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Page 1 of 18GRADE 12: MATHEMATICSPRELIM EXAM 2016 PAPER II
MATHEMATICS DEPARTMENT
Grade 12: Mathematics
Paper II
Preliminary Examination 2016
Time: 3 hours 150 marks
NAME:__________________________________________________________
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY
1. This question paper consists of 18 pages and 13 questions. A separate formula sheet will be
handed out. Please check that your paper is complete.
2. Read the questions carefully.
3. Answer all the questions ON THE QUESTION PAPER. Additional space is provided at the
end of the paper.
4. Number your answers exactly as the questions are numbered.
5. Round off your answers to two decimal digits where necessary.
6. All the necessary working details must be clearly shown.
7. Approved non-programmable calculators may be used unless otherwise stated.
8. It is in your own best interest to present your work neatly.
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Page 2 of 18GRADE 12: MATHEMATICSPRELIM EXAM 2016 PAPER IISECTION A
Question 1
In the diagram below, A (−2 ;6 ) , B(8 ;−4) and C (−6 ;k ) are three points in the Cartesian
plane with point M the midpoint of AB and AB=BC.
M
C(-6;k)
B(8;-4)
A(-2;6)
x
y
1.1. Determine the coordinates of M . (2)
1.2. Determine the value(s) of k . (6)
1.3. Hence, determine the equation of the line CM if k=−6. (5)
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Page 3 of 18GRADE 12: MATHEMATICSPRELIM EXAM 2016 PAPER II1.4. Determine the value of :
1.4.1. θ if k=−6 (2)
1.4.2. α (2)
1.4.3. C A B (2)
[19]
Question 2
2.1. Simplify without the use of a calculator: (6)
−cos225 ° sin60 °cos765°
+sin (−210 °)tan 210°
2.2. Solve for x if sin ( x−20 ° )=tan 140° for x∈(0 ° ;360) (4)
[10]
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Page 4 of 18GRADE 12: MATHEMATICSPRELIM EXAM 2016 PAPER IIQuestion 3
O is the centre of the circle through A , B ,C and D. AB=BC and A OC=2 x .
2x
12
1
2
321
O
D
C
B
A
Express the following in terms of x, giving reasons for all statements:
3.1. A2 (2)
3.2. A BC (3)
3.3. D (3)
[8]
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Page 5 of 18GRADE 12: MATHEMATICSPRELIM EXAM 2016 PAPER IIQuestion 4
Use the sketch given below to prove the opposite angles of a cyclic quadrilateral are
supplementary.
x
OD
CB
A
[6]
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Page 6 of 18GRADE 12: MATHEMATICSPRELIM EXAM 2016 PAPER IIQuestion 5
In the diagram below, AC is a diameter of circle ABC. C2=C3 , AC⊥ FG, AD∥CG ,
BC=6 units and AC=10 units .
G
21
321
32
1
F
C
D
B
A
5.1. Prove that ∆ CAD is isosceles. (3)
5.2. Prove that BCAC
=BGAG (3)
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Page 7 of 18GRADE 12: MATHEMATICSPRELIM EXAM 2016 PAPER II
5.3. Prove that ∆ AFG∨¿∨∆ ABC (3)
5.4. Determine the length of AB (2)
5.5. Determine the length of AG (2)
5.6. Determine the length of AF (2)
[15]
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Page 8 of 18GRADE 12: MATHEMATICSPRELIM EXAM 2016 PAPER IIQuestion 6
Fifteen people took part in a board game tournament. The points scored by each person are
given below.
33 17 20 34 24 52 31 32 18 33 22 36 43 52 28
6.1. Determine the median of the data set. (2)
6.2. Determine the inter quartile range for the data. (3)
6.3. Draw a box and whisker diagram to represent the data. (3)
6.4. Use the box and whisker diagram to comment on the points scored by the
participants of the tournament. (2)
[10]
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Page 9 of 18GRADE 12: MATHEMATICSPRELIM EXAM 2016 PAPER IISECTION B
Question 7
7.1. Use the sketch given to prove tanθ × 1sin θ
−cosθ= 8 pp2−4
(6)
B
A
O
p+2
p-2
y
x
7.2. If cosθ+2cosθ−2
=rs
7.2.1. Prove that cosθ=2 s+2 rr−s (4)
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Page 10 of 18GRADE 12: MATHEMATICSPRELIM EXAM 2016 PAPER II
7.2.2. Hence determine the solution for θ if cosθ+2cosθ−2
= 3−5 and θ∈[0° ;540 ° ] (5)
[15]
Question 8
The graph of f ( x )=a tan x for x∈[−180 °;180 ° ] is given below.
