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Contents
Introduction ........................................................................................................................................................................ 4
Marking Criteria ........................................................................................................................................................... 5
Learning Outcomes Checklist ................................................................................................................................. 7
Useful Links .................................................................................................................................................................... 7
ALGEBRA - Apply algebraic methods in solving problems, 91261 .............................................................. 9
Expanding Brackets .................................................................................................................................................... 9
One bracket .................................................................................................................................................................... 9
Two brackets ................................................................................................................................................................. 9
Three brackets .............................................................................................................................................................. 9
ACTIVITY 1 .................................................................................................................................................................. 10
Factorising ................................................................................................................................................................... 10
Two terms .................................................................................................................................................................... 10
Two terms (difference of two squares) ........................................................................................................... 10
Three terms ................................................................................................................................................................. 11
Three terms (with coefficient of x2, with no common factor) ................................................................ 11
ACTIVITY 2 .................................................................................................................................................................. 12
Solving Equations and Inequalities ................................................................................................................... 13
Equations ..................................................................................................................................................................... 13
Inequalities .................................................................................................................................................................. 13
Simplifying Rational Expressions ...................................................................................................................... 14
ACTIVITY 3 .................................................................................................................................................................. 14
Changing the Subject of a Formula .................................................................................................................... 15
ACTIVITY 4 .................................................................................................................................................................. 15
Indices ........................................................................................................................................................................... 16
Rules............................................................................................................................................................................... 16
Solving exponential equations ............................................................................................................................ 17
Logarithms .................................................................................................................................................................. 17
Rules............................................................................................................................................................................... 17
Solving Logarithmic Equations ........................................................................................................................... 18
ACTIVITY 5 .................................................................................................................................................................. 18
Solving polynomial equations ............................................................................................................................. 19
Solving a polynomial in factorised form ......................................................................................................... 19
ACTIVITY 6 .................................................................................................................................................................. 19
1|NCEA Level 2 Mathematics
The Quadratic Formula .......................................................................................................................................... 20
ACTIVITY 7 .................................................................................................................................................................. 21
Quadratic Formula Theory ................................................................................................................................... 22
Completing the square: .......................................................................................................................................... 23
Collated Questions – 2015 Exam............................................................................................................................. 24
Question set 1 ............................................................................................................................................................. 24
Question set 2 - 2017 Exam .................................................................................................................................. 27
Question set 3 - Exam Paper – 2018 ................................................................................................................. 33
Calculus - Apply calculus methods in solving problems, 91262 ................................................................ 41
Differentiation and the Derivative: ................................................................................................................... 41
ACTIVITY 1 .................................................................................................................................................................. 42
Gradient of a Tangent.............................................................................................................................................. 43
ACTIVITY 2 .................................................................................................................................................................. 44
Finding the equation of a tangent to a curve: ............................................................................................... 45
Turning Points ........................................................................................................................................................... 46
Second Derivatives ................................................................................................................................................... 47
ACTIVITY 3 .................................................................................................................................................................. 47
Sketching the gradient function.......................................................................................................................... 48
Integration (Anti-differentiation) ...................................................................................................................... 49
What is the c?.............................................................................................................................................................. 49
Definite Integrals ...................................................................................................................................................... 50
ACTIVITY 4 .................................................................................................................................................................. 51
Rates of Change ......................................................................................................................................................... 52
Finding Maximum/Minimum Volumes/Areas ............................................................................................. 52
ACTIVITY 5 .................................................................................................................................................................. 53
Collated Questions ........................................................................................................................................................ 54
Question set 1 - 2015 exam .................................................................................................................................. 54
Question set 2 - 2017 Exam .................................................................................................................................. 58
Question set 3 - 2018 Exam .................................................................................................................................. 67
Probability - Apply probability methods in solving problems, 91267 .................................................... 77
Data activity ................................................................................................................................................................ 77
Measures of Centre: ................................................................................................................................................. 77
ACTIVITY 1 .................................................................................................................................................................. 77
ACTIVITY 2 .................................................................................................................................................................. 78
ACTIVITY 3 .................................................................................................................................................................. 78
2|NCEA Level 2 Mathematics
Measures of Spread.................................................................................................................................................. 79
Standard Deviation .................................................................................................................................................. 79
Normal distribution ................................................................................................................................................. 81
ACTIVITY 4 .................................................................................................................................................................. 83
The Standard Normal Distribution.................................................................................................................... 84
ACTIVITY 5 .................................................................................................................................................................. 86
ACTIVITY 6 .................................................................................................................................................................. 86
Comparing Distributions ....................................................................................................................................... 87
Relative Frequency .................................................................................................................................................. 90
Risk and Relative Risk............................................................................................................................................. 90
ACTIVITY 7 .................................................................................................................................................................. 91
Probability Trees and Two Way Tables .......................................................................................................... 92
ACTIVITY 8 .................................................................................................................................................................. 93
Two-way Tables ........................................................................................................................................................ 94
Collated Questions ........................................................................................................................................................ 96
Centre and Range questions ................................................................................................................................. 96
Normal distribution questions ............................................................................................................................ 96
Normal distribution questions ............................................................................................................................ 97
Table and Tree questions ...................................................................................................................................... 98
Question set 1 - 2017 Exam ................................................................................................................................ 100
Question set 2 - 2018 Exam ................................................................................................................................ 112
Study Tips ....................................................................................................................................................................... 121
Study Habits .............................................................................................................................................................. 121
Study Methods ......................................................................................................................................................... 121
Study Techniques ................................................................................................................................................... 122
Study Mind-set ......................................................................................................................................................... 123
Pre-Exam Tips .......................................................................................................................................................... 123
Sitting the Exam ...................................................................................................................................................... 124
3|NCEA Level 2 Mathematics
ALGEBRA - Apply algebraic methods
in solving problems, 91261
Expanding Brackets
When you expand an equation, you remove all brackets from the equation.
