ST STITHIANS COLLEGE
Department of Mathematics
GRADE 12
MATHEMATICS CORE – PRELIM PAPER 1
DATE:
July 2015
TOPICS:
All Paper 1 Topics
TIME:
3 Hours
TOTAL MARKS:
150
EXAMINER: Cluster North 143
MODERATOR: Cluster North 143
INSTRUCTIONS:
1. The paper consists of 10 questions. Answer ALL the questions.
2. This question paper consists of 12 pages, including the cover page.
3. Clearly show ALL calculations you have used to determine the answers.
4. An approved scientific calculator (non-programmable and non-graphical) may be used,
unless otherwise specified.
5. If necessary, answers should be rounded off to TWO decimal digits, unless stated
otherwise.
6. Diagrams are not necessarily drawn to scale.
7. Number your answers correctly according to the numbering used in this question paper.
8. It is in your own interest to write legibly and to present your work neatly.
This Page is intentionally blank
Page 2
QUESTION 1
[13]
Solve for x rounded off to two decimal places where necessary:
1.1.
 x  3  2 x  1  0
(2)
1.2.
7   x  2 x  3  0
(4)
1.3.
log3  2 x 1  log3 5  log3 6
(3)
1.4.
Consider the inequality:
3x 2  15 x  5
Determine the solution to this inequality if x integers .
QUESTION 2
(4)
[15]
1 8 27
; ;
; ...
2 9 16
2.1.
Determine the nth term in this sequence of fractions:
2.2.
The 2nd term of a geometric sequence is 6 and the 5th term is 162.
(3)
2.2.1. Determine the 10th term of the sequence.
(5)
2.2.2. Which term in the above sequence has a value of 86 093 442?
(2)
2.3.
The 1st term of an arithmetic sequence is 2 and the 3rd term is 10.
Find the sum of the second 20 terms  i.e. T21  T22  ...  T40  .
(5)
Page 3
QUESTION 3
3.1.
Given: f ( x) 
[15]
2
x3
3.1.1. Write down the domain of f ( x) .
(2)
3.1.2. Write down the equations of the asymptotes of f ( x)  2 .
(2)
3.1.3. Write down the equation of the graph of h( x) formed if f is shifted
3 units up and 2 units right.
(2)
3.1.4. Determine the equation of f 1 ( x) , the inverse of f ( x) , in the form
f 1 ( x) : y 
3.2.
(2)
The graphs of f ( x)  3x and g ( x)  ax  q are drawn. A(1;3) is the point of
intersection of the two graphs. B(0;5) is the y-intercept of g and C(0;1) is
the y-intercept of f .
y
5
B
f(x)
 A(1;3)
1
g(x)
C
x
3.2.1. Find the values of a and q.
(2)
3.2.2. Write down the domain of f 1 ( x) , the inverse of f.
(2)
3.2.3. For which value(s) of x is f 1 ( x).g ( x)  0 ?
(3)
Page 4
QUESTION 4
4.1.
[18]
A survey was conducted of 200 visitors to the Kruger Game Reserve, to ask them
what interested them most in the game reserve, the Trees, the Animals or the Birds.
The following results were obtained:
Trees
Animals
12
30
4
6
41
14
x
y
Birds
4.1.1. If 156 visitors said they were interested in Animals, what are the values
of x and y ?
4.1.2. How many visitors were NOT interested in Trees at all?
(4)
(1)
4.1.3. What is the probability that a visitor was only interested in one of the sights
of Kruger?
4.2.
(2)
The probability that the Springbok rugby team has all its players fit to play
in the World Cup is 65%.
The probability that the team will win the World Cup if all the players are fit
is 75%.
When the players are not all fit, the probability of the team winning the
World Cup drops to 50%.
Calculate the probability that the Springbok rugby team will win the
World Cup.
(5)
Page 5
4.3.
A South African computer company, Hangalot, has been having problems
with one of their laptop models. They had previously outsourced the assembly
of some of these units to a company in Nigeria.
A sample of 483 laptops revealed the following:
Assembled in Nigeria
Assembled in South Africa
Total
Faulty
14
x
98
Non Faulty
y
330
385
Total
69
414
483
4.3.1. Complete the table by determining the values of x and y.
(2)
4.3.2. By testing for independence, decide whether Hangalot should continue
outsourcing the assembly of some units to the company in Nigeria.
Justify your answer based on suitable calculations.
(4)
Page 6
QUESTION 5
5.1.
[16]
Eugene wishes to invest a sum of money so that after three years the initial
amount of the investment has doubled. Determine the annual rate of interest
for this investment. (Round your answer to the nearest percentage).
5.2.
(3)
Brenda takes out a twenty year loan of R400 000. She repays the loan by
means of equal monthly payments starting one month after the granting of
the loan. The bank interest rate is 18% per annum compounded monthly.
5.2.1. Show using an appropriate formula, that Brenda’s monthly repayments
are R6 173,25.
(3)
5.2.2. Calculate the amount still owed on her loan account at the end of the first
year (after the 12th payment).
(4)
5.2.3. If Brenda chose to pay an instalment amount of R7 000 per month instead
of the amount calculated in 5.2.1., determine how long, in months, it will
take for her to amortise (repay) the loan in full.
(6)
Page 7
QUESTION 6
[17]
6.1.
Determine f ( x ) from first principles if f ( x)  1  x 2 .
6.2.
Evaluate: lim
(3)
6.3.
2 