8.1. Sketch the graph of g ( x )=sin(x+45° ) on the axis provided above. (3)
8.2. Determine the value of a. (1)
8.3. The graph of h ( x )=cos bx+c a period of 180 ° and a y-intercept at (0 ;3). Determine
the values of b and c. (2)
[6]
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Page 11 of 18GRADE 12: MATHEMATICSPRELIM EXAM 2016 PAPER IIQuestion 9
The diagram shows points B, C, D and E on the ground such that B, D and C lie in a straight
line. AP is a flagpole. The angle of elevation of A from E is x and from C it is y. The length
of DC is r and AB=AE.
r
y
x
E
D CB
A
9.1. Show thatAE= r tan ysin x (4)
9.2. Calculate the distance of BE if E D C=120° and DE=32 m. (4)
[8]
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Page 12 of 18GRADE 12: MATHEMATICSPRELIM EXAM 2016 PAPER IIQuestion 10
Given below is the circle with centre A given by the equation (x+2)2+(x−6)2=r2 and the
circle with centre C (a ;b) that lies in quadrant 2. DE is a common tangent to circles A and C
at the point B(−3 ;5). Circle F has the equation x2+ y2=−1+2 x+6 y .
F
E
D
A
B(-3;5)
C(a;b)
x
y
10.1. Write down the coordinates of A. (2)
10.2. Determine the radius of circle A. (2)
10.3. Determine the equation of the tangent DE. (4)
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Page 13 of 18GRADE 12: MATHEMATICSPRELIM EXAM 2016 PAPER II
10.4. If ABBC
=13 , determine the coordinates of C. (8)
10.5. Write down the coordinates of F. (4)
[20]
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Page 14 of 18GRADE 12: MATHEMATICSPRELIM EXAM 2016 PAPER IIQuestion 11
In ∆ ACE, A C E :C A E : A E C=16 :15:5. B and F are the midpoints of AC and AE. ABC is
the diameter of the circle with centre B and G is a point on the circle.
3
12
1
2
1 2
12
1 2
2
1G
B
A
F
DC
E
11.1. Giving reasons for all statements, determine the size of:
11.1.1. C A E (3)
11.1.2. E (1)
11.1.3. F1 (3)
11.2. Show that AG=GD (2)
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Page 15 of 18GRADE 12: MATHEMATICSPRELIM EXAM 2016 PAPER II11.3. Prove that ∆ ACG ≡ ∆ DCG (4)
11.4. Hence, show that CD=2 AB. (2)
[15]
Question 12
In the diagram below, ABCD is a square with sides 7cm long. FC=x ,BF⊥ EF and
∆ BCF∨¿∨∆ FDE.
x
1
2
1
F
E D
CB
A
Show that the area of ∆≝¿ x ¿¿
[4]
Question 13
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Page 16 of 18GRADE 12: MATHEMATICSPRELIM EXAM 2016 PAPER II
x 10 20 30 40 50 60 70 80 90 100
y 2 4 3 5 5 6 8 7 9 10
13.1. Dra
w a scatter plot of the data above. (3)
13.2. Determine the equation for the line of best fit for the data. (3)
13.3. Draw the line of best fit on your scatter plot. (2)
13.4. Determine the correlation coefficient of the data. (2)
13.5. “The corresponding y-value for an x-value of 35 is 4,13.” Do you agree with this
statement and if so, is the value reliable? (4)
[14]
TOTAL FOR THIS PAPER: [150]
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Page 17 of 18GRADE 12: MATHEMATICSPRELIM EXAM 2016 PAPER II__________________________________________________________________________________
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Page 18 of 18GRADE 12: MATHEMATICSPRELIM EXAM 2016 PAPER II
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