Download Packet 1-24 - Amundsen High School

S-01
The Number System
1. Put T (true) or F (false) beside each statement.
(a) -3 is not an integer
(b) 6 is a rational number
(c)
7
is a rational number
9
(d)
(e)
7
is a natural number
9
(f)
(g)
7
is a real number
9
3 is irrational
(h) a natural number can’t be negative
4 is irrational
(i) an integer is always rational
(j) 0.2777654 is irrational
(k) -7 is in
(l) 150% is an integer
2. Place each number from the box in its correct place in the diagram below:

-5
0.2
7
2
3

19
1
7
3
0
6
R
Q
Z
N
3. Give an example of:
(a) an irrational number which lies between 1 and 2
(c) a number which is in
but not in
(b) a number which is in
but not in
(d) a rational number which lies between
4. Two natural numbers are chosen at random. Put true or false beside each statement:
(a) If the two numbers are added the answer must be a natural number.
(b) If one number is divided by the other, the answer must be a natural number.
(c) If one number is subtracted from the other, the answer must be a natural number.
1
2
and
3
2
S-02
Accuracy and Estimation
When you’re doing a calculation on you r calculator and the answer is not exact the
rule is to give answers
general
correct to 3 significant figures
unless, of course, the question gives other instructions such as “correct to 4 decimal places”.
1. Use the 3 significant figures rule for these:
(a) 11.32
(b)
2.3
4.286 11.77
(c)
232  382
(c) 7.382
(d)
(e) 37.6  5.33
(f) 2.083
4. Give the answer correct to the specified level
of accuracy. “dp” means “decimal places” and
“sf” means “significant figures”.
(a) Find the area. [2sf]
(d) 17.7% of 10.89
(b)
1 2
(g) 120  49
(h) A motor-cyclist travels 18.5km in 11
minutes. Find the average speed
(i) in km per minute
(ii) in kmh-1
(i) Find the volume of a cuboid which is
4.6cm by 6.9cm by 8.2cm.
(j) (3  3)
1.85m
83cm
(e) Find the area of a rectangular field that
237m by 124m.
(f)
77.5
3
3 (cube root!)
[6dp]
(c) A school fund-raising event raises 226,432
Japanese yen. There are 487 students in the
school. Find the average amount raised per
student. [nearest whole number]
(d) 1.277  4389
(e)  71.7 2  24.3
(f) 0.2755
[4sf]
[2sf]
[3sf]
4
One situation where the 3 significant
figures rule doesn’t make much sense is
with MONEY. Often it is reasonable to give
2 decimal places : €12.56.
2. (a) When would it not make much sense to
give 2 decimal places for money?
Give 2 decimal places here :
(b) Find 3.7% of $583.25
(c) €130  17
(d) Petrol costs €1.19 a litre. Find the cost of
8.55 litres.
3. Estimate these without a calculator. Give your
estimate correct to 1 significant figure.
(a) 9.7  4.6
(b) 12.6  1.95
(g) There are about 270 million people in the
USA. Estimate the number who are female
and left-handed and aged over 70 years old.
[1sf]
(h)
1
(1  0.9856)3
[nearest thousand]
5. Find the percentage error (correct to 2
significant figures).
true value
approx value
68.544
107.4
2
69
110
1.41

22
7
50
49
% error
S-03
Standard Form : a  10k (where 1  a  10 and k 
1. (a) One of the statements in (b) – (g) below is
wildly wrong! Which one?
In (b) to (g) put the number used in the
statement into the form a 10k , where
1  a  10 and k  .
(b) A sheet of normal A4 paper weighs about
0.000005 tonnes.
(c) The total population of China and India is
about 2.1 billion people.
3.
These can be done purely by brain-work
or purely by calculator :
p  4  106
q  6  105
r  3  108
Find the value of these, giving answers in the
form a 10k , where 1  a  10 and k  :
(a) pr
(b)
q
p
(c)
p
(d)
r
20
th
(d) On your 17 birthday you are about
536,000,000 seconds old.
(e) There are about 500,000,000,000 atoms in
a slice of toast.
(e) q 2
(f) p + r
(f) A normal sized novel might contain about
120,000 words.
(g) pqr
(h)
(g) If you throw four normal 6-sided dice,
your chance of getting four 6s is about
0.00077.
r
q
(i) 6% of q
(j) qr + p
2. Put these numbers in the form a 10 , where
1  a  10 and k  :
k
4. Given
f  5.5  105
g  1.6  107 :
(a) 427000
(b) 0.000042
(c) 108,000,000,000,000
(a) Which is larger, 7f or
(b) Solve for x :
g
?
5
fx = g
(d) 0.00000000000000155
1
in the form a 10k , where
fg
1  a  10 and k  .
In these, perform the calculation then make
sure the value of a (in 1  a  10 ) is given
correct to three significant figures:
(d) True or false: g + 100f = 71000000
(c) Write
(e) 1.270
(f) 0.2275  0.00986
)
(e) Calculate f 20 giving your answer in the
form a 10k , where 1  a  10 and k  .
