JUNE MATRIC 2014
MATHEMATICS: PAPER I
Time: 3 hours
150 marks
Reading Time: 10 Minutes
Examiner: R. Bourquin
Moderators: M Brown and R Karam
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY
1.
This question paper consists of 12 pages, an Answer Sheet (1 page) and an
Information Sheet. Please check that your paper is complete.
2.
Question 4(b) must be completed on your Answer Sheet. Please make
sure your answer sheet is placed in your answer book and is handed in.
Please make sure that your name is on your answer sheet.
3.
All other questions must be answered in your answer book.
4.
Read the questions carefully.
5.
You may use an approved non-programmable and non-graphical
calculator, unless otherwise stated.
6.
All the necessary working details must be clearly shown.
7.
Round off your answers to one decimal digit where necessary, unless
otherwise stated.
8.
It is in your own interest to write legibly and to present your work neatly.
9.
No diagrams are drawn to scale.
JUNE 2014 EXAMINATIONS:
Page 2 of 12
MATRIC MATHEMATICS: PAPER I
SECTION A
QUESTION 1
Solve for 𝑥:
(a)
2𝑥 2 − 11𝑥 + 12 = 0
(2)
(b)
3𝑥 2 + 5𝑥 − 2 ≥ 0
(4)
(c)
92𝑥+1 =
(d)
log⁡(𝑥 + 2)2 = 2
1
(4)
27
(3)
[13]
QUESTION 2
99
(a)
Given:
 (3t - 1)
t 0
(b)
(1)
Write down the first three terms of the series.
(1)
(2)
Calculate the sum of the series.
(4)
Determine the first three terms of the geometric sequence of which the
6
3
6th term is − 16 and the 9th term is
.
256
(7)
[12]
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JUNE 2014 EXAMINATIONS:
Page 3 of 12
MATRIC MATHEMATICS: PAPER I
QUESTION 3
(a)
Botle’s Porsche Cayenne depreciates at a reducing balance rate of 10% p.a.
Determine how many years and months it will take for her Porsche to halve
in value.
(4)
Accumulated Amount in Rands
(b)
200 000
Number of Years
The graph above illustrates the growth of a lump sum of money that
Thato invested in Ámandla Bank.
Determine the annual interest rate she received as a percentage using
the graph above.
(c)
(4)
Bianca buys a house for R2 000 000.
She puts down a deposit of R300 000.
She makes the remainder of the payment by means of a loan from the bank at
a fixed interest rate of 9% p.a. compounded monthly.
She agrees to pay back the loan by means of equal monthly payments over
15 years starting one month after she loaned the money.
(1)
Determine the equal monthly payments.
(5)
(2)
Assuming Bianca’s monthly repayments are R17 242,53 calculate the
outstanding balance of her loan after 7 years (immediately after she
makes her 84th repayment).
(5)
Calculate the total amount of interest Bianca would have paid to the
bank once she has fully paid back her loan at the end of 15 years.
(3)
(3)
[21]
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JUNE 2014 EXAMINATIONS:
Page 4 of 12
MATRIC MATHEMATICS: PAPER I
QUESTION 4
(a)
𝒚
𝒙
Refer to the figure above.
𝐴(1; 0) ; 𝐵(5; 0) and 𝐶(0; 10) lie on 𝑓.
Determine:
(1)
the equation of the axis of symmetry of 𝑓.
(1)
(2)
the equation of 𝑓 in the form 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐.
(4)
(3)
the domain and range of 𝑓.
(4)
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JUNE 2014 EXAMINATIONS:
(b)
Page 5 of 12
MATRIC MATHEMATICS: PAPER I
ANSWER THIS QUESTION ON YOUR ANSWER SHEET
𝒚
𝒙
1 𝑥
Refer to the figure above of 𝑔(𝑥) = ( )
3
On your answer sheet:
(1)
draw a labelled sketch of ⁡⁡𝑔−1 on the axes provided.
(3)
(2)
determine the equation of ⁡⁡𝑔−1 .
(2)
(3)
write down the equation of the line of reflection of ⁡⁡𝑔 and ⁡⁡𝑔−1 .
(1)
(4)
𝑔 is reflected about the 𝑥 – axis and then the 𝑦 − axis to form graph ℎ.
Determine the equation of ℎ.
(2)
[17]
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JUNE 2014 EXAMINATIONS:
MATRIC MATHEMATICS: PAPER I
Page 6 of 12
QUESTION 5
(a)
Determine 𝑓 ′ (𝑥) from first principles if 𝑓(𝑥) =
(b)
Find
(c)
𝑑𝑦
𝑑𝑥
(1)
𝑦=
(2)
𝑦=
1
𝑥
(5)
given:
𝑥 2 −5𝑥+6
𝑥−2
3
3
√𝑥 2
;
𝑥≠2
+⁡2√𝑥⁡ ⁡; ⁡𝑥 > 0 .
Determine the equation of the tangent to 𝑦 = 𝑥 2 at 𝑥 = 3.
(2)
(4)
(4)
[15]
78 marks
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JUNE 2014 EXAMINATIONS:
Page 7 of 12
MATRIC MATHEMATICS: PAPER I
SECTION B
QUESTION 6
(a)
Given:
𝑨=√
𝒙−𝟐
𝒙+𝟑
and
𝑩 = √(𝒙 − 𝟓)(𝒙 + 𝟒)
Determine:
(1) the value(s) of 𝑥 for which A is undefined.
(1)
the value(s) of 𝑥 for which B is real.
(3)
(2)
(b)
(3) the smallest natural number that⁡𝑥 can be when B is irrational.
