St Stithians College
HERZLIA SENIOR HIGH SCHOOL
“If you will it, it is no legend”
MARKS: 150
TIME: 3 HOURS
This question paper consists of 12 pages and 1 information sheet.
INSTRUCTIONS AND INFORMATION
Read the following instructions carefully before answering the questions.
1. This question 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 and graphs that you have used in determining your answers.
5. Answers only will NOT necessarily be awarded full marks.
6. If necessary, answers should be rounded off to TWO decimal places, unless stated otherwise.
7. Diagrams are NOT necessarily drawn to scale.
8. You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise.
9. An INFORMATION SHEET, with formulae, is included.
10. Write neatly and legibly.
QUESTION 1
1.1 Solve for x:
1.1.1 [pic] (2)
1.1.2 [pic] (correct to TWO decimal places) (3)
1.1.3 [pic] (6)
1.1.4 [pic] (4)
1.1.5 [pic] (3)
1.2 Solve simultaneously for x and y:
[pic] and [pic] (6)
[24]
QUESTION 2
2.1 Given the geometric sequence: 3 ; 2 ; k ; …
2.1.1 Calculate the value of k. (1)
2.1.2 Calculate the value of n if [pic]. (3)
PLEASE TURN OVER
2.2 The first two terms of an arithmetic series, A, and an infinite geometric
series, B, are the same.
A: –2 + x + … and
B: –2 + x + … are given.
2.2.1 Write down in terms of x the third term of the:
(a) Arithmetic series A (1)
(b) Geometric series B (1)
2.2.2 If the sum of the first three terms in the arithmetic series A is
equal to the third term of the geometric series B, then calculate
the value of x. (4)
2.2.3 If [pic], does the geometric series B converge? Show ALL
your workings to support your answer. (1)
2.3 In a Mathematics competition, the total prize money for the finalists is
R30 500. Each finalist will receive a part of the prize money according
to his/her position at the end of the competition. The table below shows
the position of the finalists at the end of the competition and the prize
money received.
|POSITION OF THE FINALIST AT |PRIZE MONEY |
|THE END OF THE COMPETITION | |
|Last |R100 |
|Second from last |R250 |
|Third from last |R400 |
|Fourth from last |R550 |
|. |. |
|. |. |
|. |. |
|. |. |
|First |Rx |
Calculate x. Show ALL your workings. (5)
2.4 The sum of the first p terms of a sequence of numbers is given by:
[pic]
Calculate the value of [pic]. (2)
[18]
QUESTION 3
Given the quadratic sequence: 0 ; 17 ; 32 ; …
3.1 Determine an expression for the general term, [pic], of the quadratic
sequence. (4)
3.2 Which terms in the quadratic sequence have a value of 56? (3)
3.3 Hence, or otherwise, calculate the value of [pic]. (3)
[10]
QUESTION 4
Given: [pic] and [pic]
4.1 On the same set of axes, sketch the graphs of f and g. Clearly indicate
all intercepts with the axes and turning points. (5)
4.2 Determine the equation of the tangent to f at [pic]. (1)
4.3 Determine the value(s) of k for which [pic] will have two unequal
positive real roots. (2)
4.4 A new graph h is obtained by first reflecting g in the x-axis and then
translating it 7 units to the left. Write down the equation of h in the
form [pic]. (2)
[10]
PLEASE TURN OVER
QUESTION 5
The diagram below shows the hyperbola g defined by [pic] with
asymptotes [pic] and [pic]. The point Q is the point of intersection of the
asymptotes of g. The graph of g intersects the x-axis at T. The line [pic]
intersects the hyperbola in the first quadrant at S.
[pic]
5.1 Write down the values of p and q. (2)
5.2 Determine the coordinates of T. (2)
5.3 Give the range of [pic]. (2)
5.4 Write down the equation of the vertical asymptote of the graph of h,
if [pic]. (1)
5.5 Calculate the length of OS. (5)
5.6 Determine the coordinates of [pic] the point which is symmetrical to T
about the point Q. (2)
[14]
QUESTION 6
In the sketch below, P is the y-intercept of the graph of [pic]. T is the
x-intercept of graph g, the inverse of f. R is the point of intersection of f and g.
O is the origin. Straight lines are drawn through O and R and through P and T.
R lies on PT.
