Download Handout 1 - Birkbeck

Birkbeck College
School of Economics, Mathematics and Statistics
BSc Financial Economics, BSc Economics with
Accounting
First Year Students’
Preparatory Mathematics Course
Monday, 17th Sept., 2012 – Friday, 28th Sept., 2012
Lecturer: Ranch Patel
Email: [email protected]
Pre-Test : 1 Hour
Insert your answers in the answer sheet provided.
Q1. Fill in ? below:
(i)
12 - 3(?) = 3
(iv) ?3 = –8
(vii)
– (3 + ?) = 5
(x)
√? =  9
(ii) 4(1+ ?)= 0
(v) 2 x 32 =?
(viii) ?  11 =-11
Q2. Write expressions for
(i)
the sum of a, b, c and d.
(iii)
the reciprocal of x.
(v)
the cube of the product of x and y.
(vii) the modulus of the sum of x and y.
(viii) the sum of absolute values of x and
(ix)
the arithmetic mean of p, q and r.
(x)
the geometric mean of p, q and r.
(iii) 8 – 2? = 0
(vi)
3(? - 6) = -15
(ix)
? – 8 = -7
(ii) the product of m and n.
(iv) e raised to the power x.
(vi) m taken away from n.
y.
Q3.
(i) A plant of height 10cm increases by 5%. What is the new height?
(ii)
A plant increases in height by 4% in a week and grows to a height of 108
cm. What was the height at the beginning of the week?
(iii)
A investor invests £10000 for 20 years at a compunded rate of 6% per
annum. What is the value of the investment after the 20 years?
Q4. The Present Value, PV of an annuity of £S per annum for n years when the
interest rate is r is given by the formula:
  1 n


 1
S  1 r 

PV 


1 r  1 



1
  1  r 



Find the Present value of an annuity of £10,000 per annum for 10 years when the
interest rate is 7%.
Q5.(a)
(i)
(ii)
(iii)
(iv)
Sketch the graph of the straight line y = 2 – 3x
what is the gradient?
what is the y intercept?
what is the x intercept?
(b) In a linear relationship when x increases from 3 to 6, y increases from 5 to 10.
(i) Calculate x and y
(ii)
What is the ratio of the increase in y to the increase in x?
(iii)
What is the ratio of the proportional increase in y to the proportional increase
in x ?
Q6.
(i)
(iv)
Solve the equations :
x3=3
(ii)
x2 – 2x – 8 = 0
Q7
Simplify each of the following:
(i)
3(3 – 2x) – (-2 + x)
3x 2 y3z
9(xyz)3
(iii)
(v)
2(√x – 3) =10
(iii) – 4 + 2x2 = 2
(v)
(x – 3)2 = 1
(ii)
x2y + 4xy2 – xy(x + y)
4x2 + y2 – (x – 2y)2
(iv)
3
1

