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GANSBAAI ACADEMIA
EXAM P1
MATHEMATICS June 2014
Grade 12 Total: 150
Time: 3 hours

EDUCATOR L. Havenga
MODERATOR A. van Wyk

INSTRUCTIONS
1. This question paper consists of 11 questions. Answer ALL questions.

2. Clearly show ALL calculations, diagrams, graphs, et cetera, which you have used in determining
the answers.

3. An approved scientific calculator (non-programmable and non-graphical) may be used,

unless stated otherwise.

4. If necessary, answers should be rounded off to TWO decimal places, unless stated otherwise.

5. Number your answers correctly according to the numbering system used in this question paper.

6. Diagrams are not necessarily drawn to scale.

7. It is in your own interest to write legibly and to present your work neatly.

PAPER 1
QUESTION 1

1.1 Solve for x correct to two decimal places, if necessary:

1.1.1  x  4 x  3  2 (3)

1.1.2  x 2  2   x  3  x 3  6 x (5)

1.1.3 x 2  2 x  15 (4)

1.2 Solve simultaneously 𝑥 and 𝑦:

x  3 y  5 and xy  y 2  3 (7)

1.3 If f ( x)  4 x and g ( x)  x 2 , determine f ( g (9)) . (3)

1.4 Simplify without using a calculator:

3 3
133002 133002
 (4)
3
133003  3 133000 3
133000

[26]

1
QUESTION 2

1 1
2.1 Consider the series: 4  2  1   ... 
2 128
2.1.1 How many terms are in the series? (3)
2.1.2 Write the series in sigma-notation. (2)
2.1.3 Calculate the value of the series. (2)

2.2 Consider the following sequence: 2; 5; 2; 9; 2; 13; 2; 17; ...
Calculate the sum of the first 100 terms of the sequence. (4)
[11]

QUESTION 3

3. Given the series:

2(3)6  2(3)5  2(3)4  2(3)3  k

3.1 Show that the series will converges. (2)

3.2 Calculate the sum to infinity. (2)

3.3 Calculate the sum of the first 7 terms of the series. (3)

3.4 Use your answers from 3.2 and 3.3 to calculate the following:

 2.(3)
n 8
8 n
(3)

[10]

QUESTION 4

6; 5  x; 6 and 6x are the first four terms of a quadratic sequence.

4.1 Calculate the value of x . (4)

4.2 Hence, determine a formula for the nth term of the sequence. (4)
[8]

2
QUESTION 5

5.1 Calculate the annual compound interest rate that will have the same outcomes over three years
at a simple interest rate of 22% per year. (5)

5.2 Thuso is a young farmer. He has just bought his first tractor for R500 650. Due to inflation, the
value of the tractor depreciates at a rate of 7% p.a. on a reducing balance. Thuso knows that he
will have to replace the tractor in four years’ time. The price of a new tractors appreciates at 9%
per annum.
5.2.1 Calculate the scrap value of his tractor after four years. (2)
5.2.2 Determine the cost of a new tractor in four years’ time. (2)
5.2.3 He plans to trade in the old tractor after four years. In his budget, he makes provision
for R50 000 unforseen expenses that might occur during the transaction. How much
money will he need in the sinking fund in four years’ time? (2)
5.2.4 Thuso immediately starts to pay equal monthly payments into the sinking fund. The
fund earns interest at 9% per annum, calculated monthly. His last payment is made at
the end of the four-year period. How much does he pay every month? (3)
[14]

QUESTION 6

The following equations are represented graphically.

f ( x)  ( x  2)2  9 AND g  x   mx  k

D is the turning point of f ( x ) .

3
6.1 Write down the range of f . (2)

6.2 Calculate the coordinates B, the intercept of f . (3)

6.3 Calculate the values m and k. (2)

6.4 Calculate the distance from E to F, where E and F lies on the symmetry axis of f . (3)

6.5 Determine the equation of the tangent to the curve f and goes through C. (3)

[13]

QUESTION 7
a
Given f ( x)   q . The point A (2, 3) is the intersection asymptote of f .
x p

The graph of f goes through x axis at (1; 0)

D is the y -interception of f .

7.1 Write down the equations of the asymptote of f . (2)

7.2 Determine the equation of f . (4)

7.3 Write down the coordinate of D. (2)

7.4 Write down the equation of g , if g is ‘n straight line that connects A and D. (3)

7.5 Write the coordinates of the other intersections of f and g . (4)

[15]

4
QUESTION 8

 1
The graph h  x   a x is outlined below. A  1;  is a point on the graph h .
 2
𝑦 h

A(-1; ½) Q
𝑥
O

8.1 Explain why the coordinates of Q is Q  0;1 . (2)

8.2 Calculate the value of a. (2)

8.3 Write down the equation of the inverse function, h -1 , in the form of y  .... (2)

8.4 Draw a graph on DIAGRAM SHEET 1 of h -1 . Show on the graph the coordinate of the
two points that lies on this graph. (3)

8.5 Read from your graph the values of x which log 2 x  1 . (2)

[11]

QUESTION 9

Calculate f   x  using the first principle f  x   2 x  1 .

2
9.1 (4)

9.2 Evaluate:
 2 x4 
9.2.1 Dx    (3)
 x 8
dy 3
if x y  7 x  x
2 5
9.2.2 (4)
dx 2
d
9.2.3  f ( x)  g ( x) if f ( x)  3x3  x and g ( x)  x  1 (3)
dx
[14]

5
QUESTION 10

The graph of f  x    x3  ax 2  bx  c is sketched below. The x -intercept are indicated.

10.1 Determine the values of a, b and c. (4)

10.2 Calculate the x -coordinate of A and B, the turning point of f . (5)

10.3 For which values of x be f   x   0 ? (3)

[12]

QUESTION 11

11.1 A feeding scheme conducted a survey about the preferences for mutton, beef and chicken
among 150 learners at a school. The findings are as follows:
 54 eat mutton
 75 eat chicken
 66 eat beef
 6 eat all three types
 30 do not eat any meat at all
 23 eat chicken and beef but not mutton
 16 eat mutton and beef but not chicken
 𝑥 number of learners eat chicken and mutton but not beef

11.1.1 Represent the results in a Venn diagram. (2)

11.1.2 How many learners eat mutton only? (2)
11.1.3 What is the probability that a learner, if randomly selected, would eat only two
types of meat? (2)

6
11.2 In July 2012, the Western Cape government started a campaign against people using a cell
phone while driving a vehicle. Cell phones are confiscated and returned to the owner upon
paying a fine. This action is based on their assumption that speaking on a cell phone is more
likely to contribute to an increase in the number of accidents. The Grade 11 learners of a private
school conducted a survey among 650 drivers in their area to test this assumption. The results
of the survey are captured in the following table.

Never
Involved in
involved in Total
accidents
accidents
Drivers using cell phones while
13 169
driving
Drivers not using cell phones while
444
driving
Total 650

11.2.1 Copy and complete the table. (3)

11.2.2 Use the table to determine the following probabilities:

a) P(drivers never involved in accidents) (1)
b) P(drivers using cell phones while driving or drivers never involved in accidents) (2)
c) P(drivers using cell phones while driving and involved in accidents) (1)

11.2.3 Contrary to the government’s belief, most of the public argue that using cell phones
while driving and being involved in accidents are two independent events. Which
perception is supported by the survey? (3)
[16]

TOTAL: 150

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DIAGRAM SHEET 1

NAME AND SURNAME: ……………………………………………………………………………………..