10 - Linear Law

St Joseph’s Institution Secondary Four Mathematics TOPIC − Linear Law Name:_____________________________________ (

Q1)

) Class: ___________

Kepler’s third law states that the period of a satellite around a planetary body (T) is related to the distance of the satellite from the body (R) by the formula: T = aR b An experiment was conducted on a planet in a galaxy far far away and the following table gives the values of T corresponding to values of R. T(/h)

8

125

353

649

1000

R(km)

40

250

500

750

1000

Use the data above in order to draw, on graph paper, the straight line graph of lg T against lg R. [4] Use your graph to estimate (i) (ii) (iii) Q2.

the values of a and b the value of T when R = 100 km; the value of R when T = 750 h.

The table shows experimental values of two variables, x and y. x 1.0 1.4 1.8 2.2 2.6 y 2.12 2.31 2.63 3.05 3.57 2 3 It is known that x and y are connected by the equation ay − bx = 1 . By plotting y 2 against x 3 . Obtain a straight line to represent the above data. Use your graph to estimate the value of a and of b. By drawing a suitable straight line, find the value of x and of y which satisfy the simultaneous equations. ay 2 − bx 3 = 1, y 2 − x 3 = 1.

© Jason Ingham 2009

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Q3. Answer the whole of this question on a sheet of graph paper. The table shows experimental values of two variables x and y. x y

0.20 0.059

0.25 0.071

0.50 0.125

1 0.200

2 0.286

It is known that x and y are connected by the equation

a 1 + = b where a x y

and b are constants.

(a) Plot

1 1 against and obtain a straight line graph y x

(b) Use your graph (i) to find the value of a and of b. (ii) to estimate the value of x when

Q4.

1 = 7. y

The table below shows experimental values of x and y which are known to be related by the equation y = ax 2 + bx . x y

(a)

1 27

Plot

2 60

3 101

4 148

5 200

y against x and use the graph to estimate the value of x

a and b .

2

6 258

Q5)

The figure below shows a straight line graph obtained by plotting

y −1 x

against x . i) ii)

Express y in terms of x . When y = 2 , find the value(s) of x correct to 2 decimal places.

y −1 x

(-2,3)

x (-3,0)

3

Q6)

(a) The table shows experimental values of two variables x and y. x y

1 −0.5

2 0

3 0.86

4 2

5 3.35

It is known

that x and y are connected by an equation of the form y = cx x + d x , where c and d are constants. y (i) By plotting against x, obtain a straight line graph to represent x the above data. (ii) Use your graph to estimate the value of c and of d. (iii) Hence, obtain the value of x when 4y = x (b) The variables x and y are related in such a way that when xy is plotted against y, a straight line is obtained passing through the points  4, 6  and  8,3 . Find the value of the gradient of the straight line obtained when

1 is plotted against x. y

Q7) Answer The Whole Of This Question On A Sheet Of Graph Paper. The table shows experimental values of two variables x and y. x y

1.0 0.83

1.5 0.61

2.0 0.50

It is known that x and y are related by the equation

2.5 3.0 0.42 0.38 x  ax  b x , where a y

and b are constants. 1 (a) Plot against x to obtain a straight line. y (b) Use your graph to estimate the value of a and of b. (c)

On the same diagram, draw the straight line representing the equation y x 1 and write down the value of x given by the point of intersection of the two lines.

(d)

Evaluate the value of the gradient of the straight line obtained when 1 1 is plotted against . y x x

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