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ADvTECH Copyright © 2021 Prelim Exam 2021: Mathematics Paper 1 of 25

CRAWFORD INTERNATIONAL PRELIMINARY EXAMINATION

SEPTEMBER 2021

MATHEMATICS: PAPER I

Time: 3 hours 150 marks

NAME OF CANDIDATE:

NAME OF EDUCATOR:

MARKER TO ENTER MARKS:

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12

TOTAL

18 13 15 12 12 12 6 8 7 20

14

13

150


PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY

1. This question paper consists of 25 pages and an Information Sheet of 2 pages (i–ii).

Please check that your question paper is complete. Spare pages have been left for

rough/extra working.

2. Read the questions carefully.

3. Answer ALL the questions on the question paper and hand this in at the end of

the examination. Remember to write your name in the space provided.

4. Diagrams are not necessarily drawn to scale.

5. You may use an approved non-programmable and non-graphical calculator, unless

otherwise stated.

6. Ensure that your calculator is in DEGREE mode.

7. All the necessary working details must be clearly shown. Answers only will not

necessarily be awarded full marks.

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

9. Round off to two decimal places unless otherwise stated.


SECTION A

QUESTION 1 [18 Marks]

(a) Solve for x. Show all working:

( 1) 30x x − = (3)

(b) 7 1x x+ − = (4)

(c) 23 3 72x x+ − = (3)


(d) ( 1)

112

x x − −

− (5)

(e) log log4 log( 4)x x− = − (3)


QUESTION 2 [13 Marks] Lydia and Abdul are investigating the following sequence : 4 ; 8 ; 16 ; …

(a) Lydia claims that the fourth term of the sequence is 32.

Abdul claims that the fourth term is 28.

Explain why it is possible that Lydia and Abdul could both be correct. (2)

(b) Show that the general term of Abdul’s sequence is 22 2 4nT n n= − + (3)


(c) Determine the largest term in Abdul’s sequence which is less than 268. (4)

(d) Using a formula, determine nn

T20

3=

, if nT is the general term of Lydia’s sequence. (4)


QUESTION 3 [15 MARKS]

(a) Paul wishes to go on a trip through Africa in 5 year’s time.

He is able to make a total of 20 quarterly deposits of R5 000 into a savings account.

The interest rate is 9,5% per annum, calculated quarterly throughout.

Calculate the total amount he is able to save. (4)

(b) Chris purchases a house for R780 000 on a mortgage of 20 years at an interest rate of

17% per annum, compounded monthly.

(1) Calculate his monthly payments over the 20-year period. (3)


(2) Calculate the outstanding balance after 8 years. (4)

(c) After 8 years the interest changes to 19% per annum compounded monthly.

Calculate the new monthly payment until the end of the 20-year period. (4)



(a) If 2( ) 2 1f x x= − + , determine the derivative, '( )f x , from first principles. (5)

(b) Determine dy

dx of the following, leaving your answers with positive exponents:

(1) 4xy = (3)

(2) 2 2 2( )y x x−= + (4)

ADvTECH Copyright © 2020 VCAA 2020: Mathematics Paper 1 of 25


(a) On the same set of axes, sketch the graphs f with equation ( ) 2xf x =

and g with equation 2

( ) 11

g xx

= −−

Clearly show all intercepts with the axis and asymptotes. (4)

y

x

ADvTECH Copyright © 2021 Prelim 2021: Mathematics Paper 1 of 25

(b) Determine the domain of g . (1)

(c) Determine the equation of 1f − , the inverse function of f . (2)

(d) Determine the equation of the function which is obtained by translating g

2 units to the right and 1 unit up. (2)

(e) The function ( )p x mx c= + is a decreasing function and is an axes of symmetry of g.

Determine the values of m and c. (3)



`

In the diagram the functions of f and g are shown, where 2( ) 2 4 6f x x x= − + + and ( ) 4 12g x x= − +

Q is a point on f and P is a point on g. PQ is parallel to the y-axis.