2x 4x
Q9 Simplify and write the following expressions with the exponents in the
numerator:
i.
3
2x 2
ii
8x 2 y 2 x
2x 4 y 3
2
1
iii.
5
x2 y
iv.
x3 y
3
xy
2
v.
3
x 1
Q10
(a) Factorise the following:
(i)
6x2y – 3xy2
(ii) ac + bc - ad - bd
(b) Simplify the following: (Hint: Factorise numerator and then cancel out)
x 2  3x  2.
(x - 1)
Q11 Consider the relation y: = x2 – 2x – 8
(i)
(ii)
(iii)
(iv)
(v)
Find the y intercept. (Hint: Put x=0 and evaluate)
Find the x intercepts (Hint: Put y=0 and solve for x)
Find the axis of symmetry
Find the minimum value of y (Hint: use your answer from (iii)
Use the above information to sketch the graph.
Chapter 1
Number Types & Basic Operations
Real Numbers, 
Irrational Numbers
Rational
numbers
Integers, 
Natural numbers, N
Natural Numbers
1.1
The set of whole numbers {1, 2, 3, 4, 5,…….} is called the set of Natural numbers. 0
and negative numbers such as -4 are not Natural numbers.
E1.1
1.2
Which of these is not a natural number?
12 yes/no; 3.4 yes/no; -4 yes/no; 0 yes/no
Integers
Extending the Natural Numbers by including 0 and negative whole numbers gives
the set of Integers.
The set of numbers { ….,-3, -2, -1, 0, 1, 2, 3, …} is denoted as .
E1.2
Which of these is not an Integer?
12 yes/no; 3.4 yes/no; -4 yes/no; 0 yes/no;
A1.2
E1.3
A1.3
E1.4
A1.4
1.5
2
yes/no; -5.0 yes/no
5
Yes; no; yes; yes; no; yes
Are all natural numbers also Integers? Yes/No
Yes
Are all integers also natural numbers?
No. The set of Natural numbers is ‘properly nested’ inside the set of integers.
Rational Numbers
Extending the Natural Numbers by including decimals and negative numbers gives
the set of Rational Numbers.
The number 2.5 lies between the integers 2 and 3. But 2.5 is not an integer. It is a
Rational number.
E1.5
Which number lies exactly between 2.5 and 3.5?
Is this number a natural number? Is it an integer?
A1.5
E1.6
A1.6
1.7
The number 3 (or 3.0) It is a natural number and an integer.
Are all natural numbers Rational numbers? Yes/No
Are all integers Rational numbers? Yes/no
Are all rational numbers integers? Yes/no
Yes, all natural numbers are Rational numbers.
Yes, all integers are Rational numbers.
No: the set of integers is ‘properly nested’ in the set of rational numbers.
Every decimal can be expressed as a fraction and every fraction can be expressed as
a terminating or recurring decimal.
A recurring decimal is non terminating with repeating digits.
For example 0.123123123… is a recurring decimal. We write it more formally as a
terminating decimal with dots over the recurring digits viz:
0.123123123… = 0.1 2 3
E1.8
There a calculator key that toggles the display of a rational number between a
fraction and a decimal. Find it on your calculator and ensure you know how to use it.
Complete the table using your calculator
Decimal Fraction Type of decimal
0.5
Terminating
1
2
Terminating
5
8
0.525
Terminating
Recurring
1
3
A1.8
1.9
Decimal Fraction Type of decimal
0.5
Terminating
1
2
0.525
Terminating
5
8
0.525
Terminating
5
8