1
(f) Write as a normal decimal:
g
S-04
Arithmetic Sequence
1. Write down the next two terms in these
arithmetic sequences:
(a)
(b)
(c)
(d)
(e)
7 , 11 , 15 , ...
3.8 , 5.1, 6.4 , ...
20 , 18 , 16 , ...
33 , 19 , 5 , ...
1 1 1
, , ,...
6 3 2
6. Find the sum of the first 10 terms of each
arithmetic sequence:
(a)
(b)
(c)
(d)
7. Here you can see in the square brackets how
many terms of the arithmetic sequence to sum.
2. The first four terms of some sequences are
given below. Which ones are not arithmetic
sequences?
(a)
(b)
(c)
(d)
(e)
1 , 2 , 4 , 8 , ...
2 , 4 , 6 , 8 , ...
1 , 3 , 6 , 10 , ...
5 , 1 , –3 , –7
40 , 20 , 10 , 5
3. In this question you are given the first two
terms of arithmetic sequences. That’s enough
to be able to work out the term shown in
square brackets.
(a)
12 , 19 , ...
(b)
4.6 , 5.2 , ...
(c)
30 , 26 , ...
(d)
1 , 3 , ...
(e)
36 , 34.5 , ...
(f)
100 , 97 , ...
(g)
22 , 22.8 , ...
 u 20 
 u13 
 u10 
 u 40 
 u 25 
 u 22 
 u 25 
4. The three sequences below are all arithmetic.
Find the values of the letters a to g:
4 , a , 16 , b
c , 21 , d , e , 27
–3 , f , 10 , g
5. In each arithmetic sequence below, find the
least value of n such that u n  100 .
(a)
(b)
(c)
17 , 25 , 33 , ...
–25 , –18 , –11 , ...
24 , 33 , 42 , ...
4 , 7 , 10 , ...
2 , 2.3 , 2.6 , ...
8 , 6.5 , 5 , ...
5 , 10 , 15 , ...
(a)
13 , 16 , 19 , ...
(b)
1.2 , 1.6 , 2 , ...
(c)
–20 , –14 , –8 , ...
(d)
1 3 1
, , , ...
4 8 2
(e)
2 , 4 , 6 , ...
S14 
S30 
S15 
S25 
S100 
8. The nth term of an arithmetic sequence is given
by the formula:
u n  5  3n
(a) Find u1 , u 2 and u 3 .
(b) Find S16 .
9.
13 + 17 + 21 + ..... + 49 + 53
(a) How many terms are there in this sum?
(b) Now find the sum.
10. Find: (a) 11 + 18 + ..... + 81 + 88 + 95
(b) –3 + 1 + 5 + ..... + 45 + 49
11. A child’s parents put $100 into a bank account
for her on her first birthday, then $120 on her
second birthday, and so on, increasing the
amount by $20 each year. How much will they
have paid into her account the day after her
18th birthday?
12. The second term of an arithmetic sequence is
32 and the fourth term is 26. Work out:
(a) the common difference
(b) the first term
(c) S18 .
S-05
1.
2.
3.
Geometric Sequences
Give the next two terms of each geometric
sequence:
(a)
3 , 6 , 12 , 24 , …
(b)
1 , 3 , 9 , 27 , …
(c)
0.2 , 1 , 5 , 25 , …
(d)
80 , 40 , 20 , 10 , …
(e)
2 , –6 , 18 , –54 , …
5.
20 , 30 , 45 , …
(b)
2 , 5 , 12.5 , …
(c)
54 , 36 , 24 , …
(d)
48 , 36 , 27 , …
1.5 , 3 , 6 , 12 , … [ u10 ]
(b)
0.2 , 0.6 , 1.8 , …
[ u12 ]
(c)
4 , –12 , 36 , …
[ u8 ]
6.
Give answers correct to 4 significant figures:
4.
(b) 0.2 , 0.6 , 1.8 , …
S9 
(c) 3 , 6 , 12 , …
S8 
(d) 64 , 96 , 144
S10 
(e) 4 , 5 , 6.25 , …
S20 
(f) 4 , –6 , 9 , …
S8 
S15 
Give these correct to 4 decimal places
(a)
(d)
4,6,9,…
[ u15 ]
(e)
12 , 4 , 4
[ u15 ]
(f)
1 , 0.5 , 0.25 , …
(g)
10 , 8 , 6.4
,…
S10 
(g) 8 , 6 , 4.5 , …
For each geometric sequence find the term
given in square brackets:
3
(a) 1 , 2 , 4 , …
Give these correct to 3 significant figures:
For each geometric sequence below find (i)
the common ratio, and (ii) the next term:
(a)
Find the sum of the required number of terms
of each geometric sequence:
,
1
9
,…
S6 
(i) 1 ,
1
3
,
1
9
,…
S18 
Find the final value of the investments,
correct to the nearest whole number. Interest
is compounded annually.
€5000
$25000
₤1000000
$1
Interest
(per
annum)
2%
1.5%
5%
5%
Time
FINAL
VALUE
10 yrs
4 yrs
18 mths
100 yrs
7.
