St Dominic’s Catholic School for Girls
Mathematics Exam
150
NAME :
Topic(s) :
Paper 1
Grade :
12
Examiner :
E. Sales
Date :
September 2012
Duration :
3 hours
Moderator(s): S. van Rooyen
INSTRUCTIONS :
1.
Number your answers clearly and exactly the same as on the
question paper.
2.
Skip a line after each answer.
3.
Start each question (Question1, Question 2 etc.) on a new page.
4.
Do not work in columns.
5.
Write on both sides of the page.
6.
Show all your calculations.
7.
An approved calculator may be used unless otherwise stated.
8.
Round off to two decimal places where applicable.
9.
It is in your own interest to write legibly and to present your work
neatly.
10.
Pencil work will not be marked. Sketches may be done in pencil.
11.
You may not use correction fluid (“tippex”).
12.
This paper contains 7 pages.
13.
You will receive a formula sheet which needs to be handed in
separately.
1
Question 1
[19]
1.1
Simplify without the use of a calculator : 2 log 27 9 − 4 log 4 8
1.2
Solve for 𝑥:
1.3
a) 5 = 2𝑥
(3)
b) 𝑥 − √5 + 𝑥 = 7
(5)
Solve for 𝑥 and 𝑦 simultaneously :
𝑥 2 + 𝑦 2 = 100 and 4𝑦 + 3𝑥 = 50.
(7)
Question 2 (rule off and start on a new page)
2.1
2.2
2.3
(4)
[14]
On Daniel’s first birthday, Mrs Sales deposits 𝑅1000 into a savings
account for him. Every year, on his birthday, she increases this amount
by 𝑅100.
a) This pattern forms an Arithmetic Series. Explain why.
(1)
b) Find an appropriate formula for 𝑇𝑛 , the amount of money she will
deposit on his 𝑛𝑡ℎ birthday.
(2)
c) Use this formula to determine the amount that will be deposited when
he turns 9.
(1)
Suppose instead of increasing the amount by 𝑅100 every year, Mrs Sales
decided to increase the amount by 10% every year on his birthday.
a) Determine the amount deposited in year 2, 3 and 4.
(2)
b) Determine the constant ratio for this Geometric Sequence.
(1)
c) Find an appropriate formula for 𝑇𝑛 , the amount of money she will
deposit on his 𝑛𝑡ℎ birthday.
(2)
The last deposit is made on Daniel’s 17th birthday. He will receive the
amount saved in this account on his 18th birthday. Which of the two
options above (2.1 or 2.2) do you think he would prefer? Motivate your
answer mathematically. (You don’t need to take into account the effect of
interest earned.)
(5)
PLEASE TURN OVER FOR NEXT QUESTION
2
Question 3 (rule off and start on a new page)
[6]
3.1
Sarah invests 𝑅18 000 for a 7 year period. For the first two years the
interest was compounded semi-annually at 12% p.a. For the next five
years it was compounded quarterly at 10% p.a. Calculate the value of the
investment at the end of the 7 years.
(3)
3.2
A lawnmower is traded in for R500 after 10 years. The depreciation rate
was calculated at 9,2% p.a. on a straight-line basis. What was the cost
of the lawnmower when it was bought 10 years ago?
(3)
Question 4 (rule off and start on a new page)
[14]
The sketch, not
drawn to scale,
shows the
parabola 𝑦 =
−𝑥 2 + 2𝑥 + 𝑘
that intercepts
with the 𝑥-axis at
𝐴 and 𝐶. A
straight line
through 𝐴 cuts
the parabola in
𝐵(3; 5).
4.1
Show by calculation that 𝑘 = 8.
(1)
4.2
Determine the co-ordinates of 𝐴.
(3)
4.3
The straight line 𝐸𝐷 cuts the parabola in 𝐸 and 𝐷(2; 𝑝). Find the
value of 𝑝.
(2)
4.4
If 𝐸𝐷 is parallel to 𝐴𝐵, determine the co-ordinates of 𝐸.
(8)
Question 5 (rule off and start on a new page)
[9]
Determine 𝑓′(𝑥) for each of the following :
1
5.1
𝑓(𝑥) = 3 𝑥 2
5.2
𝑓(𝑥) =
5.3
𝑓(𝑥) =
(1)
5𝑥 4 −3𝑥
(4)
𝑥2
3𝑥 2 −12
(4)
3𝑥+6
PLEASE TURN OVER FOR NEXT QUESTION
3
Question 6 (rule off and start on a new page)
6.1
[16]
3
If 𝑓(𝑥) = 𝑥 3 − 2 𝑥 2 , determine
a)
the 𝑥- and 𝑦-intercept(s) of 𝑓(𝑥) .
(4)
b)
the co-ordinates of the local turning points of 𝑓(𝑥) .
(5)
6.2
Sketch the graph of 𝑓(𝑥) .
(3)
6.3
Determine the equation of the tangent to 𝑓(𝑥) at 𝑥 = 2.
