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Kingsmead College
Grade 12 Preliminary Examination
July 2012
MATHEMATICS
Name: ______
Time allowed: 3 hoursExaminers : D Clogg
Total marks available:150 Moderator: G. Gouws
S. Isack
INSTRUCTIONS
1.Write your name in the space provided above and on the diagram sheet.
2.This paper consists of 8 pages, including the diagram sheet. Please check that this paper is complete.
3.Non–programmable calculators may be used, but you must show your working clearly.
4. Give your answers to 2 decimal place unless directed otherwise.
- Write neatly in pen.
SECTION A
QUESTION 1: [37 marks]
a)Solve for
i) (3)
ii) (5)
iii) (4)
iv) (3)
v) (4)
b)Evaluate, showing all working:
i) (4)
ii)…….. (3)
iii) (3)
c)Simplify:
where (4)
d)If (4)
QUESTION 2: [12 marks]
a)Determine (4)
b)The second term of a geometric sequence is , the fifth term is .
Find the term. (5)
c)For which values of will the geometric series
………. converge?(3)
QUESTION 3: [17 marks]
a)Find by first principles if (4)
b)Find if:
i)(3)
ii)(3)
iii)(4)
c)If the displacement in metres of a particle at time (in seconds), is governed by the equation , find its acceleration after 2 seconds. (Acceleration is the rate of change of velocity and velocity is the rate of change of displacement.) (3)
QUESTION 4: [16 marks]
a)Given
i)Sketch the graph of (2)
ii)Write down the equation of h, the reflection of about the y-axis in simplified form. (2)
iii)Determine the equation of the new graph formed j, if the graph of is shifted1 unit to the right and then 2 units upwards. (1)
b)Given the graph
i)Determine the value of if the graph passes through the point(2)
ii)Write down the equations of the asymptotes. (2)
iii)Write down the equation for the line of symmetry that intersects the above graph. (2)
c)If and , then sketch the graph of (5)
Total: Section A: 82 marks
SECTION B
QUESTION 5: [16 marks]
(a)Sketched (not drawn to scale) is the graph of
i)Find the lengths of OB and FD(2)
ii)Determine the co-ordinates of C(4)
iii)Find the equation of the tangent to the curve at D.(3)
iv)Determine the x co-ordinate of a point on the curve where the tangent is parallel to the line (4)
b) Below is the graph of
Draw a rough sketch of if and . (3)
QUESTION 6: [14 marks]
a)Sam is training for a fun run by running every week for 26 weeks. She runs 3km in the first week and each week after that she runs 2km more than the previous week, until she reaches 25km in a week. She then continues to run 25km each week.
i)How far does Sam run in the 9th week?(1)
ii)In which week does she first run 25km?(2)
iii)What is the total distance that Sam runs in 26 weeks?(3)
b)A rubber ball dropped from a height of 15m loses 20% of its previous height at each rebound. Determine:
i)the height to which the ball will rise after the second rebound.(1)
ii)the number of times it will rise to a height of over 3m(3)
iii)the total distance the ball travels before it comes to rest.(4)
QUESTION 7: [12 marks]
In the gardens of King Pasha Khan there are two classes of gardeners: Cutters and Sweepers. Each Cutter uses his own cutting machine and needs at least two Sweepers to clean up his cuttings. The accommodation in the grounds is sufficient for only 240 gardeners, and there are only 180 cutting machines. Because Sweepers also do other chores, it is necessary to have at least 40 of them.
A Cutter earns 8 pasheks per day and a Sweeper earns 5 pasheks per day.
Let be the number of Cutters and be the number of Sweepers in the gardens.
a)Write down the constraints for the above situation.(4)
b)Represent these constraints on the axes provided in the diagram sheet. Show clearly the feasible region. (4)
c)Write down the equation to determine the King’s cost (C)(1)
d)Determine the maximum amount that King Pasha Kahn could find himself spending. (3)
QUESTION 8: [5 marks]
A farmer wants to build three rectangular camps as shown in the given drawing. He has 600 metres of fencing available.
What will the values of and be to ensure maximum area for the three camps? (5)
Question 9: [12 marks]
A biochemist is testing the effect of a new antibiotic on a mould growing in a dish. Without the antibiotic the mould grows as a circular patch, where the radius increases as time increases according to centimetres and is measured in hours since the mould was introduced to the dish. The area of the mould is given by , the area of a disc of radius When the radius of the disc reaches 2 centimetres, the biochemist introduces the antibiotic. This causes the radius of the disc to reduce according to the formula centimetres and is measured in hours since the antibiotic was introduced.
a)Graph the radius of the disc against elapsed time through the duration of the entire experiment. (From the introduction of the mould until it disappears) (4)
b) What was the time duration of the entire experiment?(1)
c)How fast was the area of the disc increasing before the antibiotic was introduced? (3)
d)How fast was the area of the disc decreasing once the antibiotic was introduced? (4)
QUESTION 10: [9 marks]
This picture shows a tower of cards in a 3 storey triangular pattern.
a)Complete the table:
Number of storeys of the tower / 1 / 2 / 3 / 4Number of cards used / 2 / 7
(1)
b)Following the pattern in the picture, what is the most number of storeys for a tower built from one 52 card pack? (1)
c)Given that a playing card is approximately 9cm long and that we can consider each triangle to be equilateral, determine the height of the triangular tower in question (b) above? (3)
d)How many 52 card packs are needed to build a tower 12 storeys high?(4)
Total: Section B: 68 marks
Total: 150 marks
Diagram SheetName:______
Question 7:
Memo:
1
Question 1:
a)i)
ii)
iii)
iv)
v)
b)i)
ii) …
iii)
c)
d)
Question 2:
a)
b)
c)
Question 3:
a)
b)i)
ii)
iii)
Question 4:
a)i)
ii)
iii)
b)i) sub
ii)
iii)
c)
y-int:
TP:
Question 5:
a)i) units
units
ii)
iii)
iv)
Question 6:
a)
i)
ii)
iii)
b)
i)
ii)
iii)
Question 8:
Question 9:
a)
b)8
c)
d)
Question 10:
a)
1 / 2 / 3 / 42 / 7 / 15 / 26
b) 5 storey tower
c)
d) 2 7 15 26
5 8 11
3 3
Question 7:
a)
…(1)
…(2)
…(3)
…(4)
c)
Max at