If 10000 pesos are put in an account for 3
years earning 7.5% interest per annum
compounded annually, how much interest
will be earned (correct to the nearest peso)?
8.
A child’s parents put $100 into a bank
account for her on her first birthday, then
$120 on her second birthday, and so on,
increasing the amount by 20% each year.
How much (correct to the nearest dollar) will
they have paid into her account the day after
her 18th birthday?
9.
Use a GDC method.
[ u 20 ]
Use your GDC to solve these. (Trial
(a) In the geometric sequence 6 , 15 , 37.5 ,
… what is the first term that is greater than
200?
(b) In the geometric sequence 14 , 7 , 3.5 , ..
what is the least value of n such that
u n  0.01 ?
(c) In the geometric sequence 5 , 6 , 7.2 , …
what is the least value of n such that u n  50 ?
1
3
Amount
invested
[ u12 ]
and
error is an acceptable method, but show
some working!)
(h) 1 ,
$5000 was put in an
account earning r% interest per annum
compounded annually. At the end of 5 years
the value of the investment was $5600. What
is the value of r correct to 2 decimal places?
S-06
Pairs of Linear Equations
Answer on this sheet
1. These are easy to solve without a GDC:
(a) a + b = 11
a – b = 3
(b) a + b = 11
2a
– b = 1
(c) x + 2y = 15
3x
– 2y = 29
(d) p + q = 17
p – 2q = -1
(e) f – 3g = 7
2f + g = 21
(f) 2y + z = 11
y + z = 8
(g) m + 3n = 14
2m – n = 0
(h) x + y = 21
2x + 3y = 50
(i) 3a – 5b = 1
b = a - 3
2. Using the graph on the right, write down
solutions to these pairs of linear equations.
Estimate answers correct to 1 decimal place
where necessary.
3.
(a) 2x + 3y = 11
4x – y = 8
(b) x + y = –4
x – 2y = 6
(c) 2x + 3y = 11
x – 2y = 6
(d) a – 2b = 6
4a – b = 8
Use your GDC to write down solutions to these pairs of linear equations. Give answers correct to 2
decimal places where necessary.
(a) 7x + 11y = 30
x – 5y = 11
(b) 3x + 7y = 20
5x – 3y = 30
(c) 3x + 5y = 16
2x + 9y = 39
(d) y = 11x – 5 (e) 9x – 5y = 20
5x + 6y = 100
7x – 4y = 10
S-07
1.
Quadratic Equations
Write down the solutions to these quadratic
equations:
3.
(a) x 2  4
These need re-arranging then factoring then
solving:
(a) x 2  2x  24
(b) (x  2)(x  5)  0
(c) (x  3)(x  4)  0
(b) x 2  x  20
(d) (x  6)(x  6)  0
(e) (x  1)(x  4)  0
(c) x 2  x  6
(f) x(x  8)  0
(g) (x  3)2  0
(h) (x  2)(2x  5)  0
(d) x(x  3)  10
(i) (3x  2)(4x  7)  0
2.
These need factoring before you can solve
them:
4.
(a) x 2  8x  12  0
(b)
x 2  4x  3  0
(c)
x  5x  4  0
(d)
x 2  x  12  0
(e)
x 2  7x  10  0
Use your GDC or “the formula” or any other
method to solve these:
(a) x 2  4x  13  0
(b) x 2  x  100
(c) 3x 2  x  12  0
2
(d) x(x  5)  22
(e) (x  3)2  10
(f) 20  2x  3x 2  0
(g) x 2 
(f)
x 2  4x  21  0
5.
(g)
x 2  2x  15  0
(h)
x 2  4x  0
(i) x 2  x  30  0
(a)
50  3x
4
Write down a quadratic equation
whose roots are x = 2 and x = 4.
(b)
What happens if you try to solve
x 2  4x  7  0 ?
(c)
Use any method to solve
x(2x  5)  (x  1)(x  4) .
(d) Write down three roots of the equation.
(x  1)(x  4)(x  2)  0 .
S-08a
Word Problems
Involving Equations
S-08b
Word Problems
Involving Sequences
1. A coffee and three donuts will cost you $2.50,
while three coffees and five donuts will cost
you $5.30.
If a coffee costs c cents and a donut costs d
cents, make two linear equations and solve
them by any method.
How much would you pay for two coffees and
seven donuts?
2. A rectangle whose area is
80cm2 is 4cm longer
than it is wide – see
diagram.
x
x+4
1. $20000 is invested at 4% interest. Below, find
the value of the investment (i) if it is simple
interest, and (ii) if it is compounded annually.
Answer correct to the nearest dollar.
(a) after 1 year
(c) after 20 years.
(b) after 5 years
2. The human population of the earth is currently
about 6 billion. Find (in billions, correct to one
decimal place) what the population would be
in 50 years time if it grew at
80cm2
Solve the quadratic
equation suggested by
the diagram and find the perimeter of the
rectangle.
3. If you subtract twice the son’s age from the
mother’s age the answer is 12.
(a) 1% per annum
(c) 5% per annum.
