Mathematics Paper 2 Memorandum
Total Page：16
File Type：pdf, Size：1020Kb
MATHEMATICS PAPER 2 MEMORANDUM VCAA 2016
Marks: 150 Time: 3 hours
INSTRUCTIONS
1. This question paper consists of 28 pages and an Information Sheet of 2 pages. Please check that your paper is complete.
2. Read the questions carefully.
3. Answer all questions on the question paper and hand this in at the end of the examination.
4. Do not work on loose sheets of paper.
5. Extra space is provided at the end of the paper, should this be necessary.
6. Diagrams are not necessarily drawn to scale.
7. You may use an approved non-programmable and non-graphical calculator, unless otherwise stated.
8. Round off to one decimal place, unless specified otherwise.
9. Ensure that your calculator is in DEGREE mode.
10. All necessary working details must be clearly shown.
QUESTION 1 2 3 4 5 6 7 8 9 10 11 12 TOTAL
MAXIMUM 9 12 20 12 10 12 13 10 14 16 15 7 150
MARK
VCAA 2016: Mathematics Paper 2 Marking Guidelines Page 1 of 25 SECTION A
QUESTION 1
(a) Simplify the following expression:
(5)
(b) Evaluate the following expression without using a calculator:
(4)
or any other correct expansion
9 marks
QUESTION 2
The given data values reflect the masses (in kg) of 20 athletes in the school team:
40 47 52 53 55 57 57 58 60 x 2 63 64 64 65 66 67 67 68 69 73
(a) Determine the value of x if the median of the data is 61,5 kg. (2)
(b) Draw a box-and-whisker diagram of the given data, indicating the necessary values clearly. Use a scale of . (5)
(c) A value in the data set is considered to be an outlier if that value is either: less than or greater than
Determine whether 40kg is an outlier of this data set. (3)
outlier
VCAA 2016: Mathematics Paper 2 Marking Guidelines Page 3 of 25 (d) If 4 kg were added to each data value in the given set, what effect would it have on the resulting (1) mean and (2) standard deviation? (2)
(1) The mean would increase by 4 kg
(2) The standard deviation would remain the same
12 marks QUESTION 3
(a) , and are given. BA is produced to D and .
The circle with centre A passes through E, the midpoint of BA.
4 Determine: (1) the coordinates of E. (2)
(2) the equation of the circle with centre A. (4)
(3) the angle of inclination of the line BC. (3)
VCAA 2016: Mathematics Paper 2 Marking Guidelines Page 5 of 25 (4) the equation of the line CD. (3)
(b) The equation of the circle with centre K is given.
(1) Calculate the coordinates of K and the radius of the circle. (4)
6 (2) Hence, determine the equation of the tangent at the point on the circle. (4)
20 marks QUESTION 4
(a) Refer to the sketch: Q, V, T, S and R lie on the circle. RV and ST are produced to meet at P. PQ is a tangent to the circle at Q. QS is a diameter.
Complete the table giving the reason for each of the statements. (6)
STATEMENT REASON
Tan-chord theorem
VCAA 2016: Mathematics Paper 2 Marking Guidelines Page 7 of 25 Ext cyclic quad
Tan rad
in semi-circle OR tan-chord theorem
Opp cyclic quad
(b) AB is a tangent to the circle with centre O. BCOD is a straight line and .
(1) By using the given reason in each case, write a true statement that includes .(3)
STATEMENT REASON
a Tan-chord theorem
a Alt ’s,
a ’s in the same segment
(2) Giving reasons, show that . (3)
STATEMENT REASON
in semi-circle
8 Int of
12 marks QUESTION 5
(a) Without using a calculator, determine the value of if it is given that and . (5)
Sketch or Pythag
(b) If express the following in terms of p without using a calculator: (1) (2)
VCAA 2016: Mathematics Paper 2 Marking Guidelines Page 9 of 25 (2) (3)
(1)
(2)
10 marks QUESTION 6
(a) A, B, C and D are points on a circle with centre O and diameter BD.
