rustenburg girls’ high school

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Rustenburg Girls’ High School

Grade 12 Mathematics Prelim Paper 1

September 2021 Marks: 150 Time: 3 Hours

Instructions and Information:

1. This question paper consists of 9 questions,

2. Number the questions correctly according to the numbering system used

in the question paper.

3. Clearly show ALL calculations, diagrams, graphs etc. that you have used in

determining your answers.

4. Answers only will not necessarily be awarded full marks.

5. You may use an approved scientific calculator (non – programmable and

non – graphical) unless otherwise stated.

6. If necessary round off all answers to two decimal places, unless stated

otherwise.

7. Diagrams are not necessarily drawn to scale.

8. An information sheet with formulae will be provided for your use.

9. Write neatly and legibly.

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Question 1

1.1. Solve for the unknown variable in each case:

1.1.1. 2𝑥2 − 3𝑥 = 8 (to two decimal places) (4)

1.1.2. 12𝑥 − 5 × 3𝑥 = 0 (3)

1.2. Given: 𝑎 +1

𝑎= 2

1.2.1. Solve for 𝑎. (3)

1.2.2. Hence or otherwise, solve for 𝑥 in: 2𝑥2 − 𝑥 +1

2𝑥2−𝑥= 2 (3)

1.3. Solve for 𝑥 and 𝑦 in the following equations:

2𝑦 − 𝑥 = 6

4𝑥2 + 𝑦2 = 144

(7)

1.4. If 𝑓(𝑥) = √(2𝑥 + 2)(𝑥 + 3) and 𝑔(𝑥) = 4𝑥 + 3

1.4.1. For which values of 𝑥 will 𝑓(𝑥) be non – real? (3)

1.4.2. Solve for 𝑥 if 𝑓(𝑥) = 𝑔′(𝑥) (5)

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Question 2

2.1. For the following number sequence:

1 ; 3 ; 6 ; 10 ; 15 ; …

2.1.1. Show that the above sequence is a quadratic sequence. (2)

2.1.2. The general term is 𝑇𝑛 = 1

2𝑛2 +

1

2𝑛. Find the eighth term. (2)

2.2. In a linear sequence/arithmetic progression the 3rd term is −2 and

the 10th term is 19. Determine the first term and the constant

difference.

(5)

2.3. If 𝑝 + 1 ; 𝑝 − 1 ; 2𝑝 − 5 are the first three terms of a geometric

sequence, calculate the value(s) of 𝑝.

(5)

2.4. Calculate the following:

∑(3𝑡 − 1)

99

𝑡=3

(4)

2.5. The length of each side of the outer square is 8𝑥 cm. The midpoints

of each side are joined to form another square. This process is

repeated indefinitely. Calculate the sum of the areas of the squares

that are formed this way.

(6)

[24]

8𝑥

cm

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Question 3

3.1. Jamie purchases furniture for her apartment to the value of

R15 000. The furniture depreciates at 7% per annum on the reducing

balance method. Determine the value of her furniture after 5 years.

(3)

3.2. Lynne purchases a new car for R350 000. They take out a 6 year loan on 1

January 2019. The monthly instalments are paid at the end of every month.

Interest is fixed at 18% per annum compounded monthly.

3.2.1. Calculate the monthly repayment. (4)

3.2.2. Due to financial difficulty, Lynne misses the 40th , 41st and

42nd payments. Determine the balance outstanding at the end

of the 42nd month.

(4)

3.2.3. If Lynne’s new monthly repayment is R10 000. How many

months will it take them to pay back the rest of the loan?

(4)

[15]

Question 4

Given 𝑓(𝑥) = 2𝑥

𝑔(𝑥) is obtained by translating 𝑓 2 units left and ℎ(𝑥) = 𝑓−1(𝑥).

4.1. Write down the equations of 𝑔 and ℎ in the form 𝑦 = ⋯ (3)

4.2. On the same set of axes, sketch the graphs of 𝑓 and ℎ. Showing all

asymptotes and intercepts with the axes.

(6)

4.3. State the domain of ℎ(𝑥). (1)

4.4. Describe the transformation of 𝑓, that would result in the following

equation 𝑝(𝑥) = (1

2)

𝑥− 2

(2)

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Question 5

The diagram below shows the graphs of 𝑓(𝑥) = 𝑎𝑥 and 𝑔(𝑥) = 𝑎(𝑥 − 𝑝)2 + 𝑞.