(b) 3% per annum
3. 7 sections of the spiral below have been
drawn. From the center outwards the lengths
are 1 , 1 , 2 , 2 , 3, 3 and 4 units.
Find the length of the spiral after 20 sections
have been drawn.
If you double the mother’s age and add it to
the son’s age the answer is 99.
Form two linear equations and find the age of
each.
4. When two consecutive odd integers are
multiplied together the answer is 4623.
(a) If the smaller of the two integers is n, what
is the larger?
(b) Form a quadratic equation in n and solve
it.
5. A box contains x 5-cent coins and y 2-cent
coins. There are 33 coins altogether and their
total value is €1.02.
Form a pair of linear equations and solve
them.
4. A woman invested $1000 at 3% simple
interest on 1 Jan 1995, then another $1000 at
3% simple interest on 1 Jan 1996, and so on.
Find the total value of her investment on 31
Dec 2005.
5. A car is estimated to lose 20% of its value for
each year of its age.
Find (correct to the nearest hundred dollars)
the value of a car which costs $35000 new,
after:
(a) 1 year
(b) 2 year
(c) 5 years
(d) 10 years
S-09
Sets 1 : Notation
1. U = {a, b, c, d, e, f, g, h}
2. U = {x x 
and 2  x  12 }
A = {a, d, f, h}
B = {b, c, e, g}
C = {a, c, f}
D = {b, c, d, e}
List the elements of:
(a)
B C
W = {multiples of 3}
X = {factors of 24}
Y = {prime numbers}
List the elements of W, X and Y. Check your
answers are correct before continuing!
Now list the elements of:
(b)
C D
(c)
C
(d)
C  B
(e)
( B  C )
(f)
D
(g)
D  C
(h)
D  A
(i)
B
(f)
n(U )  10
(j)
B   D
(g)
{3, 6}  (W  X )
(k)
( B  D)
Find:
(a)
W
(b)
X Y
(c)
(W  X )
(d)
X  Y
(e)
W   X  Y 
True or false:
(h)
Find:
(l)
n(U )
(m)
n(C  D)
(n)
n( A  B )
True or false:
n(W  Y )
3. U = { , , , , ,  }
A = { , , ,  }
B = { ,  }
C = { ,  }
D = { ,  }
F = { , , ,  }
(o)
A B  
E = { , ,  }
(p)
g  B
(a) Which sets are subsets of A?
(q)
f  C
(b) C is a subset of two other sets (apart from
(r)
(B  C)  D
U). Which ones?
Solve these:
(c) A  X  U . Which sets could X stand for
(s) Construct a set E such that:
(apart from U)?

n( E ) = 3

E D = 

a E
(t) How many subsets of A having exactly
two elements can you find?
(u) Find B  (C  D) and ( B  C )  D .
(d) Find n( F  ( B  D)) .
(e) True or false: (i) A  D  F  
(ii)  C  D
(f) List the elements of B  E .
(g) B Y   . Which sets could X stand for?
S-10 Sets 2 :Venn Diagrams
1. Represent the sets below with a Venn
Diagram, putting each element in its correct
place in the diagram.
(a)
(b)
U
A
B
C
=
=
=
=
{1 , 2 , 3 , 4 , 5 , 6}
{4}
{1 , 5 , 6}
{2 , 5}
U
W
X
Y
=
=
=
=
{x x  and 2  x  12 }
{multiples of 3}
{factors of 24}
{prime numbers}
2. Make copies of this Venn
Diagram for each part
of the question below.
5. Students at Polyglot High School all study at
least one language other than English. The
three languages on offer are Spanish, German
and Russian.
An interview with a class of 22 students
produced the following information:
 3 students study all three languages
 nobody studies Russian only, but 2
students study German only and 4 study
Spanish only
 5 students study Spanish and German (but
not Russian)
 3 students study Spanish and Russian (but
not German).
By making a Venn Diagram find:
(a) the number of students who study German
and Russian (but not Spanish),
Shade the area
corresponding to:
(b) the number of students who study German.
(a) P  Q
(b) Q 
(c) ( P  Q )
(d) P  Q
3.
Use the Venn Diagram above to list the
elements of:
(a) X  Y
(b) Y  Z
(c) X  Y  Z
(d) ( X  Y )  Z
(e) ( X  Z )
(f) Y  X 
4. Two sets A and B have the following
properties:
n( A  B )  4
n( B )  9
n( A)  7
n(U )  20
By making a Venn Diagram or otherwise find
n( A).
6. By a curious coincidence students at Polyglot
High School can only practice three sports:
athletics, basketball and cricket. In the same
class of 22 students the following facts were
found:
 2 students practiced no sport
 the number of students who only practiced
athletics was equal to the number who
only practiced cricket. Call this number x.
 4 practiced all three sports
 5 students practiced exactly two sports
 3 students only practiced basketball.
By making a Venn Diagram find:
(a) the value of x,
(b) the number of students who practiced
exactly one sport.
S-11
Logic 1 : Propositions and Basic Symbols (      )
1. Tick the propositions:
5.
