IB Math Studies : Semester Exam, May ‘02


NAME:
Circle or box the solution to each question clearly.
Marks will be given for your working in each qu.
TIME: 2 hours
INSTRUCTIONS
The exam consists of two sections, A and B. Section A has 10 short answer questions which should take you about 45
minutes. Section B has 2 long answer questions which should take you about 45 minutes.
You will need a pencil, eraser, ruler, calculator and 1 sheet of graph paper (provided).
SECTION A
Do all working in the space provided. If you need more space use the back of the last page.
(1) Calculate 3.7  16.2  500 , writing your answer to:
2
a) 2 decimal places
b) 3 significant figures
c) in the form a  10 , where 1  a  10 , k  Z
k
(2) A rectangular garden is measured to the nearest metre. Its length is 17m and its width is 8m .
(a) Write the range of possible values for the length of the rectangle in the form: a  length  b
(b) Calculate the minimum possible area of the rectangle
(3) a) Solve the equation x  5 x  6  0
2
b) Hence find the coordinates of the points where
the graph of
y  x 2  5x  6 intersects the x-axis
(4) The Venn diagram shows the number of people at a club who take part in various sporting activities, where:
A = {members who play Archery}
B = {members who play Badminton}
C = {members who play Croquet}
Find the number of people who:
a) play only Croquet
b) play both Archery and Badminton
c) do not play Croquet
d) Shade the part of the Venn diagram that represents the set
(5) Six Venn diagrams are illustrated below:
C  B
a) ignoring the shading, which of the diagrams best illustrates:
i) A  B  A
ii) A  B  
b) in which Venn diagram does the shading represent:
i) the complement of B
ii) the complement of A  B
(6) On the diagram below,
iii) only A
O is the origin. Draw and label directed line segments to represent the vectors:
3 
OA   
 2
 0 

b) AB  
  6
a)
c) Calculate the length of
OB
L1 & L2 are parallel
a) What is the gradient of L1 ?
(7) In the diagram lines
b) Write down the eqn of
L1 in the form y  mx  c
c) Write down the eqn of
L2 in the form ax  by  c  0
(8) The height of a vertical cliff is 450m . The angle of elevation from a ship to the top of the cliff is 23 . The ship is
x metres from the bottom of the cliff.
a) Draw a diagram to show this information
b) Calculate the value of
x
(9) The diagram shows the plan of a playground with dimensions as shown. Calculate:
a) the length BC
b) the area of the triangle
ABC
(10) In the diagram, triangle
Find:
a) the size of angle
ABC is isosceles. AB  AC , CB  15cm and angle ACB  23
CAB
b) the length of AB
(11) The table shows the number of children in 50 families:
Number of Children
1
2
3
4
5
6
Frequency
3
m
12
p
5
2
T
Cumulative Frequency
3
22
34
q
48
50
a) Write down the value of T
b) Find the value of m, p and q
(12) A bag contains 2 Red, 3 Yellow and 5 Green sweets. Without looking, Mary takes one sweet out of the bag and
eats it. She then takes out a second sweet. Find the probability that:
a) if the first sweet is green, the second sweet is also green
b) if the first sweet is not red, the second sweet is not red
(13) In a school, 180 pupils are asked which is their favourite sport.
pie chart shows the result of the survey. The diagram is not drawn to
a) Calculate the angle of the sector representing rugby
The
scale.
b) Estimate the probability that a pupil’s favourite sport will be hockey
(14) The table below shows the number of left and right handed tennis players in a sample of 50 males and females:
Male
Female
Total
Left Hand
3
2
5
Right Hand
29
16
45
Total
32
18
50
If a tennis player was selected at random from the group, find the probability that the player is:
a) male and left handed
b) right handed
c) right handed, given that the player selected is female
(15) The mean of the ten numbers listed is
a) Find the value of a
b) Find the median of these numbers
5.5 : 4, 3, a, 8, 7, 3, 9, 5, 8, 3
(16) The length and width of 10 leaves are shown on the scatter diagram below
a) Plot
the point
M  (97,43) which represents the mean length and the mean width
b) Draw a suitable line of best fit
c) Write a sentence describing the relationship between leaf length and leaf width for this sample
SECTION B
(1) i) Let U = {positive integers less than 15}
X = {multiples of 2}
Y = {multiples of 3}
a) Show, in a Venn diagram, the relationship between the sets U, X and Y
b) List the elements of: i) X  Y
ii) X  Y 
c) Find the number of elements in the complement of ( X  Y )
[1 mark]
[3 marks]
[2 marks]
ii) Three propositions are defined as follows:
p: the oven is working
q: the food supply is adequate
r: the visitors are hungry
a) Write one sentence, in words only, for each of the following logic statements
i) q  r  p
ii) r  ( p  q )
[4 marks]
b) Write the sentence below using only the symbols p, q and logic connectives
[2 marks]
“If the oven is working and the food supply is adequate then the oven is working or the food supply is adequate”
c) A tautology is a compound statement which is always true. Use the truth table below to determine whether or not
your answer to part (b) is a tautology.
[3 marks]
p
q
T
T
T
F
F
T
F
F
(3) A marine biologist records as a frequency distribution the lengths (L), measured to the nearest cm, of 100 mackerel.
The results are given in the table below:
Length of Mackerel
(L cm)
Number of
Mackeral
27 < L  29
2
29 < L  31
4
31 < L  33
8
33 < L  35
21
35 < L  37
30
37 < L  39
18
39 < L  41
12
41 < L  43
5
f i xi  x 
Cumulative
Frequency
2
100
a) Complete the cumulative frequency column for the data in the table above
[2 marks]
b) On the grid below, draw a cumulative frequency curve, using a scale of 1cm = 2 units on the x axis and
1cm = 10 units on the y axis. Hint: Plot your Cumulative Frequency at the top of each interval
[3 marks]
c) Use the cumulative frequency curve to find:
i) an estimate, to the nearest cm, for
the median length of the mackerel
[2 marks]
ii) an estimate, to the nearest cm, for the
interquartile range of mackerel length
[2 marks]
e) the percentage of mackerel that
are less than 32 cm in length
d) the percentage of mackerel that
are more than 40 cm in length
[1 mark]
[2 marks]
e) If the Mean ( x ) length of the mackerel is 36cm, complete the last column in the table above and hence find the
standard deviation for the data:
[3 marks]
NOW GO BACK AND CHECK YOUR WORK!