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TRIAL EXAMINATION: MATHEMATICS PAPER 1
HILTON COLLEGE
TRIAL EXAMINATION
AUGUST 2017
MATHEMATICS: PAPER I
Time: 3 hours
150 marks
Examiner:
Mrs C. Padayachee
Moderator:
Mr T. Mills
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY
1. This question paper consists of 11 pages and an Information Sheet of 1 page (i).
Please check that your paper is complete.
2. Read the questions carefully.
3. Answer all the questions.
4. Number your answers exactly as the questions are numbered.
5. You may use an approved non-programmable and non-graphical calculator, unless
otherwise stated.
6. Round off your answers to one decimal digit where necessary.
7. All necessary working details must be clearly shown. Answers only will not
necessarily be awarded full marks.
8. It is in your own interest to write legibly and to present your work neatly.
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Page 2 of 11
TRIAL EXAMINATION: MATHEMATICS PAPER 1
SECTION A
QUESTION 1
(a)
Solve for x :
2 x 2 − 3x = 7
(b)
Given p + 5 =
(2)
14
:
p
(1)
Solve for p .
(2)
Hence, solve
(2)
x+5 +5 =
(c)
Sketch the graph of f ( x) =
(d)
f ( x) = x 2 + 5 x − 6
(1)
(2)
(e)
(f)
14
x+5
(3)
−3
+ 1 , showing all intercepts and asymptotes.
x−2
Without solving, describe the roots of f .
Determine x if :
f ( x) = 0
(i)
f ( x) 0
(ii)
f ( x ) 0
(iii)
3
and g ( x) = 3x −2 .
x−2
By means of a sketch explain why f ( x) = g ( x) will have only one root.
(3)
(2)
(2)
(2)
(2)
f ( x) =
(3)
The solution of a quadratic equation is given by:
−2 13 − k
3
Determine the largest integral value of k for which the x values will be
rational.
x=
(3)
[24]
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TRIAL EXAMINATION: MATHEMATICS PAPER 1
QUESTION 2
(a)
Solve for x and y :
x − 3y = 1
(5)
x 2 − 2 x + 9 y 2 = 17
(b)
Solve for x :
8 x −5 = 3210 − 2 x
(3)
[8]
QUESTION 3
Consider: f ( x) = −( x + 1)2 + 4
(a)
Sketch the graph of f , showing all intercepts clearly.
(3)
(b)
Calculate the coordinates of the point on f for which the tangent to f will
have a gradient of 1.
(4)
Write down the values of k for which f ( x ) − k will always be a negative
value.
(2)
Explain why the inverse of f is not a function.
(2)
(c)
(d)
[11]
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Page 4 of 11
TRIAL EXAMINATION: MATHEMATICS PAPER 1
QUESTION 4
(a)
(b)
R100 000 invested at 12% p.a., compounded monthly, grows to R181 669,67.
For how long was the money invested?
(3)
A house is bought for R260 000. The buyer pays R50 000 cash and secures a
loan for the balance to be repaid monthly over a period of 20 years.
(1)
Calculate the loan amount.
(1)
(2)
If interest is calculated at 19,5% p.a., compounded monthly, what will be
the equal monthly payments ?
(2)
(3) What would be the balance of the loan after 10 years, immediately after
th
the 120 payment?
(4)
(4) What would be the monthly payments if the loan were repaid over 10
years?
(3)
[13]
QUESTION 5
(a)
Given f ( x) = 7 x − 2 x 2 determine f ( x ) from first principles.
(b)
Differentiate with respect to x : y =
2 x3 − 1
x
(5)
(4)
[9]
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TRIAL EXAMINATION: MATHEMATICS PAPER 1
Page 5 of 11
QUESTION 6
Given the geometric series:
8 x 2 + 4 x 3 + 2 x 4 + ....
(a)
Determine the nth term of the series.
(2)
(b)
For what value(s) of 𝑥 will the series converge?
(2)
(c)
Calculate S if x = 32 .
(3)
[7]
QUESTION 7
Sketch the graph of f ( x) = ax 2 + bx + c if :
•
a0
•
b0
•
b 2 − 4ac = 0
[3]
TOTAL MARKS SECTION A : 75
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Page 6 of 11
TRIAL EXAMINATION: MATHEMATICS PAPER 1
SECTION B
QUESTION 8
y
Shown alongside are the graphs of
k
where x 0 and
x
g ( x) = log p x .
f ( x) =
f
(3; 2)
The point ( 3; 2 ) lies on f and the
g
point ( 2; −1) lies on g .
x
(2; −1)
(a)
Find the values of p and k .
