ST. DAVID’S MARIST INANDA CAPS 2018 Prelim...continue with the assembly of units in South Africa...

ST. DAVID’S MARIST INANDA

GRADE 12

MATHEMATICS PRELIMINARY EXAMINATION

PAPER 1

5 SEPTEMBER 2018

EXAMINER: Mrs L. Nagy MARKS: 150

MODERATOR: Mrs C Kennedy TIME: 3 hours

NAME:__________________________________________

INDICATE YOUR TEACHER’S NAME:

MRS

BLACK

MRS

GOEMANS

MRS

KENNEDY

MRS

NAGY

MRS

RICHARD

MR

VICENTE

INSTRUCTIONS:

This paper consists of 24 pages and a separate Formula sheet. Please check that your paper is complete.

Please ensure that your calculator is in DEGREE MODE.

Please answer all questions on the Question Paper and read each question carefully.

You may use an approved non-programmable, non-graphics calculator unless otherwise stated.

It is in your interest to show all necessary working details.

Work neatly. Do NOT answer in pencil.

Diagrams are not drawn to scale.

SECTION A Q1 Q2 Q3 Q4 Q5 TOTAL

MARKS 20 14 22 12 7 75

SECTION B Q6 Q7 Q8 Q9 Q10 Q11 TOTAL

MARKS 14 12 8 13 10 18 75

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SECTION A

QUESTION 1 [20 marks]

Leave your answers in surd form where necessary.

(a) Given the sequence 3a; 12a; 27a;..........

(1) Write the second and third terms in simplest surd form. (2)

(2) Write down the fourth term. (1)

(3) Determine the product of the first and third terms. (1)

(b) Solve for p: 3p 9p5 5 (4)

(c) If 2 is a root of 2mx x 3 , determine m and the other root. (4)

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(d) Show that the series 7 + 14 + 21 + …………cannot have a sum of 1400. (4)

(e) Solve for x: x 1 x 2325

3.5 4.5 (4)

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(a) p + 20; 24; p + 4 and 14 are the first four terms of a quadratic sequence.

(1) Calculate the value of p. (4)

(2) Determine nT (4)

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In this question the final answers must be rounded off to the nearest whole number.

(b) Mrs Nagy receives 10kgs of Lindt chocolate as first prize in a raffle competition. She

eats 5% of the chocolate the first day and decides that on each consecutive day she

will eat 5% of the remaining chocolate.

(1) After how many days will 6 kgs of chocolate be left? (4)

(2) How much of the chocolate will be left after 44 days? (2)

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(a) Below are the functions 2f x x x 12 , g x 2x 6 .

(1) Calculate the x-coordinates of A and B. (4)

(2) Determine the value of x for which the gradient of the tangent to f is 3. (2)

x

y

f g

A

B

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(3) Rewrite 2f x x x 12 in the form 2

f x a x p q , by completing

the square. (3)

(4) Write down ONE possible way to restrict the domain of f so that 1f will be a function. (2)

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(b) Given: 4h x log x

(1) Draw h and 1h on the set of axes provided below. Clearly show the

x-intercepts as well as one other point on each function. (4)

(2) Write down the equation of 1h in the form 1h x ......... (1)

(3) For which values of x, is 1h x h x 0 (2)

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

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(c) Given below is a hyperbola (g) which passes through the point (0; 3).

The asymptotes intersect at (-2; 2).

Determine the equation of the hyperbola in the form g x ....... (4)

x

y

(−2; 2)(0; 3)

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(a) Determine from first principles the derivative of f, if 2f x 4x x . (5)

(b) Differentiate f if x 1

f xx

(3)

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(c) Determine dy

dx, if y x 1 x 1 (4)

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The seven numbers 0, 1, 2, 3, 4, 5 and 6 are used to make four-digit codes.

(a) How many unique codes are possible if digits can be repeated? (2)

(b) How many unique codes are possible if the digits cannot be repeated? (2)

(c) In the case where digits may be repeated, how many codes are numbers that

are greater than 3000 and exactly divisible by 5? (3)

[SECTION A: 75 marks]

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SECTION B


(a) Bhavik borrows money from his father to buy a holiday home. His father agrees

that his instalments are R45 000 every three months, starting three months after

he receives the loan amount. An interest rate of 4% per annum will be charged.

