MATHEMATICS: PAPER I TRIAL EXAMINATION 28 AUGUST 2015

Transcription

1 MATHEMATICS: PAPER I TRIAL EXAMINATION 8 AUGUST 015 TIME: 3 HOURS TOTAL: 150 MARKS EXAMINATION NUMBER: PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. Write your examination number on the paper.. This question paper consists o 0 pages and an Inormation sheet. Please check that your question paper is complete. 3 Read the questions careully. 4. Answer ALL the questions on the question paper and hand this in at the end o the examination. 5. Diagrams are not necessarily drawn to scale. 6. You may use an approved non-programmable and non-graphical calculator, unless otherwise stated. 7. All necessary working details must be clearly shown. 8. Round o your answers to one decimal digit where necessary, unless otherwise stated. 9. Ensure that your calculator is in DEGREE mode. 10. It is in your own interest to write legibly and to present your work neatly.

2 Page 1 o 0 SECTION A (38 Marks) Examination Number: QUESTION 1 Solve or x (a) x 3x (3) (b) x 1 x (4) 4 (c) log x 16, without the use o a calculator (4) 3 [11]

3 Page o 0 QUESTION Simpliy the ollowing, without using a calculator: [3] QUESTION 3 Solve simultaneously or x and y i y x 3 81 and x 6x y 9 [5]

4 Page 3 o 0 QUESTION 4 I it is given that A 1 x 4x (a) For which values o x will A be non-real? (4) (b) Determine the maximum value o A. (3) [7]

5 Page 4 o 0 QUESTION 5 99 Given: 3t 1 t 0 (a) Write down the irst three terms o the series. (1) (b) Using an appropriate ormula, calculate the sum o the series. Show all your working details. (3) [4]

6 Page 5 o 0 QUESTION 6 Dierentiate ( x) rom irst principles. x [4] QUESTION 7 In an arithmetic sequence, the th n term is given as T n and the sum o the irst n terms is S n. It is given that: Find the value o T 1. T S T 9 S [4]

7 Page 6 o 0 SECTION B (36 Marks) Examination Number: QUESTION 8 In a converging geometric series S 40 3 and T 5. Calculate the possible value(s) o the irst term o the series. [6]

8 Page 7 o 0 QUESTION 9 k ( x) ax bx c and g ( x) q x p are sketched alongside. g Points A and B are symmetrical to each other in the line. and g x 1 intersect each other at 1 C 3 ; and D. B ; 4 ; A B(; 4) C 3; 1 (a) Write down the value o p (1) D H E x = 1 (b) Determine the values o k and q (4) (c) Write down the co-ordinates o A. (1)

9 Page 8 o 0 (d) Determine the co-ordinates o D. () (e) Determine the values o a and b. (4) () For which values o x is : (1) ( x) 0 () () ( x) g( x) (3) ( x) (3) 1 g ( x) where x 0 () [19]

10 Page 9 o 0 QUESTION 10 Given 1 ( x) and x ( 4) 3 Find the equation o the normal (a line perpendicular to the tangent) to the graph o at x 4. [5]

11 Page 10 o 0 QUESTION 11 Sketched are the graphs o a 0 1 and y x. ( x) x a, where (; 36) ; 36 The point is on (a) Determine the value o a. () 1 (b) Give the equation o y... 1 x in the orm (1) (c) For which value(s) o x x? (1) 1 is 0 (d) Write down the domain o h(x) 1 i h ( x) ( x ) () [6]

12 Page 11 o 0 SECTION C (35 Marks) Examination Number: QUESTION 1 Frank sets o on a camping trip. He heads south and sets up his tent in the Addo Elephant Park. He opens the inormation booklet and analyses some o the inormation about the Eastern Cape Aloe. End o First year End o second year End o third year End o ourth year Number o leaves on Aloe x x 1 4x Frank suspects that the pattern has a constant second dierence. (a) Use this act to calculate how many leaves are on the aloe at the end o the ourth year (4) (b) Determine an expression or the number o leaves on the aloe at the end o the th n year. (3) [7]

13 Page 1 o 0 QUESTION 13 ( x) x 3 5x 8x 1 x 1 (a) Show that is a actor o x 3 5x 8x 1. () (b) Find the x and y intercepts o. () (c) Calculate the co-ordinates o the turning points o. (4) (d) Show that has a point o inlection at 5 x. () 3 (e) I y x c 15 is a tangent to at b a ;, determine the value o c i a, b Z (5) [15]

14 Page 13 o 0 QUESTION 14 5 d (a) Find n x x dx n 3 (4) (b) Find dy dx x 4 x given: y 3 3 x x x 3 (6) [10]

15 Page 14 o 0 QUESTION 15 Alex bought a laptop or R It depreciated in value to R 5546,3 ater 5 years. Calculate the annual depreciation rate using a reducing balance. [3]

16 Page 15 o 0 SECTION D (41 Marks) Examination Number: QUESTION 16 A couple take a mortgage loan on a house. The plan is to repay the loan monthly over a period o 30 years. The value o the loan is R and the interest is 9% p.a., compounded monthly. (a) Calculate the monthly payment. (4) (b) What is the total amount the house would eventually cost? () (c) Ater 8 years the couple wants to clear the account. What would be the outstanding balance o the account? (4) [10]

17 Page 16 o 0 QUESTION 17 (a) Events A and B are mutually exclusive. It is given that: P( B) P( A) P ( A or B) 0, 57 Calculate P(B) (3) (b) Given the word S U M M E R Work out the actorials e.g. 4! = 4 Determine: i) the number o 6-letter arrangements that can be made () ii) the probability that a randomly selected word will have an M at each end. (3)

18 Page 17 o 0 (c) A survey was carried out to investigate the relationship between Maths results and extra Maths lessons. The results have been recorded in the table below. Extra Maths B Lessons No Extra Maths B Lessons A A 80% or More or Maths Less than 80% or Maths TOTAL TOTAL Extra Maths teachers claim that learners who take extra Maths lessons are more likely to get more than 80% or Maths than those that don t. Are they correct? Justiy your answer with the necessary calculations to test or independence. (5) [13]

19 Page 18 o 0 QUESTION 18 A bracelet is made by using 10 spheres and 10 cylinders. The radii, r, o the spheres and the cylinders are exactly the same. The height o each cylinder is h. The spheres and cylinders are to be coated in coloured paint. (Ignore the holes in the spheres and cylinders). V V r h 4 r 3 3 S S r 4 r r h (a) I to: 6 h r, show that the total surace area (S) o all the painted suraces o the bracelet is equal 10 S 60 r (3) r (b) Determine the value o r so that the least amount o paint is used. (4) [7]

20 Page 19 o 0 QUESTION 19 Andre is required in a test to ine the derivative o a unction However, by mistake he inds the inverse instead. He inds that: (x). x 7 1 ( x) 3. Find the correct answer to the problem [4] QUESTION 0

21 Page 0 o 0 3 The sketch o ( x) x 3x is given: (a) I h( x) ( x), give the equation o h(x) (3) (b) Write down the values o k or which ( x) k will have only one real root. () (c) Write down the values o x or which ( x). ( x) 0 () [7]