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Mathematics TRIALSPaper 1FORM 5

August 2014

TIME: 3 hours TOTAL: 150 marks

Examiner: Mrs A Gunning Moderated: Mrs B Philpot

PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY BEFORE ANSWERING THE QUESTIONS.

This question paper consists of 8 pages plus an information sheet. Please check that your

question paper is complete.

Read and answer all questions carefully.

It is in your own interest to write legibly and to present your work neatly.

Number your answers exactly as the questions are numbered.

All necessary working which you have used in determining your answers must be clearly

shown.

Approved non-programmable calculators may be used except where otherwise stated.

Where necessary give answers correct to 2 decimal places unless otherwise stated.

Diagrams have not necessarily been drawn to scale.

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SECTION AQUESTION 1

Solve for x in each of the following

(a) ( x−4 ) ( x+2 )=7 (3)

(b) 2 .3x=162 (2)

(c) x (x+2)<3 (4)

(d) log 12

x=−3 (2)

(e) log2 (−16 )=x (1)

[12]

QUESTION 2(a) Examine the tiling pattern below.

Stage 1 Stage 2 Stage 3 Stage 4 Stage nNumber of patterned tiles 3 5 7 9

Number of black tiles 1 4 9 16Number of white tiles 2 6 12 20 n2+nTotal number of tiles 6 15 28 45

i. Work out, showing all relevant working detail, Stage n for the number of:a. patterned tiles (2)b. the number of black tiles (1)c. the total number of tiles. (4)

ii. At which stage will the total number of tiles be 5151? (3)

(b) Given the geometric sequence 38; 3

4; 3

2;… . .

Determine which term has a value of 96. (3)(c) The first term of an arithmetic series is 3, the common difference is 2,5 and the last

term is 53. Find the sum of the series. (4)[17]

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QUESTION 3

(a) I invested a sum of money that doubled in value in 5 years. Determine the annual

compound interest rate that was used. Give your answer as a percentage correct to 1

decimal digit. (3)

(b) R150 000 was invested for a period of time at 7,8% per annum, calculated quarterly. The

accumulated amount of this investment at the end of this period was R320 470. For how

long, in years, correct to 1 decimal digit, was the money invested? (4)

(c) Ashton is planning to take a number of sports team on an overseas tour in 5 years time.

Vista Tours is currently advertising such a sports tour deal for R 42 310 per student.

1. If the rate of inflation remains at 7% p.a. compounded annually, for the next 5 years,

how much would parents have to pay for each student at that time? Give your

answer to the nearest whole Rand. (3)

2. A financial expert suggests that parents save a fixed amount each month for the next

5 years, starting at the beginning of this month and ending one month before the

trip ie 60 payments. If the bank is advertising an interest rate of 9,5% per annum,

compounded monthly, how much should each set of parents save each month in

order to be able to pay for the trip in 5 years time. (5)

[15]QUESTION 4

(a) If f (x) is any function, what is the relationship between:

1. f (x) and −f (x) (1)

2. f ( x ) and f (−x) (1)

3. f (x) and f '( x) (1)

4. f (x) and f−1(x) (1)

5. f (x) and f (x−1) (1)

(b) Given f ( x )=−2 ( x+4 )2+3

Determine the co-ordinates of the turning points of:

1. y=f ( x )−1 (3)

2. y=f (x+1) (2)

3. y=4 f (x) (2)[12]

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QUESTION 5

Refer to the figure showing the sketch of y=g ( x ) and y=g−1 ( x ) where g ( x )=bx .