N stands for a positive integer
in this question.
(a) 2 + 2 = 4
(b) 10 - 7
a :
N is even
(c) Italian carrots have blue ears.
b :
N is greater than 10
(d) Their arre know speling misstakes hear.
c :
N is less than 15
(f) Go away!
d :
N is a multiple of 4
e :
N is 14
f :
N is less than 6
(g) 3y – 7
2. Assign T (true) or F (false) to each
proposition:
Assign T or F to each proposition:
(a) 2 is the only even prime number
(a)
ea
(b)
ad
(b) The Mathematical Studies course
involves two examination papers.
(c)
cf
(d)
e  f
(e)
e  d
(f) (f  d)  N is 4
(c) –20  –2
(d) Indonesia has a larger population than
Japan.
Write in symbols:
(g) N is an even number greater than 10.
3.
p :
I am happy
(h) N is less than 6 or greater than 10.
q :
I am working
(i) N is greater than or equal to 15.
r
I am at my computer
(j) N is an odd number and less than 6.
:
Write a short sentence corresponding to:
(a)
p
(b)
qr
(c)
p  q
(d)
(e)
p  q  r
(p)
(f)
q  q
(g)
qp
(k) If N is greater than 10, then N is not
less than 6.
(l) N lies between 11 and 14 (inclusive).
(m) N is an even number, but not a multiple
of 4, and is less than or equal to 10.
Assign T or F to each proposition for the given N
value:
(n) b  c
[N = 16]
(o) a  f
[N = 11]
(p) (d  c)  f
[N = 4]
(q) (d  c)  f
[N = 40]
(r) (d  c)  f
[N = 14]
S-12
Logic 2 : Truth Tables (  )
Complete these truth tables.
1.
p
T
T
F
F
q
T
F
T
F
2.
p
T
T
F
F
p
q
T
F
T
F
p
T
T
F
F
p
T
T
F
F
5.
p
T
T
F
F
q
T
F
T
F
q
T
F
T
F
pq
p
T
T
T
T
F
F
F
F
q
T
T
F
F
T
T
F
F
r pq
T
F
T
F
T
F
T
F
r
(p  q)  r
p  q
7. Complete these truth tables. What do you
notice about the final column of each?
3.
4.
6.
q
T
F
T
F
q pq
T
F
T
F
p q
pq
p q
q
T
F
T
F
p
T
T
F
F
pq
pq
p
T
T
F
F
(p  q)
8. Construct the truth table for (p  q)  q :
p
p
(p  q)  p
Compare this table with the table in Q2 and
comment:
(p  q)  (pq)
q
pq
(p  q)  q
S-13
Logic 3 : Tautology, Contradiction and Implication (  )
1. Complete this truth table for the proposition
(p  q)  p . Use your truth table to explain
why this proposition is a tautology.
p
T
T
F
F
q
T
F
T
F
pq
5. “If I study logic, then my mind
improves.”
(p  q)  p
Give the following related implications:
INVERSE:
CONVERSE:
CONTRAPOSITIVE:
2. Complete this truth table for the proposition
(p  q)  q . Use your truth table to explain
why this proposition is a contradiction.
p
T
T
F
F
q
T
F
T
F
p
pq
p  (p  q)
3. Complete this truth table for the proposition
(p  q)  p . Use your truth table to explain
why this proposition is a tautology.
p
T
T
F
F
q
T
F
T
F
p
pq
(p  q)  p
6. Complete this truth table and comment on the
result:
p
q
T
T
F
F
T
F
T
F
: N is even
q
: N is a multiple of 5
r
: N is odd
s
: N is a multiple of 2
t
: (N+1) is odd u : N is a multiple of 10
Assign a truth value to each of these:
(a)
pt
(b)
s  r
(c)
qr
(d)
(q  r)  u
(e)
s  u
(f)
u q
q
(p  q)  q
(p  q)  q  p
7. Use the truth tables below to show that
(p  q)
and p  q are logically equivalent:
p
T
T
F
F
4. N stands for a positive integer in this question.
p
pq
p
T
T
F
F
q
T
F
T
F
q
T
F
T
F
pq
p
q
(p  q)
p  q
S-14
Probability 1 : Sample Spaces
It’s usually easiest to give probabilities as
FRACTIONS.
1. In a bag there 6 red counters, 4 green counters,
3 yellow counters and 2 blue counters. If a
counter is chosen at random, find the
probability that it is:
(a) yellow
(b) not yellow
(c) green or blue
(d) not red or yellow.
In fact the counter is yellow. It is not put back
in the bag. If another counter is now chosen,
find the probability that it is:
(e) green or yellow
2.
(f) not blue.
4. If a student is chosen at random, find the
probability that the student is:
(a) a girl
(b) aged 16
(c) a 17-year-old girl
(d) a 16-year-old or a boy.
(e) A girl is chosen at random. Find the
probability that she is not 18 years old.
