GRADE 12 FET
PRELIMINARY EXAMINATIONS 2014
BRIDGE HOUSE
MATHEMATICS DEPARTMENT
MATHEMATICS: PAPER I
Time: 3 hours
150 marks
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY
1. This question paper consists of 8 pages and 13 questions. Please check that your paper
is complete.
2. Please make sure you get a separate formula sheet.
3. Read the questions carefully.
4. Answer all the questions.
5. Number your answers exactly as the questions are numbered.
6. You may use an approved non-programmable and non-graphical calculator, unless a
specific question prohibits the use of a calculator.
7. Round your answer to two decimal digits where necessary.
8. All the necessary working details must be clearly shown.
9. It is in your own interest to write legibly and to present your work neatly.
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GRADE 12 FET: MATHEMATICS PAPER I
PRELIMINARY EXAMINATIONS 2014
Page 2 of 8
SECTION A
QUESTION 1
a. Solve for 𝑥:
i.
2𝑥(𝑥 + 20) = 88
(4)
ii.
7 × 32𝑥−1 = 15309
(3)
iii.
2𝑥 = 3 (give your answer rounded to the 2.d.p.)
(2)
iv.
12 − 3𝑥 − 𝑥 2 < 0
(3)
b. Fully simplify:
9𝑥 − 3𝑥 − 2
6𝑥 − 2𝑥+1
(5)
[17]
QUESTION 2
Consider the following sequence:
−13; −9; −5; …
a.
b.
c.
d.
Determine the general term 𝑇𝑛 .
Determine 𝑛 if 𝑇𝑛 = 71.
Calculate the 71st term.
Determine 𝑆1041.
(2)
(2)
(2)
(3)
[9]
QUESTION 3
Consider the following geometric series:
3−1+
1
−⋯
3
a. Determine the constant ratio (r).
(1)
1
b. Use an appropriate formula to calculate 𝑛 for which 𝑇𝑛 = 243
(4)
c. Use an appropriate formula to calculate 𝑆∞ .
(3)
[8]
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PRELIMINARY EXAMINATIONS 2014
Page 3 of 8
QUESTION 4
Nick buys a new motor cycle for R120 000. The motor cycle depreciates at a reducing
balance rate.
a. What is the percentage rate of depreciation, r % , if in 5 years’ time the motor
cycle has depreciated to a value of R65 147,48. Rounding your answer to two
decimal places.
(3)
b. How many years will it take for the motor cycle’s value to drop by half of its
selling price? Using 𝑟 = 11% 𝑝. 𝑎. Round your answer to the closest year.
(4)
[7]
QUESTION 5
16
Consider the sketch below. 𝑓(𝑥) = 𝑥−1 + 2 and 𝑔(𝑥) = 𝑚𝑥 + 𝑐. 𝑔(𝑥) is an axes of
symmetry for 𝑓(𝑥). Determine:
a. The equations of the asymptotes of 𝑓(𝑥).
(2)
b. The values of 𝑚 and 𝑐.
(2)
c. The co-ordinates of A.
(3)
[7]
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PRELIMINARY EXAMINATIONS 2014
Page 4 of 8
QUESTION 6
2
a. Determine 𝑓′(𝑥) from first principles 𝑓(𝑥) = 𝑥.
(6)
b. Determine:
𝑑𝑦
𝑑𝑥
4
if 𝑦 = 2√𝑥 + 𝑥 2
(2)
c. The distance (d) a particle has travelled through space in time (t) from a given
point can be modelled by the following function: 𝑑(𝑡) = 𝑡 3 − 6𝑡 2 + 9𝑡. (𝑡 in
seconds and 𝑑 in metres)
i.
Determine the average speed of the particle between 𝑡 = 0 and 𝑡 = 2.
(3)
ii.
Determine the instantaneous speed of the particle at 𝑡 = 2.
(4)
iii.
What is the acceleration of the particle at 𝑡 = 3?
(3)
iv.
Determine 𝑑(𝑡) = 0.
(3)
v.
On the graph, on the Answer Sheet provided, sketch this function
showing all intercept(s), turning point(s) and the inflection point. Place
the distance on the 𝑦 − 𝑎𝑥𝑖𝑠 and time variable on the 𝑥 − 𝑎𝑥𝑖𝑠.
