Download End of year Exam Preparation questions File

1.
If x = 3.1 × 104 and y = 2.4 × 10–7, calculate the values of the following, expressing your answers in the
form a × 10k, where 1 ≤ a < 10 and k  .
(a)
x2
(b)
x
y
(Total 4 marks)
2.
The sets A, B and C are subsets of U. They are defined as follows:
U = {positive integers less than 16}
A = {prime numbers}
B = {factors of 36}
C = {multiples of 4}
(a)
List the elements (if any) of the following:
(i)
A;
(ii)
B;
(iii)
C;
(iv)
A  B  C.
(4)
(b)
(i)
Draw a Venn diagram showing the relationship between the sets U, A, B and C.
(ii)
Write the elements of sets U, A, B and C in the appropriate places on the Venn diagram.
(4)
(c)
From the Venn diagram, list the elements of each of the following
(i)
A  (B  C);
(ii)
(A  B);
(iii)
(A  B)  C.
(3)
(d)
Find the probability that a number chosen at random from the universal set U will be
(i)
a prime number;
(ii)
a prime number, but not a factor of 36;
(iii)
a factor of 36 or a multiple of 4, but not a prime number;
(iv)
a prime number, given that it is a factor of 36.
(6)
(Total 17 marks)
1
3.
Consider the numbers 5, 0.5, 5 and –5. Complete the table below, showing which of the number sets,
, and these numbers belong to.
Answers:
5
0.5
5
–5
(Total 8 marks)
4.
The diagram shows the graph of y = x2 – 2x – 8. The graph crosses the x-axis at the point A, and has a
vertex at B.
y
A
x
O
B
(a)
Factorise x2 – 2x – 8.
(b)
Write down the coordinates of each of these points
(i)
A;
(ii)
B.
(Total 4 marks)
5.
The line L1 shown on the set of axes below has equation 3x + 4y = 24. L1 cuts the x-axis at A and cuts the
y-axis at B.
Diagram not drawn to scale
y
L1
B
L2
M
O
A
x
C
2
(a)
Write down the coordinates of A and B.
(2)
M is the midpoint of the line segment [AB].
(b)
Write down the coordinates of M.
(2)
The line L2 passes through the point M and the point C (0, –2).
(c)
Write down the equation of L2.
(2)
(d)
Find the length of
(i)
MC;
(2)
(ii)
AC.
(2)
(e)
The length of AM is 5. Find
(i)
the size of angle CMA;
(3)
(ii)
the area of the triangle with vertices C, M and A.
(2)
(Total 15 marks)
Solve the equation x2 – 5x + 6 = 0.
Find the coordinates of the points where the graph of y = x2 – 5x + 6 intersects the x-axis.
6.
(a)
(b)
7.
The diagrams below show the graphs of two functions, y = f(x), and y = g(x).
(Total 4 marks)
(a)
(b)
y
y
1.5
y = f(x)
y = g(x)
1
2
0.5
1
0
–360º
1
2
x
–180º –0.5
–1
180º
360º
x
–1.5
State the domain and range of
(a)
the function f;
(b)
the function g.
(Total 8 marks)
3
8.
The four diagrams below show the graphs of four different straight lines, all drawn to the same scale.
Each diagram is numbered and c is a positive constant.
y
y
c
c
Number 1
Number 3
x
0
x
0
y
c
y
c
Number 2
Number 4
x
0
x
0
In the table below, write the number of the diagram whose straight line corresponds to the equation in the
table.
Equation
Diagram number
y=c
y=–x+c
y=3x+c
y=
1
x+c
3
(Total 8 marks)
9.
The diagrams below are sketches of some of the following functions.
(i)
y = ax
(ii)
y = x2 – a
(iv)
y=a–x
(v)
y=x–a
(iii)
(a)
y = a – x2
(b)
y
y
x
(c)
x
DIAGRAMS NOT
TO SCALE
(d)
y
y
x
x
4
Complete the table to match each sketch to the correct function.
Sketch
Function
(a)
(b)
(c)
(d)
(Total 8 marks)
10.
Two functions are defined as follows
(a)
f (x) =
4  x for 0  x  4
x  4 for x  4
g (x) =
1
x
3
Draw the graphs of the functions f and g in the interval 0 ≤ x ≤ 9, 0 ≤ y ≤ 6 using a scale of 1 cm to
represent 1 unit on both axes.