5. A coin has its two faces marked ‘1’ and ‘2’. A
die has its six faces marked from ‘1’ to ‘6’ in
the usual way. The sample space can be shown
like this:
V I E T N A M
1 2 3 4 5 6
If one of the seven letters above is chosen at
random, find the probability that it is:
1
2
If the coin and the die are thrown together,
find the probability that the two numbers seen:
(a) a vowel
(b) in the word MEXICO.
(a) add up to 4
(b) add up to 5 or more
3. The natural numbers from 1 to 15 are written
on counters. If a counter is chosen at random,
find the probability that its number is:
(a) even
(c) are the same.
(d) Find also the probability that the number
on the die is larger than the number on the
coin.
(b) a factor of 12
6. Questions involving throwing two dice can be
solved with a larger version of the grid in Q5:
(c) a multiple of 3
(d) an even multiple of 3
1 2 3 4 5 6
(e) an even number or a multiple of 3
(f) less than 6
(g) greater than 20.
4. The ages and sexes of participants in a school
trip are summarised below:
aged 16
aged 17
aged 18
boys
9
11
4
girls
10
12
2
So, for example, there were nine 16-year old
boys on the trip.
1
2
3
4
5
6
If two dice are thrown, find the probability
that:
(a) the scores add up to 9,
(b) the scores add up to 4 or less,
(c) the score on each die is the same,
(d) the product of the scores is 18 or more,
(e) the scores differ by 2 or less.
S-15
Probability 2 : Independent Events and Tree Diagrams
1. In each question below the two events are
independent.
(a) I throw a coin and a die (dice). Find the
probability that I get a Tail and a 5.
3. The probability that a student works hard on
any day is 0.6. If she works hard, the
probability that she passes the test next day is
0.8. If she doesn’t work hard then that
probability is only 0.5. Complete the tree
diagram below, and find the probability that
she passes the test.
(b) I throw a die twice in a row. Find the
probability that the first is a 2 and the second
is 5 or 6.
pass
work
fail
(c) I throw a die twice in a row. Find the
probability that both times it shows 3 or more.
pass
not
work
(d) I pick a day of the week at random, twice
in a row. Find the probability that both are
Friday.
(e) I pick a day of the week at random, twice
in a row. Find the probability that the first is
Saturday and the second is not Saturday.
fail
4. An unbiased coin is thrown three times.
Complete the tree diagram below, and find the
probability of getting
(a) two heads and one tail
(b) all three throws the same.
(f) I pick a day of the week at random, twice
in a row. Find the probability that both days
contain the letter “r”.
2. In a bag are 6 green balls and 4 yellow balls.
A ball is picked at random and its color is
noted. It is replaced, and the process is
repeated.
(a) Complete the tree diagram below:
G
G
0.6
Y
H
T
5. The experiment in Q2 is repeated except this
time the first ball is not replaced.
(a) Think carefully about the entry 5 below
9
and complete the tree diagram.
(b) Find the probability that both balls are the
same color.
G
5
9
G
G
Y
6
10
Y
Y
(b) Write down the probabilities that:
G
(i) both balls are green
(ii) the two balls are the same color
(iii) the two balls are different colors.
Y
Y
S-16
Probability 3 : Mutually Exclusive Events and Venn Diagrams
1. Market research has been done on a group of
50 people asking them whether they were in
interested in Archery, Billiards, Cooking or
Diving. The numbers expressing an interest
are shown in the Venn Diagram below.
(a) If a person is chosen at random, find the
probability that (s)he expresses an interest in:
(i)
Archery,
(ii)
Archery, but not Cooking,
(iii)
Billiards or Diving,
(iv)
exactly one of A, B, C and D,
(v)
three or more of A, B, C and D.
3. Here are three events involving throwing two
normal 6-sided dice:
A : there is at least one 3
B : the total is 10 or more
C : there are no 5s or 6s
Which pairs of events are mutually exclusive?
4. A group of students were surveyed about
whether they studied Biology and/or
Chemistry.
In the Venn Diagrams of Q1 and Q2 the
numbers of people were shown. In the Venn
Diagram below the probabilities have been
shown.
(f) An interest in billiards is mutually
exclusive with which other interest?
(g) Name another pair of mutually exclusive
interests.
Find the probability that a student chosen at
random:
(a) studies Chemistry
(b) does not study Biology.
5. In a small international school the breakdown
of students by gender and grade is as follows:
2. A group of 30 American international business
travelers were asked if they traveled regularly
to Asia or Europe.
 12 said ‘yes’ to Asia
 19 said ‘yes’ to Europe
 4 said ‘no’ to both Asia and Europe.
(a) Thinking particularly about A  E
complete the Venn Diagram below.
(b) A traveler is chosen at random, Find the
probability (s)he says ‘yes’ to Asia and ‘no’ to
Europe.
male
gr 9 - 12 25
20
gr 6 - 8
female
30
25
(a) With F representing Female and S
representing Grade Six to Eight, complete the
Venn Diagram below, which shows
probabilities.
(b) Find p(F  S) .
6. A and B are two events. It is known that:
p(A) = 0.6 ; p(B) = 0.3 ; p(A  B) = 0.75.
Find: (a) p(A  B)
(b) p(A  B) .
