# MATRIC MATHEMATICS PAPER I: SEPTEMBER 2015Page 1 of 12

MATRIC MATHEMATICS PAPER I: SEPTEMBER 2015Page 1 of 12

**MATHEMATICS PAPER I**

**PRELIMINARY EXAMINATION**

SEPTEMBER 2015

Time: 3 HoursMarks: 150

Reading Time: 10 MINUTESExaminer: A Abatzidis

Moderator: R Bourquin

**READ THE FOLLOWING INSTRUCTIONS CAREFULLY**

1.This question paper consists of 12 pagesand anInformation Sheet.

Please check that your paper is complete.

2.Write your examination number in the blocks provided above and on your Answer Script.

3.Read the questions carefully.

4.Answer ALL the questions.

5.Number your answers as the questions are numbered.

6.All the necessary working details must be clearly shown. Answers only may not necessarily be awarded full marks.

7.Approved non-programmable and non-graphical calculators may be used except where otherwise stated.

8.Give answers correct to ONE decimal digit, where necessary.

9.Diagrams are not drawn to scale. Do not redraw given diagrams.

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

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**PUPIL MARK PUPIL % **

SECTION A

QUESTION 1

(a)Simplify fully:(3)

(b)Solve for x without using a calculator:

(1)(3)

(2)(4)

(c)Given the quadratic equation:

(1)Solve the equation, in terms of p, using the quadratic formula.

Leave your answer in **simplest surd form**.(3)

(2)Hence, state for which value(s) of p the equation will have real roots.(2)

[15]

QUESTION 2

(a)Determine:(2)

(b)Consider the following picture pattern which continues in the same way indefinitely.

Calculate the number of square blocks in the 50th pattern.(5)

(c)The sum to infinity of a geometric series is 5 times the first term.

Calculate the common ratio.(3)

(d)In an arithmetic sequence, the nth term is given as and the sum of the first n terms is .

It is given that:

Determine the value of .(4)

(e)Refer to the figure (not drawn to scale):

The lengths of the sides of a triangle are such

that they form a geometric sequence.

The longest side has a length of 9 cm.

The other two sides are of length a cm and b cm

with .

(1)Show that .(2)

(2)Given that the perimeter of the triangle is 19 cm, determine the length of the

shortest side.(5)

[21]

QUESTION 3

(a)Determine:(3)

(b)Differentiate with respect to x, leaving answers with positive exponents:

(1)(1)

(2)(4)

(3)(3)

(c)Given:

Determine:(4)

[15]

QUESTION 4

Kego invests R5000 into an account earning interest of 9,6% p.a. compounded quarterly.

Calculate how many years and months it will take for the investment to be worth R35000.[5]

QUESTION 5

Refer to the figure (not drawn to scale):

A is a point of intersection of

and .

The parabola cuts the x-axis at (–3 ; 0) and (1 ; 0)

and the y-axis at S.

g cuts the y-axis at R.

(a)Calculate the value of p.(3)

(b)Determine the equation of the parabola in the form .

Show all working.(5)

(c)Calculate the coordinates of Q, the turning point of the parabola, given that the equation

of the parabola is(2)

(d)Determine the length of SR.(3)

(e)(1)Determine the equation of in the form (2)

(2)State the domain and range of .(2)

(f)Determine the equation of which is the reflection of about the y-axis.(2)

[19]

**Section A Total: 75 marks **

SECTION B

QUESTION 6

A school buys a bus for transporting girls to the sports venues.

The bus costs R1,2 million. It is expected that the bus will depreciate on a

reducing balance to be worth R491520 in 4 years’ time.

The price of a new bus is expected to increase by 15% p.a.

(a)(1)Calculate the expected cost of a new bus in 4 years’ time.(2)

(2)Hence, calculate the overall percentage increase in the cost of a new bus over the 4 years.(2)

(b)Calculate the percentage annual rate of depreciation of the bus.(4)

(c)The bus will need to be replaced in 4 years’ time. The old bus will be traded in at the reduced

value of R491520.

