Grade 12 Mid-Year Examinations

Mathematics: Grade 12 July Paper 1 2015

GRADE 12 MID-YEAR EXAMINATIONS

June 2015
MATHEMATICS PAPER 1

Examiner: K Moffat Marks: 150 Moderator: N Vazzana Time: 3 hours
______

Instructions:

1.The question paper consists of 10 pages, including this one and a formula sheet.

2.Read the Questions and Instructions carefully, paying attention to mark allocation.

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

4.Number your answers exactly as the questions have been numbered.

5.All the necessary working details must be clearly shown.

6.Calculators MAY be used in this paper.

7.Answers must be rounded off to two decimal places or as stated in each question.

8.Diagrams are not necessarily drawn to scale.

9.Include units as part of ALL your answers.

10.Use your time wisely: spend no more than half an hour on every 25 marks.

SECTION A: [71]

QUESTION 1: [12]

(a)Solve for :

(1)(2)

(2)(2)

(b)Solve for , correct to two decimal places where necessary:

(1)(3)

(2)(5)

QUESTION 2: [7]

(a)For which real values of are the expressions defined?

(1)(1)

(2)(1)

(3)(1)

(b)If , express in terms of(4)

QUESTION 3: [9]

(a)A quadratic function has a maximum value of if
The function value is if .
Determine the function in the form (5)

(b)Draw a sketch of the function:

Show all intercepts with the axes and equations of the asymptotes.(4)

QUESTION 4: [16]

(a)For the series , determine:

(1)The first three terms of the series(2)

(2)The value(s) of = 104(6)

(b)Consider the following geometric sequence:

(1)Determine the common ratio of the sequence in terms of (2)

(2)For which values of will the sequence converge?(3)

(3)Calculate if (3)

QUESTION 5: [9]

(a)Determine in each of the following, leaving answers as positive exponents

(1)(4)

(2)(3)

(b)The cost in rand of producing ounces of gold from a certain mine is given by

and

(1)At what rate is the production cost changing when the th
ounce of gold is produced. (1)

(2)The production cost is increasing at the moment.
How do we know this?(1)

QUESTION 6: [18]

The diagram shows the graph of
and are the -intercepts. and are the turning points.

(a)Calculate the co-ordinates of and (6)

(b)Hence, use the graph to determine the values of which satisfy
the following inequalities simultaneously:
and (2)

(c)If is the image of translated two units upwards, give the equation

defining n the form . (1)

(d)Determine the equation of the tangent to at the point (5)

(e)Determine the co-ordinates of the point of inflection.(4)

SECTION B: [79]

QUESTION 7: [16]

(a)Consider the equation:

(1)Solve for (3)

(2)Hence, or otherwise solve for , if:

(i)(3)

(ii)(3)

(b)Solve the following system of simultaneous equations for and :

and(7)

QUESTION 8: [13]

(a)Consider the following sequence: 20; 1; 15; 2; 10; 4; 5; 8;…

(1)Write down the next two terms of this sequence. (2)

(2)Calculate the sum of the first 25 terms of this sequence. (6)

(b)Derive a formula to determine thenthterm in the following sequence

......

(5)

QUESTION 9: [8]

The diagram below shows the curve defined by , which passes through the points and Refer to the diagram and answer the questions that follow.

(a)Determine the value of (2)

(b)Determine the equation of the inverse of , giving your answer in the
form (2)

(c)Draw a neat sketch graph of the inverse of , showing the co-ordinates
of any intercepts with the axes, as well as the co-ordinates of the point
on the curve where (3)

(d)If is the reflected image of about the , write
down the equation defining .
Give your answer in the form (1)

QUESTION 10: [6]

Determine the area of the largest rectangle which can be drawn in the region having the curve , the and the line as boundaries.
Point has the co-ordinates

(6)

QUESTION 11: [23]

(a)Office equipment initially valued at R 6000000, is now only worth R 4000000.
If the annual rate of depreciation is 8% on a reducing balance, calculate the time
period involved. Give your answer in years and months.(5)
(b)At Vusi’s Reliable Investments, the manager is willing to offer you a range of
interest rates compounded quarterly.At Frikkie’s Foreign Investments you can
get a range of interest rates that is compounded monthly.
If Vusi’s Reliable Investments offers you an interest rate of 9,35% , what is the lowest interest rate that Frikkie’s Foreign Investments must offer you in order to match the deal that Vusi’s Reliable investments can offer you? (4)
(c)You take out a loan of R 750 000 and the bank charges you an interest rate of
p.a. compounded monthly.

You plan to pay off the loan over 20 years, in fixed monthly payments, to be paid at the end of each month, starting in one month’s time.

(1) Calculate the monthly payments you must make if you intend to pay off the
loan in exactly 20 years. Give your answer rounded off correctly to the
nearest cent.(6)

(2) Calculate the interest you will pay on the loan. Give your answer rounded
off correctly to the nearest cent. (2)

(3)Calculate the balance on the loan after 8 years(6)

QUESTION 12: [13]

(a)The probability that Johan’s favourite hiking spot is fully booked for the weekend is 30%. When his favourite hiking spot is fully booked the probability for him hiking is 20%. If his favourite hiking spot is not fully booked the probability that Johan will hike during the weekend becomes 60%. Calculate the probability of Johan hiking during the weekend. (4)

(b)The Lourens family wants to take family pictures. The photographer decides that the following set up will work the best for the three married couples, their 7 children and the one set of grandparents:

The couples need to stand husband and wife together at the back, the grandparents in the middle and the children in the other positions.

(1)How many photos are possible? (5)

(2)What is the probability of the men all appearing on the same side of their

wivesat the same time? (4)

MATHEMATICS

INFORMATION SHEET

; ;

1