Student:
Form:
Maths Teacher:
Melbourne High School
YEAR 10 MATHEMATICS 2018
Semester 2 Examination 1
November 2018
Reading time: 15 minutes
Writing time: 75 minutes
Section
Number of questions
Marks available
Section 1
Multiple Choice
8
8
Section 2
Short Answer and Extended Answer
13
42
Total: 50
•
•
Students are permitted to bring into the examination room: pens, pencils, highlighters, erasers, rulers.
Students are NOT permitted to bring into the examination room: notes of any kind, blank paper, white out
liquid/tape, calculators, mobile telephones, nor any other unauthorised electronic devices.
Materials supplied
•
•
This question and answer book
A sheet of miscellaneous formulae.
Instructions
•
•
•
•
•
•
•
•
Write in comprehensible, concise, and unambiguous English.
Set out your solutions clearly and logically. Appropriate working must be shown for full credit in any question part
worth more than one mark.
Answer all questions in the spaces provided.
Any rough working done on loose sheets of paper will not be assessed but should still be submitted.
In all questions where a numerical answer is required, an exact value must be given unless the question specifies
otherwise. Decimal responses will be assumed to be approximate.
Final answers must be fully simplified unless the question specifies otherwise.
Answers must include units where applicable.
Diagrams in the examination paper are not drawn to scale.
Year 10 Mathematics, Melbourne High School
Examination Paper 1 | Semester 2, 2018
(Reverse of covering page)
THIS PAGE HAS BEEN INTENTIONALLY LEFT BLANK
This space may be used for rough work. What is written in this space will not be marked.
Page 2 of 12
Year 10 Mathematics, Melbourne High School
Examination Paper 1 | Semester 2, 2018
Section A – Multiple choice (8 marks)
Circle the response that is correct for the question.
No marks will be given if more than one answer is circled for any question.
All diagrams shown are not drawn to scale.
1
Two coins are tossed. What is the probability that both coins show Heads given that at least
one of the coins shows Heads?
A
B
2
1
3
1
C
4
D
cannot be determined
1
2
Given the chord BC of a circle, centre O, where A is a point on the major arc with
∠𝑂𝑂𝑂𝑂𝑂𝑂 = 5° and ∠𝑂𝑂𝑂𝑂𝑂𝑂 = 40° as shown in the diagram below,
the value of x is
3
A
50
B
45
C
40
D
35
The graph of 𝑦𝑦 = 2𝑥𝑥 is translated by 3 units up (in the positive y-direction) followed by a
translation of 1 unit left (in the negative x-direction) and finally reflected in the x-axis.
What is the equation of the final transformed graph?
A
B
C
D
𝑦𝑦 − 3 = −2𝑥𝑥+1
𝑦𝑦 = 2𝑥𝑥+1 + 3
𝑦𝑦 + 3 = 2(𝑥𝑥+1)
𝑦𝑦 = −2(𝑥𝑥+1) − 3
Page 3 of 12
Year 10 Mathematics, Melbourne High School
Examination Paper 1 | Semester 2, 2018
4
Consider the triangle below.
A
60o
30o
B
C
The ratio of AB : AC is
A
B
C
D
5
√2 : 1
2 : 1
1 ∶ √3
√3 : 1
Triangle JKL has vertices as shown in the diagram below.
J (1, 5)
O
(−1, − 2) K
x
L (7, − 2)
What is the equation of the perpendicular bisector of KL?
A
y=5
B
x=4
C
x=3
D
y=3
Page 4 of 12
Year 10 Mathematics, Melbourne High School
Examination Paper 1 | Semester 2, 2018
6
7
In the figure, the line BA is a tangent to the circle at point B. Line AC is a secant that meets
the tangent at A. What is the value of angle z?
A
50o
B
85o
C
95o
D
100o
The graph with equation of the form
y a cos bx + c is shown for 0 ≤ 𝑥𝑥 ≤ 360°.
=
y
=
y a cos bx + c
4
O
−2
180
360
x
What is the equation of this graph?
A
=
y cos x + 3
B =
y cos 3 x + 1
C =
y 3cos x + 1
D =
y 4 cos x + 2
Page 5 of 12
Year 10 Mathematics, Melbourne High School
Examination Paper 1 | Semester 2, 2018
8
Which of the following is the graph of y = x( x − 1)( x + 2) ?
