ST CATHERINES CONVENT
GRADE 12
MATHEMATICS: PAPER 1
September 2015
TIME:
3 hours
MARKS:
150
EXAMINER:
Mrs V Germishuys
MODERATOR:
Mrs A Rossouw
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY:
(1)
This question paper consists of 13 questions and 8 pages including the cover sheet
and is divided into two sections, Section A and Section B.
(2)
A separate formula sheet has also been provided for you.
(3)
Read the questions carefully.
(4)
You may use an approved non-programmable and non-graphical calculator, unless
stated otherwise in any question.
(5)
Unless stated otherwise, round all answers to two decimal places where necessary.
(6)
Clearly show all details of your workings and calculations.
(7)
It is in your own interest to write legibly and to present your work neatly.
Grade 12 Preliminary Examination
St Catherines Convent
SECTION A
Paper 1 – September 2015
(75 MARKS)
QUESTION 1
[25]
1.1. Solve the following equations and inequalities:
1.1.1.
2x + 3 =
4
3- x
(4)
1.1.2.
2x 2  9x  5
(4)
1.1.3.
1.1.4.
x2  x  10  2  x  x  1
x( x  7)  16 
(6)
60
0
x  7x
(6)
2
1.2. Solve the following equations simultaneously for x and y :
33 x
 243
9y
and
2 x.4 y1  32
QUESTION 2
(5)
[17]
Given the function, f ( x)  x3  kx 2  8x  12 ,
2.1. Show that k  5 , if ( x  1) is a factor of f .
(2)
2.2. Hence, factorise f fully.
(4)
2.3. Find the coordinates of the turning points of f .
(5)
2.4. Find the coordinates of the point of inflection of f .
(3)
2.5. Sketch the graph of f , showing the coordinates of all intercepts
with the axes, turning points and point of inflection.
(3)
Grade 12 Preliminary Examination
Paper 1 – September 2015
St Catherines Convent
QUESTION 3
[14]
3.1. Using first principles, find the derivative of the function
f (x) = 6x - 4x 2 .
3.2
Given:
(5)
f ( x)  ax3  q ;
f (2)  3 and
f '(2)  12
Prove a = 1 and q = -5 and then determine equation of the tangent to the curve
of f (x) at x = 2.
QUESTION 4
4.1
(9)
[9]
The following sequence forms a convergent geometric sequence:
x3
5x ; x 2 ;
; ...
5
4.1.1 Determine the possible values of x.
(3)
4.1.2 When x = 2, determine S  .
(3)
4.2
The sum of the first n terms of a series is given by the formula:
Sn = 3n2 – 2n.
Find the tenth term.
QUESTION 5
(3)
[10]
Determine the following:
5.1
Writing your answers with positive exponents, find dy if:
dx
5.2
2
4 x

x2
4
5.1.1
y
5.1.2
y (1  2 x)  8 x3  1
(3)
(4)
d  2x2  x  4 


dx 
x

(3)
END OF SECTION A
Grade 12 Preliminary Examination
Paper 1 – September 2015
St Catherines Convent
SECTION B
(75 MARKS)
QUESTION 6
6.1
[10]
A certain school has 80 girls and 120 boys. After the exams, the headmaster
conducts a survey for the boys and the girls and notes down how many did better at
Maths and how many did better in English. The results are given in the contingency
table below:
Maths
English
Boys
75
45
Girls
50
30
Is there evidence to suggest that boys do better in Maths. Support your answer
with appropriate calculation.
(4)
6.2
6.3
Six friends sit in a row on a bench. Two of them have just had an argument
and will not sit next to each other. How many possible arrangements are
there?
(4)
P(A) = 0,6. What is the probability that A occurs at least once in 8 trials?
Round your answer off to four decimals.
(2)
QUESTION 7
7.1
[11]
A pattern proceeds as follows, with the number of occurrences of each letter
corresponding to its position in the alphabet:
A BB CCC DDDD EEEEE ........
7.1.1
Determine the number of letters in the sequence.
(3)
7.1.2
Which letter will be in the 95th position?
(3)

