Maths Task:Linear graphs&equations, Quadratics

  

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Part 1
SEND: Work for Submission
Please Note: TECHNOLOGY FREE questions have to be completed without using calculator.
You may use your CAS calculator to verify these answers but you must show all working out to
receive credit for this week’s work.
TECHNOLOGY FREE
1.
Factorise each of the following as far as possible.
2.
(a)
3×2 + 3x–6
(b)
(2y + 1)2-25
(c)
5(x + 1)2- 3y(x + 1) + 4(x + 1)
(d)
12×2 + 3x–9
(a)
X=
z +z
y
z1 + z2 + z1 z2 where y = 1 2
2
2
(
)
Evaluate X when z1 = 9 and z2 = 1.
(b)
Transpose the following formula to make t the subject:
s = tq2 – 5
(c)
l
is the formula for the time T of one complete swing of a pendulum of length l
g
T = 2?
under gravitational acceleration g.
Express g as the subject of the formula
3.
Simplify the following expressions as far as possible.
(a)
8 xy
24 x
(b)
9-a
a -9
(c)
2a 4a

3
5
(d)
x – 2y 3 – x

6y
3x
(e)
2a
1 – 3a
– 2
a – 3 a – 6a + 9
(f)
y 2 – 16 y 2 – 5 y + 4
?
y2 – 2 y
2- y
Select one answer for Question 4. Justify your answer by showing all working.
4.
(
6 +3
A
)(
)
6 – 3 simplifies to give
6 -3
B
33
C
-3
D
3
E
3
PART 2
TECHNOLOGY FREE.
You may use your CAS calculator to verify these answers but you must show all working out
to receive credit for this week’s work.
1.
Find all solutions to the following equations. Make sure you exclude possibilities of zero
denominators, and check your answers. [Hint: multiply through by L.C.D.first.]
x + 3 2x – 5
+
=5
(i)
2
5
10
2