S-17
Probability 4 : Conditional Probability
1. In a small international school the breakdown
of students by sex and grade is as follows:
gr 9 - 12
gr 6 - 8
boy
25
20
girl
40
15
4. An unbiased 6-sided die is thrown twice in
succession.
(a) Find the probability that the total is 7 (not a
conditional probability problem!).
(b) Find p (total is 7 | the first die did not show 6).
(a) Find the probability that a student chosen
at random is in Grade 6 – 8 (not a conditional
probability problem!).
(b) Find the probability that a girl chosen at
random is in Grade 6 – 8. HINT : completely
(c) Comment on you answers to (a) and (b).
5. The Venn Diagram below shows the
probabilities of students in a certain region
studying Economics and Geography.
ignore the ‘boy’ column.
NOTE
:
expressed
Question
as
(b)
follows
could
have
been
E
G
0.1 0.3
0.15
: Find the probability
that a student chosen at random is in Grade 6 –
8, given that the student is a girl.
(c) Find the probability that a student chosen
at random is in Grade 9 – 12, given that the
student is a boy.
(d) Find the probability that a student chosen
at random is a girl, given that the student is in
Grade 9 – 12.
2. A 12-sided die (numbered from 1 to 12) is
thrown . Find:
(a) p(score is 9)
(b) p(score is 9 | it’s a multiple of 3)
(c) p(score is 7 or more | it’s 10 or less)
Find:
(a) p(E)
(b) p(E | G)
(c) p(G | E)
(d) p(G | E  )
6. Use the Venn Diagram below to find:
(a) p(A | B)
(b) p(B | A)
(c) Are A and B independent? Explain!
A
A 6 x 6 grid as a sample space will help
with these.
Two unbiased 6-sided dice are thrown. Find:
B
0.49 0.21 0.09
(d) p(score is even | it’s not a multiple of 3)
3.
0.45
0.21
7. For events C and D it is known that p(C) =
0.25, p(D) = 0.35 and p( C  D ) = 0.5.
(a) Find p( C  D ) .
(a) p (total is 10 or more | there is at least one 6)
(b) Find p(C | D)
(b) p (total is 10 or more | dice show different scores)
(c) p (total is 10 or more | there are no 6s)
(d) p (total is 10 or more | there are no 1s or 2s)
(e) p (there is at least one 6 | total is 10 or more)
8. X and Y are mutually exclusive events.
p(X) = 0.4 and p(Y) = 0.3. Find:
(a) p(X  Y)
(b) p(X Y)
S-18
Functions
f (x)  3x  2
x 1
h(x) 
2
1.
g(x)  10  3x
5. Find the domain and range of each function:
(a)
6
Find the values of:
3
(a) f (6)
(b) g(1)
(c) h(5)
(d) f ( 1)
(e) g(1)
(f) h(7)
(g) f (2.5)
 2
(h) g  
 3
(i) f (2) - f (3)
(j) f (g(2))
0
5
(b)
10
1
-1
1
2. Using the functions from Q1, solve:
(a)
f (x)  7
(b)
g(x)  8
(c)
h(x)  5
-1
6. The graph below shows y  f (x) and
y  g(x) .
(a) Find
(i) f (6)
(ii) g(1.5)
(iii) g(7)
3.
f :x
2x  1
h:x
g:x
x(x  3)
(b) Solve (i) f (x)  4
sin(5x)
(ii) g(x)  5
Find the values of:
(a) h(6)
(b) f (4)
(c) g(4)
(d) f (0)
(e) f ( 1)
(f) h(36)
(g) g(3)
(h) h(18)
(i) f (10)
(j) g(0.5)
(c) Write down: (i) the domain of f (x)
(ii) the range of g(x)
(d) Write down three integers n for which
g(n)  f (n)
7
y=g(x)
4. Using the functions from Q3 solve these. Your
GDC could be useful!
(a) f (x)  7
(b) g(x)  28
y=f(x)
(c) f (x)  g(x)
0
7
S-19
Linear Functions
Answer each question on a separate sheet of graph paper.
1. Construct axes with 4  x  4 using a scale of 2cm to 1 unit, and with 20  y  20 using a scale of
1cm to 2 units.
Now construct the following lines accurately:
(a)
y  2x  3
(b)
2x  3y  12
(c)
x  4y  6  0
(d) Write down the coordinates of the point of intersection of lines (a) and (b) correct to 1 decimal place.
2. For a small car hire company charges €22 per day plus 13 cents for each kilometer driven.
(a) Find the total cost of hiring a car for a day if you travel
(i) 50km
(ii) 100km
(iii)300km.
(b) Use your answers to (a) to construct a graph showing the cost of hiring a car for any distance up to
300km. Put ‘distance’ on the x-axis using a scale of 1cm to 20km, and ‘cost’ on the y-axis using a scale
of 1cm to 4€.
(c) Using your graph find how many kilometers you traveled if your bill was €47.
Note: show lines on your graph to indicate clearly that you have used your graph!