Calculate the shortfall that will be required.(1)

(d)Calculate the amount that needs to be invested monthly into a sinking fund to cover the shortfall

that is expected in 4 years’ time. The account earns interest at 9% p.a. compounded monthly and payments will be made at the end of each month. (4)

[13]

QUESTION 7

(a)Suppose the boarders’ Sunday is structured as shown in the table.

Morning / Afternoon / EveningShopping / Play netball / Play Scrabble

Movies / IT time / Watch TV

Study / Paint the courtyard / Read

Watch TV / Study

Study

Each activity in the various categories is allocated at random.

Giving your answers as **simplified fractions**, determine the probability that a girl:

(1)studies all day.(2)

(2)watches TV all day.(1)

(3)goes shopping in the morning and plays Scrabble in the evening.(2)

(4)either goes to the movies in the morning or reads in the evening.(3)

(b)Determine the probability that a point selected at random within the large semi-circle is also

within one of the equal sized small semi-circles.(4)

[12]

QUESTION 8

Given:

(a)(1)Determine the equation of the tangent to the graph of f at .(4)

(2)Determine the x-co-ordinate where this tangent cuts the graph of f again.(3)

(b)Refer to the figure:

Consider the function above to answer the questions:

For which value(s) of x is:

(1)(2)

(2)(2)

(3)(2)

(c)Refer to the figure:

The graph of the derivative of , i.e. , is shown.

has the following characteristics:

The turning point is ( 2 ; 1 ).

( 1 ; 0 ) and ( 3 ; 0 ) and ( 0 ; – 3 ) are points on .

Use the graph to answer the following questions:

(1)Determine the x-co-ordinate(s) of the stationary point(s) of the original function, g.(2)

(2)Determine the x-co-ordinate(s) of the point(s) of inflection of the original function g.(2)

(3)Determine the values of x for which the gradient of g is negative.(2)

(4)Draw a rough sketch illustrating the shape of the original function g.(2)

(no co-ordinates are required)

[21]

QUESTION 9

The average mass of a baby in the first 30 days of life is given by the equation:

where t is the time (in days) and m is the mass (in kilograms).

(a)Write down the average mass of a baby at birth, according to this model.(1)

(b)For a short period of time after birth, it is usual for a baby to lose mass.

Determine when the baby’s mass is likely to reach a minimum.(5)

(c)Determine after how many days it takes for the baby’s mass to again be the same as at birth.(4)

[10]

QUESTION 10

(a)Refer to the figure (not drawn to scale):

The length of a rectangular prism is eight times its height. The width is four times the height.

The length of AE is 36 cm.

Find the volume of the prism.(7)

(b)Two whole numbers are squared and the squares added.

The same numbers are then added and their sum is squared.

The latter results in a value which is 72 more than the previous number.

Find **all possibilities** for the original two numbers.(5)

[12]

QUESTION 11

Consider sequences that begin with any two positive integers where each number thereafter

is the sum of the preceding two.

Famous examples of these are:

The Fibonacci sequence:1 ; 1 ; 2 ; 3 ; 5 ; 8 ; 13 ; 21 …

The Lucas sequence:1 ; 3 ; 4 ; 7 ; 11 ; 18 …

(a)For the Fibonacci sequence:F1 = 1, F2 = 1, F3 = 2.

Write down F9.(1)

(b)For the Lucas sequence:L1 = 1, L2 = 3, L3 = 4.

Write down L7.(1)

(c)The “Golden Ratio” is a famous irrational number that can be calculated by:

Find the value of correct to 5 decimal digits.(1)

(d)The nth term of a Fibonacci sequence can be calculated by:

and the nth term of a Lucas sequence can be calculated by:

(1)WITHOUT USING A CALCULATOR, simplify .(2)

(2)Hence, determine, in terms of n, which Fibonacci number will generate.(2)

[7]

**Section B Total: 75 marks **

**TOTAL: 150 MARKS**

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