MATHEMATICS PAPER I

ANTE-MATRICNOVEMBER EXAMINATIONS 2016

Marks: 150 Time: 3 hours Examiner: R. KaramReading Time: 10 Min Moderator: M. Brown ___________________________________________________________________________________PLEASE READ THESE INSTRUCTIONS CAREFULLY:

The Habits of Mind that you should be making use of in this examination are:Striving for accuracy, Applying past knowledge to new situations, Thinking and communicating with clarity and precision, Questioning and posing problems.

1. This question paper consists of 12 pages, an Answer Sheet (2 pages) and an Information Sheet. Please check that your paper is complete.

2. Read the questions carefully.

3. Answer ALL the questions. Answer Question 11 on the Answer Sheet provided and ensure that your name is written on it. Insert the Answer Sheet at the back of your Answer Script.

4. Number your answers as the questions are numbered.

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

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

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

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

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

ANTE-MATRIC MATHEMATICS PAPER I: NOVEMBER 2016 Page 2 of 12

SECTION A

QUESTION 1

(a) Solve for x: Show all working steps.

(1) (2x+1 )2−4=0 (3)

(2) −x+5−2 x2=0 (rounded to 1 decimal digit) (3)

(3) x=√2 x−1 +2 (4)

(4) ( x+2)(2 x−3)≤−3 (4)

(5) −4 x2−12 x+8=0 (by completion of the square) (5)

(6) 32 x+2+1 = 10 .3x(5)

(b) (1) Solve for x and y simultaneously:

x+ y = 2 and y + 1 = −3

x−1 (6)

(2) Hence, or otherwise, determine the value of: ( 1

x+ 1

y )(2)

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ANTE-MATRIC MATHEMATICS PAPER I: NOVEMBER 2016 Page 3 of 12

QUESTION 2

Given: the graph of f ( x )=5

x

Determine the equation of the new graph formed if the graph of f ( x )=5

x is:

(a) shifted 4 units upward. (1)

(b) shifted 2 units to the left and then 3 units down. (1)

(c) reflected about the y-axis. (1)

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QUESTION 3

(a) After how many years will the value of an item be zero, if straight line

depreciation is 12,5% p.a.? (3)

(b) R10 000 is invested for a period of 5 years.

For the first 18 months, the interest rate is 8,5% p.a. compounded monthly.

The rate then changes to 9,25 % p.a. compounded quarterly, for a further 24 months,

after which it changes to 9,1% p.a. compounded monthly, for the remaining period.

(1) Calculate the balance at the end of 5 years. (4)

(2) Calculate the effective interest rate over the five year period. (3)

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ANTE-MATRIC MATHEMATICS PAPER I: NOVEMBER 2016 Page 4 of 12

QUESTION 4

(a) Without using a calculator, evaluate each of the following:

(1) (3)

(2) (3)

(b) Simplify each of the following expressions, assume non-zero denominators:

(1) (4)

(2) (4)

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ANTE-MATRIC MATHEMATICS PAPER I: NOVEMBER 2016 Page 5 of 12

QUESTION 5

A sequence is formed from the first number in each of the rows.

1

2 3 4

5 6 7 8 9

10 11 12 13 14 15 16

: : : : : : : : :

Determine:

(a) an expression for to determine the first term in the nth row. (5)

(b) the first number in the 50th row. (2)

(c) in which row the last term will be 121. (2)

(d) how many numbers will appear in the 112th row. (3)

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SECTION A TOTAL: 71 MARKS

ANTE-MATRIC MATHEMATICS PAPER I: NOVEMBER 2016 Page 6 of 12

SECTION B

QUESTION 6

The sketch graph below shows the curve of f ( x ) = ax2+ bx+ c which passes through the

points A (–3 ; 0 ), B ( 0 ; 3 ) and C ( 1 ; 0 ).

