National Test in Mathematics Course D SPRING 2005 Directions
National Test in Mathematics Course D SPRING 2005 Directions
National Test in Mathematics Course D SPRING 2005 Directions
Concerning test material in general, the Swedish Board of Education refers to the Official Secrets Act, the regulation about secrecy, 4th chapter 3rd paragraph. For this material, the secrecy is valid until 10th June 2005.
Part I
This part consists of 9 problems that should be solved without the aid of a calculator. Your solutions to the problems in this part should be presented on separate sheets of paper that must be handed in before you retrieve your calculator. Please note that you may begin working on Part II without your calculator.
1.
Evaluate ( x 2 1) dx
1
(2/0)
2.
Determine f (x) if a) b) c)
f ( x) = 4 cos 3 x
f ( x) = (3 2 x) 6 f ( x) = x 2 e 3 x
3.
Which two of the functions F (x) below are the antiderivatives to f ( x) = 3 x 5 + 1 ? Only answer is required
(1/0)
A B C D E F
F ( x) =
3x 4 4
F ( x) = 15 x 4
F ( x ) = 0 .5 x 6 + x F ( x) = x 6 + 2 x
F ( x) =
x6 + x +1 3
x6 F ( x) = + x 14 2
4.
Arrange the following numbers according to size: a = sin 24 , b = cos 100 and c = sin 165 Justify your answer.
(1/1)
5.
The figure shows the graph of the function y = a + b sin 2 x Determine the constants a and b. Only answer is required
(1/1)
6.
Which one of the following expressions A F can be simplified to 1? Only answer is required A B C D E F
(sin x + cos x)
2
(0/1)
(sin x cos x) 2
(sin x + cos x)(sin x cos x) cos x(tan x sin x + cos x)
7.
The number of starlings in Sweden has been investigated since 1979. The results of this investigation can be described mathematically by the differential equation:
dy = 0.03 y , where y is the number of starlings at the time t years from 1979. dt
Explain, in your own words, the meaning of the differential equation in this context.
(1/1)
8.
(0/1/)
9.
The function F is the antiderivative to f The figure below shows y = F (x) Determine
f ( x ) dx
0
(0/2/)
Part II
This part consists of 8 problems and you may use a calculator when solving them. Please note that you may begin working on Part II without your calculator.
10.
In the triangle ABC the sides AC and BC are of equal length. Calculate the area of the triangle.
(2/0)
11.
Use the antiderivative to calculate the area of the region enclosed by the functions f ( x) = x 2 + x + 1 and g ( x) = 9 x (3/0)
12.
Daniel and Linda are looking at a flat. According to the information received the living-room is 31.2 m2. They want to check if this is correct so they measure the walls and draw a sketch of the living-room. They know that one corner of the room is right-angled. Their sketch looks like this:
6.08 (m)
5.25
4.50
6.02
What is the area of the living-room according to Daniels and Lindas sketch?
(2/2)
13.
(2/1)
14.
Determine the number of solutions to the equation sin 2 x = where x is measured in radians.
x2 1 , 10 (1/1)
15.
A sheet of corrugated iron is made by pressing a flat sheet into curving folds. Seen from the side, the corrugated iron in the picture has the shape of a sine curve with period 0.20 m and amplitude 0.050 m.
a)
(0/1)
There is a formula for calculating the length of a curve. According to this, the length s of a curve y = f (x) from x = a to x = b can be calculated from
s = 1 + ( f ( x)) 2 dx
a
b)
How long a flat iron sheet should you start with in order to get a piece of corrugated iron with a length of 5.0 m?
(0/3/)
16.
For which values of the constants a and b is it true that the function f ( x) = ax 2 + bx sin 3 x has a local maximum when x = 0?
(1/2/)
When assessing your work with this problem your teacher will take take into consideration: How well you carry out your calculations How well you justify your conclusions How well you present your work How well you use the mathematical language 17. The figure shows a parabola and a rectangle in a coordinate system. The shaded region is enclosed by the parabola and the x-axis. The area of the shaded region will from now on be referred to as the area of the parabola.
Two of the corners of the rectangle coincide with the points where the curve intersects the x-axis. One of the side of the rectangle touches the maximum point of the curve.
In this problem, you are going to investigate the relation between the area of the parabola and the area of the rectangle. Let the equation of the parabola be y = b ax 2 , where a and b are positive numbers. You may for example start by letting b = 9 and a = 1 and draw the graph of the function y = 9 x 2 . Then determine the relation between the area of the parabola and the area of the rectangle. Choose other examples yourself and try to formulate a conclusion based on your chosen examples. Investigate if your conclusion also holds for the general case with the parabola y = b ax 2 (3/4/)