MATHEMATICS PRELIMINARY EXAMINATION – PAPER I

20th AUGUST 2013

GRADE 12

TIME: 3 HOURSTOTAL MARKS: 150

EXAMINERS: L. le ROUX,MODERATORS: M. SOLOMONS, M. SOLOMONS, L. le ROUX,

A. LEVIN A. LEVIN

NAME:______

CLASS:______

Instructions

  1. This question paper consists of 10 pages and a Formulae Sheet.

Please check that your paper is complete.

2.Read the questions carefully.

3.Answer ALL the questions.

4.Number your answers as the questions are numbered.

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

6.Approved non-programmable and non-graphical calculators may be used.

7. Set your calculator to Degree measure where appropriate.

8.Give answers correct to TWO decimal digits, where necessary.

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

QUESTION

/ MAXIMUM MARKS / MARKS OBTAINED
1 / 17
2 / 17
3 / 25
4 / 12
5 / 5
6 / 17
7 / 38
8 / 6
9 / 13
QUESTION 1

Solve for x:

1.1 (5)

1.2 if (3)

1.3 (4)

1.4 (5)

[17]

QUESTION 2

2.1Write down the third term of the sequence if it has a constant ratio: . (3)

2.2A ladder has 11 rungs, decreasing in length from top

to bottom by 2cm. If the top runghas a length of 50cm,

calculate:

2.2.1which rung, counting from the bottom will have a length of 38cm. (3)

2.2.2 the sum of the lengths of the rungs of the ladder. (3)

2.3The figure below shows the beginning of a long row of cubes:

2.3.1What is the surface area of the 7th cube? (4)

2.3.2Give a general formula for the volume of a cube in this sequence. (4)

[17]

QUESTION 3

3.1What nominal rate, with monthly compounding, is equivalent to a nominal rate of

13% p.a. compounded semi-annually? (3)

3.2Tamsyn wishes to buy a car in 3 years’ time that currently costs R180000. The priceof the car rises with inflation at 6% p.a. The value of her current car is R125000 andit will depreciate to R68 750in 3 years’ time. Tamsyn intends to trade in her car in 3 years’ time as part payment for the new car.

3.2.1Calculate the rate of depreciation if it is calculated according to the straight line method. (2)

3.2.2How much extra money will Tamsyn need to pay for the new car in 3 years’ time? (3)

3.2.3If Tamsyn saves this amount by making equal monthly instalments starting immediately, at 15% p.a. compounded monthly, calculate the amount of the instalments. (4)

3.3Jason buys a house and takes out a loan for R875 000. The interest rate is 10,5% p.a. compounded monthly.

3.3.1 Calculate the monthly repayments if the loan is to be paid over 20 years, starting one month after Jason takes out the loan. (4)

3.3.2How much money does he pay in total over 20 years to repay the loan? (2)

3.3.3Calculate the monthly repayments if Jason starts paying the loan exactly one year from the time it was taken out, and finishes as scheduled 20 years from the time it was taken out. (5)

3.3.4How much extra will Jason pay if he starts his payments after a year instead of after the first month? (2)

[25]

QUESTION 4

Given

4.1Write down the equation of the asymptote of (2)

4.2Solve for x if (2)

4.3Draw a neat sketch graph of f, showing the x- and y-intercepts and one other point. (4)

4.4Give the equation of the reflection of f in the line in the form (3)

[12]

QUESTION 5

The straight line intersects the parabola at

and. Calculate the values of a, b, m and c. [5]

QUESTION 6

The figure shows the functions and

Determine:

6.1thevalues of a, b and c. (6)

6.2 the coordinates of D, the turning point of the parabola by writing theequation in the form:

. (3)

6.3the length of OS if TR= 10 units (4)

6.4the minimum positive value of if the graph of f is reflected in the xaxis to give h. (2)

6.5the equation of j, the reflection of g in the line (2)

[17]

QUESTION 7

7.1Given .

7.1.1Determine from first principles. (5)

7.1.2Hence, find the equation of the tangent to f at the point where . (4)

7.27.2.1Determine giving your answer with positive exponents. (5)

7.2.2If , find . (3)

7.2.3 Determine if (5)

7.3 Given

7.3.1Prove that is a root of and hence, determine the x-intercepts of g showing all working. (5)

7.3.2Determine the coordinates of the stationary points. (3)

7.3.3State the nature of these stationary points. (3)

7.3.4 Sketch g clearly labelling all significant points. (3)

7.3.5If , write down the values of k for which h has two negative

roots and one positive root. (2)

[38]

QUESTION 8

Consider the diagrams below:

Justifying your answers, match each graph to the appropriate equation. Give your answers in the form

[6]

QUESTION 9

A cell phone company makes use of 2 types of sim cards namely a normal sized (N) and a micro sim card (M). Let x represent the number of normal sim cards, and y the number of micro sim cards and the shaded area in the diagram the feasible region.

9.1Write down the constraints that lead to the feasible region above. (5)

9.2If the profit on type N is 300c and on type M 600c write down an equation for the profit function. (1)

9.3With the aid of the graph determine the number of each sim card that needs to be sold to ensure a maximum profitand determine the maximum profit. (3)

9.4It takes workers 2 minutes to assemble a normal sim card and 3 minutes to assemble a micro sim card and the factory operates for a maximum of 12 hours daily. What will the maximum profit be if the factory is closed for 3 hours due to a safety inspection on a particular day? (4)

[13]

1