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Mathematics TRIALSPaper 1FORM 5

26 August 2016TIME: 3 hours TOTAL: 150 marks

Examiner: Mrs A Gunning Moderated: Ms C Mundy

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.

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 1 decimal place unless otherwise stated.

Ensure that your calculator is in DEGREE mode.

Diagrams have not necessarily been drawn to scale .

SC Form 5 Mathematics Paper 1 26th August 2016 of 9

SECTION A

QUESTION 1

(a) Solve for x in each of the following, giving all answers, where relevant, in the simplest surd

form.

(i) x (x+4 )=4 (x+8) (3)

(ii) √2(x−1)+1=x (4)

(iii) −x2+4 ≤ 3 x (3)

(b) Given that x=2t−1 and y=23 t

(i) find an expression in terms of t for:

1) xy (1)

2) 2 y2 (1)

(ii) Hence or otherwise find the value of t for which 2 y2−xy=0 (2)

(c) Given f ( x )=x2+4 x+m,

(i) by completing the square, solve x2+4 x+m=0, giving your solutions

in terms of m . (3)

(ii) For which values of m will x2+4 x+m=0 have real roots? (1)

(d) Given f ( x )=x2+2, determine all possible values of f ' (18 )+ f−1(18) (3)

[21]

QUESTION 2

(a) You are given f ( x )=2 x2−4 x+5

(i) Express f (x) in the form a ( x−b )2+c (3)

(ii) Hence write down the coordinates of the turning points of:

1) y= f ( x ) (1)

2) y=f ( x+3 )−2 (2)


(b) The diagram shows the curve with equation y=2x−5 which intersects the x and y axes at the

points B and A respectively.

Find the length of AB, giving your answer correct to 2 decimal places. (5)

(c) The diagram shows the graph of y=f ( x ) which is defined for −2 ≤ x ≤2.

Labelling the given axes in a similar way, sketch the graph of

y=f −1 ( x ) stating the domain of f−1(x) (3)

[14]

QUESTION 3

Mrs Abernathy wants to buy a house which has been advertised for R 2850 000.She plans to use the

money she received from her parents on her 21st birthday as a deposit.

(a) On the day she was born, her parents opened a savings account. They immediately deposited

R 1000 and then continued to deposit R 1000 every month with the final deposit being made on

her 21st birthday. The interest rate was fixed at 5% pa compounded monthly.

Determine how much money Mrs Abernathy received from her parents on her 21st birthday.

(3)

(b) At that time, Mrs Abernathy invested the money into an account which earned her 6,95% pa

compounded quarterly. She left it in that account for 10 years until she needed it as the deposit on

her new home. What was her deposit and hence how much will she need to borrow from the bank

to be able to buy the house. (4)

(c) She agrees to repay the loan over 20 years. If the bank charges an interest rate of 6,95 % p.a.

compounded monthly, what will her monthly repayment be? (3)


(d) Mrs Abernathy bought the house and paid the monthly instalments as specified. What was the

balance outstanding on her loan after 12 years (immediately after her 144th payment). (3)

[13]QUESTION 4

(a) A quadratic sequence has a third term equal to 2, a fourth term equal to -2 and a sixth term equal

to -16. Calculate the second difference of this quadratic number pattern. (5)

(b) How many terms in the series ∑ 2 m−3 must be added to give a sum greater than 2000?

(5)

(c) A student programs a computer to draw a series of straight lines with each line beginning at the

end of the previous one, and at right angles to it. The first line is 4 mm long and thereafter each

line is 25% longer than the previous one, so that a spiral, such as the one shown here is formed.

(i) Find the length, in mm, of the eighth straight line drawn by the programme. (3)

(ii) Find the total length of the spiral, in metres, when 20 straight lines have been drawn.

(3)

[16]

QUESTION 5

(a) f ( x )=x−x2, determine f ' (x) using the definition ie from first principles. (4)

(b) Determine

f '( x) if f ( x )= 3√x− 13 x (leave your answer with positive exponents.) (3)

(c) A curve has the equation y=2+3 x+k x2−x3 where k is a constant. Given that the gradient of the curve is

−6 at the point P where x=−1 ,find the value of k (4)

[11]


m=1

n

SECTION B

QUESTION 6

(a) An initial investment of R1000 is placed in a savings account which gives 2,2% compound

interest per quarter. How long, to the nearest year, will it take for the initial investment to

double in value if interest is compounded quarterly. (4)

(b) The first and fourth terms of an arithmetic series are x and (2 x+3 ) respectively.

(i) Find an expression in terms of x for the common difference of the series (3)

(ii) You are given also that the 20th term of the series is 52. Find the value of x . (2)

[9]

QUESTION 7

The graphs of f ( x )=log4 x and g →2 y=2 x are sketched below.

(a) Sketch the graph of f−1(x) on the set of axes given in the answer booklet. (3)

(b) Write down the equation of h which is the reflection of f−1 about the y axis. (1)

(c) Determine the coordinates of k , the point of intersection of f and g. (Hint: write each of f ∧¿g in

exponential form.) (5)

[9]


QUESTION 8

Refer to the figure below where f is a cubic function with A (−3 ;0 ) , D(0 ;−27) and a stationary

point at C(-1; -32); and g which is a quadratic function such that g ( x )=g(−x ).

B is an x-intercept of g and a stationary point of f .

x

y

(a) Write down the coordinates of B. (1)(b) Find the equation of f . (4)(c) Select which of the following is the possible sketch of f ' (x) (2)

(d) Use the graphs given above in (a) to determine the values of x for each of the following conditions.

(i) f ( x ) . g ( x )<0 (2)(ii) f ' ( x ) and g(x ) are both negative. (2)

[11]

QUESTION 9

(a) On the given set of axes in the answer booklet, sketch the graph of

f ( x )= 1x+1 ,

showing clearly the coordinates of any points of intersection with the axes and the equations of any asymptotes. (3)

(b) By sketching another suitable curve on your diagram in part (a), show that the equation

x3− 1x+1

=2

has one positive and one negative real root. (6)

[9]


A(-3; 0) B

C(-1; -32)

D(0; -27)

QUESTION 10

(a) You are given f ( x )=2 x3+5 x2−1.

Find the set of values of x for which f ( x ) is increasing. (5)

(b) The curve C has equation y=x3−x2+2 x−4. Prove that the curve C has no stationary points.

(4)

(c) Show that x=−b3 a is the x coordinate of the point of inflection of

h ( x )=a x3+b x2+cx+d. (4)

[13]QUESTION 11

(a) If you are given 3 identical green cards, 1 red card, 1 blue card and 1 yellow card, determine the

number of different ways in which you can arrange these cards in a single row. (2)

(b) A 7 unit bar code is designed in such a way that the first 4 places are letters (excluding vowels)

and the last 3 are filled with the digits 1 to 9. Letters and numbers may not be repeated.

(i) How many different bar codes can be generated? (2)

(ii) Find the probability that the barcode will begin with an A and the digits be divisible by 5.

(3)

(c) Given letters of the word DECIDED, in arranging the letters what is the probability of getting an

arrangement of letters which starts with a C and ends with an I. (No repetitions) (4)

(d) Let A and B be two events in a sample space such that and . If A and B are

not mutually exclusive, but are independent, determine . (4)

[15]


QUESTION 12

The cross section of a trapezium based prism is drawn. The length of the prism is 2 x+5.

(a) Show that the area of the trapezium base is 12+22 x−4 x2. (2)

(b) Determine the volume of the prism in terms of x . (3)

(c) Hence determine the value of x for which the prism has a maximum volume. (4)

[9]


3x-4

6-x

5x+8

2x+5

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