Gr 12 Mathematics Paper 1 (LO 1 & LO 2) Record Exam 2008 Page 2 of 11

Record Exam 2008

Grade 12 Mathematics Paper 1 (LO1 & LO2)

Date: /2008 Time: 3 h Total: 150

INSTRUCTIONS: Read the following instructions carefully before answering the questions:  This question paper consists of 10 pages, including a Diagram Sheet. An information sheet has been provided. Please ensure that you use it but DO NOT write on it. Please check that your paper is complete.  Read the questions carefully.  Answer ALL questions.  Questions 1 - 15 must be answered on lined paper provided.  Question 5.3, 14.1 and 15.5 must be answered on the diagram sheet.  You may use an approved non-programmable and non-graphical calculator, unless a specific question prohibits the use of a calculator.  Round off answers to TWO decimal digits where necessary.  Read through the whole paper before starting.  All the necessary working details must be clearly shown.  Number your answers exactly as the questions are numbered.  Diagrams are not necessarily drawn to scale.  It is in your own interest to write legibly and to present the work neatly.  You may NOT use Tippex.  You may not borrow ANYTHING during the examination Gr 12 Mathematics Paper 1 (LO 1 & LO 2) Record Exam 2008 Page 2 of 10

Section A

QUESTION 1

Given the arithmetic sequence: 3; 7; 11; ….

1.1 Calculate the sum to twenty terms of the sequence. (3)

1.2 If the n-th term is 191, determine the value of n. (3)

[6 marks]

QUESTION 2

Calculate the following:

n 15  2  81  (4) n0  3  [4 marks]

QUESTION 3

Without the use of a calculator:

log 2  log3 3.1 Simplify: (3) log 72  log 2

3.2 Solve for x: log(x 10)  log x 1 (4)

3.3 Determine the value of the following in lowest surd form, showing all intermediate steps:

75  48 (4) 12

9 x  32x1 3.4 Evaluate: (4) 18x 2x

[15 marks] Gr 12 Mathematics Paper 1 (LO 1 & LO 2) Record Exam 2008 Page 3 of 10

QUESTION 4

Sue wants to save R10 000 in five year’s time to give to her son as a 21st birthday present. She is offered two different investment plans, namely the Ruby Plan where she makes a payment every 12 months for 5 years and earns 8% per annum compounded annually, or the Emerald Plan in which she pays a monthly payment at the end of each month and earns 7.2% interest per annum compounded monthly for 5 years.

4.1 Calculate the payments she needs to make in each case. (6)

4.2 Which is the better option? Motivate your answer. (2)

[8 marks]

QUESTION 5

f (x)  x 3  x 2  x 1

5.1 Determine the x- and y-intercepts. (5)

5.2 Determine the co-ordinates of the local turning points. (6)

5.3 Draw a sketch graph of f, clearly labelling all intercepts with the axes and the turning points. (5)

[16 marks]

QUESTION 6

6.1 If f (x)  4x2  4 calculate f ' (x) from first principles (5)

dy 3 3 6.2 Determine in y  x  (2) dx x

6.3 Find the equation of the tangent to the graph y  g(x)  3x 2  2x 1 at the point x  1 (4)

QUESTION 7

3 Given g(x)  x  3

7.1 What type of graph is this? (1)

7.2 Give the equation(s) of any asymptote(s). (2)

7.3 Determine x if g(x)  1 (2)

7.4 Give the equation of a new graph, f(x), which is g(x) shifted one unit in a positive direction horizontally. (2)

7.5 Give the equation of a new graph, h(x), which is g(x) shifted up one unit vertically. (2)

7.6 Give the equation of the inverse function g 1 (x) in the form g 1 (x) = …. (3)

[12 marks]

Section B Question 8

Simone deposits R8 000 into a savings account and 3 years later she deposited R12 000. The interest rate is 12% p.a. compounded quarterly for the first two years and then 10.5% p.a. compounded monthly. The money is left in the account for 6 years. Use a time line to calculate how much money will she have in the account at the end of the 6 years? [6 marks]

Question 9

9.1 Determine for which values of x the given geometric series will be convergent  n 6(2x 1) (3) n2

9.2 An unlimited number of rectangles are drawn. The area of the first rectangle is 108cm2 .

108cm2

Each rectangle has the length of both its sides reduced by 20% to get to the next rectangle. Find the maximum possible total area of all the rectangles. (5)

Question 10

You are starting an SMS chain letter in which you send a letter to 5 different friends. They in turn will also send the letter to 5 different friends, and so on.

10.1 Write down the first 7 terms of the sequence that shows the number of letters that are sent at each mailing. (2)

10.2 How many letters are sent on the 10th mailing, provided everybody sends the required 5 letters? (4)

10.3 After how many mailings will a total of 61 035 155 letters be sent? (5)

[11 marks] Question 11

Peter must solve the following equation by completion of the square. Examine his working and indicate any mistakes, giving an explanation of the problem next to each error. Then write out a correct solution. (5)

px x 2   9  0 2 px Step 1 x 2   p 2  p 2  9  0 2

Step 2 (x  p)2  9  p 2

Step 3 x  p   9  p 2

Step 4 x  p  3 p

Step 5 x  3 [5 marks] Gr 12 Mathematics Paper 1 (LO 1 & LO 2) Record Exam 2008 Page 6 of 10

Question 12

A group of kids went out for ice cream. They were to share their expenses equally. The total bill was R120. Ten of the kids did not have money, so each of the others had to pay an extra R2. Determine the number of the kids in the group. Let the number of kids be x and the amount spent, y [5 marks]

Question 13

A rectangular box has a volume of 243 cm3 . Its width is x cm and the length is twice the width. Its height is h.

13.1 Find h in terms of x (3)

 729  13.2 Show that the surface area is given by A  4x 2   cm2 . (5)  x 

13.3 Calculate the value of x for which this surface area is a minimum. (5)

[13 marks]

Question 14

14.1 Draw the graph of y  f (x)  4 x using x {1;0;1} on a system of axes (3)

14.1.1 Determine the inverse of f and write it in the form f 1 (x)  K (2)

14.1.2 Sketch the graph of f 1 on the same system of axes as f (2)

14.2 Use the same function f (x)  4 x . Let g(x)  f (x 1)  2

14.2.1 Describe the transformation of f to g in words (2)

14.2.2 Sketch the graph of g (4)

Question 15

The owner of a pleasure boat is prepared to take a school group consisting of learners and adults on a cruise, provided that the group consists of not more than 60 people. In addition:

(i) There must be at least 35 people in the group

(ii) There must be at least 6 adults in the group

(iii) There must not be more than 14 adults

Let x be the number of learners and y the number of adults

15.1 Give all the constraints in terms of x and y (5)

15.2 If the group has 25 learners, what is the minimum number of adults that must accompany them? (1)

15.3 Eight adults offer to go on the cruise. What is the maximum number of learners that can be accommodated on the boat? (1)

15.4 If T is the amount in Rand paid by the whole group, what is the cost per learner if T =30x+50y (2)

15.5 Represent the constraints graphically and indicate the feasible region clearly (5)

15.6 How many learners and adults should go on the cruise, to ensure a maximum income? (3)

[17 marks]

TOTAL [150] Gr 12 Mathematics Paper 1 (LO 1 & LO 2) Record Exam 2008 Page 8 of 10 NAME: ______Diagram Sheet Question 5.3

Gr 12 Mathematics Paper 1 (LO 1 & LO 2) Record Exam 2008 Page 9 of 10 Question 14.1

Question 15.5

Gr 12 Mathematics Paper 1 (LO 1 & LO 2) Record Exam 2008 Page 10 of 10