Name:
Teacher:
Unit 1 Maths Methods (CAS) Exam 2015
Wednesday June 3 - 1.50 pm
Reading time: 10 Minutes
Writing time: 80 Minutes
Instruction to candidates:
Students are permitted to bring into the examination room: pens, pencils, highlighters, erasers, sharpeners,
rulers, a single bound exercise book containing notes and class-work, CAS calculator.
Materials Supplied:
Question and answer booklet, separate multiple choice answer sheet.
Instructions:
Write your name and that of your teacher in the spaces provided.
Answer all short answer questions in this booklet where indicated.
Always show your full working where spaces are provided.
Answer the multiple choice questions on the separate answer sheet.
Section A
Section B
/20
Total exam
/36
1
/56
Section A – Multiple choice questions (20 marks)
Question 1
The solution to the equation
!
!!!
!
= 4! + 1 is:
a) ! = − !!
b) ! = −3
c) ! = −
d) ! =
!!
!
!!
!
!
e) ! = − !!
Question 2
Which of the following correctly shows the transformation of the equation !"
subject?
= !"# to make ! the
PV
RT
RT
n=
PV
n = PV + RT
n = PV − RT
n = PVRT
a) n =
b)
c)
d)
e)
Question 3
A straight line has the equation
a)
b)
c)
d)
e)
4y − 3x = 8 . Another form of this equation is:
3x
+4
8
y = 8x + 3
3x
y=
+2
4
3x
y=
+4
4
3x
y=
+8
4
y=
Question 4
A straight line segment joins the points (1,5) and (6,10). The distance between the points is:
a)
c)
5 2
2 5
274
d)
e)
50
1
b)
2
Question 5
The equation of the linear function graphed here is:
a)
b)
c)
d)
e)
y = 2x + 7
y = −2x + 7
2x
y=
−2
7
2x
y=− +2
7
7x
y=
−2
2
Question 6
Two points on an architectural plan have coordinates (-2, 3) and (4, 8). The midpoint of these points is:
a) (2, 11)
b) (6, 7)
c) (-1, 7)
d) (6, 11)
e) (1, 5.5)
3
Question 7
The straight lines
a)
b)
c)
d)
e)
y=−
3x
+ 4 and y = 2x − 3 intersect at the point:
2
(1,2)
(2,1)
(1, 3)
5
( , 2)
2
1
(7, )
2
Question 8
Which one of the following lines is perpendicular to the line
a)
b)
c)
d)
e)
y=
x
+ 3?
5
x
+5
3
x
y=5−
3
x
y= −3
15
y = 5x
y = −5x + 3
y=
Question 9
The solutions to the quadratic equation
a)
b)
c)
d)
e)
! = ! ! − 11! + 24 are:
x = 6 or x = 4
x = -6 or x = -4
x = -3 or x = -8
x = 3 or x = 8
x = 2 or x = 12
Question 10
In order to factorise the equation ! = ! ! − 8! + 4 using algebra, the best technique to use would be:
a) find a highest common factor
b) the difference of two squares rule
c) the perfect squares rule
d) factorising by inspection (the “crossed swords”)
e) completing the square
4
Question 11
The quadratic function !
a) (-2, -1)
b) (2, -1)
c) (-4, -3)
d) (4, -3)
e) (4, 3)
= ! ! + 4! + 3 has a turning point at:
Question 12
The equation
k>6
k > −6
k≥6
d) k ≤ −6
e) k = ±6
x 2 − kx + 9 = 0
has only one solution if:
a)
b)
c)
Question 13
A parabola is shown below. Which of the following quadratic equations correctly defines the parabola?
a) y = −2(x +1)(x + 3)
b)
c)
d)
e)
y = 2(x −1)(x − 3)
y = 2(x +1)(x + 3)
y = (x −1)(x − 3)
y = (x +1)(x + 3)
5
Question 14
What are the x intercepts of the quadratic function
a)
b)
c)
d)
e)
y = x 2 + ( f − g)x − fg ?
x = f and x = g
x = f and x = −g
x = − f and x = g
x = − f and x = −g
f
g
x = and x =
g
f
Question 15
Which of the following correctly lists all of the factors of the cubic function
a)
b)
c)
d)
e)
y = 2x 3 + 3x 2 − 23x −12 ?
