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MathematicsTrials Paper 1
FORM 529 August 2019
TIME: 3 hours TOTAL: 150 marks
Examiner: Miss Eastes Moderated: Mrs. Gunning
NAME:
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY BEFORE ANSWERING THE QUESTIONS.
This question paper consists of 12 questions plus an information sheet.
Answer all questions on this question paper.
Read and answer all questions carefully. Write legibly and present your work neatly.
All necessary working which you have used in determining your answers must be clearly shown.
Approved non-programmable calculators may be used except where otherwise stated. Where
necessary give answers correct to 1 decimal place unless otherwise stated.
Ensure that your calculator is in DEGREE mode.
Diagrams have not necessarily been drawn to scale .
Give reasons for all Euclidean Geometry questions unless otherwise stated.
Question 1 2 3 4 5 6 7 8 9 10 11 12
Mark
Out of 18 16 14 9 10 10 24 14 10 9 10 6
Total (150) Percentage
2
SECTION A
Question 1:
a) Solve for x:
(1) x−6+ 2x=0 ; x≠0 Show all working. (4)
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(2) √2x−1−x+2=0 (5)___________________________________________________________________________
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3
(3) x2−3 x−4<0 (3)___________________________________________________________________________
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b) Solve simultaneously for x and y:3y−81x=0 and y=x2−6x+9 (6)
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[18]
4
Question 2:
a) Given the following Venn Diagram:
What is the value of x if A and B are independent? (5)
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b) 1) How many unique arrangements can you make with the word:
ASHTONIAN (Leave your answer in factorial form) (3)
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2) How many unique arrangements can you make with the word:
ASHTONIAN if the arrangement is to start with two N’s or end with two N’s?
(Leave your answer in factorial form) (3)
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5
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c) Alex is training for a triathlon. Some days she chooses to go surfing after training.
The table below shows the various probabilities for the different training events
on any given day.
Event Probability
Running (R) 0,65
Cycling (C) 0,3
Swimming (S) 0,05
After running, the probability that she will go surfing is 0,4. The probability
that she will go surfing after a swim is 0,5 and after cycling, the probability
of not surfing is 0,8. The tree diagram is shown.
Complete the tree diagram above and then determine the probability that she goes surfing on any particular day.
[ Correct to 3 decimal places] (5)
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6
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[16]
Question 3:
a) Belinda buys a BMW car for R389 000.
1) Using the reducing balance method, the dealership estimates that the value of the car will be R159 000 in 4 years time. Calculate the annual rate of depreciation over the 4 years. (3)
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2) Belinda sets up a sinking fund (savings account) on the day she buys the car to replace the car in 4 years time. The inflation rate is expected to be 11% p.a. Belinda also plans to sell her current car for R159 000 in 4 years time and use that money towards buying the new car. Calculate how much money is needed in the sinking fund. (3)
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3) Calculate the monthly payments needed, if the sinking fund needs to cover at least R432 000. The sinking fund earns interest at 9% p.a. compounded monthly. All payments are made at the beginning of the month, starting immediately and the last payment is made at the end of the 4th year.
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(4)b) Belinda buys a property for R1 500 000. After paying a deposit, she takes a
loan for the remaining amount of R1 275 000, at an interest rate of 9,2% p.a. compounded monthly. The monthly installment on the loan is R11 636,02. How many years will it take her to pay the loan off?
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(4)
8
[14]
Question 4:
Given: f ( x )= 1x+2
+ 12
a) Determine the equations of the asymptotes. (2)
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b) Give the domain of f . (2)
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c) If the graph of f is symmetrical about the line y=−x+c,determine the value of c. (2)
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d) Determine, if any, the x value of the point(s) of intersection of the graph f (x) and the line y=3. (3)
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[9]
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Question 5:
Given that f ( x )=ax with a>0, and point Q (−2; 9 ) lies on the graph of f .
a) Show that the value of a=13. (2)
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b) Write down the equation of f−1, the inverse graph of f .Leave your answer in the form y=… (2)
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c) Draw a rough sketch of f and f−1. (4)
d) For which values of x is f−1(x)<1. (1)___________________________________________________________________________
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e) Determine the range of the graph of f ( x )−2. (1)___________________________________________________________________________
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[10]Question 6:
a) Given f ( x )=4 x−x2, determine f '( x) from First Principles. (5)___________________________________________________________________________
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b) Determine : [Leave answer with positive exponents.]
dydx
if y=( 2x−√ x)
2
(5)
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[10]
SECTION A = 77 MarksSECTION B
Question 7:
a) The sum of the first five terms of an arithmetic series is zero.The 5th term is 6. Determine the first term and the common difference. (5)
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b) Determine n without using the calculator. Show all working.
∑k=0
n
2 (3 ) k=3100−1
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c) If Sn=3n2+2n, then determine T 15 (4)___________________________________________________________________________
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d) The first, third and thirteenth terms of an arithmetic sequence are the first 3 terms of a geometric sequence. If the first term of both sequences is 1, determine the first three terms of the geometric sequence if r>1. (6)
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13
e) The length of one side of an equilateral triangle is 28cm.The midpoints of the sides of this equilateral triangle are joined to form another equilateral triangle. The process continues indefinitely, hence determine the total sum of the perimeters of the triangles.
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[24]
14
Question 8:
In the diagram below: ( Diagram not drawn to scale.)
f ( x )=2x2+7 x+3
g ( x )=2− x+2
Point A is the turning point of f . Points B and C are the x intercepts of f .Point D is the y intercept of f and g. Points A, B and C form a triangle ABC.
a) Determine the coordinates of A, B and C. (4)_________________________________________________________________________
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b) Determine the area of ∆ BAC. (2)_________________________________________________________________________
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c) Determine the values of x for which f ( x ) . g (x)≤0. (2)___________________________________________________________________________
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d) Use any method of your choice to determine the nature of roots of: p ( x )=−f ( x )−2. (4)
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e) Determine graphically the value(s) of k if 2 x2+7 x+3=khas non-real roots. (2)
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[14]
16
Question 9:
a) The equation f ( x )=−x3+10 x2−17 x−28 is sketched below.
Points on the curve A, B, E and F(5 ; 12) are shown, where A and B
are x-intercepts and E the turning point.
1) Determine the coordinates of point E. (4)
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2) Determine the equation of the tangent to the curve at point F. (4)
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3) If the equation was changed to
y=−( x−3 )3+10 ( x−3 )2−17 ( x−3 )−28, what would the
new coordinates of point F be? (2)
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[10]
18
Question 10:
The graph represents the function y=f ' (x), where f ( x )=ax2+bx+c
a) Write down the x-coordinates of the turning point of f (1)
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b) For what values of x will f increase. (1)
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c) State, with a reason, whether the turning point is a maximum
or minimum. (2)
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d) Determine the equation of f if f (0 )=5. (5)
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Question 11:
An open-top box is to have a square base and a volume of 10 m3.
The cost per m2 of material is R5 for the bottom and R2 for each of the sides of the box.
Let x and y be lengths of the boxes’ width and height respectively.
Let C be the total cost of the material required to make the box.
a) Express the volume of the box in terms of x and y. (1)
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b) Express the total cost of the box in term of x. (3)
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c) What is the minimum cost to make the box? (6)
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x
x
y
20
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Question 12:
The functions f ( x )=2x2+3 px−3 and g ( x )=2 x2+( p−2 ) x−1 have a common factor
of ( x−r ) .
Prove that: r=1p+1 [6]
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SECTION B = 73 Marks