The point N (1;1

2) is the point of intersection of 𝑓 and 𝑔. D is the point (0 ; 1).

H is the turning point of 𝑔 and B and C are the 𝑥 – intercepts of 𝑔.

5.1. Calculate the values of 𝑎 , 𝑝 and 𝑞. (4)

5.2. Determine the length of DE. (3)

5.3. Explain why the inverse of 𝑔 is not a function. (2)

5.4. Write down two ways in which the domain of 𝑔 can be restricted

such that 𝑔−1 is a function.

(2)

5.5. For which values of 𝑥 is 𝑓(𝑥) ≤ 𝑔(𝑥) (2)

5.6. Determine the value(s) of 𝑘 for which 𝑔(𝑥) + 𝑘 = 0 has one real root. (1)

[14]

Question 6

6.1. From first principles, determine the derivative of 𝑘(𝑥) = −𝑥2 + 2 (5)

6.2. Determine the following:

6.2.1. 𝐷𝑡 (2√𝑡3+𝑡4

4𝑡3 ) (4)

6.2.2. 𝑑𝑢

𝑑𝑥 if 𝑢 = ℎ′(𝑥) and ℎ(𝑥) = 3𝑥5 + 𝑥2 − 𝑥 (3)

[12]

𝒇

𝒈

𝐃 𝐍

𝐄

𝐇 (−𝟐 ; −𝟒)

𝐁 𝐂

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Question 7

The graph 𝑓 is defined by 𝑓(𝑥) = −𝑥3 + 𝑏𝑥2 + 𝑐𝑥 + 𝑑. The turning point B has

coordinates (2 ; 0). A sketch is given below.

7.1. Show that 𝑏 = 1 ; 𝑐 = 8 𝑎𝑛𝑑 𝑑 = −12 (3)

7.2. Determine the equation of the tangent to the curve at E. (3)

7.3. Determine the point of inflection of 𝑓. (3)

7.4. Determine the 𝑥 value of the turning point of 𝑔 if 𝑔(𝑥) = 𝑓′(𝑥 − 2) (3)

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−3

A

E

B (2; 0)

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Question 8

A construction company has been assigned to build a tunnel and a bridge in a

mountainous area. They have determined that the cross section can be modelled

by a function: 𝑦 = 2𝑥3 − 17𝑥2 + 35𝑥 ; 0 ≤ 𝑥 ≤ 5 where 𝑥 is the distance (in

hundreds of metres) from a point where the tunnel will start and 𝑦 is the height

(in hundreds of metres) above the proposed bridge.

Giving answers to the nearest metre:

8.1. Determine the length of the tunnel and of the bridge. (4)

8.2. Calculate the height of the top of the mountain. (5)

8.3. The cost of labour to complete the construction of the tunnel and

bridge, 𝐂 in thousands of rands, is dependent on the number of

workers, 𝑤, according to the formula 𝐶 =1

3𝑤3 −

35

2𝑤2 + 300𝑤 + 50

8.3.1. Determine the number of workers that must be employed to ensure a

minimum cost, if at least 15 workers are to be employed.

(3)

8.3.2. Determine the minimum cost of labour. (2)

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Tunnel

Mountain

Valley

Bridge 𝒙

𝒚

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Question 9

9.1. In a survey, 800 grade 11 and 12 learners were polled to see which

pop artist they preferred. The results are tabulated in the

contingency table below.

Grade 11 Grade 12 Total

Ed Sheeran 200 364

Taylor Swift 196

Total 440

9.1.1. Copy and complete the table. (3)

9.1.2. Determine the probability of randomly selecting a grade 12

learner.

(1)

9.1.3. Determine the probability of randomly selecting a learner

that prefers Ed Sheeran.

(1)

9.1.4. Is a learner’s preference for Ed Sheeran independent of

being in Grade 12?

(5)

9.2. Three boys and four girls are to be seated randomly in a row.

9.2.1. Calculate the total number of possible seating arrangements. (2)

9.2.2. Determine the probability that the row has boys and girls

sitting in alternate positions.

(3)

9.3. Consider the letters in the word INTERESTING. The letters are

arranged in any order without repetitions, but using all 11 letters to

form different words. What is the probability that the new word

formed will start with the letter S?

(4)

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END

rustenburg girls’ high school

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