[pic]
6.1 Determine the equation of g (in terms of b) in the form y = … (2)
6.2 Write down the equation of the line passing through O and R. (1)
6.3 Write down the coordinates of point P. (1)
6.4 Determine the equation of the line passing through P and T. (2)
6.5 Calculate the value of b. (4)
[10]
PLEASE TURN OVER
QUESTION 7
7.1 After 7 years of reducing balance depreciation, an asset has a [pic] of its original
value.
Calculate the depreciation interest rate, as a percentage. (3)
7.2 Neil takes a personal loan from a bank to buy a motorcycle that costs R46 000.
The bank charges interest at 24% per annum, compounded monthly.
How many monthly payments will it take Neil to repay the loan, if the monthly instalment is R1 900 and the first instalment is at the end of the first month? (4)
7.3 Julia set up an investment fund. Exactly 3 months later and every 3 months
thereafter she deposited R3 500 into the fund. The fund pays interest at
7,5% per annum, compounded quarterly. She continued to make quarterly
deposits into the fund for [pic] years from the time that she originally set up
the fund.
Julia made no further deposits into the fund, but left the money in the same
fund at the same rate of interest. Calculate how much she will have in the
fund 10 years after she originally set it up. (5)
[12]
QUESTION 8
8.1 Given: [pic]
Determine [pic], using first principles. (5)
8.2 Determine [pic] if:
[pic] (4)
8.3 Find the value(s) of x for which the tangent to [pic] is parallel
to the tangent to the curve [pic]. (3)
8.4 The function [pic] has a point of inflection at [pic].
Calculate the values of b and c. (5)
[17]
QUESTION 9
The sketch below shows the graph of [pic].
The x-intercepts of f are [pic], [pic] and [pic]. A and B are the turning
points of f and D is the y-intercept of f.
[pic]
9.1 Show that [pic]. (1)
9.2 Determine the coordinates of A and B. (5)
9.3 Determine the value of x where the concavity of f changes. (2)
9.4 Determine the coordinates of the point on f with a maximum gradient. (2)
9.5 Determine for which value(s) of x is [pic]. (3)
[13]
PLEASE TURN OVER
QUESTION 10
The figure above shows the design of a theatre stage which is in the shape
of a semicircle attached to a rectangle. The semicircle has a radius r and
the rectangle has a breadth b. The perimeter of the stage is 60 m.
10.1 Determine an expression for b in terms of r. (2)
10.2 Hence, show that the area of the stage in terms of r is
[pic]. (3)
10.3 For which value of r will the area of the stage be a maximum? (3)
[8]
QUESTION 11
11.1 It is given that two events, A and B, are independent.
• P(A) = [pic]
• P(B) = [pic]
Calculate P(A or B). (3)
11.2 Grade 12 learners in a certain town may choose to attend any one of
three high schools. The table below shows the number of Grade 12
learners (as a percentage) attending the different schools in 2016 and
the matric pass rate in that school (as a percentage) in 2016.
|SCHOOLS |NUMBER OF LEARNERS ATTENDING (%) |MATRIC PASS RATE (%) |
|A |20 |35 |
|B |30 |65 |
|C |50 |90 |
If a learner from this town, who was in Grade 12 in 2016, is selected
at random, determine the probability that the learner:
11.2.1 Did not attend School A (1)
11.2.2 Attended School B and failed Grade 12 in 2016 (2)
11.2.3 Passed Grade 12 in 2016 (3)
PLEASE TURN OVER
11.3 A FIVE-digit code is created from the digits 1 ; 2 ; 4 ; 6 ; 8
How many different codes can be created if:
11.3.1 Repetition of digits is NOT allowed in the code (1)
11.3.2 Repetition of digits IS allowed in the code (1)
11.4 A group of friends decide to plan a trip to Europe with the intention
of visiting Rome, Madrid, Florence, Milan, Geneva and Paris. They
choose the order of their visits randomly.
11.4.1 If Rome, Madrid, and Florence are grouped together
in that order, determine the number of different
orders of their visits. (1)
11.4.2 What is the probability that they will visit Rome,
Madrid and Florence one after the other in any
order? (2)
[14]
TOTAL : 150
-----------------------
PRELIMINARY
EXAMINATION
GRADE 12
MATHEMATICS PAPER 1
FRIDAY 1ST SEPTEMBER 2017
Q
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