Determine the gradient of g ( x)  3 x  2 x   at x  3 .
5x 

(3)
6.4.
Determine
dy
given: yx  y  3x 2  4 x  1
dx
(3)
6.5.
Determine the equation of the tangent to the curve p( x)  x3  x 2 at the point
(4)
x2  2 x
x 2 x 2  4
where the curve cuts the negative x-axis.
(4)
QUESTION 7
7.1.
[24]
Given the following sketch of f ( x)  2 x3  6 x  4 , with A and B the
x -intercepts, C the y -intercept, and A and D the turning points:
y
A
B


C

x

D
7.1.1. Determine the coordinates of A, B and C, showing all your working.
(4)
7.1.2. Determine the coordinates of D.
(3)
Page 8
7.1.3. Determine graphically, the values of p for which the equation: 2 x3  6 x  4  p
will have only one real solution.
(2)
7.1.4. Determine the value(s) of x where f ( x)  0 .
(2)
7.1.5. Determine the average gradient of the curve between points B and D.
(2)
7.2.
Refer to the figure below, showing the graphs of f ( x)  2 x 2  bx  30
and g ( x)  2 x  10 . A and B are the x-intercepts and C is the y-intercept of f ( x) .
G is the turning point of f ( x ). A is also the x-intercept of g ( x ) and D is the
y-intercept of g ( x ) .
y
G
 C
D
J


A
g ( x)
O

B K
x
L
f ( x)
7.2.1. Show by performing the necessary calculations, that b  4 .
(3)
7.2.2. If JL  60 units, determine the length OK, where K is the point on the x-axis
made by the vertical line JL.
7.2.3. Determine the line of symmetry of p ( x ) if p( x)   f ( x  2) .
(4)
(4)
Page 9
QUESTION 8
[14]
y
8.1.
The function defined by
x
f ( x)  x3  ax 2  bx  4 is sketched
alongside.
P(1; 3) and R are the turning points of f.
P ( 1;  3)


Determine the values of a and b.
R
(6)
8.2.
A cubic function f has the following properties:

1
f    f  3  f  1  0
2

 1
f   2  f      0
 3
 1 
 f decreases for x    ; 2  only.
 3 
Draw a possible sketch graph of f, clearly indicating the x-coordinates
of the turning points and ALL the x -intercepts.
8.3.
(4)
A cannon fires a projectile onto the top of a hill, such that the height
s (in metres) reached by the projectile t seconds after it is fired from the
cannon is given by the parabola equation, s  200t  5t 2 .
If the velocity of the projectile, when it hits the hill on the way down, is
170 metres per second, what is the height of the hill where the projectile
hits?
(4)
Page 10
QUESTION 9
9.1.
[10]
A function in the form Tn  an 2  bn  c is sketched below. Points on this
graph can also be modelled using the formula Tn  an 2  bn  c , where n  1
and n integers. Four points on the graph are given.
Use the points to determine the general formula for the sequence represented.
(4)
Tn
 (4; 2)
n
 (3;  2)
 (1;  4)  (2;  4)
9.2.
The following sequence forms a convergent geometric sequence:
7 x ; x2 ;
x3
; ...
7
9.2.1. Write the series in sigma notation if there are an infinite number of terms.
(2)
9.2.2. If the constant ratio (r) for the sequence satisfies the equation 7r  4  0 ,
determine the value of x .
(2)
9.2.3. Hence, calculate the value of S .
(2)
Page 11
QUESTION 10
[8]
A rectangle is inscribed in the closed region bounded by the x-axis, the y-axis,
and the graph of f ( x)  8  x3 as shown in the sketch below:
The base of the rectangle is drawn along the x-axis.
10.1. If the base of the rectangle is 1,5 units long, determine the height of
the rectangle.
(2)
10.2. Determine the dimensions of the rectangle such that the area of the
rectangle is a maximum.
(6)
Page 12