(5)
(b)
(i)
Mark the intersection points A and B of f (x) and g (x) on the graph.
(ii)
Write down the coordinates of A and B.
(3)
(c)
Calculate the midpoint M of the line AB.
(2)
(d)
Find the equation of the straight line which joins the points M and N.
(4)
(Total 14 marks)
11.
The mean of the ten numbers listed below is 5.5.
4, 3, a, 8, 7, 3, 9, 5, 8, 3
(a)
Find the value of a.
(b)
Find the median of these numbers.
(Total 4 marks)
5
12.
The table shows the number of children in 50 families.
Number of
children
Frequency
Cumulative
frequency
1
3
3
2
m
22
3
12
34
4
p
q
5
5
48
6
2
50
T
(a)
Write down the value of T.
(b)
Find the values of m, p and q.
(Total 4 marks)
6
13.
A marine biologist records as a frequency distribution the lengths (L), measured to the nearest centimetre,
of 100 mackerel. The results are given in the table below.
Length of mackerel
(L cm)
Number of
mackerel
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
100
(a)
Construct a cumulative frequency table for the data in the table.
(2)
(b)
Draw a cumulative frequency curve.
Hint: Plot your cumulative frequencies at the top of each interval.
(3)
(c)
Use the cumulative frequency curve to find an estimate, to the nearest cm for
(i)
the median length of mackerel;
(2)
(ii)
the inter quartile range of mackerel length.
(2)
(Total 9 marks)
7
14.
The sports offered at a retirement village are Golf (G), Tennis (T) and Swimming (S). The Venn diagram
shows the numbers of people involved in each activity.
2
G
T
11
6
1
3
4
4
8
S
(a)
(b)
How many people
(i)
only play golf?
(ii)
play both tennis and golf?
(iii)
do not play golf?
Shade the part of the Venn diagram that represents the set G  S.
(Total 4 marks)
15.
The table below shows the relative frequencies of the ages of the students at Ingham High School.
(a)
Age
(in years)
Relative
frequency
13
0.11
14
0.30
15
0.23
16
0.21
17
0.15
Total
1
If a student is randomly selected from this school, find the probability that
(i)
the student is 15 years old;
(ii)
the student is 16 years of age or older.
There are 1200 students at Ingham High School.
(b)
Calculate the number of 15 year old students.
(Total 4 marks)
8
16.
The table below shows the number of left and right handed tennis players in a sample of 50 males and
females.
Left handed
Right handed
Total
Male
3
29
32
Female
2
16
18
Total
5
45
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.
(Total 4 marks)
17.
In each of the Venn diagrams, shade the region indicated.
(a)
(b)
(c)
(d)
AB
A
B
A
B
A
B
A
B
The complement of (A  B)
The complement of (A  B)
A  (B  C)
C
(Total 4 marks)
9
18.
Three propositions p, q and r are defined as follows:
p: the water is cold.
q: the water is boiling.
r: the water is warm.
(a)
Write one sentence, in words, for the following logic statement:
(b)
( p   q)  r
Write the following sentence as a logic statement using symbols only.
"The water is cold if and only if it is neither boiling nor warm"
(Total 4 marks)
19.
Let
ξ = {positive integers less than 15};
X= {multiples of 2};
Y = {multiples of 3}.
(a)
Show, in a Venn diagram, the relationship between the sets ξ, X and Y.
(1)
(b)
List the elements of:
(i)
XY
(1)
(ii)
X  Y.
(2)
(c)
Find the number of elements in the complement of (X  Y).
(2)
(Total 6 marks)
20.
Let the propositions p, q and r be defined as:
p : Matthew arrives home before six o'clock
q : Matthew cooks dinner
r : Jill washes the dishes
(a)
(i)
Express the following statement in logical form.
If Matthew arrives home before six o'clock then he will cook dinner.
(1)
(ii)
Write the following logic statement in words.
qr
(1)
10
(b)
(i)
p q r
Copy and complete the truth table below.
pq
q  r r
(p  q)  (q  r)  r
p
[(p  q)  (q  r)  r]  p
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
(5)
(ii)
Explain the significance of the truth table above.
(2)
(Total 9 marks)
11