UNIVERSITY OF CANTERBURY
Mid-Course Test, Semester 2 2022
Prescription Number:
MATH101-22S2 (C)
Paper Title:
Methods of Mathematics
Time allowed:
90 MINUTES
Number of questions:
FIVE
Answer all FIVE questions.
There is a total of 40 marks.
Write your answers in the spaces provided.
Use blue or black ink only.
MARKS
Office use only
Q1
Show all your working.
Marks will be lost for poorly presented or incomplete answers.
No calculators permitted.
Family Name/
Surname:
Q2
Q3
Q4
...........................................................
Q5
First Name/
Given Name:
...........................................................
Student ID:
...........................................................
Total
Mid-Course Test, Semester 2 2022
MATH101-22S2 (C)
Blank page for rough working.
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Mid-Course Test, Semester 2 2022
MATH101-22S2 (C)
Question One (9 marks)
(a)
Solve the following equation:
3x + 1
=8−x
2
(b)
Rearrange the following equation to make x the subject:
1
2 3
= +
x
a b
(c)
Evaluate the following expression and simplify your answer.
18(3)−2 + 82/3
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Mid-Course Test, Semester 2 2022
(d)
MATH101-22S2 (C)
Consider the following system of linear equations
5x − 6y = 1
−10x + 3y = −8
(a) Solve the system of linear equations.
(b) Describe the relationship between the lines 5x − 6y = 1 and −10x + 3y = −8.
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Mid-Course Test, Semester 2 2022
(e)
(i)
MATH101-22S2 (C)
Describe the activity in your favourite MATH101 workshop so far.
(ii) Write one or two sentences describing one thing you learnt from this workshop,
or how it changed your view of mathematics.
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Mid-Course Test, Semester 2 2022
MATH101-22S2 (C)
Question Two (8 marks)
x−
√
x−3=5
(a)
Solve the following equation.
(b)
Solve the following inequality and give your answer using interval notation.
| 7 − 2x |≤ 8
(c)
3
Give the slope of a line that is perpendicular to y = x − 3.
2
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Mid-Course Test, Semester 2 2022
(d)
MATH101-22S2 (C)
Draw the graph of the following relationship on the axes below.
3
y = x−3
2
Spare axes are included below but you must clearly specify which graph is to be marked.
y
7
6
5
4
3
2
1
−7 −6 −5 −4 −3 −2 −1
−1
−2
−3
−4
−5
−6
−7
1 2 3 4 5 6 7 x
y
7
6
5
4
3
2
1
−7 −6 −5 −4 −3 −2 −1
−1
−2
−3
−4
−5
−6
−7
1 2 3 4 5 6 7 x
Page 7 of 16
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Mid-Course Test, Semester 2 2022
MATH101-22S2 (C)
Question Three (10 marks)
(a)
Consider the following equation
0 = 2x2 − 4x + 3.
(i)
Use the discriminant ∆ = b2 − 4ac to determine how many distinct real
solutions this equation has.
(ii) Use your answer from part (i) to select the most appropriate graph for the
equation y = 2x2 − 4x + 3. Provide a reason for your answer.
Graph A
Graph B
Graph C
Graph D
Page 8 of 16
Mid-Course Test, Semester 2 2022
(b)
(i)
MATH101-22S2 (C)
Use the method of completing the square to rearrange the following quadratic
into the form y = a(x − h)2 + k.
y = x2 − 10x + 7
(ii) What is the vertex of the parabola given by y = x2 − 10x + 7?
Page 9 of 16
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Mid-Course Test, Semester 2 2022
(c)
MATH101-22S2 (C)
Consider the cubic given here in factorised form:
y = (x − 2)(x + 1)(x − 4)
(i)
What is the y-intercept of the cubic given above?
(ii) Sketch the graph of this cubic on the axes below. Show the x-intercepts and
y-intercept clearly, but otherwise a sketch is sufficient.
Spare axes are included below but you must clearly specify which graph is to be marked.
y
10
9
8
7
6
5
4
3
2
1
−5 −4 −3 −2 −1
−1
−2
−3
−4
−5
−6
−7
−8
−9
−10
1
Page 10 of 16
2
3
4
5x
Mid-Course Test, Semester 2 2022
MATH101-22S2 (C)
y
10
9
8
7
6
5
4
3
2
1
−5 −4 −3 −2 −1
−1
−2
−3
−4
−5
−6
−7
−8
−9
−10
1
Page 11 of 16
2
3
4
5x
TURN OVER
Mid-Course Test, Semester 2 2022
MATH101-22S2 (C)
Question Four (7 marks)
(a)
Consider the following piecewise function.


1 − x, x < −2
f (x) = x2 ,
−2 ≤ x < 2


5,
x≥2
Evaluate the following.
(i)
f (−3)
(ii) f (2)
(iii) f (0)
(b)
Give the domain of the function g(x) =
3
.
x+5
Page 12 of 16
Mid-Course Test, Semester 2 2022
(c)
MATH101-22S2 (C)
Give the equations that correspond to the following graphs.
(i)
Equation 1:
y
5
4
3
2
1
−6 −5 −4 −3 −2 −1
−1
1
2
3
4
5
6 x
−2
(ii) Equation 2:
y
4
3
2
1
−4
−3
−2
−1
1
2
3
4 x
−1
−2
−3
−4
Page 13 of 16
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Mid-Course Test, Semester 2 2022
MATH101-22S2 (C)
Question Five (6 marks)
(a)
The following diagrams are not to scale.
(a) Evaluate the side length x in the triangle below. Give your answer exactly.
6
30◦
x
(b) Evaluate the angle A in the triangle below. Give your answer exactly, in
radians.
√
3 2
3
A
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Mid-Course Test, Semester 2 2022
(b)
MATH101-22S2 (C)
Determine whether or not the following triangle contains a right-angle. Justify your
answer.
Image not to scale.
√
3
30
5
END OF TEST
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Mid-Course Test, Semester 2 2022
MATH101-22S2 (C)
Blank page for rough working.
END OF PAPER
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