SENIOR CERTIFICATE EXAMINATION
MATHEMATICS P1
STANDARD GRADE
2014
MARKS: 150
TIME: 3 hours
This question paper consists of 7 pages and a formula sheet.
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INSTRUCTIONS AND INFORMATION
Read the following instructions carefully before answering the questions.
1.
This question paper consists of SEVEN questions. Answer ALL the questions.
2.
Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in
determining your answers.
3.
You may use an approved calculator (non-programmable and non-graphical), unless
stated otherwise.
4.
If necessary, answers should be rounded off to TWO decimal places, unless stated
otherwise.
5.
Number the answers correctly according to the numbering system used in this
question paper.
6.
Diagrams are NOT necessarily drawn to scale.
7.
A formula sheet is included at the end of the question paper.
8.
Write neatly and legibly.
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QUESTION 1
1.1
1.2
Solve for x:
1.1.1
x 2  9 x  8
(3)
1.1.2
2 x 2  7 x  4  0 (round off correct to TWO decimal places)
(4)
1.1.3
x  2  5( x  2)
(5)
Solve for x and y if they satisfy the following equations simultaneously:
y  2x  1  0
x 2  y 2  2x  4 y  5
(7)
[19]
QUESTION 2
2.1
For which value(s) of a will the roots of ax 2  6 x  3  0 be equal?
(3)
2.2
Prove that the roots of the equation x 2  (2  k ) x  k  0 are real and unequal for
ALL real values of k.
(5)
2.3
When x 3  3x 2  mx  2 is divided by x  1 , the remainder is 10.
Determine the value of m.
2.4
(3)
Given: f ( x)  x 3  3x 2  4
2.4.1
Factorise f completely.
(5)
2.4.2
Hence, or otherwise, solve for x if f ( x)  0.
(2)
[18]
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QUESTION 3
3.1
The diagram below (not drawn to scale) represents the graphs of:
 A parabola f which cuts the y-axis at  2,
P(1 ; 1) and Q(2 ;  2) , and has turning point R
 A straight line g that passes through Q and R
 A hyperbola h , x  0 , that passes through point P.
passes through points
y
h
f
P(1;1)
x
O
g
2
Q(2 ;  2)
R
3.1.1
Show that the equation of the parabola is y  x 2  2 x  2 .
(6)
3.1.2
Show that R is the point R (1 ;  3).
(3)
3.1.3
Determine the equation of g in the form ax  by  c  0 .
(3)
3.1.4
Write down the equation of the hyperbola h.
(2)
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3.2
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The sketch below shows the graphs of a parabola g , which cuts the x-axis at A and
B and passes through point C(2 ; 10), and a semi-circle f which also cuts the x-axis
at A and B.
y
g
 C(2 ; 10)
3
A(3 ; 0)
f
O
B(3 ; 0)
x
3.2.1
Write down the domain of the semi-circle f.
(2)
3.2.2
Determine the equation of f.
(3)
3.2.3
Derive the equation of the parabola g.
(4)
3.2.4
Write down the range of g.
(2)
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QUESTION 4
6 2 n 3
is independent of n.
2 n 2.18 n1
4.1
Show that
4.2
Simplify the following without using a calculator:
4.2.1
4.3
1
27
log 2 8  log 3
3
4.2.2
(4)
(3)
16  3 54
3
250
(5)
Solve each of the following equations for x:
4.3.1
5 x1  (0,2) x2
(4)
4.3.2
2 x  2 x1  6
(3)
4.3.3
log 2 ( x  6)  log 2 x  4
(5)
4.3.4
3x  6
(3)
[27]
QUESTION 5
5.1
The first term of a sequence is 4. The fourth term is
1
. Determine the numerical
2
value of the third term in each of the following cases:
5.2
5.1.1
The sequence is geometric.
(4)
5.1.2
The sequence is arithmetic.
(4)
The first three terms of an arithmetic sequence are 2 x  1; 3x and 5x  2 .
5.2.1
Determine the value of x.
(3)
5.2.2
Calculate the sum of the first 21 terms of the sequence.
(4)
n
5.3
Solve for n, given that
2
k 1
 4 092 .
(5)
k 1
5.4
A loan of R10 000,00 is taken over 5 years at an interest rate of 10,8% per year,
compounded monthly. How much is repaid altogether?
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QUESTION 6
6.1
6.2
Given: f ( x)   x 2
6.1.1
Determine the average gradient of f between x  1 and x  2 .
(3)
6.1.2
Determine f (x) from first principles.
(5)
Differentiate the following with respect to x:
6.2.1
y  x3 
6.2.2
y  x2  x

1
3x 3
(4)

2
(4)
[16]
QUESTION 7
7.1
7.2
Given: f ( x)  x 3  6 x 2  9 x  4
7.1.1
Determine the intercepts with the axes of the graph of f.
(5)
7.1.2
Determine the coordinates of the turning points of the graph of f.
(5)
7.1.3
Draw a neat sketch graph of f. Clearly indicate the intercepts with the
axes and the coordinates of the turning points.
(4)
4
The sketch below shows the graph of a hyperbola g ( x)  , x  0 and a straight
x
line h( x)  8  x. Point A lies on h and point B is on g.
y
A
g
h
B
O

x
Determine the largest possible vertical distance AB between the graphs of h and g.
TOTAL:
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Mathematics Formula Sheet (HG and SG)
Wiskunde Formuleblad (HG en SG)
x
 b  b2  4ac
2a
Sn 
Tn  a  ( n  1 ) d
Sn 
Tn  ar
S 
n1
n
a  Tn 
2
Sn 
n
2a   n  1d 
2
Sn 

a 1rn
1r


r 

A  P 1 

100 

n
r 

or / of A  P  1 

100 

f ( x  h)  f ( x )
h
h0
f ' ( x )  lim
d  ( x2  x1 )2  ( y2  y1 )2
y  mx  c
y  y1  m( x  x1 )
y1
x1
m  tan 
 x  x2 y1  y2 
( x 3 ; y3 )   1
;

2
2 

x2  y2  r 2
( x  p)2  ( y  q)2  r 2
a
b
c
In ABC:


sin A sin B sinC
a 2  b 2  c 2  2bc .cos A
area Δ ABC  1 ab.sinC
2
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
a rn 1
(r≠1) Sn 
(r≠1)
r 1
a
( r  1)
1r
y 
m 2
x2 
or / of
n
n
a  
2