SECTION A
[81 MARKS]
QUESTION 1
a)
[18]
Solve for x :
2
b)
1
(1)
( x 3  4)( x 3  2)  0
(2)
(2)
x2  1
 0
x 1
(1)
(3)
2 2 x  1  3 (2 2 x  1 )  4 x  12
(4)
(4)
x 2  4x  1 
2
x  4x  2
(5)
2
There are four solutions to the equation ( x 2  5)( 2 x 2  3x  9)  0
Solve for x if:
c)
1)
x
(2)
2)
x  Q│
(2)
The roots of a quadratic equation are 3 
12  3a 2 . Determine the value(s) of a
for which the roots will be equal.
(2)
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QUESTION 2
a)
Consider the number sequence 2; 5; 2; 9; 2; 13; 2; 17; …
(1)
(2)
b)
c)
[16]
Write down the next two terms of the sequence, given that the pattern
continues.
(1)
Calculate the sum of the first 100 terms of the sequence.
(4)
The seventh term of a geometric sequence is
3 645
135
and the fourth term is
,
64
8
determine the first term.
(4)
The sum of 5 + 15 + 45 + … to n terms is 605. Determine the value of n .
(3)
Grade 12 Mathematics – Paper 1
d)
21 July 2014
Page 2 of 8
Show that:
∞
3
2𝑘 − 3
∑(𝑘 − )𝑛 =
2
5 − 2𝑘
𝑛=1
(4)
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QUESTION 3
a)
[27]
f ( x)  
Given:
4
1
x
y
A
B
O
x
4
g
f
(1)
Give the equation of the asymptote through point B.
(2)
Determine the coordinates of point A, the point where the straight line
g ( x)   x  4 intercepts the hyperbola.
(3)
(1)
(4)
Calculate the length of AB when B is the reflection of point A in the line y   x .
Leave the answer in surd form.
(3)
(4)
Give the equation of h if h is the translation of f by two units to the left.
(1)
(5)
Hence give an equation for any of the axis of symmetry for h .
(1)
(6)
Determine the values of x for which f ( x)  g ( x)
(2)
Grade 12 Mathematics – Paper 1
b)
21 July 2014
Page 3 of 8
The sketch shows f ( x)  ax 2  bx  c and g ( x)  k x . Both pass through the point
(1; 3).
y
f
8
(1; 3)
g
O
c)
x
4
2
(1)
Determine the values of a, b and c .
(4)
(2)
Determine the value of k .
(1)
(3)
Determine the range of
(2)
(4)
Consider the graph of
f (x ) .
f (x ) .
(i)
Explain why f (x) does not have an inverse.
(1)
(ii)
What are the values of x for which f (x) will have an inverse?
(1)
The graph of f ( x)  log 1 x and h ( x)   3 are given below:
3
y
f
O
A
x
h
B
Grade 12 Mathematics – Paper 1
(1)
(2)
21 July 2014
Page 4 of 8
Determine the coordinates of:
(i)
A
(1)
(ii)
B
(1)
Use the graph to solve for x if log 1 x   3
(2)
3
1
(3)
Determine the equation of f
( x) .
(4)
Determine the new equation of f (x) , if f (x) is reflected about the x-axis.
(1)
(1)
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QUESTION 4
a)
Given:
b)
Find
(1)
[10]
f ( x)  x 2  3x , find f │ (x ) from first principles.
dy
in each of the following:
dx
4
y  2 x  x3  
9
(4)
(2)
1
(4)
2
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(2)
8x 3  2 xy  y  1  0 ; x 
QUESTION 5
[10]
A quadratic pattern has a second term equal to 1, a third term equal to -6 and a fifth term equal to 14. Calculate the second difference of this pattern and hence calculate the first term.
(10)
Grade 12 Mathematics – Paper 1
SECTION B
21 July 2014
Page 5 of 8
[73 MARKS]
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QUESTION 6
[14]
A new car currently costs R250 000.
a)
The value of the vehicle depreciates at 18 % p.a. on a reducing balance basis.
In which year will the car be valued at 30 % of its original value?
(5)
b)
What will a new car cost in 5 years time if inflation is calculated at 8 % p.a.?
(2)
c)
A man wants to start saving so that he can afford to buy a car in 5 years’ time.
He opens a savings account with an interest rate of 9,5 % p.a. compounded monthly.
What must his monthly payments into the savings account be, so that he can afford
to buy the car in 5 years time?
d)
(4)
Calculate the effective annual interest rate as a percentage and correct to 2 decimal
places for the interest rate in 7 c).
(3)
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Grade 12 Mathematics – Paper 1
QUESTION 7
21 July 2014
Page 6 of 8
[19]
The shown below is defined by the equation: f ( x)  x 3  ax 2  bx  12. The point
S (1; 0) is an x - intercept and T (4; ‒36) and Q are stationary points.
Q
R
P
S
U
f
T (4; ‒ 36)
a)
Show that a = ‒5 and b = ‒8.
(7)
b)
Find the co-ordinates of Q.
(4)
c)
Find the co-ordinates of P and U.
(2)
d)
Determine the equation of a tangent to f (x) at point P.
(3)
e)
Use the graph to find x if f (x) . f │ (x ) ≤ 0
(3)
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QUESTION 8
[8]
Two particles are projected simultaneously towards each other from the opposite ends of a
straight tube, 1 000 cm long. Particle A travels 51 cm in the first second, 49 cm in the second
second, 47 cm in the third second, etc. Particle B travels 25 cm in the first second, 24 cm in the
second second, 23 cm in the third second, etc.
1 000 cm
a)
How far does particle A travel in t seconds?
(3)
b)
Determine how long it takes the particles to meet.
(5)
Grade 12 Mathematics – Paper 1
21 July 2014
Page 7 of 8
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QUESTION 9
[14]
a) Given 𝐴 =
i)
ii)
3𝑛 −4
6𝑛 − 2𝑛+2
Simplify A
(3)
𝑛
Hence determine √𝐴
(1)
b) Refer to the figure below, where a rectangle has sides; √5 − 1 & √5 + 1 ; calculate the
length of the diagonal
√5 − 1
√5 + 1
(3)
c) In January 1999, by working out the value of 23021377 − 1, a computer took 46 hours to find
the largest prime number. It contained 909526 digits. Suppose you were to write out this
digit (time in seconds), with each set of 10 digits taking up 5cm of space
i)
Calculate the time it would take to write out the number showing that it would take
more than a week
(3)
ii)
Find the length of the number, write the final answer in kilometres
(2)
[14]
QUESTION 10
[7]
A soccer ball is kicked in the air. Its height h in metres,
t seconds after it has been kicked, is given by the
formula h  20(t 2 
1 3
t ).
3
a)
What is the velocity of the ball after 1,5 seconds?
(2)
b)
For what value of t is the velocity a maximum?
(2)
c)
For what value of t is the height of the ball a maximum? Calculate the
greatest height from the ground.
(3)
Grade 12 Mathematics – Paper 1
21 July 2014
Page 8 of 8
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QUESTION 11
Given:
f ( x) 
[7]
1
. The tangent at D where x  a is drawn.
x
B
D (a ; f (a))
f
O
a
C
a)
Show that the equation of the tangent at D is x  a 2 y  2a .
(4)
b)
Calculate the area of  OBC.
(3)
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