Please Read the Following Instructions Carefully

GRADE 12: ADVANCED PROGRAMME MATHEMATICS: TRIAL EXAMINATION: AUGUST 2009 Page 8 of 9

ST MARY’S DSG, KLOOF

GRADE: 12 AUGUST 2009

ADVANCED PROGRAMME MATHEMATICS

TIME: 3 HOURS TOTAL: 300 MARKS

PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY

1. This question paper consists of 9 pages and a formula sheet. Please check that your question paper is complete.

2. This question paper consists of four modules, of which two must be answered.

MODULE 1: CALCULUS AND ALGEBRA (210 MARKS) IS COMPULSORY.

Choose ONE of the THREE Optional Modules:

MODULE 2: STATISTICS OR

MODULE 3: FINANCE AND MODELLING OR

MODULE 4: MATRICES AND GRAPH THEORY (90 MARKS)

3. Non-programmable and non-graphical calculators may be used, unless otherwise indicated.

4. All necessary calculations must be clearly shown and writing should be legible.

5. Diagrams have not been drawn to scale.

6. Write all your answers in the separate Answer Book provided.

NAME:

PLEASE TURN OVER

MODULE 1 CALCULUS AND ALGEBRA (200 MARKS)

QUESTION 1

1.1 (a) Factorise into rational factors (3)

(b) Hence solve for (5)

1.2 If and are both zeros of ,

Solve for fully for , if . (8)

[16 marks]

QUESTION 2

2.1 Write the following as a natural logarithm: (3)

2.2 Solve for :

(a) (4)

(b) , correct to 2 decimal places (4)

(d) (8)

[29 marks]

PLEASE TURN OVER

QUESTION 3

3.1 Given and

Sketch the functions, clearly indicating intercepts and asymptotes (8)

3.2 The sketch shows the curve of intersecting the line at P

and Q.

Determine the -coordinates of the points P and Q, the points of intersection of

and the line . (6)

[14 marks]

QUESTION 4

4.1 If , where and are constants, and , find the values of

and . (12)

4.2 Find (5)

[17 marks]

PLEASE TURN OVER

QUESTION 5

Given the function :

5.1 Determine if is continuous at , and explain fully. If not continuous, name the type of

discontinuity. (8)

5.2 Determine if is differentiable at . (2)

5.3 Assuming that the function is continuous at , determine, with a full explanation, if is

differentiable at , using algebraic methods. You do not have to use first principles. (9)

[19 marks]

QUESTION 6

6.1 Determine the intercept(s) of the graph of . (2)

6.2 Find, and simplify, an expression for , the derivative of . (6)

6.3 Determine the turning points of , and state if they are local maxima or minima. (6)

6.4 Determine the equations of any vertical, horizontal and oblique asymptotes. (12)

6.5 Sketch the graph of , showing all and intercepts and asymptotes. (7)

[33 marks]

QUESTION 7

7.1 Find if (5)

7.2 Find if (7)

[12 marks]

PLEASE TURN OVER

QUESTION 8

8.1 Find if (6)

8.2 Find the equation of the tangent

to the curve at

the point (-1; 2).

(5)

[11 marks]

QUESTION 9

The function is defined by for all .

Starting with , use Newton’s method to find the value of (correct to 4 decimal places) where a turning point occurs. (8)

[8 marks]

QUESTION 10

A cubic curve of the form, , has a point of inflection at (2; -22).

The graph passes through the origin and has a gradient of -3 at the origin.

10.1 Show that . (5)

10.2 By finding the values of and determine the equation of the graph. (7)

[12 marks]

PLEASE TURN OVER

QUESTION 11

A and B are points on the arc of a semi-circle

with equation . AB // -axis. O is the origin.

11.1 If , give the co-ordinates of A.

(2)

11.2 Determine the arc AB if

(4)

11.3 Determine the area of the sector AOB if the

radius was 3 and length from A to B was 3 units.

(5)

[11 marks]

QUESTION 12

12.1 Without the use of a calculator, determine the value of the integral correct to 2

decimal places.

(10)

[10 marks]

QUESTION 13

Given the parabola

and the straight line .

13.1 Determine an approximation (using Upper

Sum mathod) for the area

enclosed by the parabola, the axis and

the lines and ,

by using 4 rectangles of equal width.

(6)

13.2 Calculate the co-ordinates of

point A and B. (4)

13.3 Use integration to find the area of the region enclosed

by the parabola and the straight line.

(6)

13.4 If the parabola , where is rotated about the x-axis through 3600,

determine the volume of the solid which is formed. (6)

[18 marks]

[TOTAL FOR MODULE 1: 210 MARKS]

PLEASE TURN OVER

EITHER

MODULE 2 STATISTICS OR

MODULE 3 FINANCE AND MODELLING

MODULE 4 MATRICES AND GRAPH THEORY (90 MARKS)

QUESTION 1

1.1 Describe the transformation, write down the matrix which maps :

(a) shape A on to shape B. (4)

(b) shape C on to shape D. (5)

1.2 (a) Describe two transformations which, when combined, will map shape

C on to shape B. (5)

(b) Find the single transformation matrix which maps C onto B. (6)

1.3 Determine the image of the point (8; 6) under a rotation of 30º anticlockwise.

Leave your answer in surd form. (6)

[26 marks]

PLEASE TURN OVER

QUESTION 2

2.1 Solve the system of equations, using Gaussian elimination:

(7)

2.2 Given that and ,

a) Find (5)

b) Find the matrix of P, where (8)

[20 marks]

QUESTION 3

The network above shows the “major” dirt roads that are to be graded by a local council in the Karoo. The number on each edge is the length of the road in kilometres.

3.1 List the vertices that have an odd order. (2)

3.2 Starting and finishing at A, find a route of minimum length that covers every road at

least once. You should clearly indicate which, if any, roads will be travelled twice. (8)

3.3 State the total length of your shortest route. (4)

3.4 Use Dijkstra’s shortest path algorithm to find the shortest path from point

B to G. Show your working on the diagram provided and give this length. (8)

[22 marks]

PLEASE TURN OVER

QUESTION 4

The management of a zoo asks for your help to network five computers, one computer at the office and one each at the four entrances. Laying computer cable is expensive so they require you to find the minimum total length of cable required to network the computers.

The adjacency matrix shows the shortest distance, in metres, between the various sites.

OFFICE / ENTRANCE1 / ENTRANCE 2 / ENTRANCE 3 / ENTRANCE 4
OFFICE / - / 938 / 975 / 500 / 1630
ENTRANCE 1 / 938 / - / 2110 / 2505 / 1698
ENTRANCE 2 / 975 / 2110 / - / 606 / 852
ENTRANCE 3 / 500 / 2505 / 606 / - / 532
ENTRANCE 4 / 1630 / 1698 / 852 / 532 / -

4.1 Starting at THE OFFICE, demonstrate the use of Prim’s algorithm and hence find a minimum spanning tree. You must communicate your method fully, indicating the order in which you selected the edges of the network. (10)

4.2 Calculate the minimum total length of the cable required. (2)

4.3 By removing the office, use the LOWER BOUND ALGORITHM, to find a lower bound. Communicate your method fully, indicating in which order you selected the edges of the network and give the total length. (10)

[22 marks]

[TOTAL FOR MODULE 4: 90 MARKS]

[TOTAL: 300 MARKS]