A
y
O
x
1
y
B
O
1
x
y
C
−1
2
O
x
y
D
−1
O
2
x
Page 6 of 12
Year 10 Mathematics, Melbourne High School
Examination Paper 1 | Semester 2, 2018
Section B – Short answers (42 marks)
Answer all questions in the spaces provided.
Where a numerical answer is required, an exact value must be given unless otherwise specified.
In questions where more than one mark is available, appropriate working must be shown.
Diagrams shown are not drawn to scale.
1
Sketch the graphs of the following equations on separate axes, clearly labelling all
intercepts, turning points and equations of any asymptotes.
a. 𝑦𝑦 = 1 − 12𝑥𝑥
[2]
b. 𝑦𝑦 = 𝑥𝑥 2 − 12𝑥𝑥
[3]
c. 𝑦𝑦 = 𝑥𝑥 2 (2 − 𝑥𝑥)3 , stating the number of turning points. You DO NOT need to find
[3]
the coordinates of the turning points for this part of question.
Page 7 of 12
Year 10 Mathematics, Melbourne High School
Examination Paper 1 | Semester 2, 2018
2
3
(2𝑥𝑥−3)2 −4
Fully simplify the expression
2𝑥𝑥 2 +𝑥𝑥−15
.
By completing the square, solve for x in the equation
[3]
[3]
𝑥𝑥 2 − 5𝑥𝑥 + 3 = 0
4
Fully simplify
�6+�5��6−�5�
√31
.
[2]
Page 8 of 12
Year 10 Mathematics, Melbourne High School
Examination Paper 1 | Semester 2, 2018
5
Solve for x in the following equations.
[2]
a. log 3 𝑥𝑥 − 1 = log 3 27
𝑥𝑥
b. �√10� = 1002−𝑥𝑥
6
Evaluate the following expression.
1
50 − (8)3 ×
7
[3]
[2]
5
4−2
Ian wrote down six whole numbers 3, 4, 7, a, 3, b where b > a.
[3]
If the mode of these numbers is 3, the mean is 6 and the median is 5, find the value of a
and of b.
Page 9 of 12
Year 10 Mathematics, Melbourne High School
Examination Paper 1 | Semester 2, 2018
8
4
Given that x is an acute angle and that cos (180o + x) = − 5 , find the value of the
following giving your answer as a fraction.
a. cos x
[1]
b. cos (180o – x)
[1]
c. tan x
[2]
9
y is inversely proportional to d 2 . If y = 2 for a certain value of d .
Find the value of y when this value of d is increased three times.
[2]
10
Given that sin 28° = 0.469 , solve for 𝜃𝜃 in the equation
[2]
for 𝜃𝜃 ∈ [0, 360°].
sin 𝜃𝜃 = −0.469
Page 10 of 12
Year 10 Mathematics, Melbourne High School
Examination Paper 1 | Semester 2, 2018
11
Three pipes are stored on horizontal ground as shown in the diagram.
[2]
Each pipe has a circular cross section with radius 1 metre.
Calculate the height, h metres, of the stacked pipes,
giving your answer in exact form.
(Ignore the thickness of the pipes)
12
Two unbiased six-sided dice are rolled once. The number on the upper face of the first
die determines b while the number on the upper face of the second die determines c in
the quadratic equation 𝑥𝑥 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐 = 0.
[3]
With the aid of a table or otherwise, calculate the probability that the quadratic
equation will have two real and different roots.
Page 11 of 12
Year 10 Mathematics, Melbourne High School
Examination Paper 1 | Semester 2, 2018
13
BC is a diameter of circle with centre O. The tangent at E meets CB produced at A.
[3]
Given that ∠𝐵𝐵𝐵𝐵𝐵𝐵 = 3∠𝐵𝐵𝐵𝐵𝐵𝐵, prove that ∠𝐶𝐶𝐶𝐶𝐶𝐶 = 𝑘𝑘 ∠𝐵𝐵𝐵𝐵𝐵𝐵,
where 𝑘𝑘 is an integer that you need to find.
(Proof by accurate drawing will not be accepted.)
~ End of paper ~
Page 12 of 12