ST STITHIANS GIRLS COLLEGE
MATHEMATICS: PAPER 1
CORE MATHS: GRADE 12
DATE: July 2011
NAME:
TIME: 3 hours
____________________________TEACHER:
MARKS: 150
_________________
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY
1. This question paper consists of 7 pages, excluding the front cover. A formula sheet is attached
at the end which you may detach.
2. Read the questions carefully.
3. Question 7 must answered in its entirety on the specific page provided.
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 otherwise
stated.
7. Unless otherwise stated, round answers to two decimal places 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.
Page 1 of 7
Question 1
a)
b)
[29]
Solve for x, correct to TWO decimal places, where necessary:
i)
𝑥 ( 𝑥 − 1 ) = 12
(3)
ii)
2𝑥 2 + 3𝑥 − 7 = 0
(3)
iii)
𝑥 = 10
iv)
log 3 𝑥 = 2
(2)
v)
7𝑥 2 + 18 𝑥 − 9 > 0
(4)
(2)
log 27
Solve for x and y simultaneously:
2𝑥 − 𝑦 = 7
(7)
𝑥 2 + 𝑥𝑦 = 21 − 𝑦 2
c)
Simplify completely, without the use of a calculator. Show all calculations.
2
i)
(√3 − 2)
(3)
ii)
1
3log2  2log6  log16
2
+ 2√12
(5)
Question 2
[14]
a)
Using first principles, differentiate 𝑓 ( 𝑥 ) = 1 − 3 𝑥 2
(5)
b)
Determine
c)
Determine f '( x) if 𝑓 ( 𝑥 ) = (𝑥 − 1 )( 𝑥 + 2 )
d)
Determine the following limit:
lim
x 3
x2  9
x 2  3x
Page 2 of 7
𝑑𝑦
𝑑𝑥
if
𝑦 =
3
𝑥2
− 4𝑥 + 1
(2)
(3)
(4)
Question 3
a)
[16]
Given the geometric sequence:
3 3
6 3   
2 4
i)
Find the sum of the first eleven terms of the series, to 3 decimal places.
(3)
ii)
Find the sum to infinity.
(2)
b)
Determine
n
 (2k  4)  500
(4)
k 1
c)
i)
ii)
If the following numbers form a sequence: -1 ; 5 ; 15 ; 29 ; …
determine what kind of sequence it is.
(1)
Determine the general term of the sequence.
(6)
Question 4
a)
Determine the equation of the tangent to the curve y 
where x   1
b)
[8]
1
x
at the point
(5)
The gradient of the tangent to the curve of f  x   2x 2 at a point
( a ; b ) on the curve is 6. Determine the value of a.
Question 5
(3)
[11]
Melissa buys a car for R 140 000 and she pays a cash deposit of 15%. The balance is paid
using a bank loan. Interest is charged at 13,5% p.a. compounded monthly over 5 years.
a)
Determine her monthly repayments.
(6)
b)
Determine the outstanding balance on the loan after 36 months.
(5)
Page 3 of 7
Question 6
[8]
Given the four sketch graphs below, (not drawn to scale), answer the following questions:
y
y
g
f
x
0
0
y
y
k
h
0
0
x
0
0
x
0
0
x
State whether the following are TRUE or FALSE:
a)
b)
c)
d)
e)
f)
g)
h)
𝒇 is a many-to-one function.
The inverse of 𝒈 is a function.
𝒉 is not a function.
𝒌 is a many-to-one function.
The inverse of 𝒇 is a function.
𝒈 is not a function.
𝒉 is a one-to-one function.
The inverse of 𝒌 is not a function.
Question 7
Answer this entire Question 7 in the space provided in the answer booklet, using the
page and grid provided.
[18]
Kineta writes a Mathematics test that consists of algebra and trigonometry questions.
Candidates may answer a maximum of 12 questions from each section. The time for the
test is 60 minutes. Each algebra problem usually takes Kineta 3 minutes while she takes
6 minutes for each trigonometry problem. An algebra problem is worth 9 marks and a
trigonometry problem is worth 16 marks.
Let x represent the number of algebra questions Kineta answers and y represent the
number of trigonometry questions.
a)
b)
c)
Write the above scenario as a set of constraints in the space on the answer sheet. (6)
Illustrate these constraints on a Cartesian Plane and shade the feasible region.
(7)
Determine, in the space provided on the answer sheet, the number of questions from
each section that Kineta should answer in order to maximize her marks and give this
maximum.
(5)
Page 4 of 7
Question 8
[16]
Below are the graphs (not drawn to scale) of 𝑓 ( 𝑥 ) = 𝑥 2 + 4𝑥 + 3 and 𝑔 ( 𝑥 )
being a cubic function. The two functions have the roots at A and B and 𝑔( 𝑥 )
1
has another root at 𝑥 = 2. The length of DE = 6 units.
y
f (x)
K
G
S
G
D
G
A
g (x)
0
G
B
T
G
E
G
C
G
x
a)
Find the roots at A and B.
(3)
b)
Determine the equation of the axis of symmetry of 𝑓 ( 𝑥 ).
(2)
c)
Determine the co-ordinates of E.
(2)
d)
Find the equation of the function 𝑔 ( 𝑥 ). Show all working.
(4)
e)
If 𝑔 ( 𝑥 ) = 2 𝑥 3 + 7 𝑥 2 + 2 𝑥 − 3 , determine the co-ordinates of K,
where the two functions intersect.
(5)
Turn Over for Section B
Page 5 of 7
SECTION B
Question 9
[8]
a ; 4 ; b form a geometric sequence and (a – 2 ) ; 4 ; b form an arithmetic
sequence. Find the values of a and b.
Question 10
a)
[5]
Incorporate ALL THREE of the following pieces of information and draw ONE
neat sketch graph of the curve of 𝑝 ( 𝑥 ) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 if:

the roots of 𝑝 ( 𝑥 ) differ by 6.

the value of 𝑥 =

the range of the function is 𝑦 ≤ 4
−𝑏
2𝑎
is 3.
Indicate on your graph the intercepts with the x axis, the axis of symmetry and
the turning point.
b)
(3)
Write down the co-ordinates of the turning point if 𝑦 = 𝑝 ( 𝑥 ) becomes:
i)
𝑦 = 𝑝(𝑥)+ 3
(1)
ii)
𝑦 = 𝑝(𝑥−3)
(1)
Page 6 of 7
Question 11
[13]
The profits of a gold mine can be modeled according to the formula
P ( t )   5 t 3  110 t 2  200 t (See sketch graph below), where t is the time in years.
P is the profit in millions of rands and 0  t  20 .
a)
b)
c)
What does the curve below the x-axis represent?
For how many years does the ‘mine’ show a profit?
If it is advisable to sell the mine after it reaches a maximum profit, when
should the mine be sold, and what is the maximum profit, in millions of rands?
Question 12
A water Hyacinth plant currently covers about 300 m2 of a dam surface, with the total
dam surface being 3 000 m2. This particular plant increases in size (the surface area that
it covers) by approximately 7% daily. Determine how many days it will take to cover the
entire dam.
Page 7 of 7
(1)
(6)
(6)
[4]