(4)
Question 7 (rule off and start on a new page)
[7]
7.1
Draw sketch graphs of 𝑓(𝑥) = 2𝑥 and 𝑔(𝑥) = 2−𝑥 on the same
set of axes, where 𝑥, 𝑦 ∈ ℝ.
(2)
7.2
Write down the equation of the axis of symmetry between 𝑓 and 𝑔.
(1)
7.3
Will the inverse of 𝑔(𝑥) = 2−𝑥 represent a function? Give a reason
for your answer.
(1)
7.4
Find the equation of ℎ(𝑥), the inverse of 𝑔.
(3)
Question 8 (rule off and start on a new page)
[3]
1
If 𝑓(𝑥) = 9𝑥 , express 𝑓(𝑥 + 2) in terms of 𝑓(𝑥) .
(3)
Question 9 (rule off and start on a new page)
[8]
A woman borrows 𝑅275 000 to buy a new car. She agrees to make
36 equal monthly payments, with interest compounded monthly at
11,75% per annum.
9.1
How much will she repay each month?
(4)
9.2
How much will she still owe the bank at the beginning of the second year?
(4)
PLEASE TURN OVER FOR NEXT QUESTION
4
Question 10 (rule off and start on a new page)
[11]
A new school establishes a computer centre. The total budget available is
𝑅115 200 and the space in the centre allows for a maximum of 20
computers. Two types are available, the Standard and the Super model.
A computer supplier is prepared to sell computers to the school, provided
the school buys at least twice as many expensive models as cheaper
models. The special price offered is 𝑅4 800 for the Standard model and
𝑅7 200 for the Super model. At least three of the Standard models must
be bought. Let the number of Standard models bought be 𝑥 and the
number of Super models bought be 𝑦.
10.1
Write down the implicit constraints.
(1)
10.2
Write down the explicit constraints as a set of inequalities.
(4)
10.3
Draw the inequalities on a system of axes and indicate the feasible region.
(6)
Question 11 (rule off and start on a new page)
[11]
The sketch shows 𝑓(𝑥) = 𝑥 2 − 6𝑥 + 8 and 𝑔(𝑥) = 3𝑥 . Both pass
through the point (1; 3). The turning point of 𝑓(𝑥) is (3; −1).
11.1
If ℎ(𝑥) = 𝑝 𝑥 represents the reflection of 𝑔(𝑥) in the 𝑦-axis, determine
the value of 𝑝.
11.2
Consider 𝑓(𝑥).
11.3
(3)
a) Explain why 𝑓 (𝑥) does not have an inverse.
(1)
b) What are the value(s) of 𝑥 for which 𝑓 (𝑥) will have an inverse?
(2)
By using the graphs above, determine the value(s) of 𝑥 for which:
a) 𝑓(𝑥). 𝑔(𝑥) > 0
(2)
b) 𝑓′(𝑥) > 0
(1)
c) 𝑓′(𝑥). 𝑓(𝑥) < 0
(2)
5
Question 12 (rule off and start on a new page)
[8]
A brick wall is built by using 2 bricks less in each layer than the previous
one. The wall has 𝑚 layers and in total 21𝑚 + 40 bricks are used. If
the top layer has only 4 bricks, determine :
(HINT : the TOP layer is your first row)
12.1
The value of 𝑚.
(5)
12.2
The total number of bricks in the wall.
(1)
12.3
The number of bricks in the first layer.
(2)
Question 13 (rule off and start on a new page)
[6]
The numbers from 1 to 700, namely 1, 2, 3, . . . , 700 are written down. The
multiples of 7, namely 7, 14, 21, . . . , 700 are then removed from the list
and the remaining numbers are added. Calculate this sum.
(6)
Question 14 (rule off and start on a new page)
[7]
A transport company has to transport a load of sugar cane 400𝑘𝑚 to a
client.
14.1
If the vehicle carrying the sugar cane travels at 𝑘 kilometres per hour,
write down a formula, in terms of 𝑘, for the time it takes to complete the
journey.
14.2
If the cost per hour is (10000 + 100) rands, show that the cost 𝐶, in rand,
𝑘3
𝑘2
for the journey is 𝐶 = 25 +
14.3
40000
𝑘
(1)
(1)
.
Calculate, correct to the nearest 𝑘𝑚/ℎ , the most economical speed for
this trip.
PLEASE TURN OVER FOR NEXT QUESTION
6
(5)
Question 15 (rule off and start on a new page)
[8]
The diagram below illustrates the constraints under which a manufacturer
makes chairs (𝑥) and tables (𝑦).
15.1
Write down the inequalities represented by 𝑃𝑄 and 𝑃𝑇 in the form
𝑦 ≥ . . . or 𝑦 ≤ . . .
(6)
15.2
Determine the value(s) of the gradient of the search line in order for the
profit to maximise at 𝑄.
(2)
Question 16 (rule off and start on a new page)
[3]
What will the units digit be of the value of 22012 ? Show your
working.
(3)
END OF PAPER
7