Question 1
Suppose A = {x E JR : -2 ::; x < 5} and B = (-l, 6].
Draw A and B on two separate number lines. (6)
Find A U B and give the answer in interval notation. (7)
Write A n B as one set using set builder notation. (4)
(17]

Question 2
Calculate

5 3 + 2 – 2 -4 + 9

2 4
without using a calculator. Write the answer:

(a) as an improper fraction (6)
(b) as a mixed fraction (1)
(c) as a decimal fraction correct to two decimal places. (1)
(8]
Question 3

(a) A certain bank offers an interest rate of l2, 5% on a one-year fixed deposit and the interest is compounded at the end of the year. Suppose you invest p rands for one year, and at the end of the year the investment is worth R9 000. Calculate p. (5)
(b) Find the time (in years) that it will take an initial investment of R2250 to double in value at an interest rate of 8,75% per annum, if the interest is compounded quarterly. (Give the answer correct to two decimal places by using a calculator.) (8)
(c) In the year 2000 the population of the world was 6,1 billion. The doubling time of the world population is 20 years. In which year will the world population reach 100 billion if it continues to grow at the same rate? (You will need to use a calculator.) (9)
(22]

21

Question 4

Solve the following inequality for x:

2×2 + 5x < 3. (6)

Without solving the equation 2x2 + 5x = 3, how would you know that it has two rational

solutions? (4)

(10]

Question 5
Use percentages to find out in which assignment Tebogo performed the best. She obtained
64 for Assignment 1,
52 for Assignment 2, and
36 for Assignment 3. (5]

Question 6
Find the solution set of

Question 7
Find the solution set of

Question 8

Solve for x:

J(x + l)2 + 3x = 0. (8]

32(x+l) – 8.3x+l = 9. (6]

log (x + 3) + log x = l. (5)

Show that the above equation will have an irrational root if the base is changed to 2, and a rational root if the base is changed to 4. (6)

(11]

Question 9
The sizes of the angles of a triangle is in the proportion 1:2:3. Find the size of each angle. (4]

TOTAL: (100]

Question 1
Suppose the functions f, g, h, r and £ are defined as follows:

1 1

f (x) –

3 log3 4 + log3 x

g (x) – ✓(x – 3)2
h (x) – 5x – 2×2
r (x) – 23x+l – 2x-2 1
£ (x) – /x

Write down Df , the domain of f and then solve the equation f (x) - log 1 /3 x. (6)

3

x
Write down Dg and then solve the equation g (x) – 2 . (6)
Write down Dh and then solve the inequality 2 � h (x) . (6)
Write down Dr and then solve the equation r (x) – 0. (6)

Write down Dr f_ without first calculating (r • £) (x) .    (2)


Write down D f

g

without first calculating f_ (x) . (4)

(30]

Question 2

The sketch shows the graph of a function f, which is a straight line defined by y – mx + k such that the point V lies on the line, and the graph of a function g, which is a parabola with vertex R. The straight line and parabola intersect at P and Q. The points S and T lie on the parabola and straight line, respectively, between P and Q. The line ST is parallel to the y-axis.

Calculate m and k, and hence write down the equation for f. (3)
Find the equation of the parabola.  (4)
Find the maximum length of ST. At what value of x will ST be at its longest?    (5)
What is the equation of the circle with centre P and radius equal to the distance from P to the origin?

(4)
What is the equation of the hyperbola through the point V ? (1)
Use the graphs to solve the inequality (f • g) (x) � 0. (3)
(20]

TOTAL: (50]

Always keep your rough detailed working so that you can compare your solutions with those that will be sent to you. Also keep a copy of your answers/options. You may need it later.
Show all the steps of your working and give your reasons clearly. Give a proper conclusion to your answers where applicable.

Question 1
Write the repeating decimal number 3,1• 4• 2•

as an improper fraction. [5]

Question 2
A father is four times as old as his daughter. In 6 years he will be three times as old as she is then. How old is the daughter now? (6)
The distance between two towns A and B, is 90 km. John cycles from A to B whereas Harry cycles from B to A. They start cycling towards each other at the same time. John cycles twice as fast as Harry does. If they meet 2 hours later, at what average speed is each cyclist traveling? (8)
The area covered by water weed on a dam increases exponentially according to the formula
A – AOekt,

where A is the area, in square metres, covered by weed after t days. The initial area AO of water weed is 200m2. After 10 days the area is 300m2.
(a) Determine the value of k. Leave your answer in terms of ln. (4)
(b) Use the formula above, and the value of k in terms of ln, to determine the area covered by water weed after 30 days. (5)

[23]

Question 3
Suppose the functions f, g, h and l are defined as follows:
f (x) – 4×2 – 5x + 1

g (x) – 2 2
1

x

• 2 – x

h (x) – -2 x + 3
l (x) – log4 (x + 3) – log4 (x – 2) .

Write down DJ and solve the inequality f (x) ::; 0. (6)
Write down Dg, Dh and Dg+h. (5)
Solve the equation g (x) - -4.  (7)
Solve the equation  4h(x) - 8.  (4)
Write down Dl and solve the equation l (x) - 1 .    (7)

[29]

Question 4

x

The sketch shows a circle, a parabola, which is the graph of f, and a straight line, which is the graph of g. The parabola has x-intercepts 2 and 6, and y-intercept 6. Its turning point is C. The circle has its centre at the origin and it passes through the point A, which has coordinates ( 2, 0) . At point B both the circle and the straight line cut the x-axis. The straight line has y-intercept 1.

What are the coordinates of B?  (1)
What is the equation of g?  (5)
Write down the equation of the line that is perpendicular to the graph of g and passes through

B. (4)
Find the equation that defines f. (5)
Show that C has coordinates (2; 8) . (1)
(a) What is the distance between A and C? (1)

(b) Find the midpoint D of AC. (1)
(c) Write down the equation of the circle that has AC as diameter. (4)
Find the maximum vertical distance between the graphs of f and g on the interval x E [0; 6] .
Hint: First find an expression which defines this vertical distance. (5)
Use the graphs of f and g (not calculations) to solve

f (x) • g (x) 2″ 0 (3)
[30]

Question 5
A glass tank is 25cm long, 20cm wide, 30cm high and contains water. The surface of the water is 5 cm below the top of the tank. After a solid metal spherical ball B1 has carefully been placed into the tank, the surface of the water is 3cm below the top of the tank.

5 cm 3 cm

30 cm

25 cm

20 cm

(The formula for the volume of a sphere (ball) is 4 ?Tr3.)

Calculate the volume of the ball.

Hint: The volume of the ball is equal to the volume of water displaced by the ball. (4)
Calculate the radius of the ball. Leave your answer in terms of ?T and a surd if necessary. (4)
Suppose the radius of a second solid metal ball, B2, is half the radius of ball B1. Suppose ball B2 was put into the tank of water instead of ball B1. Would the surface of the water be 4cm below the top of the tank? Explain your answer. (5)

Answers to Above Questions by Maths Expert

Expert Answer 1: In order to separate A and B into two separate lines, it is important to understand the sets first.

Set A:

Suppose A = {x E JR : -2 ::; x < 5}

Here it means the real numbers should be between 2 and 5

and B = (-l, 6].

Here it means real numbers should be greater than 1 but it should include 6. This can be represented through number line as follows:

—◯=================◯—
2 5

The open cricles in the above line means these values of 2 and 5 are not included.

—◯=================●—
1 6

The open circle in the able line is not included while the dotted circle is included.

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