(2)
(b)
Write the equation of g −1 in the form y = ... .
(2)
(c)
Sketch the graph of g −1 , showing coordinates of one point.
(2)
[6]
QUESTION 9
Cell phones in South Africa have 10 digits.
(a)
(b)
(c)
How many different cell phone numbers are
possible if all numbers must start
with a zero and digits may be repeated?
(2)
A new service provider has come into the country and has decided that their
10 digit cell phone number will begin with a zero, but their numbers will have
4 even digits, followed by 5 odd digits. How many possible numbers can be
created in this way, if digits can be repeated?
(2)
Determine the probability that a service provider randomly offers you the number
0321456897 and digits may not be repeated.
(3)
[7]
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Page 7 of 11
TRIAL EXAMINATION: MATHEMATICS PAPER 1
QUESTION 10
In the diagram the graph of y = f ( x) is given where f ( x) = ax3 + bx 2 + cx
represents a cubic graph. It is shown that f (1) = f (2) = 0 and f (0) = 12 .
y
f
12
1
2
x
(a)
Show that the equation of f ( x) = 6( x − 1)( x − 2) .
(3)
Determine:
(b)
the x values of stationary points of f ( x ) .
(2)
(c)
the values of x for which f ( x ) is increasing.
(2)
(d)
the equation of f ( x ) .
(3)
[10]
QUESTION 11
(a)
The sum to n terms of a sequence is Sn = n(n + 2). Determine T10 .
(b)
The seventh, eighth and thirteenth terms of an arithmetic sequence are
3 x − 1; x and −8x + 20 respectively.
(3)
(1)
Determine the common difference in terms of x .
(2)
(2)
Hence determine the numerical value of x.
(4)
[9]
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Page 8 of 11
TRIAL EXAMINATION: MATHEMATICS PAPER 1
QUESTION 12
−1
Part of the graph of a cubic graph f is shown below with f = 0 . The
2
x intercepts are ( −5; 0) and (3; 0) the coordinates of the turning points are
(2; b) and ( −3; a ).
(a)
y
O
-5
x
(1)
For which values of x is f ( x ) = 0 ?
(1)
(2)
Draw a sketch graph of f ( x ) clearly indicating the x intercepts on
the graph.
(2)
The equation f ( x) = k has only one real root. What are the possible
values of k ?
(2)
For what value(s) of x is the graph concave up?
(2)
(3)
(4)
(b)
3
The equation of a tangent to the curve of f ( x) = ax3 + bx is y − x − 4 = 0 . If
the point of contact is (−1;3) , calculate the values of a and b .
(4)
[11]
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Page 9 of 11
TRIAL EXAMINATION: MATHEMATICS PAPER 1
QUESTION 13
The graph of f −1 is shown below, with turning point ( 3;1) and an x intercept of (1;0 ) .
y
( 3;1)
1
(a)
(b)
x
Find the equation of f , given that the graph of f is a parabola.
(Do not simplify your answer)
(4)
Restrict the domain of f ( x ) so that f −1 is a function.
(2)
[6]
QUESTION 14
The arithmetic sequence given below is the sequence of the first differences of a quadratic
sequence.
3 ;
7 ; 11 ; ...
If the first term of the quadratic sequence is −1 , determine T38 of the quadratic sequence.
[5]
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Page 10 of 11
TRIAL EXAMINATION: MATHEMATICS PAPER 1
QUESTION 15
A toy wooden block is made as shown in the diagram. The ends are right angled triangles
having sides 3x , 4x and 5x . The length of the block is y . The total surface are of the
block is 3600cm 2 .
3x
4x
y
5x
300 − x 2
.
x
Find the value of x for which the block will have a maximum volume.
Show that y =
(a)
(b)
(3)
(4)
[7]
QUESTION 16
Given that A and B are independent events. Calculate the value of x and y if:
•
•
•
•
P ( B only ) = 0,3
P ( A and B ) = 0, 2
P( A only ) = x
P ( A or B) = y .
[4]
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TRIAL EXAMINATION: MATHEMATICS PAPER 1
Page 11 of 11
QUESTION 17
The common difference of an arithmetic sequence is 4. Nick’s first line of working was
.
Line 1:
Sn =
n
( 2a + n − 1(4) )
2
Use his first line of working to prove that:
S 2 n − 2 S n = 2n
[5]
QUESTION 18
A solid iron cylinder with height 26cm and diameter of 12cm has a hole with diameter 2cm
drilled through the centre. Determine the weight of the pipe if iron weighs 8 g / cm3 .
12cm
26 cm
[5]
TOTAL MARKS SECTION B : 75
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