Recurring
1
0.3
3
The last row shows the fraction ‘one-third’ as a non-terminating decimal, 0.3333...
19
as a decimal.
99
The calculator will show this as a ‘recurring decimal’, 0.191919…
i.e. a decimal that is not terminating but has repeating digits ‘19’ in this case.
Square Numbers
Use your calculator to express the fraction
1.10
Consider the sequence of integers 0, 1, 4, 9, ....
They are called square numbers because
0x0 = 0, 1x1 = 1, 2x2 = 4, 3x3 = 9, ...
E1.10 What are the next three square numbers?
A1.10 4x4 = 16, 5x5 = 25, 6x6 = 36. So the next three square numbers are 16, 25, 36.
We use brackets to denote multiplication e.g. (4)(4) = 16
1.11
Learn this sequence of square Numbers:
0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361,
400
E1.11 Complete the following:
(i) (?)(?) = 9, (ii) (?)(?) = 361 (iii) 13x13 = ?
A1.11
(i) 3 (ii) 19 (iii) 169
1.12
We have a special form of notation for expressing square numbers.
We write 32 = 9 to mean 3x3 = 9. Also 72 = 49.
E1.12 Replace the ? in the following:
(i) 42 = ? (ii) ?2 = 100 (iii) 8? = 64
A1.12
(i)
16 (ii) 10 (iii) 2
E1.13 Which is the missing square number in the sequence of square numbers?
9, 16, 25, ? , 49
A1.13 36
Square Roots
1.14
The inverse of squaring is called square rooting, for which we use the symbol √.
For example: The statment, 62 = 36 i.e. the square of 6 is 36 can be inverted to:
36  6 , which is read as the ‘the square root of 36 is 6’.
Here are more examples
Using Square Using Square root
42 = 16
16  4
2
9 = 81
81  9
152 = 225
E1.14
225  15
Complete the table below
Using Square Using Square root
122 = 144
92 =
144 
A1.14
Using Square Using Square root
122 = 144
144  12
2
9 = 81
81  9
122 = 144
144  12
E1.15
A1.15
E1.16
A1.16
1.17
Fill in the ? in the following
(i) ?2 = 100 (ii) 62 = ? (iii) 144  ?
(i) 10 (ii) 36 (iii) 12
Find the value of
81 , i.e. what is the square root of the square root of 81.
81  9  3
Irrational Numbers
Consider the two adjacent square numbers 16 and 25.
We know that:
the square root of 16 is 4 i.e. 16  4
the square root of 25 is 5 i.e. 25  5
So what is the square root of 20 which lies between the square numbers 16 and 25?
Clearly the answer lies between 4 and 5, which are adjacent integers.
So the answer is not an integer. Is it a rational number?
Use your calculator to find
20.
Your calculator will show 4.472135955…
This is non-terminating decimal but it is not recurring. It is an IRRATIONAL
number.
E1.17
Which of these roots are irrational numbers?
25 ii. 250 iii. 2500 iv. 2.5 v. 0.25
i.
A1.17
i. No ii. Yes iii. No iv. Yes v. No
Number Line
1.18
A number line, which shows the relative sizes of the numbers is a useful graphical
aid. For example it allows to rank numbers and compare their sizes.
4
3
2
1
0
-1
-2
-3
-4
Note that when we write the number 4 we really mean the number, positive +4 but
we drop the + sign. So 4 has a silent or redundant + in front of it.
1.19
Note that we have used the symbol + to mean ‘positive’ sign and to represent the
ADD operation.
For example: +4 means positive 4.
Mathematics is a language that uses notation that is totally unambiguous and is
also as economical as possible. Dropping the (positive) + sign is economy of
notation.
You may argue that + is ambiguous because it means the sign of a positive number
and also the operation of ADD.
However the ambiguity is removed by the context.
Writing 2 + 3, clearly uses + as ADD and writing +4 clearly uses + as the positive
sign. It is DUALITY of usage not ambiguity in usage.
+3 + +4 = +7 : This means positive 3 ADD positive 4 equals positive 7.
For economy we write: 3 + 4 = 7 i.e. we remove the redundant 3 (Positive) +
symbols.
E1.19
Remove any redundant + in the following expression:
+2 + +3 + 5 + 6 = +16.
A1.19 2 + 3 + 5 + 6 = 16.
1.20
The above number line enables us to order numbers in increasing value or
decreasing value.
We use the notation:
> for “greater than” ; e.g. 6 > 4 (six is greater than 4)
< for “less than” ; e.g. -6 < -4 (minus six is less than minus 4)
≤ for “less than or equal to” and
≥ for “greater than or equal to”.
E1.20
a) Rewrite the set with its members listed in order of increasing value from
smallest to largest.
{ 7, 0, -3, -5, 4, 3 }  { , , , , , }
b) State whether the following statements are true or false. Where the statements
are False rewrite them correctly.
Statement
5 > 10
6 < -3
-6 > -3
4 ≤ 3+1
True/False
Statement
0 > -3
-6 > -3
7 > -4 < 0
5≥5>0
True/false
A1.20 a) { 7, 0, -3, -5, 4, 3 }  { -5 ,-3, 0, 3, 4, 7}
