Max Mark / 8 / 13 / 8 / 15 / 14 / 12 / 6 / 6 / 7 / 23 / 16 / 9 / 13
Mark
Name: ......
TOTAL
DIAGRAM SHEET 1 for Question 2.2
Mass (in kg) / Frequency / Cumulative Frequency80 ≤ x < 90 / 1
90 ≤ x < 100 / 2
100 ≤ x < 110 / 9
110 ≤ x < 120 / 2
120 ≤ x < 130 / 1
DIAGRAM SHEET 2 for Question 2.3
DIAGRAM SHEET 3 for Question 3.1
DIAGRAM SHEET 4 for Question 9.1
DIAGRAM SHEET 5for Question 13.5
RONDEBOSCH BOYS’ HIGH SCHOOL
Mathematics Grade 12
Tuesday 25thSeptember 2012 3 Hours ; 150 Marks
Set by: R Harmuth Moderated by: D Geldenhuys
PAPER TWO
INSTRUCTIONS
- Calculators can be used, unless otherwise stated, with answers corrected to two decimal places.
- All necessary working MUST be shown.
- When necessary, leave answers with positive exponents.
- Number your answers as the questions are numbered.
- Untidy work will be penalised.
- Only blue and black pens may be used.
- Sketches may be done in pencil.
- This exam contains thirteen questions.
Question One
The table below gives a breakdown of the Western Union Currie Cup Rugby log standings for the top 8 teams at the end of the 2001 season in Wales:
TEAM / POINTSHarlequins / 56
Wasps / 52
Sparrows / 50
Robins / 50
Wonders / 44
Sharks / 40
Dolphins / 32
Golden Men / 24
1.1 Determine the median (number of points scored). (2)
1.2 Determine the lower AND the upper quartile. (2)
1.3 Draw a box and whiskerto represent the points scored. (3)
1.4 Use your box and whisker to comment on the spread of the points scored by the teams. (1)
[8]
Question Two
The masses (in kg) of the Springbok Rugby Team to play New Zealand in September are given below:
95 105 100 118 109 126 107 105 102 82 108 116 109 98 100
2.1 Calculate the mean mass of the Springbok team. (2)
2.2 Complete the following table on DIAGRAM SHEET 1.
Mass (in kg) / Frequency / Cumulative Frequency80 ≤ x < 90 / 1
90 ≤ x < 100 / 2
100 ≤ x < 110 / 9
110 ≤ x < 120 / 2
120 ≤ x < 130 / 1
(2)
2.3 Draw an ogive (cumulative frequency curve) of the above information on the grid provided on DIAGRAM SHEET 2. (4)
2.4 Calculate the percentage of the team who have masses within one standard deviation of the mean. Show ALL calculations. (5)
[13]
Question Three
A group of 12 learners was randomly selected from a class. The marks scored in a standardised English test (out of 100 marks) and the average number of hours they spent playing TV games (on a computer) per week was recorded.
English Marks / 80 / 65 / 50 / 30 / 70 / 60 / 85 / 45 / 45 / 90 / 35 / 75TV hours / 10 / 20 / 35 / 45 / 15 / 25 / 10 / 37 / 23 / 5 / 40 / 18
3.1 Represent this data as a scatter plot on the grid provided on DIAGRAM SHEET 3. (4)
3.2 Draw a line of best fit for your scatter plot. (1)
3.3 What conclusion can you make about the learners’ marks and the average number of hours spent on TV games? (1)
3.4 Another learner from the class spends 30 hours playing TV games. Predict his English mark. (2)
[8]
Question Four
In the diagram below, ∆ PQR with vertices P(3 ; 1), Q( 8 ; 2) and R(2 ; 3) is given:
4.1 Calculate the length of QR, leaving your answer in surd form. (2)
4.2 Determine the co-ordinates of M, the mid-point of QR. (2)
4.3 Determine the equation of the line parallel to PR, passing through M. (4)
4.4 If PQTR is a parallelogram find the co-ordinates of T. (2)
4.5 Calculate PRQ. (5)
[15]
Question Five
5.1 The equation of a circle is given as
5.1.1 Prove that the point P( 1; 9) lies on the circumference of the given circle. (2)
5.1.2 Determine an equation of the tangent to the circle at the point P( 1; 9). (7)
5.2 Calculate the length of the tangent DE, drawn from the point D(2 ; 6) to the circle with equation . E is a point on the circumference of the circle. (5)
[14]
Question Six
The circle with centre C and equation is drawn below. A is a y-intercept of the circle.
6.1 Determine the value of at A. (4)
6.2 The circle is enlarged by a scale factor of 2½ about the origin. Write down the equation of the new circle in the form . (3)
6.3 In addition to the given circle with centre C and equation , another circle with centre P and equationis now given.
6.3.1 Calculate the distance between C and P. (2)
6.3.2 Do these two circles cut once, twice, or not at all? Justify your answer. (3)
[12]
Question Seven
The point A(;) is rotated about the origin through an angle of 120° in an anti clockwise direction to give the point B(;). Calculate the values ofand. [6]
Question Eight
Consider thepointP(14 ; 5). The point is reflected about the -axis to PꞋ.
8.1 Write down the co-ordinates of PꞋ. (1)
8.2 An alternative transformation from P to PꞋ is a rotation about the origin through an angle of °, where < 180°. Calculate . (5)
[6]
Question Nine
In the diagram below, ∆ MNR is drawn with vertices M(6 ; 2), N( 6 ; 8) and R(1 ; 6). ∆ MNR is now enlarged by a factor of 2 to ∆ MꞋNꞋRꞋ.
9.1 Draw ∆ MꞋNꞋRꞋ on the grid provided on DIAGRAM SHEET 4. (3)
9.2 Write down the values of:
9.2.1 (2) 9.2.2 (2)
[7]
Question Ten SHOW ALL NECESSARY STEPS IN THIS QUESTION
10.1 If , and , determine the following in terms of and/or :
10.1.1 (3)
10.1.2 (3)
10.1.3 (2)
10.1.4 (3)
10.2 Without using a calculator, find the value of A if:
10.2.1 (8)
10.2.2 (4)
[23]
Question Eleven
11.1 Given:
Determine the general solution. (8)
11.2 Consider the expression
11.2.1 For which values of , [0° ; 180], will this expression be undefined? (3)
11.2.2 Prove that (5)
[16]
Question Twelve
A person standing at point P looks up to the top, Q, of a double- storey house. P is 17m from the foot of the house. The angle of elevation of Q from P is 33°. He turns around and walks in the opposite direction from the house at an inclination of 8° for a distance of 12m to a point M. Let PQM .
12.1 Calculate PQ. (2)
12.2 Calculate the length of QM (4)
12.3 Hence, calculate . (3)
[9]
Question Thirteen
The graph of
13.1 Write down the period of . (1)
13.2 For which values of is (2)
13.3 Write down the - intercepts of if. (2)
13.4 Write down the amplitude of if (2)
13.5 Draw ON DIAGRAM SHEET 5 PROVIDEDshowing intercepts with axes and turning points. (4)
13.6 For which values of and? (2)
[13]