(a) Determine the equation of the tangent to f, which is parallel to the straight line g. (5)


(b) Calculate the value of x for which the length of PQ will be at a minimum (4)

(c) Determine the value(s) of x for which 1( ) '( )g x f x− = . (3)


SECTION B


Solve for x and y if:

1xy = − and 1 1

3y xx y

− + − =

(6)



In the diagram below, the function of f passes through point ( 9; 2)−A .

The function of h with equation ( ) logah x x= is the reflection of f about the line 0x = ,

(a) Determine the equation of h. (3)

(b) Determine the values of x for which ( ) 2 0f x − (2)

(c) The point ( 9; )B p− is the corresponding point on the graph ( ) 1y f x= − + .

Determine the value of p. (2)

(d) Write down which of the following statements is TRUE:

(1) 3log 3loga ax x=

(2) ( )3log 3loga ax x= (1)



Katy loves roller coasters that dip into underground tunnels during the ride.

Her favourite coaster is modeled by, f x x x x3 2( ) 2 23 59 24= − + − + , where f x( )

represents the height above the ground in metres and x represents time in seconds.

The diagram below models the 9 seconds of the ride after the coaster comes out

of a loop.

(a) Determine the period of time for which the roller coaster will be

underground for the first time. (4)

(b) Determine at what time is the roller coaster is at its steepest. (3)

B(6,04; 66,02)

A(1.63; -19,72)

Ground

𝑓

𝒙

y



(a) In a certain memory experiment, the number of words (M) a person is able to

memorize after t minutes, is represented by the following equation:

3 20,001 0,1M t t= − +

(1) Determine at what rate the number of words are memorized immediately

after 10 minutes. (2)

(2) Calculate the average number of words memorized per minute from the time

immediately after 2 minutes to immediately after 10 minutes. (4)

(3) Determine the maximum number of words a person in this experiment

is able to memorize. (3)


(b) For a certain function f , the first derivative is f x x x/ 2( ) 3 8 3= + − .

The diagram below represents the function of f / .

f /

(1) Calculate the x-coordinates of the stationary points of f . (3)

(2) Determine the values of x for which the graph of f is concave down (2)

(3) Determine the values of x for which f is increasing. (2)


(4) If it is further given that 3 2( )f x ax bx cx d= + + + and (0) 18f = − ,

determine the equation of f. (4)



(a) A box has two balls, one white and one red. One ball is selected.

It is placed back in the box, and then a second ball is selected.

Event F = the event of selecting the white ball twice.

Event G = the event of selecting two balls of different colours.

Event H = the event of selecting the white ball on the first pick.

Say whether the following events are mutually exclusive or not.

(1) F and G. (1)

(2) G and H. (1)

(b) Five netball players line up to get onto a College bus.

(1) Determine in how many ways the five netball players can line up? (2)

(2) In how many ways can these same netball players line up if

two of the players refuse to stand next to each other? (4)


(c) Mo plays the following game:

• He flips a coin.

• If it comes up tails, he gets 2 points and the game ends.

• If it comes up heads, he gets only 1 point, but he can flip the coin again.

• He can flip a coin up to three times.

Determine the probability that he can get exactly 3 points at the end of

the game. (6)



The diagram below represents a rectangular sheet of paper, 1(D ) , has dimensions 120cm x 84cm.

Sheet 1D is divided along its length to form Sheet 2D .

Sheet 2D is divided along its breadth to form Sheet 3D .

The process continues indefinitely.

Perimeter of a rectangle: 2(length) + 2 (breadth),

The perimeters of the sheets form a number pattern as indicated below:

•

•

•

Perimeter of Sheet 20D : 20

9P 2(d) 2(e)

16= + =


(a) Determine the values of a; b and c. (3)

(b) Determine the values of d and e. Leave your answer in fraction form. (4)

(c) Write down the first three terms of the number pattern as a sequence,

1 2 3;D ; D ; D ;.......representing the AREAS of sheets (3)

(d) Calculate the sum of all areas, if dividing the sheets continues indefinitely. (3)


EXTRA SPACE FOR WORKING

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