Bhavik has to repay the loan over a period of 12 years.

(1) Convert the given effective interest rate to an annual nominal rate,

compounded quarterly. Round your answer of to two decimal places. (3)

(2) Calculate the loan amount, using your answer from the question above. (4)

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(b) Matthew trades in his car for a new one every 5 years. He has just bought a new

car for R820 000 and has already decided that he will replace it with the same

model in 5 years’ time, when its trade-in value will be R315 000. The replacement cost of a new car is expected to increase by 10,2% per annum.

(1) Using the car he has just bought as a trade-in, Matthew wants to pay cash

for the new car in 5 years’ time. Calculate how much extra cash he will

need. (3)

(2) Matthew starts a savings fund to make provision for the shortfall.

He deposits x Rands into an account immediately after the purchase

and continues to deposit the same amount at the end of every month for

five years. Calculate the amount that he needs to deposit each month if the interest rate is 4,5% per annum compounded monthly. (4)

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(a) Determine the value of a, if a 4 b 1

n 1 b 1

2n 1 24 2

(6)

(b) If b 1

b 1

a.r 7

, give ONE possible value for a and r for this to be true. (2)

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(c) In a Life Science experiment, a particular kind of bean had sprouted to a height

of 1 cm when the recording of data started. After a further 24 hours its height

was 3 cm. Students found that thereafter its height increased by three quarters

of the growth of the previous 24 hours. How high will the plant grow? (4)

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Given below is xg x a k .

A(-3; 9) is on g.

The equation of the asymptote of g is y = 1.

Determine:

(a) the values of k and a. (3)

(b) the value of x, if 1g x 0. (3)

(c) Write down the domain of 1g . (2)

x

yA(−3; 9)

y=1g

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(a) A South African computer company, Cool-digital, has been having problems with one

of their laptop models. They previously outsourced the assembly of some of these

units to a company in Bangladesh, but are now assembling the units in South Africa.

A sample of 483 laptops has revealed the following:

Assembled in South

Africa

Assembled in

Bangladesh Totals

Faulty 14 x 98

Non-Faulty y 330 385

Totals 69 414 483

(1) Write down the values of x and y. (2)

(2) By testing for independence, decide whether Cool-digital should

continue with the assembly of units in South Africa or outsource to

Bangladesh. (6)

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(b) A table has 3 rows and 5 columns, as shown.

Counters are placed randomly so that there is one counter in each cell of the table.

There are 5 identical black counters and 10 identical white counters. Determine the

probability that there is exactly one black counter in each column. (5)

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In the diagram below, the graph of 2f x x 6 x is drawn. R is the local

maximum and P is the point of inflection. Q is on the x-axis.

(a) Show that P lies on the line which passes through the origin and R. (8)

(b) For which values(s) of k will 2x 6 x k 0 have only ONE real root? (2)

x

y

Q

R

P

f

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(a) For a few seconds during a practice race, Lewis’s and Max’s racing cars are moving

along parallel straight lanes of a racing track.

For these few seconds during the practice race, their positions at time t, for t [0;5]

are given by:

The distance is measured in centimetres and time in seconds.

(1) How far is Lewis ahead of Max at t = 0? (2)

(2) Calculate Max’s average speed between t = 1 and t = 4. (3)

2

2

L t 30t 20t 40

M t 10t 80t

MAX (M)

LEWIS (L)

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(3) When are they racing at the same speed? (3)

(4) Who is ahead at this instant? (2)

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(b) On a sheep farm, an enclosure is to be built next to two existing walls.

These walls meet at o135 and they do not need to be fenced.

138 metres of fencing is available for the enclosure.

If AB = x metres and AD = y metres, determine the dimensions of the largest

possible area that can be enclosed. (8)

enclosure

135°

wall

wall

y

x

D

C

BA

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[SECTION B: 75 marks]

TOTAL: 150 MARKS

ST. DAVID’S MARIST INANDA CAPS 2018 Prelim...continue with the assembly of units in South Africa...

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