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ST TERESA’S HIGH SCHOOL
GRADE 12 EXAMINATION
September 2016
MATHEMATICS PAPER 1
Time: 3 Hours
Marks: 150
Examiner: K Moffat
Moderator: A Milton
__________________________________________________________________________________
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY
1.
This question paper consists of 10 pages, including a 2 page formula sheet.
2.
The question paper consists of two sections.
3.
Work on both sides of the paper of the lined paper answer booklet, starting each question on a
new page.
4.
Please note that diagrams are not necessarily drawn to scale.
5.
All necessary working details must be shown.
6.
Approved non-programmable and non-graphical calculators may be used, unless otherwise
stated.
7.
If necessary answers should be corrected to two decimal digits, unless otherwise stated.
8.
It is in your own interest to write legibly and to present your work neatly.
St Teresa’s High School Grade 12 Preliminary Examination September 2016
SECTION A:
[74]
QUESTION 1:
[30]
(a)
x
 y 
Simplify:   y   1
y
 x 
(2)
(b)
 7x  x 
Evaluate: a  

 7x  x 
(3)
(c)
Solve for x:
2
0
(1)
12 x 3  4 x 2  5 x
(5)
(2)
32 x 1  4.3 2 x  21
(4)
(3)
2 x 2  3x  0
(4)
x4 4
y
 x
8
(d)
Determine dy
(e)
Find f (x) from first principles, given f ( x)  x 2  3
(f)
Evaluate:
if
dx
6
 (n
4
(leaving positive exponents)
 2)
(4)
(4)
(4)
n2
QUESTION 2:
(a)
4k  2 ; k  1 and k  3 are the first three terms of a geometric sequence.
(1)
(2)
(3)
(b)
[19]
Determine the value of k if k is an integer and hence show that the first three
terms of the sequence are 18, 6 and 2.
Write down the nth term (Tn) of the sequence and determine the value of n if
2
Tn 
243
Determine the sum to infinity of the terms of the sequence.
(5)
(6)
(3)
Mrs Nortje deposits R750 at the end of each month into a savings account paying 13,5% p.a.
compounded monthly. Calculate the amount in her account 18 months after the first deposit was
made.
(5)
Page 2 of 10
St Teresa’s High School Grade 12 Preliminary Examination September 2016
QUESTION 3:
(a)
[7]
The graph of the parabola given by y  ax 2  bx  c is drawn.
Then
(1) a  0 ; b  0 ; c  0
(2) a  0 ; b  0 ; c  0
(3) a  0 ; b  0 ; c  0
(4) a  0 ; b  0 ; c  0
(b)
Write just the number that corresponds with the parabola drawn.
(2)
Determine the equation of the parabola in the form y  ax 2  bx  c which passes
through 1;13 and has a turning point at  1;5 .
(5)
QUESTION 4:
[8]
x
 1 . Determine f 1 ( x) and g 1 ( x) , writing each in the form
3
y  ....... and stating the domain and range for each inverse function.
Given f ( x)   log 3 x and g ( x ) 
QUESTION 5:
[10]
The function y  f (x) is illustrated below:
f(x)
f(x)
Page 3 of 10
St Teresa’s High School Grade 12 Preliminary Examination September 2016
Sketch the graphs of the following function on separate axes in your answer booklet:
(a)
y  f ( x)  2
(3)
(b)
y  f ( x  2)
(3)
Label each graph clearly.
(c)
Give the equations of the axes of symmetry of y  f ( x  1)  3
(4)
SECTION B:
[76]
QUESTION 6:
[12]
(a)
Solve for x if: log x  log x 2  log x 3  ......  log x10  110
(b)
In a raffle, the tickets are numbered consecutively from 1 to 100. Customers draw a
ticket at random and pay the amount, in Rand, equal to the number on the ticket times ten,
except for those tickets with numbers exactly divisible by five, which are free.
If all tickets are disposed of, how much money is collected?
(7)
(5)
QUESTION 7:
[9]
Mrs Huckle has taken out a loan of R75 000 to help pay for a trip to visit a friend in Australia. The
loan is repayable in equal monthly instalments over 5 years at a rate of 14,75% per annum,
compounded monthly.
(a)
Calculate her monthly repayment.
(2)
(b)
Calculate the balance on her loan after 3 years.
(3)
(c)
If she doubles her monthly repayment, how long will it take her to repay the loan?
(4)
Page 4 of 10
St Teresa’s High School Grade 12 Preliminary Examination September 2016
QUESTION 8:
(a)
(b)
[12]
How many flags, of the style shown below, can be made if the material is available in eight
different colours, and if:
(1)
Each stripe is a different colour?
(2)
(2)
No restriction is placed on the use of colour?
(2)
Andrew is first team rugby player who eats a lot of food. When making his lunch-time
sandwiches he randomly picks which toppings to apply to either white or whole-wheat bread.
The probability that he will choose ham and cheese is ¼ and the probability that he will choose
Peri-naise and ham is ¾. The probability he will choose whole-wheat bread is 2/3.
(1)
Draw a tree diagram to represent the above information.
(3)
(2)
What are the chances that Andrew will have Ham and Cheese on White Bread?
(2)
(3)
Given that Andrews sandwich had Peri-naise on it what the chances that the sandwich was
on whole-wheat.
(3)
QUESTION 9:
[19]
Two numbers x and y are such that 2 x  y  12 . The product P is formed by multiplying the first
number by the square of the second number, so that P  xy2
(a)
Show that P  4 x 3  48 x 2  144 x
(b)
Find the stationary points of P
(2)
(5)
2
(c)
The values of x and y must both be positive. Find the value of
d p
at this stationary
dx 2
value and hence show that it gives a maximum value.
(3)
(d)
Calculate the point of inflection
(2)
(e)
Sketch the graph of P for x  
(3)
Page 5 of 10
St Teresa’s High School Grade 12 Preliminary Examination September 2016
(f)
If P / x   k has one repeated real root. Write down the value of k.
(4)
QUESTION 10:
[8]
Mrs De Bod hits another pothole driving home and takes off from the top of a slope.
x
The slope can be modelled by y  2,3 
while the car’s path can be modelled by
5
y  0,1( x  4) 2  3
(a)
Calculate the coordinates of where the car takes off and lands.
(6)
(b)
Determine the highest point that the car reaches while airborne.
(2)
QUESTION 11:
(a)
[16]
Find the integer that is the closest approximation, without the use of a calculator, to:
10 2008  10 2010
10 2009  10 2009
Page 6 of 10
(5)
St Teresa’s High School Grade 12 Preliminary Examination September 2016
(b)
If f ( x)  x 2  3x  4 ,
(1)
(2)
(3)
Show that
f (3)  f (1) is divisible by 2
f (5)  f (1) is divisible by 4
f (7)  f (4) is divisible by 3
(4)
Complete the following generalisation of the above results:
f (a)  f (b) is divisible by ……….
(2)
Prove this generalisation is true.
(5)
Total: 150 marks
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St Teresa’s High School Grade 12 Preliminary Examination September 2016
THIS PAGE CONTAINS NO QUESTIONS
Page 8 of 10
St Teresa’s High School Grade 12 Preliminary Examination September 2016
MATHEMATICS
INFORMATION SHEET
x 
–b 
b 2 – 4ac
2a
n
1 
n
n
i 1