b)
Statement
5 > 10
6 < -3
-6 > -3
4 ≤ 3+1
True/False
F:5 < 10
F:6 > -3
F: -6 < -3
True
Statement
0 > -3
-6 > -3
7 > -4 < 0
5≥5>0
True/false
True
F: -6 < -3
True
True
1.21
Every integer has a sign and magnitude. The sign is either + or .
E1.21
The magnitude is its size.
Integer
Sign
-11
+16
+
-8
7
+
0
- or +
What is the sign of -5? What is its size?
Magnitude
11
16
8
7
0
A1.21 Sign is –ve; Size is 5
1.22
The magnitude is also called the absolute value or modulus. The notation for this is
two vertical bars enclosing the integer viz:
|-4| = 4.
This is read as: ‘the absolute value of -4 is 4’ or
‘the modulus of -4 is 4’ or ‘the magnitude of -4 is 4’.
E1.22
Complete the following table. Rewrite the false statements correctly.
Statement
True/False
Statement
True/false
|5| = 5
|-6| > |5|
|-6| = 6
-3 > |-3|
Sign(-6) = 6
7 > |-4| < 0
A1.22
1.23
Statement
True/False
|5| = 5
True
|-6| = 6
True
Sign(-6) = 6
F:Sign of -6 is –ve
Multiplication
Multiplication is just repeated addition.
Statement
|-6| > |5|
-3 > |-3|
7 > |-4| < 0
True/false
True
F: -3 < |-3|
F: 7 > |-4| > 0
4 + 4 + 4 is the addition of 3 copies of 4.
So we can ‘shorten’ 4 + 4 + 4 to 3 x 4 or (3)(4).
Or even more concisely as 3(4).
Note the use of the bracket to imply multiplication.
Note also that 3 + 3 + 3 + 3 = 4(3) which we can also write as 3(4).
Note also 2(5) = 2+2+2+2+2 or 2(5) = 5+5.
Is this ambiguity? Maybe but the result of both is the same.
E1.23
Write the following expressions switching ADD with MULTIPLY.
5+5+5+5=
4(5) =
-3 + -3 + -3 + -3 =
3(-5) =
-3(4) =
A1.23 5 + 5 + 5 + 5 = 4(5)
4(5) = 5 + 5 + 5 + 5 or 4 + 4 + 4 + 4 + 4
-3 + -3 + -3 + -3 = 4(-3)
3(-5) = -5 + -5 + -5
-3(4) = -3 + -3 + -3 + -3
2.1
CHAPTER 2 – Directed Numbers
Addition and Subtraction
Directed numbers are positive and negative numbers. They can be put on a
number line as shown below.
+3
+2
+1
0
-1
-2
-3
+
3
So counting upwards from -3 we get -2, -1, 0, 1, 2 …
Writing this as a sequence, we get:
-2, -1, 0, 1, 2, 3
2.2
Increasing 5 by 3 takes us to 8.
We can write this mathematically as 5 + 3 = 8
Decreasing 5 by 7 takes us to -2
We can write this as 5 - 7 = -2
Increasing -2 by 2 takes us to 0.
We can write this as -2 + 2 = 0.
E2.2
Now complete the following table:
Action
Increase -2 by 3
Decrease -2 by 5
Result Mathematical statement
5+4=9
Increase 0 by 2
3
Increase -4 by … 4
Decrease … by 5 -1
-6 + … = ….
A2.2
2.3
Action
Result Mathematical statement
Increase -2 by 3 1
-2 + 3 = 1
Decrease -2 by 5 -7
-2 – 5 = -7
Increase 5 by 4
9
5+4=9
Increase 0 by 2
2
0+2=2
Increase -6 by 9 3
-6 + 9 = 3
Increase -4 by 8 4
-4 + 8 = 4
Decrease 4 by 5 -1
4 – 5 = -1
Now compare the following two actions:
Decrease 5 by 2 and Decrease 5 by -2.
The first result is 3 so the second result cannot be 3 as well.
The second result is 7.
Mathematically the second is written as: 5 - -2 = 7.
To decrease by a negative amount is to increase by a positive amount.
So 7 - - 3  10 is identical to7  3  10
We use the symbol ≡ to mean ‘is identical to’.
≡
So we write: 7 - -3 = 10
E2.3
7 + 3 = 10
Complete the rows of the following Table. The first has been done for you.
Action
Result
Decrease 2 by -3
Decrease -2 by -5
Decrease 5 by -4
Decrease -2 by -5
5
Mathematical
statement
2 – -3 = 5
Identical
Statement
2+3=5
5+4=
-2 – -5 = 3
-6 + 6 = 0
A2.3
Action
2.4
Result
Mathematical
statement
2 – -3 = 5
-2 – -5 = 3
5 – -4 = 9
-2 – -5 = 3
-6 – -6 = 0
Identical
Statement
2+3=5
-2 + 5 = 3
5+4=9
-2 + 5 = 3
-6 + 6 = 0
Decrease 2 by -3
5
Decrease -2 by -5
3
Decrease 5 by -4
9
Decrease -2 by -5
3
Decrease -6 by -6
0
Multiplication and Division
With multiplication and division we have the rule that ‘two negatives make a
positive’ We can summarise the rules as:
(positive)(positive) = positive e.g. (+4)(+3) = +12 or (4)(3) =12
(positive)(negative) = negative e.g. (+4)(-3) = -12 or (4)(-3) = -12
(negative)(positive) = negative e.g. (-4)(+3) = -12 or (-4)(3) = -12
(negative)(negative) = positive e.g. (-4)(-3) = +12 or (-4)(-3) =12
E2.4
Evaluate the following expressions:
(-5)(-7) =
(-5)(7) =
(5)(-7) =
(5)(7) =
(-5)(-7)(-2) =
(-5)(2)(-7) =
(3)(-5)(-7) =
-2(-5)(-3) =
A2.4
2.5
(-5)(-7) = 35
(-5)(-7)(-2) = -70
(-5)(7) = -35
(-5)(2)(-7) = 70
(5)(-7) = -35
(3)(-5)(-7) = 105
(5)(7) = 35
-2(-5)(-3) = -30
Similarly for division:
positive
 positive
positive
positive
 negative
negative
negative
 negative
positive
negative
 positive
negative
For example:
 10
10
 2 or
2
5
5
 10
10
 2 or
 2
5
5
 10
 10
 2 or
 2
5
5
 10
 2
5
E2.5
Evaluate the following quotients:
12

