Download VCE Mathematical Methods CAS Examination Unit 1 and more Study notes Mathematical Methods in PDF only on Docsity!
STUDENT NAME:
SUBJECT TEACHER: TUTOR GROUP:
VCE Mathematical Methods CAS Unit 1
(b1mMC1)
Written examination 1
(Calculator free)
XXday XX
th
of June 2013
Reading time: 11:35 am – 11:50 am (15 minutes)
Writing time: 11.50 am – 12.50 pm (60 Minutes)
QUESTION AND ANSWER BOOK
Structure of book
Section Number of Questions
Number of questions to be answered
Number of marks
Suggested time
Section A 10 Vocabulary
Questions
5 5 5 mins
Section B 8 Short answer
Questions
Total: 50 marks Total: 60 mins
- Students are allowed to bring into examination room: pens, pencils, highlighters, erasers,
sharpeners, rulers
- No calculators of any type or pages of notes are permitted
- A formula sheet is provided
Instructions
- Please ensure that you write your name, your teacher’s name and tutor group in the spaces
provided above on this page.
- Answer all questions in the spaces provided in this booklet
- A decimal approximation will not be accepted if an exact answer can be obtained
- In questions where more than one mark is available, appropriate working must be shown
Students are NOT permitted to bring mobile phones, mp3 players and/or any other
unauthorised electronic devices into the examination room.
Section A: Vocabulary ( 5 marks)
Write the correct answer from the list below in the spaces provided.
1 If two linear equations have the same gradients, then they are -
---------------------.
For a matrix [ ], the value of ( ad – bc) is known as the ---
3 In a polynomial, the highest power of x is called the -----------
of the polynomial.
4 A ------------------ is considered to be a rule where for every
value of x there is only one value of y. This is tested using the
vertical line test.
5 A----------------- matrix is one which has no inverse matrix.
Word list
Function Relation Perpendicular Parallel
Determinant Discriminant Degree Singular
Question 2
Consider the matrix equation [ ] [ ] [ ]
Using inverse matrices, solve the equation to find the values of x and y.
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
3 marks
Question 3
Consider the function f ( x ) =
a) Sketch the graph of this function for {x: -5 < x < 5} showing any intercepts and asymptotes.
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
3 marks
b) Find the point(s) of intersection between the function above and the line y = 5
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
2 marks
x
y
Question 5
A polynomial is defined as P(x) = 3 x
3
2
a) Use the remainder theorem to find a factor of P ( x )
__________________________________________________________________________
__________________________________________________________________________
1 mark
b) Find all solutions of P x ( )= 0
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
3 mark
c) Rewrite P ( x ) in factorized form.
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
1 mark
c) Give the values for x : P ( x ) < 0
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
1 mark
Question 6
The function f is defined as follows:
f : f ( x ) = (^) √
a) Find the maximal domain and the range of f ( x )
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
2 marks
b) Find the rule of the inverse function
1 f ( ) x
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
3 marks
c) Sketch below the graph of f ( x ) and
1 f ( ) x
label the graphs and important points.
3 marks
d) Give the range of the inverse function
1 mark
x
y
Question 8
Consider the simultaneous equations
m x + 3 y = 27
27 x + m y = m
2
a) Solve for x using an algebraic method
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
3 marks
b) Find the value(s) of m for which there;
i) is no solution for x
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
1 mark
ii) is a unique solution for x
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
1 mark
Formula sheet for Mathematical Methods 1&2 CAS Examination- Semester 1
Linear Relations
Gradient of straight line joining two points A(x 1 ,y 1 ) and B (x 2 , y 2 ):
Gradient 2 1
2 1
x x
y y m
The general equation of a straight line is y mx c
Equation of a line passing through two given points A(x 1 ,y 1 ) and B (x 2 , y 2 ) is
y y 1 m ( x x 1 )
The tangent of the angle of slope ( )can be found with:
2 1
2 1 tan( ) x x
y y m
Where ( )is the angle the line makes with the positive direction of the x-axis.
Two lines are perpendicular if m 1 m 2 1
Two lines are parallel if m 1 (^) m 2
The midpoint M between two points A(x 1 ,y 1 ) and B (x 2 , y 2 ) is ) 2
x 1 x 2 y 1 y 2 x (^) m ym
Distance between 2 points A and B is
2 1 2
2 AB x 1 x 2 y y
The angle, , between intersecting straight lines: 2 1
Quadratics
Quadratic equation forms:
1- General : y ax bx c
2 with a
b xt (^) p 2
..
2- Turning point form (Basic): y a x h k
2
Where (h,k) are the coordinates of the turning point.
3- Factors form: y a x b x c
Factorising Quadratics:
a- With Four Terms
5 5 5 ( ) ( ) ( )( 5 )
2 x y x xy x y xx y x y x
b- With Three Terms
5 6 { 3 2 6 3 2 5 } ( 3 )( 2 )
2 x x and x x
c- Difference of two squares: ( )( )
2 2 a b a b a b
d- Perfect Square:
2 2 2 ( a b ) a 2 ab b
e- Quadratic Formula 0
2 For ax bx c a
b b ac x 2
2
Cubic functions Forms:
a- General: y ax bx cx d
3 2
b- Basic (Inflection point) y ax h k
3 ( ) where (h,k) are the inflection point coordinates.
c- Factors y a ( x b )( x c )( x d )
d- Sum of cubes Difference of cubes
e- ( )( )
3 3 2 2 a b a b a ab b ( )( )
3 3 2 2 a b a b a ab b
f- Rational Functions