3. If you change euros for dollars at the Royal Island Principal Office of Fair Finance bank, they calculate
as follows:
 deduct 5 euros as the bank’s charge
 convert the rest to dollars at the rate 1€ to 1.2$.
18
So if you change €20 you will receive 15  1.2 = $18, and the effective rate you have received is
=
20
0.9€ per dollar.
(a) Complete the table:
€ given
$ received
10
20
18
100
(b) Using your own scale construct a graph to show what you will receive in dollars for any amount up
to 100 euros.
(c) Using your graph find how many dollars you will receive for 65 euros, and calculate the effective
rate.
(d) Using your graph find out how many euros you need to change to receive $100.
4. In Tokyo two taxi companies charge as follows:
Company A : 250 yen per kilometer traveled.
Company B : a fixed charge of 500 yen, plus 180 yen per kilometer traveled.
(a) Construct a graph showing the cost for each company for journeys up to 15km.
(b) Using your graph find how much more expensive Company A is (compared to Company B) for a
journey of 5.5km.
(c) Someone asks you which company is cheaper. What would your answer be?
S-20
Quadratic Functions
1. Using your GDC (or otherwise) sketch each
quadratic function below. Mark the x- and yintercepts and the coordinates of the vertex:
3. Find the coordinates of the vertex of each
quadratic in Q2 (a) to (c).
(a)
y  x  4x  3
2
(a)
(b)
(c)
4. Find the x-intercepts of:
-2
4
(a)
y  x 2  2x  15
(b)
y  x 2  6x  8
y  6  x  x2
(b)
5. (a) The graph of y  x 2  2x  c passes
through the point (3, 11). Find the value of c.
-4
4
(b) The graph of y  x 2  bx  12 passes
through the point (2, –2). Find the value of b.
6. Each sketch below shows a quadratic graph of
the form y  x 2  bx  c . Find the values of b
and c.
y  (x  4)(x  2)
(c)
(a)
-6
6
2. Find the equation of the line of symmetry:
(a)
y  x 2  6x  10
(b)
y  x 2  2x  5
(c)
y  10  4x  x 2
(d)
y  2x 2  3x
(b)
S-21
Exponential Functions
1. The points below all lie on the graph of the
exponential function y  4x . Find the value of
a , b , c , d and e:
(3 , a)
(-1 , b)
(0.5 , c)
(d , 16) (e , 1)
2. The population, P animals, of an endangered
mammal on an island at time t years after 1990
appears to be modeled by the exponential
function
P  1800  0.96t
(a) Write down the population in 1990.
(b) Find the population (correct to the nearest
10) in 2005.
(c) By GDC and/or trial and error find in
which year the population will first drop
below 500.
(d) Find the approximate number of years it
takes for the population to halve.
3. Each sketch graph below shows y  k  b x
(k, b  ). Find the values of k and b.
4. (a) For the year 2010 find the estimated:
(i) gross domestic product
(ii) population
(iii)gross domestic product per capita
G
  Don’t forget 3 significant figures!
P
(b) Repeat the calculations of (a) for the year
2025.
(c) Use a GDC to find in which year the gross
domestic product per capita is estimated to
exceed $20000 for the first time.
5. During an epidemic the number of new cases
(per day) of an illness reaches a peak
(maximum) of 45 on June 1st. From then on
the number of new cases per day, N, is
predicted to decline according to the formula
N  45 1.17  t
where t is the number of days after June 1st.
(a) If the prediction is correct, how many new
cases (correct to the nearest integer)
should be expected on:
(i) June 7th (ii) June 20th ?
(b) Use a GDC method to find the date on
which the number of new cases should first
drop below 0.5.
(a)
(c) Explain the significance of “below 0.5” in
part (b).
not to scale
(b)
6. The temperature, T˚ Celsius, of a cooling
object t minutes after the start of an
experiment is given by the formula
T  22  (78  20.2t )
(a) Find the value of T (correct to the nearest
integer) when:
4. The gross domestic product (G billion dollars)
of a small nation is estimated to be growing
exponentially according to the formula
G  45 1.04n
where n is the number of years after 2000.
The population (P million people) of the
country appears to be increasing exponentially
according to the formula
P  3.2 1.015n
(i) t = 0
(ii) t = 10
(iii) t = 30
(b) After how many minutes does the
temperature reach 30˚C? Answer correct to 1
decimal place.
(c) When t is very large, what value does T
approach? Physically, what do you think this
temperature represents?
S-22
Trigonometric Functions
S-23
Accurate Graphing
S-24
Graph Sketching and GDC Skills
Function
Coordinates of key points
(correct to 2 decimal places)
Sketch – include an approximate y-scale
Domain
4  x  4
1.
x-intercepts:
y  2  3x
x
y-intercept:
minimum:
-4
4
4  x  4
2.
x-intercept:
8
y   x2
x
maximum:
-4
4
2  x  4
3.
x-intercepts:
y  x  3x  x  2
3
2
y-intercept:
-2
4
minimum:
maximum:
5  x  3
4.
y-intercept:
y
1  2x
1 x
2
minimum:
maximum:
-5
3 equation of vertical
asymptote: