MATHEMATICS PAPER 1

GRADE 12

Examiner: Mrs Harrison Date: 22nd July 2020

Moderator: Ms Barclay Time: 3 Hours

Marks: 150

Name: Class:

Teacher:

Please read the following instructions carefully

1 This examination consists of 22 pages, 2 of which are for additional working space if

necessary. Please check that your question paper is complete.

2 Answer ALL the questions in the spaces provided. No extra rough work paper will

be provided.

3 Read the questions carefully before answering.

4 It is in your own interest to write legibly and to set out your work neatly.

5 An approved calculator may be used, unless otherwise stated.

6 Show all your working out. Answer ONLY will get ZERO marks.

7 Round off your answers to 2 decimal places, unless otherwise stated.

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QUESTION 1 [12]

a) The fourth term of an arithmetic series is 108 and the eleventh term is 80.

Determine the sum of the first four terms. (5)

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b) In a geometric series the sum of all the terms is given by 𝑆𝑛 = 81 − 81(3)−𝑛

1) Determine the sixth term. (3)

2) Calculate the sum to infinity, if it exists. (4)

QUESTION 2 [16]

a) Determine 𝑓′(𝑥) from first principles if 𝑓(𝑥) = 𝑥2 − 2𝑥. (5)

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b) Determine the following:

1) 𝑑

𝑑𝑥[

3√𝑥− √𝑥3

2𝑥] (4)

2) 𝑑𝑦

𝑑𝑥 if 𝑦 = 2𝑥2 + 𝑥𝑦 − 2𝑥 ; 𝑥 ≠ 1 (4)

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c) Given that 𝑠 = 𝑢𝑡 +1

2𝑎𝑡2.

Show that √2 ×𝑑𝑠

𝑑𝑎=

𝑑𝑠

𝑑𝑢 (3)

QUESTION 3 [18]

a) Determine 𝑛, if (6)

∑(3𝑖 − 2) = 651

𝑛

𝑖=1

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b) 1) Determine the roots, in terms of 𝑘 if 𝑥 (𝑥 −1

𝑘) = 𝑘(𝑥 −

1

𝑘) (3)

2) For what values of 𝑘 will the roots be equal? (2)

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c) If 3𝑥3 − 4𝑥2 − 5𝑥 + 2 is divisible by (𝑥 − 2),

1) determine the other two factors. Show ALL working out. (3)

2) Hence, or otherwise solve for 𝑥 if 33𝑥+1 − 4. 32𝑥 − 5. 3𝑥 + 2 = 0, (4)

correct to 2 decimal places if necessary.

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QUESTION 4 [18]

After Jason graduates as an actuarial scientist he purchases a house. The selling

price of the house is R 1 300 000. He pays a 15% deposit and obtains a loan from

the bank to pay back the balance. Because of his degree the bank charges him

interest of 11% per annum, compounded monthly.

a) Determine the amount of the loan. (1)

b) If Jason starts repaying the loan one month after the loan is granted, calculate his

expected monthly instalment if the loan is for a 20 year period. (4)

c) How much interest will Jason pay over the 20 years? (3)

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d) If Jason decides to rather pay R11 500 per month, calculate the balance on the

loan immediately after his 75th instalment. (5)

e) If the balance outstanding on the loan at the end of the 75th month is

R958 068,80 and Jason decides to increase his payments to R12 000 per month

from the end of the 76th month. How many more months will it take to repay his

bond with the new instalment of R12 000 per month? (5)

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QUESTION 5 [7]

Consider the graph of 𝑔(𝑥) =3

𝑥−1− 2

a) Sketch the graph of 𝑔(𝑥). Clearly showing any intercepts with the axes, any

asymptotes and points of accuracy. (5)

b) Determine the equation of the line of symmetry of 𝑔(𝑥) with a negative gradient.

(1)

c) Determine the value(s) of 𝑥 for which 𝑥. 𝑔′(𝑥) > 0 (1)

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QUESTION 6 [8]

Given 𝑓(𝑥) = 2𝑥 and 𝑔(𝑥) = (𝑥 − 1)2

a) Restrict the domain of 𝑔(𝑥) and hence determine the equation of 𝑔−1(𝑥). (4)

b) Determine ℎ−1(𝑥) if 𝒉 is 𝑓 translated 5 units left and 2 units down. (4)

SECTION A = 79 MARKS

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SECTION B

QUESTION 7 [6]

Each passenger on Flight X can chose exactly one beverage from either tea, coffee

or juice.

The results are shown in the table below:

MALE FEMALE TOTAL

TEA 20 40 60

COFFEE 𝑏 𝑐 80

JUICE 𝑑 𝑒 20

TOTAL 60 100 𝑎

a) Write down the value of 𝑎. (1)

b) What is the probability that a randomly selected passenger is female? (1)

c) Given that the event of a passenger choosing coffee is independent of being

male, calculate the value of 𝑏. Show ALL working out. (4)

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QUESTION 8 [7]

A conical cup of height 12cm and radius 4cm is shown in the diagram below. The

cup is being filled with water. When the height of the water is ℎ, the radius of the

water surface is 𝑟.

1) Show that 𝑟 =1

3ℎ (3)

2) What is the height of the water when the rate of change of the volume of water is

16𝜋 𝑐𝑚2? {𝑉 =1

3𝜋𝑟2𝐻} (4)

4 cm

r cm

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QUESTION 9 [18]

Sketched below is the graph of 𝑔(𝑥) = −2𝑥3 + 𝑎𝑥2 + 𝑏𝑥 + 𝑐.

A and T are the stationary points of 𝑔. The 𝑥 −intercepts are −2 and 5

2.

P is a point on the graph and has an 𝑥 value of −3.

a) Determine the values of 𝑎, 𝑏 and 𝑐. (4)

𝑥

𝑦

𝑃

𝑇

𝑂 𝐴 𝐵

𝑔

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b) If 𝑔(𝑥) = −2𝑥3 − 3𝑥2 + 12𝑥 + 20, determine the co-ordinates of 𝑇. (4)

c) Determine the equation of the tangent to 𝑔 at 𝑃. (4)

d) Determine the values of 𝑘 for which −2𝑥3 − 3𝑥2 + 12𝑥 − 𝑘 = 0 has 3 distinct

roots. (2)

e) Using your graph determine for which values of 𝑥 is 𝑔′′(𝑥) × 𝑔′(𝑥) > 0 (4)

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QUESTION 10 [8]

a) Corbin decides he wants to complete a triathlon at the end of the year. He starts

a training schedule. Some days he chooses to go to the gym after training.

The table below shows the various probabilities for the different training events

on any given day.

Event Probability

Running (R) 0,65

Cycling (C) 0,3

Swimming (S) 0,05

After running, the probability that he will go to gym is 0,4. The probability

that he will go to gym after a swim is 0,5 and after cycling, the probability

of not going to gym is 0,8. Determine the probability that Corbin goes to the gym on

any particular day. [ Correct to 3 decimal places] (4)

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b) 𝐸 and 𝐹 are two independent events. 𝑃(𝐸) = 𝑥 and 𝑃(𝐹) = 𝑦. 𝑃(𝐸 𝑎𝑛𝑑 𝐹) =1

3

and 𝑃(𝐸 𝑜𝑟 𝐹) =9

10. Show that 30𝑦2 + 10 = 37𝑦. (4)

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QUESTION 11 [9]

The sketch (not drawn to scale) shows the graphs of the following functions:

𝒇(𝒙) = 𝒂𝒙𝟑 + 𝒃𝒙𝟐 − 𝟓𝒙 + 𝟓𝟎

𝒈(𝒙) = 𝒇′(𝒙) = 𝒑𝒙𝟐 + 𝒒𝒙 + 𝒕

The 𝑥 − intercepts of 𝑓 are (−5 ; 0) and (2 ; 0)

The 𝑥 − intercepts of 𝑔 are (−5 ; 0) and (−1

3 ; 0)

Use the given information and sketch to answer the following questions:

a) Determine the value of 𝑡. (2)

b) Determine the value of 𝑓(𝑔(−5)) (2)

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c) Calculate the average gradient of 𝑓 over the interval 𝑥 ∈ [−5 ; 0] (2)

d) Determine the values of 𝑥 for which 𝑓′′(𝑥) < 0 (3)

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QUESTION 12 [10]

In the diagram below, the graph of 𝑓(𝑥) = 8 − 𝑥3, 𝑥 ≥ 0 and 𝑦 ≥ 0 is represented.

The graph intersects the 𝑥 −axis at B and the 𝑦 −axis at A. D is a point on 𝑓(𝑥).

P(𝑥; 0) lies between O and B.

DP ꓕ OB.

a) Determine the length of OA and PD. (2)

b) Hence, or otherwise prove that the area of OADP = −1

2𝑥4 + 8𝑥. (3)

A

B P O

D

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c) Calculate the maximum area of OADB? (5)

QUESTION 13 [8]

a) Kyle has come up with a genius theory: 1

√𝑥+√𝑥+1= √𝑥 + 1 − √𝑥

Matthew is at a loss when faced with the following problem:

1

1 + √2+

1

√2 + √3… … … … … … … . . +

1

√2499 + √2500

As well as losing his calculator, he has no idea where to start.

He needs help to find the exact value of the above expression. Using Kyle’s theory,

show Matthew how to find the exact value of the above problem. (3)

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b) Given: (𝑥 − 1)2 ≥ 𝑝

Solve for 𝑥, in terms of 𝑝 where necessary; if

1) 𝑝 < 0 (1)

2) 𝑝 > 0 (4)

QUESTION 14 [5]

In a quadratic sequence, the 12th term is 555 and 𝑇𝑛 − 𝑇𝑛−1 = 6𝑛 + 8. Determine the

value if the first term of the sequence.

SECTION B = 71 MARKS

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ADDITIONAL WORKING SPACE

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