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Np MaE ht 1996

Directions

Test period Dec 6 - 18 1996.

Test time 240 minutes without a break.

Resources Calculator (not symbolic computation) and table of formulas.

Test material The test material should be handed in together with your

solutions.

Write your name, gymnasium programme/adult education and

date of birth on all the papers you hand in.

Test The test is made up of 14 problems.

Most of the problems are of the long-answer type. With theseproblems, it is not enough to give a short answer, it requires

• that you write down what you do

• that you explain your train of thought

• that you draw figures when needed

• that you show how you use your calculator in numerical and

graphical problem solving.

For some exercises, (where it says ”Only an answer is

required”) only the answer needs to be given.

Try all of the problems. It can be relatively easy, even towards

the end of the test, to earn some points for a partial solution or

presentation.

The grading levels The teacher responsible will explain the grade levels which are

required for ”Passed” and ”Passed with Distinction”. On the test

one can attain a maximum of 57 points.

This material is confidential until the end of March 1997

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1. Find the values or expressions for A, B, C and D in the table.

Only an answer is required . (4 p)

2. Write in polar form the complex number z = + ⋅4 4 3 i (3p)

3. The number of young white-tailed eagles on the Swedish east-coast has shown a

tremendous increase since 1985. The number of young eagles y is given by the

differential equation

where t is the time in years after 1985.

a) Find how many young eagles one expects there to be on the Swedish east-

coast in 1995, according to the model. (3p)

b) What does the expressions above tell you about the number of young

eagles? (2p)

z arg z z z

1 + i A

2i B

C D 3 3i+

d

d

y

t y y= =017 0 19. , ( )

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4. a) Solve the initial-value problem¢

- = = y y y8 0 0 10, ( ) (2p)

b) Find the general solution of the following differential equation

′′ + ′ + = y y y4 13 0 (2p)

5. Find the real number t such that z t t = + + − −( )( ) ( )1 1 1 2i i is real. (3p)

6. In a brake-test, the speed of a car is measured every other second.

The results are presented in the table and diagram below.

Time in seconds 0 2 4 6 8 10 12 14

Speed in m/s 18.0 11.14 6.89 4.26 2.64 1.47 1.01 0.70

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0

t s

v m s

a) Find how far the car has gone during the first 10 seconds of

the test. (2p)

b) The speed of the car is modelled by the equation v t t ( ) .= ⋅ −18 0 24e .

Find an expression for how far the car has gone during the first 10 seconds

of the test. Using this expression, calculate the distance the car has goneduring this time. (3p)

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7. Describe the change in argument and modulus of a complex number z when it is

multiplied by1

2i (2p)

8. A function y f x= ( ) satisfies the differential equation ′′ + ′ + = y y y7 10 0 .

Also, y( )0 0= and ′ = y ( )0 3 .

Determine y. (4p)

9. A plastic box without a lid has a square bottom made of thick plastic which costs

2.00 kr/m2. Thinner plastic which costs 0.90 kr/m

2is used for the four other faces

of the box. The box has a volume of 0.020 m3.

Find the dimensions of the box which minimizes the cost of material. (4p)

10. The complex number z = 1−2i is a root of the equation z az b2 0+ + = , where a

and b are both real.

Find the values of a and b. (3p)

r

Re z

Im z

z

v

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11. The concentration of carbondioxide in the atmosphere is known to be increasing.

This is partly due to the burning of fossil fuels. Before the era of industrialisation,

the concentration of carbondioxide was 280 ppm (parts per million). In 1960 it

was 310 ppm and has since increased at a rate of 0.40% per year.

a) Write down a differential equation describing the carbondioxideconcentration y( x) ppm at time x years after 1960. (2p)

b) Some scientists believe a doubling of the pre-industrial value of

280 ppm will result in an increase in the average temperature of the earths

surface between 2 - 5 °C.

Will this doubling occur during the next century? (2p)

c) The result in b) is based on a mathematical model for the concentration of

carbondioxide in the atmosphere.

What arguments can you give against using the model for this purpose? (1p)

12. Find a function g x( ) , such that

g x dx( ) =∫ 51

4

and ′ =g ( )0 2 . (3p)

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13. a) Describe in your own words and with the help of a figure, a numerical

method for solving first-order differential equations. (4p)

b) Use the method you have described to find an approximate value for y(2),

given that

d

d

y

x y=

2 and y( )1 2=

. Use a step length equal to 0.5. (2p)

14. A scented sphere having the volume 3.0 cm3

is used to give a room a pleasant

smell. The material in the sphere evaporates gradually and the volume decreases in

a way that the rate of decrease is proportional to the area of the sphere. In one

month, the volume has decreased to 2.0 cm3.

a) Show that the rate of change for the radius r measured in cm is ddr t

k = ,

where k is a constant. (3p)

b) Calculate the volume of the sphere after 4 months. (3p)

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