linear law (addmaths form 5)

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