T is the point of intersection of the 2 graphs. The graph of g contains the point K (−1 ;2)

(a) Show that b=12 (1)

(b) Determine the equation of g−1(x ) in the form g−1 ( x )=…. (2)

(c) The x coordinate of T is 0,64 (correct to 2 decimal digits). Give the y co-ordinate of T

(correct to 2 decimal digits). (2)

(d) Write down the value(s) of x for which g−1 ( x )>0 (2)

(e) h(x ) is the reflection of g−1(x ) about the y-axis. Give the equation of h(x ) stating

the relevant domain. (2)

[9]

QUESTION 6

(a) Given ( x )=−x2

4 , determine f '( x) from first principles. (4)

(b) Find dydx for y=15 x2+x−2

3 x−1 x≠

13 (3)

(c) Given f ( x )=√x+ 1x2 −3 x>0

Evaluate f '(4) (4)

(d) Given f ( x )=x3−3x2+kx+8 where k is a constant. The graph has a turning point at x

= 1. Find the value of k , (3)

[14]

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SECTION B

QUESTION 7

Refer to the figure showing the graphs of f ( x )= x2

2−7

2x+3 and g ( x )=−x+6.

D is the x-intercept of both f and g.

(a) Determine the co-ordinates of A (5)

(b) Write down the values of x for which f ( x )≤g( x). (2)

(c) Determine the equation of the tangent to f at C, an x-intercept of f . (6)

(d) PQ⊥CD, P is on g and Q is on f such that P lies between A and D.

i. Show that PQ=52x−1

2x2+3 (2)

ii. Determine the maximum length of PQ. (4)

[19]

QUESTION 8

(a) Consider the digits 1, 2, 3, 5, 6 and 7

i. How many different arrangements of these digits are there if there can be no

repetitions of digits? (1)

ii. What is the probability that the different arrangements, with no repetitions, will

not be divisible by 5. (3)

(b) What is the probability, when the letters A, B, C, D and E are arranged to form 5 letter

‘words’, with no repetitions, that the vowels will be together. (3)

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(c) In the Western Cape, it was found that the probability, if it was raining on any morning,

that it would rain the following day was 0,6; and that if it was fine on any morning the

probability it would be fine the next day was 0,75.

Draw a tree diagram to assist in answering the following question.

If it is raining on Thursday morning, what is the probability that it will be a fine day for

the international cricket game at Newlands on Saturday morning? (Begin the tree with

rain on a Thursday morning. (5)

(d) It is given that P (C )=0,4 , P (D )=0,25 and P (C∧D )=0,1. Showing all relevant working

detail and giving reasons,

i. State whether or not C and D are independent. (2)

ii. Hence, or otherwise, give the value of P(C∪D). (2)

[16]

QUESTION 9

A number of circles touch each other as shown below.

The area of the smallest circle is 4 π c m2and each consecutive circle has an area 94 times that

of the previous one. (a) What is the radius of the first 2 circles? (2)

(b) Write down a sequence of numbers representing the length of the diameters. (2)

(c) If the distance AB=665

8cm, how many circles are there? (3)

[7]

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QUESTION 10

In the diagram, the tangent to the curve y=f ( x )=x3+ax+b at A (1; 4) meets the x-axis at

E (5; 0).

f(x)=x^3 -4x + 7

f(x)=-x+5

f(x)=x+3

x

y

A (1; 4)

E (5; 0)D

(a) Find the equation of the tangent to the circle, AE. (2)

(b) What is the gradient of f (x) at the point A. (1)

(c) Find the values of a and b. (5)

[8]

QUESTION 11

(a) Consider f ( x )=3 x3+4 x+7

i. Write down f '( x) (1)

ii. Hence, or otherwise, explain why f (x) is always increasing. (2)

(b) Function h is defined by h ( x )=x2+4 x for x≥ k and it is given that the inverse of h is

also a function.

i. What is the mapping of h ? (1)

ii. State the smallest possible value of k. (2)

(c) The equation of a curve is such that the second derivative d2 yd x2 =2 x−1 . It is also

given that the curve has a minimum value at x = 3.

i. Find the expression for dydx (4)

ii. Give the x co-ordinate of the maximum point. (3)

[13]

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QUESTION 12

(a) Given f ( x )= 2x−3

+1

i. Write down the equations of both asymptotes of the graph of y=f (x ). (2)

ii. Determine the lines of symmetry of y=f (x+5) (3)

(b) Given f ( x )= x2 when x is rational but f ( x )=x2 when x is irrational.

Evaluate f (√4 )+ f (√8) (3)

[8]

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