ST STITHIANS GIRLS’ COLLEGE
GRADE 12
MATHEMATICS: PAPER 1
July 2012
TIME: 3 hours
EXAMINER: Mr Schaerer
MARKS: 150
MODERATOR: Mrs Germishuys
NAME:
TEACHER: Ms du Plessis
Mrs Germishuys
Mr Schaerer
Mrs van Rooy
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY
(1)
This question paper consists of 5 pages excluding the cover sheet.
(2)
Read the questions carefully.
(3)
Answer question 8 on the answer sheet provided.
(4)
You may use an approved non-programmable and non-graphical calculator,
unless otherwise stated.
(5)
Unless otherwise stated, round answers to two decimal place where necessary.
(6)
All the necessary working details must be clearly shown.
(7)
It is in your own interest to write legibly and to present your work neatly.
Grade 12
St Stithians Girls’ College
July 2013
Question: 1
Solve for x:
(a)
x 5 x 3
(5)
(b)
log3( x 1) 2
(2)
(c)
4.3x 3x
(3)
(d)
3x 2 7x 5 (2 decimal places, show all working)
1
81
(4)
[14]
Question: 2
Simplify, without the use of a calculator:
(a)
(b)
(52 x ) 2 .20x 1.125x
21 2 x
x2
1 x
if x 1
1
(5)
(5)
3
[10]
Question: 3
dy
from 1st principles.
dx
(a)
Given y x 2 x . Determine
(b)
Determine lim
(c)
Determine g'( x ) , leaving your answers with positive exponents:
(d)
x
x2 1
1 1
x
2x 3 7 x 2 3x
x 3
(1)
g( x )
(2)
g( x ) 2 x
If f ( x )
1
2x 2
(5)
(3)
x 3
(5)
x 0
(4)
x 2 3x 2 . Find the equation of the tangent at x 3 .
Page 1 of 5
(5)
[22]
Grade 12
St Stithians Girls’ College
July 2013
Question: 4
Given f ( x )
4
3 and g( x )
x 2
x 2
g
f
(a)
What is the domain of f ?
(2)
(b)
Find the values of x for which f ( x ) g( x ) .
(5)
(c)
Using the graphs, find the values of x where f ( x ) g( x ) ?
(2)
[9]
Question: 5
Given g( x )
(a)
1
4
x
(1)
What is the equation of h(x), the function of g symmetrical about the
y-axis?
(1)
(2)
Draw a rough sketch of h(x).
(2)
(b)
Find g 1( x ) and draw a rough sketch on the same set of axes as above.
(2)
(c)
What is the asymptote of g 1( x 3) ?
(1)
[6]
Question: 6
(a)
(b)
Given the geometric sequence 4 ;
8 16
;
…
3 9
(1)
Determine the nth term.
(3)
(2)
If the above forms a series, determine the sum to infinity.
(2)
Given Sn 3n2 2
(1)
S6
(2)
(2)
T6
(3)
Page 2 of 5
Grade 12
(c)
St Stithians Girls’ College
July 2013
Susie comes to school with Flu and infects two friends. The next day, they each
then spread the flu to two friends and so on. Assume that this pattern continues
on a daily basis.



   
 
 
(1)
How many girls will contract Flu on the 5th day?
(2)
How long will it take for all 420 of the girls in the School to be infected?
Give your answer correct to the nearest day.
(1)
(3)
[14]
Question: 7
A small factory produces two types of radios, A and B. x of A, and y of B are
produced each day. The table below shows details with regard to costs of components
and costs of assembly of each type of radio as well as the maximum capital available
each day.
Type A
Type B
Maximum
Cost of components
R200
R150
R2400
Cost of assembly
R100
R150
R1800
Further to this, owing to demand, at least 3 of type A and 4 of type B must be produced
per day. Write down four inequalities in terms of x and y which represent the above
constraints.
[4]
Question: 8
(ANSWER ON THE SHEET PROVIDED)
(a)
Sketch the following constraints on the axes provided, showing the feasible
1 y 4
y
x 7
y 2x 1
region.
(b)
If P 3x 4 y , determine the x and y values which will give a maximum
value for P. (Show all working)
Page 3 of 5
(5)
(5)
[10]
Grade 12
St Stithians Girls’ College
July 2013
Section: B
Question: 9
Mr Schaerer decides he wants to buy his daughter, Riley, a car when she turns 18.
When she is 2 years old a Polo cost R180 000. The price of a polo increases with
inflation at 5% p.a.
(a)
What will the car cost when she turns 18?
(3)
(b)
If Mr Schaerer decides to make monthly deposits into a savings account,
starting 1 month after Riley’s 2nd birthday. The last payment is made on her
18th birthday. Assuming the price will be R400 000 and the interest rate on the
savings account is 8,5% p.a. compounded monthly, what will these deposits be? (5)
(c)
(1)
Mr Schaerer never set up a savings account. He therefore has to buy the
car on a loan agreement with an interest rate of 6% p.a. compounded
monthly. Assuming the car will cost R400 000 and his monthly
instalments are R5 900, how many payments are made to pay off the car? (6)
(2)
After 2 years of payments what is the balance still owing on the loan?
(4)
[18]
Question: 10
Refer to the figure:
The graphs of f ( x ) x3 ax b and g( x ) x 2 are drawn. The curves intersect at P and Q.
L lies on g and K Lies on f such that LK y -axis. f has turning points at R and ( 1;0) .
(a)
Show that the values of a and b are −3 and −2 respectively.
(5)
(b)
Calculate the coordinates of the turning point R.
(4)
(c)
Find the value(s) of x for which f '( x ) 0 .
(2)
(d)
Find the maximum length of LK. (LK lies between P and Q)
(5)
[16]
Page 4 of 5
Grade 12
St Stithians Girls’ College
July 2013
Question: 11
A tunnel has a parabolic cross-section with maximum height of 8 metres and a width
of 8m.
8
m
8m
1
m
(a)
Give the equation of the parabola.
(3)
(b)
The roof of the bus just touches the top edge of the tunnel when its wheels are
1 metres from the side of the tunnel. Determine the height of the bus.
(2)
[5]
Question: 12
Alfred takes painkiller tablets after an operation. The levels of painkiller ingredients in
the bloodstream are directly related to the effect on pain. The levels of painkiller
ingredients present in his bloodstream t hours after taking the tablet is given by
P 4t 3 3t 2 36t 4 . Determine when the tablets have the maximum effect for him.
[4]
Question: 13
(a)
The sequence 1 ; 5 ; 11 ; 19 ; … gives the number of squares in each pattern
below.
Pattern 1
Pattern 2
Pattern 3
Pattern 4
(1)
Calculate a formula for the nth pattern of the sequence in simplest form. (4)
(2)
Calculate the smallest pattern number, n, for which the number of
squares will be greater than 505.
(4)
(b)
Two times the sum of first 8 terms of an arithmetic sequence is eight times the
second term and the 9th term is 12 more than the 6th term. Find the first 3
terms of the sequence.
(7)
(c)
Find m if
2m
(6)
2k 3 280
k m 1
[21]
Page 5 of 5