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Kingsmead College
Grade 12 Examination
July 2011
MATHEMATICS I
Name: _____________________________
Time allowed: 3 hours
Total: 150 marks
Examiner: D. Clogg
INSTRUCTIONS:
1.
Write your name in the space provided above as well as on all your
answer sheets.
2.
This paper consists of 10 pages including the diagram sheet. Please
check that this paper is complete.
3.
Answer all questions in the workbooks provided. Detach diagram
sheet, with your name printed in the space provided, and insert in
the front of your booklets.
4.
Non–programmable calculators may be used, but you must show all
your working clearly.
5.
Give your answers correct to 2 decimal place unless directed
otherwise.
6.
Write neatly in pen.
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SECTION A
QUESTION 1: [21 marks]
Solve for 𝑥: (Show all steps- answer only will be 1 mark)
a) (𝑥 − 7)(𝑥 − 5) = 3
(3)
b) √2𝑥 − 9 + 6 = 𝑥
(5)
c) 3𝑥 − 3𝑥−1 = 54
(4)

d)
 8x( 12 ) k 1  x 2
(5)
k 1
e) 𝑥(𝑥 − 1) < 6
(4)
QUESTION 2: [11 marks]
a) Simplify:
𝑎−1 𝑏−𝑏 −1 𝑎
(4)
𝑎−1 +𝑏−1
b) Evaluate:
3
∑
𝑟=0
2𝑟
2𝑟
(2)
1
c) If 𝑓(𝑥) = 𝑥 2 − 𝑥 determine 𝑓 (𝑥) and for which values of 𝑥
1
𝑓(𝑥)
1
+ 2𝑓 (𝑥) = 0
(5)
QUESTION 3: [10 marks]
a) For what value(s) of 𝑥 is the sequence 𝑥 − 2 ; 20 ; 5𝑥 ; …
i)
Arithmetic
(2)
ii)
Geometric
(3)
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b) 5; 𝑎; 𝑏; … are the first three terms of an arithmetic sequence and 𝑏; 𝑎; 9; … are
the first three terms of a geometric sequence. Find the values of a and b. (5)
Question 4: [8 marks]
Look at the pattern of the tiles shown and then complete the table below in your
answer book.
Fig(i)
Fig(ii)
Number of grey tiles
Number of white tiles
Number of black tiles
fig(i)
3
2
1
fig(ii) fig(iii) fig(iv) fig(n)
5
6
4
Question 5: [6 marks]
The graph of y = f(x) is sketched below.

1

1
1
2
Sketch each on a separate set of axes
a) 𝑦 = 𝑓(−𝑥)
(2)
b) 𝑦 = 𝑓(𝑥 − 3)
(2)
c) 𝑦 = 𝑓 ′ (𝑥)
(2)
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Question 6: [9 marks]
Durban intends to bid for the 2020 Olympic games. Luckily many structures are in
place due to the 2010 world cup. Below is a picture of The Moses Mabhida Stadium
in Durban while under construction.
The arch above the stadium is approximated by the parabola below.
 106
( x  175) 2  106
The equation of the parabola is y =
2
(175)
y
B
C
D
N
A
A
x
a) How high above the ground is the highest point, B, of the parabola?
(1)
b) What is the distance (on the ground) between the two endpoints of the arch?
(1)
c) If CN is a crane positioned 55 metres to the left of AB, how tall is the crane
(correct to the nearest m)?
(2)
d) Find the average gradient between B and C.
(2)
e) Find the area of ∆DBC.
(3)
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QUESTION 7: [11 marks]
A farmer wishes to stock his dam with Brown Trout and Rainbow Trout subject to the
following conditions:

The dam can carry a maximum of 1000 trout in total.

He would like to stock at least 100 brown trout.

Due to a shortage at the hatchery he is unable to buy more than 800 rainbow
trout.