=4
x-2 x+4
(ii)
2.
Find the exactdistance between the points (5, -10) and (12, -11).
3.
Find the midpoint of the line segment joining the points (-3.5, -1) and (16, -10).
4.
Find the equation of the line which passes through the point (-2, 6) and is perpendicular
1
to the line y = ( x – 4 ) .
2
5.
Use the substitution method to solve the following simultaneous equations.
Show ALL working out.
y = 2x + 2
y = -2x + 4
Confirm your answer by sketching the graphs to show the point of intersection. (Include
your sketch graph as part of your answer.)
6.
Use the elimination method to solve the following simultaneous equations.
Show ALL working out.
2x + 5y = 3
7x – 2y= -9
Confirm your answer by sketching graphs and showing the point of intersection.
Show all working associated with your sketch graph.
(Include your sketch graph as part of your answer.)
7.
Solve the following inequations:
(a) 5 – 2x> 7x + 3
(b)
3 + 5x 4 – x 2
+
?
2
3
3
8.
Use the ‘Break It Down and Solve It’ process (as shown in Example 17) to solve the
following problem.
“If the price of 5 adults’ and 3 children’s tickets to the cinema is $96, and the price of 2
adults’ and 8 children’s tickets is $86, find the cost of an adult’s ticket and a child’s
ticket.”
Show ALL working out.
9.
CAS Calculator
For the following question, show screen capture of your calculator solution.
If you are unable to get screen captures, carefully draw the screens that you get when
answering these questions. Contact your teacher if you have any questions or concerns.
Verify your answer to question 7 by using the Calculator application on your CAS
calculator (rather than the Graphsapplication). (Refer to the instructions on pages
25/26.)
Include a copy of your calculator screen with your working.
10. Determine the angle between the line y = 3x- 2 and the x-axis
Give your answer in degrees correct to 2 decimal places, and then change the decimal
degrees to degrees and minutes correct to the nearest minute.
PART 3
TECHNOLOGY FREE
1.
Solve the following equations by first factorising each quadratic expression:
(a)
2.
x2- 5x + 6 = 0
(b) 2×2+ 9x- 5 = 0
Use the completing the square method to write the following equations in turning
point form.
State the co-ordinates of the turning points.
(a)
y = x2 – 4x+ 8
(b) y = x2 + 5x – 1
3.
Use the quadratic formula to find the exact solutions to x2 + 7x – 3 = 0.
Leave the answer in exact surd form (i.e. Do not write your answer as an approximate
decimal)
4.
After examining their discriminants, state the number of solutions to the following
equations:
5.
(a)
x2 + 4x – 2=0
(b) x2 – 6x + 10 = 0
(c)
3×2 + 9x – 2 = 0
(d) x2 – 10x + 25 = 0
Without the use of a calculator, sketch each of the following parabolas.
Indicate on your sketch, the coordinates of all intercepts and turning points. Show how you
found them.
Remember to:
• rule axes
• label axes
• label each axis with an accurate scale
(a)
6.
y = 2(x- 3)2 + 2
(b) y = x2- 2x – 3
Solve the following quadratic inequations.
Include a well labelled graph for each question.
(a)
x2 – 5x – 6 < 0 (b) x2 - 5x – 6 ? 0 7. Usethe ‘Break It Down and Solve It’ method to solve this problem. Arch Bridge
A civil engineer is designing a bridge which is 101 metres long, 30 metres high and is to have four identical
parabolic arches along its length. Each arch is 20 metres high and there is one metre between the base of
each arch as shown in the diagram.
(a)
How wide is each arch at its base?
(b)
A set of Cartesian axes is placed as shown in the diagram with the origin at the left hand end of the
base of the first arch. Find the equation of the first arch. Use the general equation y = a(x-h)2 + k.
(Hint – Substitute the x and y coordinates of known points on Arch 1 to determine the values of a, h
and k.)
(c)
Find the equation of the second arch.
(d)
A river will flow through the third arch. Barges on the river are a standard size: 3 metres across and
with the top of the barge 0.5 metres above the water. How far below the top of the arch can the water
level rise and still allow a barge to pass through?
30 m
Arch 1
1m
Arch 2
1m
Arch 3
1m
101 m
Arch 4
1m
1m
x
PART 4
TECHNOLOGY FREE
1.
Solve algebraically these pairs of simultaneous equations. Show all working, giving exact
answers (Use surds if necessary. NO approximate decimal values).
y=x – 3
y=-x2+5x+2
(a)
2.
(b)
y=2x-1
y=x2+x-4
Solve the pair of simultaneous equations (below) using two different methods:
(i)
Graphically (Sketch the graphs without using aCAS calculator) and find the
point of intersection.
Check your solution by solving the same equations algebraically
(ii)
y = -6x + 22
y = -x2 + 3x + 4
3.
Determine the equation of :
(a) the parabola with y-interceptat (0,2) and the turning point at (1,9)
(b) the parabola with x-intercepts at x = 1 and x = – 5 and the y-interceptat(0,-3)
4.
For what values of m does the line with equation y=mx-1
(a)
touch
(b)
intersect
(c)
not intersect
the parabola with equation y=2×2 ?
5.
Use a difference table to determine the equation that generated the following results.
x
y
6.
0
5
1
9
2
15
3
23
4
33
CAS Calculator
The profit $P made when a particular make of car is manufactured in thours is given by the
equation P = 33t2 – 7t – 4000. In order to just “break even”, how long must the manufacturing
process take?
This problem could be solved using a couple of different methods. You are required to use a
graphical approach using your CAS calculator.
Include a graph in your solution. Use your CAS calculator to sketch the graph and copy the
screen onto your page of working.
Be sure to include your WINDOW settings.
Give your answer in hours correct to 2 decimal places.
7.
Solve the following equation by using iterations. Show all working out.
x2 + 5x- 10 = 0
Confirm your answers using a CAS calculator.

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