Determine, showing all calculations:

(a) the equation of the graph, and show that it is f ( x ) = −x2− 2 x + 3 (4)

(b) the equation of the straight line through D, (the turning point of f ), which is

perpendicular to the straight line through A and D. (6)

(c) the value(s) of x for which the function f is decreasing. (1)

ANTE-MATRIC MATHEMATICS PAPER I: NOVEMBER 2016 Page 7 of 12

(d) the range of the f. (1)

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QUESTION 7

(a) Given: 2 x2+q=0 , where q∈ {−2 ; −1 ; 0 ; 1 ; 2}

Determine the value(s) of q:

(1) given that one root is equal to –1. (2)

(2) such that the roots are:

(i) rational and unequal. (3)

(ii) irrational. (2)

(b) Given p is one of the roots of x+4 p=20

x , determine the possible value(s) of p. (3)

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ANTE-MATRIC MATHEMATICS PAPER I: NOVEMBER 2016 Page 8 of 12

QUESTION 8

Below is a contingency table of fashion models on an agency’s books.

Tall Short Total

Blond 40 10 50

Not Blond 20 5 B

Total 60 A 75

(a) Determine the values of A and B. (2)

(b) Given that every model has the same chance of being chosen:

(1) find the probability of choosing a short, non-blond model. (1)

(2) find the probability of a blond being chosen, if only the tall models are

allowed to be selected. (1)

(c) Determine whether tall and blond are independent, using appropriate calculations.

Show all your working. (4)

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Hair Colour

Height

ANTE-MATRIC MATHEMATICS PAPER I: NOVEMBER 2016 Page 9 of 12

QUESTION 9

At a fun-fair, 45 children have the choice of riding the merry-go-round (M) and/or

the roller-coaster (R).

35 want to ride the roller-coaster

15 want to ride the merry-go-round

10 want to ride both

(a) Illustrate the given information in a Venn Diagram (4)

(b) For a child chosen at random from the group, determine:

(1) P( only M ) (2)

(2) P( M' ) (2)

(3) P( M'

and R ) (2)

(4) P( M'

or R' ) (2)

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ANTE-MATRIC MATHEMATICS PAPER I: NOVEMBER 2016 Page 10 of 12

QUESTION 10

(a) Each year the

frangipani tree

grows two new

branches on each

current branch.

The diagram

shows that

the tree will have four new branches in the third year.

Determine:

(1) the expression for . (2)

(2) the number of new branches which will grow in the seventh year. (2)

(b) For which values of x is the expression defined? (4)

(c) Calculate the sum of the digits of the product of 22020

and 52016

. (4)

(d) Determine the value of: (1+ 1

2 ) (1+ 13 )(1+ 1

4 )(1+ 15 ). .. . ..

given there are 98 terms. (4)

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ANTE-MATRIC MATHEMATICS PAPER I: NOVEMBER 2016 Page 11 of 12

QUESTION 11 ANSWER ON YOUR ANSWER SHEET

A rocket is fired upwards.

It follows a path that fits the equation h( t ) = 120 t − 3 t2, 0 ≤ t ≤ 40

where h is the height in metres and t is the time in seconds from launch time.

(a) Sketch the graph of : h( t ) = 120 t − 3 t2; 0 ≤ t ≤ 40 on the grid provided

on the ANSWER SHEET. Clearly show all turning points and intercepts with axes. (4)

(b) Use your graph to determine:

(1) the time when the height of the rocket is zero. (1)

(2) the flight time of the rocket. (1)

(3) the maximum height of the rocket. (1)

(4) how long after launching does the rocket reach its peak height. (1)

(5) how long the rocket is above 800 m. (2)

(c) Explain why there is a restriction on t. (1)

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ANTE-MATRIC MATHEMATICS PAPER I: NOVEMBER 2016 Page 12 of 12

QUESTION 12

(a) A certain company sells x units of a product.

Its profit in rands is given by: P( x )=−3( x−17 )2+560 .

Write down the number of units that must be sold to earn a maximum profit, and

hence determine the maximum profit. (2)

(b) Smurf sets aside R1 800 for his holiday.

He budgets to spend x rands per day.

However, he spends R10 more per day and finds that he has to reduce his

holiday by 2 days.

(1) Using appropriate equations and calculations determine his:

(i) budgeted expenditure per day.

(ii) actual expenditure per day. (6)

(2) Hence, determine the actual number of days he spends on holiday. (2)

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SECTION B TOTAL: 79 MARKS