Paper 1
FORM 5
26August 2016
TIME: 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.
SECTION A
QUESTION 1
(a)Solve for x in each of the following, giving all answers, where relevant, in the simplest surd form.
(i)(3)
(ii)(4)
(iii)(3)
(b)Given that and
(i)find an expression in terms of t for:
1)(1)
2)(1)
(ii)Hence or otherwise find the value of for which (2)
(c)Given
(i)by completing the square, solve , giving your solutions
in terms of (3)
(ii)For which values of will have real roots?(1)
(d)Given , determine all possible values of (3)
[21]
QUESTION 2
(a)You are given
(i)Express in the form (3)
(ii)Hence write down the coordinates of the turning points of:
1)(1)
2)(2)
(b)The diagram shows the curve with equation 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 which is defined for
Labelling the given axes in a similar way, sketch the graph of
stating the domain of (3)
[14]
QUESTION 3
Mrs Abernathy wants to buy a house which has been advertised for 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 and then continued to deposit every month with the final deposit being made on her 21st birthday. The interest rate was fixed at 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 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 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 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), determine using the definition ie from first principles.(4)
(b)Determine
if (leave your answer with positive exponents.)(3)
(c)A curve has the equation where is a constant. Given that the gradient of the curve is at the point P where find the value of (4)
[11]
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 and respectively.
(i)Find an expression in terms of 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 (2)
[9]
QUESTION 7
The graphs of and are sketched below.
(a)Sketch the graph of on the set of axes given in the answer booklet.(3)
(b)Write down the equation of which is the reflection of about the y axis.(1)
(c)Determine the coordinates of , the point of intersection of and . (Hint: write each of g in exponential form.) (5)
[9]
QUESTION 8
Refer to the figure below where is a cubic function with and a stationary point at C(-1; -32); and which is a quadratic function such that .
B is an x-intercept of and a stationary point of .
(a)Write down the coordinates of .(1)
(b)Find the equation of (4)
(c)Select which of the following is the possible sketch of (2)
(d)Use the graphs given above in (a) to determine the values of for each of the following conditions.
(i)(2)
(ii)and are both negative.(2)
[11]
QUESTION 9
(a)On the given set of axes in the answer booklet, sketch the graph of
,
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
has one positive and one negative real root.(6)
[9]
QUESTION 10
(a)You are given .
Find the set of values of for which is increasing.(5)
(b)The curve C has equation Prove that the curve C has no stationary points. (4)
(c)Show that is the coordinate of the point of inflection of
.(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
(a)Show that the area of the trapezium base is .(2)
(b)Determine the volume of the prism in terms of (3)
(c)Hence determine the value of for which the prism has a maximum volume.(4)
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SC Form 5 Mathematics Paper 1 26thAugust 2016 Page 1 of 9