10 and
Calculate, with reasons, the sizes of the following angles:
(1)
(4)
(2)
(3)
STATEMENT REASON
(1)
at centre = 2 at circle
Alternate
in semi-circle
at centre = 2 at circle
(2)
’s in same segment
’s opp equal sides
Alternate
Int ’s of
’s opp equal sides (b) In it is given that
and
VCAA 2016: Mathematics Paper 2 Marking Guidelines Page 11 of 25 Calculate the values of m and k by referring to the lengths on the diagram. (5)
STATEMENT REASON
Line one side;
Line one side;
12 marks TOTAL SECTION A: 75 SECTION B
QUESTION 7
An athlete’s ability to take and use oxygen effectively is called his / her VO2-max.
12 Twelve athletes with pre-recorded VO2-max readings ran for one hour. The distances (in kilometres) that they each covered are represented in the table:
VO -max 2 20 55 30 25 40 30 50 40 35 30 50 40 reading Distance run 8 18 13 10 11 12 16 14 13 9 15 12 (km)
(a) Draw a scatter plot of the data. (Place VO2-max on the horizontal axis.) (4)
(b) Use the correlation coefficient to describe the correlation between the two sets of data. (2)
Strong positive linear
(c) Determine the equation of the least square line of best fit and draw it on the graph. (5)
)
On graph
VCAA 2016: Mathematics Paper 2 Marking Guidelines Page 13 of 25 (d) Use the method of interpolation to predict the distance run if an athlete has a VO2-max value of 26. (2)
On graph
OR
13 marks
QUESTION 8
(a) Without using a calculator, determine the general solution of:
(6)
Alternate
14 (b) Prove that = 2 (4)
VCAA 2016: Mathematics Paper 2 Marking Guidelines Page 15 of 25 10 marks QUESTION 9
(a) The circle with centre T is given. Calculate the value(s) of m if the length of the tangent from point outside the circle to the point P on the circumference of the circle is . (6)
16 (b) Refer to the diagram alongside and determine the equation of the circle, with centre M, that touches the x-axis at and passes through the point
(8)
VCAA 2016: Mathematics Paper 2 Marking Guidelines Page 17 of 25 14 marks
QUESTION 10
(a) The curves of and are sketched for .
(1) Write down the values of a, b, c and d. (4)
(2) Write down the new equation of f if it is translated to the right and 3 units down. (2)
18 (b) A is the foot of a vertical tower AD. B and C are two points in the same horizontal plane as A. It is given that and , while the angle of elevation of D from B is y°.
If prove that: (6)
Alternate:
VCAA 2016: Mathematics Paper 2 Marking Guidelines Page 19 of 25 (c) A line is drawn from the origin O to the point .
The line OP is rotated about the y-axis to form a cone.
where s is the slant height and r the radius.
Determine the shaded outer surface area of the resulting cone. (4)
16 marks QUESTION 11
20 (a) O is the centre of the circle and The radius of the circle is and .
Calculate the length of AM (7) STATEMENT REASON
radii Midpoint Th Pythag Alternate Given in semi-circle is commom AAA Pythag
Alternate Given in semi-circle
Pythag Line one side
(b) KLMN is a trapezium with
and
VCAA 2016: Mathematics Paper 2 Marking Guidelines Page 21 of 25 Prove:
(8)
STATEMENT REASON
similarity
Given
Alt ’s,
AAA
Given
15 marks QUESTION 12
In the figure P, A, R, C, Q and B are points on the circle.
Prove that . 22 (7)
STATEMENT REASON
Opp ’s cyclic quad
Opp ’s cyclic quad
Opp ’s cyclic quad
Int ’s of
7 marks TOTAL SECTION B: 75
VCAA 2016: Mathematics Paper 2 Marking Guidelines Page 23 of 25