He would like to stock at least three times as many rainbow trout as brown
trout.
a) If he stocks 𝑥 brown trout and 𝑦 rainbow trout then clearly 𝑥 ≥ 0 and 𝑦 ≥ 0.
Write down four additional constraints.
(4)
b) Draw the constraints on the axes provided on your diagram sheet. You should
clearly indicate the feasible region.
(4)
c) If brown trout costs R60 each and rainbow trout costs R30 each then
determine how many of each he should stock in order to meet his conditions
at the lowest cost.
(3)
Question 8: [14 marks]
a) If 𝑓(𝑥) = 2𝑥 2 , find 𝑓′(𝑥) by first principles.
b) 𝑦 = (𝑥 2 − 𝑥)2 , find
(4)
𝑑𝑦
(3)
𝑑𝑥
c) Find f / ( x) if f ( x) 
2x  x 2
(3)
3x 3
d) For which values of x will the gradient of f ( x)  13 x 3  x 2  3x  1 be negative.
(4)
Total: Section A: 90 marks
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Question 9: [8 marks]
A Gymnasium needs to be built next to the soccer stadium. The
gym is in the shape of a cuboid with a square floor (see diagram below).
If the Volume is given as 500𝑚3 :
h
x
x
a) Show that the surface area of the external walls and ceiling, A, is given by
𝐴 = 𝑥2 +
2000
(4)
𝑥
b) Find the dimensions of the gym to ensure that the external surface area, including
the ceiling, is a minimum.
(4)
Question 10: [7 marks]
When the Vaal Dam is full, a sluice gate is opened. The depth of the water at a given
1
1
point is given by 𝑃(𝑡) = 32 − 16 𝑡 − 8 𝑡 3 , where P is measured in metres and time in
hours.
a) How deep is the dam when it is full ?
(1)
b) If the sluice gate was opened at 9h00 hours, at what rate is the depth of the water
decreasing at 11h30 ?
(3)
1
c) At what time will the depth of the water be decreasing at a rate of 16 6 metres per
hour?
(3)
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Question 11: [9 marks]
A gym/warm-up room (separate from the stadium) for the athletes is also to be
constructed. In keeping with the parabolic design of the stadiums roof, the gym’s
parabolic roof is shown below:
y
-1
O
2
x
The roof, represented by 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 crosses the y-axis at (0; 3).
With the current energy crisis, two solar heating panels are installed. These are
represented by the tangents at 𝑥 = −1 and 𝑥 = 2. The tangent at 𝑥 = −1 makes an
angle of 45° with the positive direction of the x-axis and the tangent at 𝑥 = 2 makes
an angle of 135° with the positive direction of the x-axis.
a) What is the value of 𝑐?
(1)
b) Find the values of 𝑎 and 𝑏.
(8)
Question 12: [15 marks]
a) The following information is given for 𝒇(𝒙) = 𝒂𝒙𝟑 + 𝒃𝒙𝟐 + 𝒄𝒙
f (3)  f (8)  0
f / (1)  f / (5)  0
f (x) is increasing for  1  x  5
i) use the above information to draw a rough sketch of f (x) (Label any relevant
points)
ii) On a separate set of axes draw f / ( x) and name any relevant points
(4)
(3)
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b) Look at the graph below:
f
f ( x)  x 2  10
( a ; f (a ) )
(1;2)
g is a tangent to f at the point
( a ; f (a ) ) and this tangent
passes through the point (1 ; 2 ),
find the point of contact of the
tangent to the curve.
(8)
g
Question 13: [16 marks]
a) A mathematical insect travels in an arithmetic spiral. Starting at P, it moves 3cm
North, 7cm East, 11cm South, 15cm West and so on.
7
3
P
11
15
In what direction will the insect be travelling when it covers 20 metres.
(5)
x1
b)
Solve for 𝑥 if
 (3k  1)  445
(6)
k 2 x
c) The radius of the first circle of an infinite sequence of circles has a radius of
4
6cm. The radius of each of the following circles is 5 of the previous one. Show
that the total area is 100𝜋.
(5)
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QUESTION 14: [5 marks]
Sally adds up the first 𝑛 positive odd integers on her calculator but she accidently
typed in one of the numbers twice. She obtains the result 3 643. Find the value of 𝑛
and the number she added twice.
Total: Section B: 60 marks
Total: 150 marks
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