7.2
ZZ
Evaluate:

n2
 2 
3n 1
n
(5)
Grade 12 Preliminary Examination
St Catherines Convent
QUESTION 8
Paper 1 – September 2015
[10]
The sketch below represents the graph of the function f ( x)  b x  1 .
Y
A(−1;5)
f
X
8.1. Calculate the value of b , if A(1;5) is a point on f .
(2)
8.2. Write the equation of f 1 in the form y  .....
(2)
8.3. What are the domain and range of f 1 ?
(2)
8.4. For what values of x is f 1 ( x)  1 ?
(3)
8.5. Write down the equation of g , the reflection of f 1 in the x -axis.
(1)
Grade 12 Preliminary Examination
St Catherines Convent
Paper 1 – September 2015
QUESTION 9
[10]
Two friends, Paige and Tammy, are both 19 years old. Paige starts a savings programme,
investing R1 000 at the beginning of each year for each of 7 years at 10% compound interest
per annum.
At this point (seven years later), Tammy decides to start saving in the same way, by investing
R1 000 at the beginning of each year at the same interest rate in the hope that, one day, her
savings will catch up with Paige’s.
Paige does not put any more money into her account, but she lets what she has saved to this
point grow at 10% per annum.
9.1
9.2
Calculate how much money Paige has in her bank account after 7 years.
How old will Tammy be when her bank balance catches up with Paige’s?
QUESTION 10
(4)
(6)
[11]
The diagram shows a plan for a rectangular park ABCD, in which AB = 40 m and AD = 60 m.
Points X and Y lie on BC and CD respectively and AX, XY and YA are paths that surround
a triangular playground. The length of DY is x m and the length of XC is 2x m.
10.1
Write an expression for the length BX, in terms of x.
10.2
Show that the area, A m2, of the playground is given by
A = x2 − 30x + 1200
10.3
Given that x can vary, find the minimum area of the playground.
(1)
(5)
(5)
Grade 12 Preliminary Examination
St Catherines Convent
Paper 1 – September 2015
QUESTION 11
[8]
11.1
If f ( x)  a , where a is a constant, find f '( x) from first principles.
11.2
A particle moves along a straight line in such a way that its displacement S (in
metres) from a fixed point after any time t (in seconds) is given by the formula
11.3
(3)
S (t )  t 2  4t  4 .
What is the velocity of the particle after 2 seconds? What does this mean?
(3)
What is the average gradient of g(x) = x² + 2x between the points x = -1 and
x=1
(2)
QUESTION 12
[5]
1+2 =3
4+5+6 =7+8
9 + 10 + 11 + 12 = 13 + 14 + 15
16 + 17 + 18 + 19 + 20 = 21 + 22 + 23 + 24
25 + 26 + 27 + 28 + 29 + 30 = 31 + 32 + 33 + 34 + 35
12.1
Give the first term of the expression in the 20th row
(1)
12.2
Determine the term that appears right after the equal sign in the
n-th row
(4)
Grade 12 Preliminary Examination
Paper 1 – September 2015
St Catherines Convent
QUESTION 13
[10]
The sketch below represents the graph of the function f ( x) 
2x  8
.
x2
Y
B
f
X
A
13.1
13.2
13.3
Determine the coordinates of A and B , the x and y intercepts of f .
(4)
A
 q , determine the equations
x p
of the horizontal and vertical asymptotes of f .
(4)
Give the equations of the asymptotes of h( x)  f ( x  1)  3 .
(2)
By writing f in the form f ( x) 
END OF SECTION B
Grade 12 Preliminary Examination
Paper 1 – September 2015
St Catherines Convent
ST CATHERINES CONVENT
GRADE 12
MATHEMATICS: PAPER 1
September 2015
TIME:
3 hours
MARKS:
150
EXAMINER:
Mrs V Germishuys
MODERATOR:
Mrs A Rossouw
NAME: ______________________________________
Algebra
Q1
Q2
Q3
Q4
Q5
Q6
Q7
Q8
Q9
Q10
Q11
Q12
Q13
Totals
TOTAL
Calculus
Sequence Probability
and Series
Functions
Finance
/25
/17
/14
/9
/10
/10
/11
/10
/10
/11
/8
/5
/25
/43
/14
/150
/21
/10
/37
/10
%