Assignment 2010 Answers Final

8/8/2019 Assignment 2010 Answers Final

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(c) Step Law applied

(d) Step Law applied


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(e)

Step Law applied

Question 3

(a)Let A, B, C and D be matrices defined by

Which of the matrices are defined and compute the matrices which are defined? If not

defined, explain why.


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Question 4

(a) Evaluate the determinant of the following matrix using the cross-multiplicationmethod.


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(ii) the cofactor matrix;

(iii) the adjoin matrix;

(iv) the determinant; (v)

inverse matrix.


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Question 5 Consider the system of linear equations

(a)Write down the system in matrix form.

(b)Use the inverse matrix method to solve the system


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(c)Confirm your answer to part (b) using the Cramers rule.


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----------------------------------------------------------------------------------------------------------------------Question 6

Consider the functions defined by

, and (a)

Find

(b)Express in the form where and are integers.


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(c)Use your answer in part (b), or otherwise, to write down the domain, denoted by

and range of, denoted by .(d)Find the domain and range of the functions and

(e)Find

and simplify your answer.


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(f) Sketch the graphs of and Clearly label the points where the graphs

cut the x-axis, y-axis and the turning point (if any) on your diagram.

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

24 11 0 -9 -16 -21 -24 -25 -24 -21 -16 -9 0 11 24

-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

-25

-20

-15

-10

-5

5

10

15

20

x

y

(3,0)

(2,-25)


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x -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 0.0 1.0 1.4 1.7 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.3 3.5

Question 7

Points A(-1, -2), B(7, 2) and C(k, 4), where is a constant, are the vertices of a triangle

-12 -10 -8 -6 -4 -2 2 4 6 8 10 12 14 16

-14

-12

-10

-8

-6

-4

-2

2

4

6

8

10

12

14

x

y

(Turning pointis when X value is

.

(The graph nevercutsthrough neitherx

axisnory

axisin any otherpoint.

(0,

1.

(0,

The graph neithercutsthrough x

axis, norhave a turning point.

-

-

.

-

-

.

-

-

.

-

-

.

.

.

.

.

.

.

.

7 7.

.

-

-

-

-

-

7

x

y


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ABC. The gradient of the line joining BC is -2.

(a)Calculate the value of k.

(b)Find the gradient of AB.

(c)Show that .

(d)Find an equation of the line joining AB.

(e)Find an equation of the line joining BC


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(f) Find the exact area of the triangle ABC.

Question 8

(a)Find an equation of a line that passes through the point (1 , 3) and is parallel to

(b)Consider the graph below

(i) Find the gradient of PQ.

(ii) Find the equation of OP.

y

xx

P(1 , 4)

Q(0, 6)

R O(0 , 0)


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(iii) Find the x-intercept of QP.

(iv) Find the equation QR if QR is parallel to OP.

Question 9

Long Life is a group ofcompanies specialized in the production ofalkaline batteries. The

following information is extracted from the company for the year2008.

Cost ofrent $25000

Cost ofmaterials per unit $1.25

Cost of labour per unit $0.25

Other variable costs per unit $0.5


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Selling price per unit $5

Let be the quantity produced.

(a)Obtain the cost function in terms of

(b)The cost of producing 15000 units.

(c)What is the revenue function? (d)Find the break-even point.

(e)Sketch the revenue function and cost function on the same set ofaxes and clearly label

the break even point, the region in which the business is gaining profit and the region in

which the business is suffering loss.


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In 2009, it costs the Long life company $234for the production of 26 units and $1404 for

producing 156 units. The company realizes that the cost is following a linear trend and would

like to derive a mathematical formula relating cost and number units produced.

(f) Derive the linear relation for the year 2009.

(g)Predict the cost of producing 75000 units under this trend.

5000 10000 15000 20000 25000 30000 35000 40000 45000 50000 55000 60000 65000

5000

10000

15000

20000

25000

30000

35000

40000

45000

50000

55000

60000

65000

70000

75000

80000

85000

90000

95000

100000

105000

110000

115000

quantity

$

Break e

en po in

Loss

Profi

!

otal Re

enue

Total Cost

Q9"

e


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Question 10

(a)Factorize the expression completely

(b)Solve the equation using the factor method

(c)Sketch the graphs with equations

and on the same set of axes and find the coordinates of the intersection point(s). 0 1 2 3 4 5 6 9 4 1 0 1 4 9

0 1 2 3 4 5 6 -1 1 3 5 7 9 11

1 2 3 4 5 6 7 8 9 10 11 12 13 14

-1

1

2

3

4

5

6

7

8

9

10

11

12

13

x

y

y=

y=

6

+


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Intersection points:


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(d)Find the quantity for which the cost , is minimized and hence find theminimum cost. Also sketch the cost function.

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170

500

1000

1500

2000

2500

3000

x

y

x

-150x+6125


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Question 11

(a)Simplify the following without using a calculator.

(b)Find if

(c)Solve for x (d)Solve the equation

(e)Solve the equation

(f)E

xpress the expression in terms of and

(g)Express the expression as a single logarithm in base b.

Question 12

(a)Azeeza wants to save money to buy a motor cycle for her son. Shewants the 10000 cycle in 5 years. The bank is offering 3% interest on

savings compounded annually. What amount of money will she need to

invest now in order to have the cycle in 5 years?


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(b)An investment of 100 pays 7.50 percent compounded quarterly. Themoney is left in the account for 15 years. How much money is in the

bank after 15 years?

p = $100

r = .50%

n = 15

k = 4

S =?

(c)The initial population of a group of animals is 20000. Due to some

disease, the animals die out at a rate of 3% per day. What is the

population of the animals in 20 days time? How long does it take for

the population to decrease half of its initial size?

i)

animals after20 days


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ii)

23(d)The price of an acre of land increases according to an exponential

growth model. The price in 2005 was 35000 per acre and the price in

2010 is 64000. What is the growth rate in percentage? How much will

it be the price in 2020?


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END OF QUESTIONS

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Assignment 2010 Answers Final