Grade 8 Extension Problems – Linear Functions and Models
1.
10 000 people attended a sports match. Let x be the number of adults attending the sports match
and y be the number of children attending the sports match.
(a)
Write down an equation in x and y.
(1)
The cost of an adult ticket was 12 AUD. The cost of a child ticket was 5 AUD.
(b)
Find the total cost for a family of 2 adults and 3 children.
(2)
The total cost of tickets sold for the sports match was 108 800 AUD.
(c)
Write down a second equation in x and y.
(1)
(d)
Write down the value of x and the value of y.
(2)
(Total 6 marks)
2.
The diagram below shows the line PQ, whose equation is x + 2y = 12. The line intercepts the
axes at P and Q respectively.
diagram not to scale
(a)
Find the coordinates of P and of Q.
(3)
b)
A second line with equation x – y = 3 intersects the line PQ at the point A. Find the
coordinates of A.
(3)
(Total 6 marks)
IB Questionbank Mathematical Studies 3rd edition
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3.
A student has drawn the two straight line graphs L1 and L2 and marked in the angle between
them as a right angle, as shown below. The student has drawn one of the lines incorrectly.
y
3
L2
90°
2
1
–4
–3
–2
–1
0
1
2
3
x
4
–1
L1
Consider L1 with equation y = 2x + 2 and L2 with equation y = –
1
x + 1.
4
(a)
Write down the gradients of L1 and L2 using the given equations.
(b)
Which of the two lines has the student drawn incorrectly?
(c)
How can you tell from the answer to part (a) that the angle between L1 and L2 should not
be 90°?
(d)
Draw the correct version of the incorrectly drawn line on the diagram.
(Total 8 marks)
4.
The four diagrams below show the graphs of four different straight lines, all drawn to the same
scale. Each diagram is numbered and c is a positive constant.
y
y
c
c
Number 1
Number 3
x
0
y
c
x
0
y
c
Number 2
0
Number 4
x
0
x
In the table below, write the number of the diagram whose straight line corresponds to the
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equation in the table.
Equation
Diagram number
y=c
y=–x+c
y=3x+c
y=
1
x+c
3
(Total 8 marks)
5.
The following diagram shows the lines l1 and l2, which are perpendicular to each other.
Diagram not to scale
y
(0, 7)
l2
(0, –2)
(5, 0)
x
l1
(a)
Calculate the gradient of line l1.
(b)
Write the equation of line l1 in the form ax + by + d = 0 where a, b and d are integers, and
a > 0.
(Total 8 marks)
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6.
A plumber in Australia charges 90 AUD per hour for work, plus a fixed cost. His total charge is
represented by the cost function C = 60 + 90t, where t is in hours.
(a)
Write down the fixed cost.
(1)
(b)
It takes 3
1
hours to complete a job for Paula. Find the total cost.
2
(2
(c)
Steve received a bill for 510 AUD. Calculate the time it took the plumber to complete the
job.
(3)
(Total 6 marks)
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SOLUTIONS
1.
(a)
x + y = 10000
(A1) (C1)
(b)
2 × 12 + 3 × 5
(M1)
39 (39.0, 39.00) (AUD)
(A1) (C2)
(c)
12x + 5y = 108800
(A1) (C1)
(d)
x = 8400, y = 1600
(A1)(ft)(A1)(ft) (C2)
Notes: Follow through from their equations.
If x and y are both incorrect then award (M1) for attempting to
solve simultaneous equations.
[6]
2.
(a)
0 + 2y = 12 or x + 2(0) = 12
(M1)
P(0, 6)
(accept x = 0, y = 6)
(A1)
Q(12,0)
(accept x = 12, y = 0)
(A1) (C3)
Notes: Award (M1) for setting either value to zero.
Missing coordinate brackets receive (A0) the first time this
occurs. Award (A0)(A1)(ft) for P(0,12) and Q(6, 0).
(b)
x + 2(x – 3) = 12
(M1)
(6, 3)
(accept x = 6, y = 3)
(A1)(A1) (C3)
Note: (A1) for each correct coordinate.
Missing coordinate brackets receive (A0)(A1) if this is the first
time it occurs.
[6]
3.
1
.
(A1)(A1) (C2)
4
Note: Award (A0)(A1)ft if the order of the gradients is reversed
or both signs are wrong or both are reciprocals of the correct
answer.
(a)
L1 has gradient 2 and L2 has gradient 
(b)
L2 is drawn incorrectly.
(c)
The product of the gradients is 2 × 
(d)
The drawing should show a straight line passing through
x and y intercepts at (4, 0) and (0, 1) respectively.
(A1)(A1) (C2)
Note: Award (A1) for each intercept. If these are wrong but
(A2) (C2)
1
1
   –1.
(M1)(A1) (C2)
4
2
Note: Award (M1) for looking at product of gradients,
(A1) for comparing something to –1.
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gradient is 
1
then (A1). If correct line is very poorly drawn
4
then (A1).
[8]
4.
Equation
Diagram number
y=c
2
(A2)
y = –x + c
3
(A2)
y = 3x + c
4
(A2)
1
(A2) (C8)
y=
1
x+c
3
[8]
5.
(a)
Gradient of l 2 
=
(M1)
2
5
Gradient of l1 
(b)
0  (2)
50
(A1)
5
2
5
x+7
2
2y = –5x + 14
5x + 2y – 14 = 0
y=
(A1) (C3)
(A1)(A1)
(A1)(A1)(A1) (C5)
[8]
12.
UP
Note: Unit penalty (UP) applies in part (c)
(a)
AUD 60
(A1) (C1)
(b)
C = 60 + 90(3.5) = AUD 375
Note: Award (M1) for correct substitution of 3.5.
(M1)(A1) (C2)
(c)
510 = 60 + 90t
(M1)(A1)
t = 5h (hours, hrs)
Note: Award (M1) for setting formula = to any number.
(A1) for 510 seen.
(A1) (C3)
[6]
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