(6)
[27]
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Page 5 of 8
SECTION B
QUESTION 7
A bag contains 7 red and 5 blue marbles. A marble is drawn at random and then not
replaced. A second marble is then drawn.
a. Draw a tree diagram showing all the possible outcomes.
(3)
b. Determine:
i.
𝑃(𝑓𝑖𝑟𝑠𝑡 𝑚𝑎𝑟𝑏𝑙𝑒 𝑑𝑟𝑎𝑤𝑛 𝑖𝑠 𝑟𝑒𝑑)
(2)
ii.
𝑃(𝑏𝑜𝑡ℎ 𝑚𝑎𝑟𝑏𝑙𝑒𝑠 𝑎𝑟𝑒 𝑏𝑙𝑢𝑒)
(2)
iii.
𝑃(𝑜𝑛𝑒 𝑚𝑎𝑟𝑏𝑙𝑒 𝑖𝑠 𝑟𝑒𝑑 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑜𝑡ℎ𝑒𝑟 𝑚𝑎𝑟𝑏𝑙𝑒 𝑖𝑠 𝑏𝑙𝑢𝑒)
(2)
[9]
QUESTION 8
The probability that a person drinks coffee is 0,5. The probability of a person drinking
tea is 0,4. The probability of a person neither drinking tea nor coffee is 0,2. Determine
the probability:
a. of a person drinking coffee and tea. (use a Venn-diagram)
(5)
b. of a person only drinking coffee.
(2)
[7]
QUESTION 9
a. Determine 𝑛:
𝑛
∑
2
1 𝑥−1
7
(7 ) = 80066
12
12
(6)
b. For which value(s) of 𝑥 will the series
3(2𝑥 + 1) + 3(2𝑥 + 1)2 + 3(2𝑥 + 1)3 + ⋯ converge?
(4)
[10]
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PRELIMINARY EXAMINATIONS 2014
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QUESTION 10
Sean buys a new apartment in Stellenbosch for R880 000. He takes out a loan over 20
years with repayments done monthly. The bank charges an annual interest rate of
12,68% 𝑝. 𝑎.
a. Determine the nominal interest rate the bank charges per month.
(3)
b. If the nominal interest rate per month is 12% 𝑝. 𝑎. compounded monthly,
determine the monthly instalment.
(4)
c. Calculate the outstanding balance of his loan after 8 years.
(5)
[12]
QUESTION 11
a. Consider the function 𝑓(𝑥) = 2𝑥 + 1 in the sketch, below
i.
On the graph provided on the Answer Sheet provided, sketch 𝑓 −1 (𝑥).
(2)
ii.
Determine 𝑓 −1 (𝑥) = ⋯.
(3)
b. Given the following equation, determine 𝑘 in terms of 𝑥.
3 × 2𝑥 = 2𝑘
(2)
[7]
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QUESTION 12
a. The graph of 𝑓(𝑥) = −𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 − 12 has the following properties.
𝑓(−3) = 0; and a turning point at 𝑥 = 2.
Determine the values of 𝑏 and 𝑐.
(7)
b. Determine the value(s) of 𝑝 if the line 𝑦 = 3𝑥 + 𝑝 is a tangent to the graph of
𝑔(𝑥) = 2𝑥 3 − 3𝑥 − 1.
(8)
[15]
QUESTION 13
a. Consider the sketch of the functions of 𝑘(𝑥) = −2𝑥 2 + 6𝑥 and 𝑔(𝑥) = 2𝑥.
Line segment RQ is parallel to the
𝑦 − 𝑎𝑥𝑖𝑠. RQ varies in length between
the two points of intersection.
i.
Determine the length of RQ in
terms of 𝑥.
ii.
(2)
Determine the maximum length
of RQ between the two points
of intersection.
(5)
b. Given the right circular cone with height, ℎ mm, and a
base radius of 𝑟 mm. With ℎ = 12 − 𝑟.
i.
Determine an expression for the Volume of the
1
cone in terms of 𝑟 if 𝑉𝑐𝑜𝑛𝑒 = 3 𝜋 × 𝑟 2 × ℎ. (3)
ii.
Now determine the length of the radius for
which the cone will have a maximum volume.
Also give the maximum volume for this
radius.
(5)
[15]
TOTAL FOR THIS PAPER: 150 MARKS
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PRELIMINARY EXAMINATIONS 2014
ANSWER SHEET
Page 8 of 8
NAME:___________________________
Q6.c.v.
Q11.a.i.
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