4
6

3
12

3
A2.5 Evaluate the following quotients:
12
 3
4
6
2
3
12
 4
3
Exercise 2
a.
Evaluate the following expressions:
-2 ─ -3 ─ -4 =
-2 ─ 3 + -4 =
-2 + 3 ─ -4 =
-2 ─ 3 ─ 4 =
2 + -3 + -4 =
2 ─ -3 + -4 =
-2 + -3 ─ -4 =
2 ─ -3 ─ -4 =
b.
Evaluate the following products:
(-2)(3)(-4) =
(2)(3)(4) =
(-2)(-3)(4) =
(2)(3)(-4) =
c.
(2)(3)(-4) =
(-2)(-3)(-4) =
(2)(-3)(4) =
(2)(-3)(4) =
Evaluate the following quotients:
12

4
12

4
 12

4
 12

4
d.
Four negative numbers and five positive numbers are all multiplied together.
What is the sign of the resultant number?
e.
Fill in ? below:
(i)
3(?) = -12
(iv) -4(?) = –8
(vii)
– 3 + ? = -5
(x)
-3(-2)(?) = -3
(xiii) (|(-3)(4)|) (|-3|) =
(ii) 4(?)= 0
(iii) 8 – ? = 10
(v) 2 + -3 ─ ? = 4
(viii) ?  11 = -11
(xi) |-4| + 5 =
(xiv) |-3||-4 |=
(ix)
(vi)
3(?)(-4) =24
? – 8 = -7
(xii) |-3|(-4) =