GRADE 12
MATHEMATICS
PAPER 1
Name
Time:
Total:
Teacher
3 hours
150 marks
Set
September 2015
MAV/JAC
INSTRUCTIONS AND INFORMATION
Read the following instructions carefully before answering the questions.
1.
This paper consists of 11 questions.
2.
Answer ALL the questions.
3.
Number the answers correctly according to the numbering system used in this
question paper.
4.
Clearly show ALL calculations, diagrams, graphs et cetera that you have used in
determining your answers.
5.
Answers only will not necessarily be awarded full marks.
6.
You may use an approved scientific calculator (non-programmable and
non-graphical), unless stated otherwise.
7.
If necessary, round off answers to TWO decimal places, unless stated otherwise.
8.
Diagrams are NOT necessarily drawn to scale.
9.
An information sheet with formulae is printed on the back of this page.
10.
Write neatly and legibly.
PLEASE PUT YOUR NAME, TEACHER’S INITIALS AND SET AT THE TOP OF THIS PAGE.
YOU WILL BE HANDING IN THE QUESTION PAPER WITH YOUR SCRIPT.
2015 September | Grade 12 | Paper 1 | Page 2
INFORMATION SHEET: MATHEMATICS Grade 11 and 12 CAPS
b  b2  4ac
x
2a
A  P(1  i)n
A  P(1  i) n
A  P(1  ni)
A  P(1  ni)
Tn  a  (n  1)d
n
Sn  {2a  (n  1)d}
2
a  r n  1
a
; r  1 S 
; 1  r  1
Sn 
r 1
1 r
Tn  ar n1
n
x 1  i   1

F 
i
P
x 1  (1  i)  n 
i
f ( x  h)  f ( x )
h 0
h
f '( x)  lim
d  ( x2  x1 )2  ( y2  y1 )2
 x  x2 y1  y2 
M 1
;
2 
 2
m  tan θ
y  mx  c
y  y1  m( x  x1 )
m
 x  a
2
y2  y1
x2  x1
  y  b  r 2
In ABC:
2
a
b
c


a 2  b2  c 2  2bc.cos A
sin A sin B sin C
1
area  ABC  ab.sin C
2
sin  α  β   sin α.cosβ  cosα.sinβ
cos  α  β   cosα.cosβ  sin α.sinβ
sin  α  β   sin α.cosβ  cosα.sinβ
cos  α  β   cosα.cosβ  sin α.sinβ
cos 2 α  sin 2 α

cos 2α  1  2sin 2 α
 2cos 2 α  1

sin 2α  2sinα.cosα
n
x
 fx
n
σ2 
 x
i 1
P(A or B) = P(A) + P(B) – P(A and B)
i
 x
2
P(A) 
n
b
n(A)
n  S
 ( x  x )( y  y )
(x  x )
2
ŷ  a  bx
2015 September | Grade 12 | Paper 1 | Page 3
QUESTION 1
1.1
Solve for x in each of the following:
1.1.1
 x  2 3x  4  0
(2)
1.1.2
 x  2 3x  4  0
(2)
1.1.3
3x  13  x  1
(5)
3x
1.2
2
x
y1
Solve for x and y given that y  32 and 3  9  243
4
1.3
Given 4  x  3 and 5  y  4 , find
1.3.1 the largest possible value of x
1.4
2
(7)
(1)
1.3.2 the smallest possible value of xy
(2)
1.3.3 the value of y if y 2  25
(1)
In the quadratic equation ax  bx  c  0 , a, b and c are positive real
numbers which form a geometric sequence.
2
State the nature of the roots of the equation.
(4)
[24]
2015 September | Grade 12 | Paper 1 | Page 4
QUESTION 2
2.1
A quadratic sequence is indicated below where each ___ shows where a number
should appear. Some of the 1st and 2nd differences are given.
SEQUENCE:
2
1st DIFFERENCES:
___
___
2nd DIFFERENCES:
2.2
2.3
___
7
___
___
___
4
___
___
___
2.1.1 Redraw the diagram above and fill in all the missing values.
(3)
2.1.2 Determine a formula for the sequence you have given in 2.1.1.
(3)
Hagrid is building with identical cubic blocks. After building a few rows he
has the structure below.
2.2.1 Calculate the number of blocks he will use to build his 15th row.
(2)
2.2.2 How many blocks will he use in total if he continues to 30 rows?
(3)
The geometric series 81  x  y  3  ... is given.
2.3.1 Determine the value of x
(4)
2.3.2 Why is the series convergent?
(1)
2.3.3 Determine S  of the series.
(2)
2015 September | Grade 12 | Paper 1 | Page 5
2.4


