GRADE 12 MATHEMATICS PAPER Mark Scheme

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Date

INSTRUCTIONS AND INFORMATION

Read the following instructions carefully before answering the questions.

1.This question paper consists of 9 questions.

2.Answer ALL the questions.

3.Clearly show ALL calculations, diagrams, graphs, et cetera that youhave used in determining your answers.

4.Start each question on a clean side of paper.

5.Answers only will NOT necessarily be awarded full marks.

6.You may use an approved scientific calculator (non-programmableand non-graphical), unless stated otherwise.

7.If necessary, round off answers to TWO decimal places, unlessstated otherwise.

8.Diagrams are NOT necessarily drawn to scale.

9.An information sheet with formulae is included at the end of thequestion paper.

10.Number the answers correctly according to the numbering systemused in this question paper.

11.Write neatly and legibly.

Question 1

1.1 Solve for 𝑥 in the following:

1.1.1 

1.1.2 

1.1.3 

1.2 For what values of x is  imaginary?

1.3 Solve simultaneously for 𝑥 and 𝑦, where:

  and  

1.4 Simplify, without using a calculator:

Question 2

2.1 The sum of the first 𝑛 terms of a sequence is given by 

2.1.1 Calculate 

2.1.2 Calculate 

2.2 Consider the geometric series:

2.2.1 For which value(s) of 𝑥 will the series converge?

2.2.2 Determine the sum to infinity if 𝑥=5.

2.3 Given the first four terms of a quadratic sequence: 𝑥+20 ; 24 ; 𝑥+8 and 14

Calculate 𝑇𝑛. Show ALL working.

2.4 Evaluate 

Question 3

3.1 Determine the values of 𝑎, 𝑝 and 𝑞.

3.2 What is the range of the graph?

3.3 Determine the equation of the axis of symmetry of 𝑓 in the form 𝑦=𝑚𝑥+𝑐 if 𝑚<0.

3.4 The graph is rotated through 90° anti-clockwise about the origin. Give the new equation of the graph.

Question 4

4.1 Show algebraically that 

4.2 Calculate: .

Question 5

5.1 Show that 𝑞=2.

5.2 Determine the co-ordinates of 𝑇, the turning point of 𝑔.

5.3 For what values of 𝑥 is 𝑔(𝑥)−𝑓(𝑥)>0 ?

5.4 Write down the equation of 𝑓−1(𝑥) in the form 𝑦=⋯

5.5 Give one way in which the domain of 𝑔 can be restricted, without  affecting the range, so that 𝑔−1 will be a function.

5.6 Give the equation of 𝑘 if 𝑘(𝑥) is the image of 𝑓(𝑥) after it has  been reflected across the 𝑥- axis and then translated 3 units  down.

Question 6

6.1 A vehicle depreciates at 22% p.a. on the reducing balance method. Calculate how long, rounded off to the nearest year, it will take for the vehicle to decrease to 30% of the original value.

6.2 Andrew takes out a loan of R50 000. He wishes to pay back the loan with equal monthly payments over the course of 4 years. Interest on the loan is charged at 17% p.a. compounded monthly. Determine his monthly instalment.

6.3 Jack inherits R86 500, which he invests in a savings account. For the first 21 months interest is 5,4% p.a. compounded monthly. It is then increased to 6,1% p.a. compounded quarterly for the remainder of the investment. Three years after he invests the initial amount, he withdraws R20 000.

Calculate how much money he has accumulated in the savings account after 6 years.

Question 7

7.1 If  , determine 𝑓′(𝑥), from first principles.

7.2 Find, leaving your answers with positive exponents where  applicable:

7.2.1    if 

Question 8

8.1 Given 

8.1.1 Calculate the 𝑥-intercepts of 𝑓(𝑥).

8.1.2 Calculate the coordinates of the stationary points of 𝑓(𝑥).

8.1.3 Calculate the 𝑥-coordinate of the point of inflection.

8.1.4 𝑦=−11𝑥+3 is a tangent to 𝑓(𝑥) at (𝑎;𝑏).

Determine the values of 𝑎 and 𝑏 if 𝑎 is a whole number.

Question 9

The Coca-Cola Company recently reduced the volume of their South  African ‘standard’ can to 330𝑚𝑙. In order to reduce costs, they are  looking to minimise the amount of aluminium required to produce the  new can, with a radius of 𝑟 cm and a height of ℎ cm.

Calculate the values of 𝑟 and ℎ, for which the total surface area  of the can is a minimum.