One bracket
Multiply all the terms inside the bracket with the term outside the bracket.
Note: Remember the negative sign belongs to the term directly after it.
Eg. 4(2a-8) = (4 x 2a) + (4 x -8)
= 8a - 32
Two brackets
Multiply every term inside one bracket with every term inside the other bracket.
Simplify by adding the ‘like’ terms together (all the x2 terms, x terms, constants etc.)
Eg.
(3 - y)(4y + 5) = (3 x 4y) + (3 x 5) + (-y x 4y) + (-y x 5)
= 12y + 15 -4y2 -5y
= -4y2 + 7y + 15
Three brackets
What do you think would happen when you have three brackets?
Similar principals apply:
-
Expand out two of the brackets
Then multiply each of the resulting terms by each term in the third bracket.
Eg.
(w + 5)(w-6)(2w + 8) = ( (w x w) + (w x -6) + (5 x w) + (5 x -6) )(2w + 2)
= ( w2 - 6w + 5w – 30) (2w + 2)
= ( w2 - w – 30) (2w + 2)
= ( w2 x 2w) + (w2 x 2) + (- w + 2w) + (-w x 2) + ( - 30 x 2w) + (-30 x 2)
= 2w3 + 2w2 – 2w2 – 2w – 60w – 60
= 2w3 – 62w – 60
9|NCEA Level 2 Mathematics
ACTIVITY 1
Have a go at expanding these equations
Expand: 6b(2b-3)
Expand: (2x + 3)2
Expand: (x + 5)(x + 2)(x – 3)
Factorising
Factorising is the opposite of expanding.
We want to put the expanded equation back into brackets.
Two terms
-
Ask yourself ‘What is the largest term that ‘goes in’ to all terms?’
Take this term out of the bracket.
For the terms inside the bracket, ask yourself “what multiplied by the outside term will
give the original term?”
Eg. 6y2 - 2y = 2y (3y – 1)
o
o
o
The largest term that ‘goes in’ to all the terms into 6y 2 - 2y is 2y, as both 6y2 and 2y
can be divided by 2y
Therefore, 2y is outside the bracket
The terms inside the bracket and (6y2 /2y = 3y) and (2y/2y = 1)
Two terms (difference of two squares)
A special case occurs when we have one squared term minus another squared term.
In this case our answer is simply:
(√ of first term - √ of second term)( (√ of first term + √ of second term)
Eg.
9y2 - 4 = (3y – 2)( 3y + 2)
REMEMBER: This only works when the terms are being SUBTRACTED.
10 | N C E A L e v e l 2 M a t h e m a t i c s
Three terms
Step One: First look for a common factor and take this out of the bracket.
Step Two: Set up two brackets
Step Three:
Look at the constant term in the expression (the number by itself with no x, y or any other
variable attached).
Also look at the number in front of x, y, or whatever variable we are using. This is called our
coefficient of the variable.
We then think of two numbers that MULTIPLY to give the constant and ADD to give the
coefficient of our variable.
Put each of these numbers in the brackets.
Eg. 9x2 - 3x - 6
Step One: Take out common factor
3x2 - 3x - 6 = 3(x2 - x – 2)
Step Two: Set up two brackets
= 3(x
)(x
)
Step Three:
We can see that -2 and 1 MULTIPLY to give -2 (our constant) and ADD to give -1 (our
coefficient of x).
Put each number in the brackets.
= 3(x – 2)(x + 1)
Three terms (with coefficient of x2, with no common factor)
In our previous case, the number in front of x2 was removed as it was a common factor.
So what do we do if there is a coefficient of x2 that cannot be removed due to a common
factor?
This is a trickier case.
-
We could just ‘guess and check’ by trying different combinations and check by
expanding the brackets out again.
-
Alternatively, we could try the following method:
11 | N C E A L e v e l 2 M a t h e m a t i c s
Eg. 2x2 + 7x + 3
Look at the constant and the coefficient of x2.
We have 3 and 2. Multiplying these numbers gives 6.
The factors we are looking for must MULTIPLY to give 6 and ADD to give 7 (our coefficient
of x by itself).
The numbers 6 and 1 will meet these requirements.
We split our middle term, 7x, into 6x and 1x.
= 2x2 + 6x + 1x + 3
We can then factorise by taking out common factors [in this case it is (x + 3)]
= 2x (x + 3) + 1( x + 3)
= (2x + 1) (x + 3)
ACTIVITY 2
Have a go at expanding these equations
Factorise: 12x + 16xy
Factorise: 16a2 – 9c2d2
Factorise: 2x2 – 6x + 4
Factorise: 2x2 + 5x - 3
12 | N C E A L e v e l 2 M a t h e m a t i c s