Simplify  x  1 x  x  x  x  x  x  x  x  1 without
8
7
6
5
4
3
2
multiplying out the terms. Show all your working.
2
2.5
Determine the value of a for which
  2k  1 
k 0
(4)
1
a
p
(5)
p 1
[27]
QUESTION 3
f  x 
a
 q and g  x   b x  c are sketched below.
x p
3.1
For which value(s) of x is f  x   g  x  ?
(1)
3.2
Write down the equations of the asymptotes of f  x 
(2)
3.3
Determine the equation of f  x 
(3)
3.4
Determine h  x  , the equation of the axis of symmetry of f  x  for which
f  x   h  x  will have two solutions.
(2)
3.5
Determine the coordinates of R, the x  intercept of f  x 
(3)
3.6
Determine the value of c
(1)
[12]
2015 September | Grade 12 | Paper 1 | Page 6
QUESTION 4
Handre Pollard takes a penalty in a world cup rugby match for the Springboks.
2
The ball follows the exact path of a parabola with equation y  a  x  p   q .
The origin is taken to be the point at which the ball is kicked. The ball reaches its
maximum height of 8 metres when it is 15 metres away from where the ball was kicked.
The cross-bar is at a standard height of 3 metres.
4.1
Determine the values of a, p and q
4.2
If a  
4.3
Using your answers from 4.1, what is the maximum distance from which
the ball could have been struck so that the ball would have cleared the
cross-bar?
(4)
8
and the ball is struck at a distance of 24 metres from the goal,
225
determine if the ball will clear the cross-bar.
(3)
(2)
[9]
2015 September | Grade 12 | Paper 1 | Page 7
QUESTION 5
The figure below represents the graphs of f  x  and g  x  .
5.1
Calculate the value of g  3  f  0 
(2)
5.2
State a common factor of f  x  and g  x 
(1)
5.3
State the values of x for which
5.4
State a possible restriction so that g  x  has an inverse that is a function.
(1)
5.5
Determine the equation of g 1  x  in the form y  ... based on your
restriction in 5.4.
(3)
Sketch g 1  x  , using your answers from 5.4 and 5.5.
(2)
5.6
f  x 
x
0 ?
(3)
[12]
2015 September | Grade 12 | Paper 1 | Page 8
QUESTION 6
6.1
6.2
Harry deposits R600 per month at 3% p.a. compounded monthly in order
to save for an overseas trip in four years’ time.
6.1.1 Calculate how much money he will have saved in four years’ time.
(4)
6.1.2 After four years, because of the declining exchange rate, he decides
not to travel, but instead invest the money he has saved at a fixed
interest rate of 6,5% p.a. compounded semi-annually.
How long does it take for his money to double?
(4)
Ronald managed to secure a bond of R350 000 at 9,5% p.a. compounded
monthly over a period of 20 years to buy a house.
6.2.1 What is the amount of his monthly payment?
(4)
6.2.2 What is the total amount that he paid over the 20 year period?
(2)
6.2.3 If he makes his first payment six months after taking out the loan,
how much would he need to pay each month if he still completed his
payments by the agreed date?
(4)
[18]
2015 September | Grade 12 | Paper 1 | Page 9
QUESTION 7
7.1
Given that f  x   2 x  x 2 , determine f   x  from first principles.
(4)
7.2
2
dy
x
2
Determine
if y 
 2
dx
2 x
(3)
7.3
 2x3  4x 
Determine Dx  5

x 

(4)
7.4
If f  x   ax 3  bx 2  cx  5 and the gradient at any point  x; f  x   is given
by 6 x  24 , find the values of a, b and c
2
(4)
[15]
2015 September | Grade 12 | Paper 1 | Page 10
QUESTION 8
The parabola in the figure below represents the curve of f   x  . The parabola is the
derivative of the cubic function f  x   ax  bx  cx  d .
3
2
8.1
Write down the gradient of the tangent to f  x  at the point where x  0
(1)
8.2
Write down the x  coordinates of the turning points of the curve of f  x 
(2)
8.3
For what values of x is f  x  strictly decreasing?
(2)
8.4
Show that x  
b
is the x-coordinate of the point of inflection of f  x 
3a
(3)
[8]
2015 September | Grade 12 | Paper 1 | Page 11
QUESTION 9
A chapel window consists of four equal rectangles and a semi-circle.
The length of the metal that is being used for the frame is 36 metres.
9.1
Prove that the area for the frame is given by
A  24 x  4 x 
2
9.2
 x2
6
Determine the length of the base PQ for a maximum area of the window.
(6)
(5)
[11]
2015 September | Grade 12 | Paper 1 | Page 12
QUESTION 10
The probability that the Springbok Team has all its players fit to play is 70%.
The probability that they will win a game if all their players are fit is 90%.
When they are not fit the probability of them winning becomes 45%.
Draw a tree diagram to illustrate this situation and hence calculate the probability
of them not winning their next game.
[6]
QUESTION 11
The calculator below shows the first few digits of pi.
IGNORE the decimal comma for ALL questions that follow.
11.1 How many different numbers, using ALL the displayed digits, can
be created?
(4)
11.2 What is the probability, using ALL the displayed digits, that a
number created has all the repeated digits next to each other, in
order of size?
(4)
[8]
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