Sn 
n
 2 a  n  1 d 
2
S 
a
1 r
i 1
Tn  a  n  1 d
Tn  a r n  1
Sn 
;
f  x   lim
h 0
a r n  1
r  1
r 1
n (n  1)
2
i
;
1  r  1
f x  h   f x 
h
A  P 1  n i 
A  P 1  n i 
A  P 1  i  n
A  P 1  i  n
 1  i n  1
F  x

i


1  1  i  n 
P  x

i


d 
( x 2 – x1 )
2
 ( y 2 – y1 )
y  mx  c
m 
y 2 – y1
x2 – x1
2
y  y2 
 x  x2
M 1
; 1

2
2


y – y1  m ( x – x1 )
m  tan 
( x – a) 2  ( y – b) 2  r 2
Page 9 of 10
St Teresa’s High School Grade 12 Preliminary Examination September 2016
In  ABC :
a
b
c


sin A
sin B
sin C
a 2  b 2  c 2 – 2 b c . cos A
area  ABC 
1
a b . sin C
2
sin (  )  sin  . cos   cos  . sin 
sin ( – )  sin  . cos  – cos  . sin 
cos (  )  cos  . cos  – sin  . sin 
cos ( – )  cos  . cos   sin  . sin 
cos 2   sin 2 

cos 2   1  2 sin 2 

2
2 cos   1
sin 2   2 sin  . cos 
n
x 

P ( A) 
f x
n
n ( A)
n (S )
𝑦̂ = 𝑎 + 𝑏𝑥
2 
 x
i 1
i
 x 2
n
P ( A or B)  P ( A)  P ( B) – P ( A and B)
𝑏=
∑(𝑥−𝑥̅ )(𝑦−𝑦̅)
Page 10 of 10
∑(𝑥−𝑥̅ )2