(x −12)3
(x − 3)(x + 4)(2x +1)
(x − 4)(2x −1)(x + 3)
(x − 2)(x − 3)(x − 23) −12
(x + 2)(x + 3)(x + 23) +12
Question 16
The complete set of solutions to the equation
a)
x=
8 = −125x 3 is:
2
5
5
2
2
c) x = −
5
2
2
d) x = and x = −
5
5
2
5
e) x = and x =
5
2
b)
x=−
Question 17
The cubic equation ! = 3(! − 2)
a)
(3, 2)
b)
(2, 5)
c)
(3, 5)
d)
(2, -5)
e)
(5, 2)
3
− 5 has a point of inflection with coordinates:
6
Question 18
The graph of a cubic function is shown below. Which of the following equations describes the graph?
a)
y = −x 2 (x + 4)
b)
y = −x(x − 4)2
c)
y = x 2 (x + 4)
d)
y = −x 2 (x − 4)
e)
y = x 2 (x − 4)
Question 19
When the expression
a) -33
b) -9
c) 0
d) 9
e) 33
! ! − 3! ! + 4! − 5 is divided by the linear factor (! + 2), the remainder is:
7
Question 20
The graph of a cubic function is shown below. Which of the following equations describes the graph?
1
y = − (x −1)3 + 2
4
1
3
b) y = (x + 2) +1
4
1
3
c) y = (x +1) + 2
4
d) y = (x +1)3 + 2
a)
e)
y = (x −1)2 + 2
8
Section B – Short answer questions (36 marks)
Question 1
Shifty car rentals charge $60.00 per day to hire a car, with an extra cost of 10 cents per kilometre.
a) Write a linear relation that describes the cost in dollars (c) as a function of the distance travelled in
kilometres (x).
(1 mark)
b) Calculate the total cost of hiring the car for a day & driving the following distances:
Distance
Cost
100 km
200 km
500 km
(3 marks)
c) Use this data to draw an appropriate graph of the cost vs distance.
(3 marks)
9
Low Budget rentals offer the same car at a fixed cost of $80 per day and 5 cents per kilometre.
d) Draw this cost on the graph.
(2 marks)
e) Calculate the distance at which the Low Budget rental becomes cheaper than Shifty.
(2 marks)
Question 2
Use long division to find the three linear factors of the cubic function
p(x) = 2x 3 − x 2 − 5x − 2 .
(6 marks)
10
Question 3
Use an appropriate algebraic technique to solve the following equations. All working must be shown.
a)
2! ! + 5! + 2 = 0
(3 marks)
b)
! ! + 9 = 6!
(3 marks)
c)
! ! + 8! − 11 = 0
(3 marks)
11
Question 4
Recognising the strength of the arch, engineers are building a parabolic shaped mine tunnel. The tunnel is 5
m wide at the bottom and 4 m high in the middle. The tunnel is designed for one single lane of traffic in one
direction.
(0,0)
a) The equation used
the tunnel is =
to model the curve of
!
!(! − 5) + 4 , where the point (0,0) is at the lower left intersection of the wall and the
ground. Show that
!
! = − !".
(2 marks)
b) Trucks moving through the tunnel are 3 m high. Using a CAS calculator or other means to find the widest
truck that can fit through the tunnel.
(2 marks)
12
Question 5
1
(x 3 − 40x 2 + 500x − 2000) , where y is the
250
height above ground level and x is the horizontal distance. The slide starts at x = 0 and ends where the
The shape of a slide is defined by the cubic function
y=−
curve goes below ground level.
a) Calculate the initial starting height of the curve.
(2 marks)
b) Calculate the height of the slide where x = 10.
(2 marks)
c) Using a CAS calculator or other means, find the value of x (to two decimal places) where the height of
the slide is 4 m.
(1 mark)
d) Using a CAS calculator or other means, find the value of x where the slide ends.
(1 mark)
13
Answer sheet for section A
Name: ______________________________
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