(1)
(4) the smallest integer that 𝑥 can be when B is non-real.
(1)
If 𝑥 is a rational number then 𝑓(𝑥) = 3.
If 𝑥 is an irrational number then 𝑓(𝑥) =
5
4
Find the value of ⁡⁡2𝑓(𝑓(√2))
(2)
[8]
QUESTION 7
(a)
The table below shows values satisfying the relationship:
𝑇𝑛 = 𝑝𝑛2 + 𝑞
n
1
2
3
4
5
6
𝑇𝑛
5
11
A
35
53
75
(1) Determine the value of A.
(2)
(2) Determine the value of 𝑝 and 𝑞.
(4)
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JUNE 2014 EXAMINATIONS:
(b)
Page 8 of 12
MATRIC MATHEMATICS: PAPER I
The sum of the first 𝑛 terms of a sequence is given by:
𝑆𝑛 = ⁡
9𝑛+5𝑛2
2
Find:
(1) the sum of the first 3 terms.
(1)
(2)
(3)
the third term.
(3) an expression for the 𝑛𝑡ℎ term of the sequence.
(c)
(4)
Find the largest 4 digit number in the sequence: 1; 4; 7; 10; 13……
g
(3)
[17]
QUESTION 8
𝒚
(a)
𝒙
The graph of 𝑓(𝑥) =
𝑓 is shifted ⁡⁡𝑎
−
2
𝑥
is illustrated above.
units to the right and ⁡⁡𝑏 units up to form graph ℎ.
𝑎 > 0 and 𝑏 > 0
Determine:
(1)
The equation of ℎ.
(2)
(2)
The equation of the axis of symmetry of ℎ, that has a negative gradient,
in terms of 𝑎 and 𝑏.⁡⁡
(3)
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JUNE 2014 EXAMINATIONS:
Page 9 of 12
MATRIC MATHEMATICS: PAPER I
(b)
𝑪
𝑪
𝟑𝒎
𝑨
𝑩
𝟏, 𝟕𝒎
𝑬
𝑫
Refer to the diagram above. This diagram is not drawn to scale.
Moteo, an enthusiastic basketball player is practising her shooting.
Each throw follows the path of a parabola.
She throws from a point A which is 1,7𝑚 above the floor.
On one of her throws, the ball reaches its maximum height of 3,1625𝑚
when it has covered a horizontal distance of 𝟑𝒎.
Unfortunately, on this throw, the ball does not go into the basket, but hits
the front of the rim of the basket which is 3𝑚 above the floor.
CD and AE are perpendicular to the ground.
Determine:
(1)
(2)
𝐴𝐵, the horizontal distance between Moteo’s hand and the front
of the rim of the basket.
(6)
𝐴𝐶, the actual distance between Moteo’s hand and the front
of the rim.
(2)
[13]
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JUNE 2014 EXAMINATIONS:
Page 10 of 12
MATRIC MATHEMATICS: PAPER I
QUESTION 9
(a)
Rebecca has either pasta or pizza for lunch. If she has pasta one day, the
probability she has pasta the next day is 0,2. If she has pizza one day, the
probability she has pizza the next day is 0,4.
Rebecca has pizza on Wednesday 11th June.
(b)
(1)
Draw a tree diagram to represent the above situation.
(2)
(2)
Hence, or otherwise, determine, to two decimal digits, the probability
that she has pasta on Friday 13th June.
(3)
Let 𝐴 and 𝐵 be two events in a sample space such that 𝑃(𝐴) =
2
5
1
.
and 𝑃(𝐵) =
3
(1)
If 𝐴 and 𝐵 are mutually exclusive, find 𝑃(𝐴 ∪ 𝐵)′.
(3)
(2)
If 𝐴 and 𝐵 are independent, but are not mutually exclusive,
find 𝑃(𝐴 ∪ 𝐵).
(4)
[12]
QUESTION 10
(a)
𝑓(𝑥) =
Given:
𝑥 3 +𝑞𝑥−𝑝𝑥 2 −𝑝𝑞
𝑥−𝑝
Determine:
(1)
𝑓(𝑝)
(2)
(2)
𝑓′(𝑝)
(4)
(3)
lim f ( x)
(2)
x p
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JUNE 2014 EXAMINATIONS:
Page 11 of 12
MATRIC MATHEMATICS: PAPER I
(b)
𝒚
𝒙
Refer to the graph of 𝑔 illustrated above.
State whether the following are positive, negative or zero:
(c)
(1) ⁡⁡𝑔(𝑎)
(2) ⁡⁡𝑔′(𝑎)
(3) 𝑔(𝑏)
(4)
𝑔′(𝑏)
(5) ⁡⁡𝑔(𝑐)
(6)
𝑔′(𝑐)
(7) ⁡⁡𝑔(𝑑)
(8)
𝑔′(𝑑)
Given:
(4)
𝑓(𝑥) = 2𝑥 3 − 4𝑥 − 6
Find the 𝑥 coordinate(s) of the point(s) on the graph of 𝑓 at which the
tangent(s) are perpendicular to the line 2𝑦 = −4𝑥 + 6.
Give answer(s) to one decimal digit if necessary.
(5)
[17]
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JUNE 2014 EXAMINATIONS:
MATRIC MATHEMATICS: PAPER I
Page 12 of 12
QUESTION 11
The two roots of the equation 4𝑥 2 + 𝑝𝑥 − 9 = 0⁡ differ by 5.
Calculate the value(s) of p.
[5]
72 marks
Total: 150 marks
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