RBHS

September 2012

Grade 12

A P Mathematics

3 hours

Ex: SC

300 marks

Instructions:

Answer all questions.

All necessary working must be shown in its proper place with the answer.

A calculator may be used unless specified otherwise.

Give answers to two decimal places, where applicable.

Blue or black pen must be used in answers although pencil may be used on diagrams.

The use of correcting fluid is not allowed.

This examination paper consists of 13 pages, a formula sheet and a 4 page answer insert.

Place the answer insert inside your answer book.

___________________________________________________________

Section A – Algebra and Calculus

Question 1

Given for and

1.1Show that (4)

1.2Hence solve for if (6)

/10/

Question 2

2.1Prove using Maths Induction that is divisible by for all (12)

2.2Use Riemann sums to calculate the exact value of the area bounded by the graph of and the -axis, between the lines and .(12)

/24/

Question 3

3.1Given

3.1.1Solve for if (4)

3.1.2Hence solve for if (5)

3.1.3For what values of will ?(5)

3.2The graph of the function , , is shown below.

3.2.1Write down the value of .(2)

3.2.2Sketch the following graphs in your answer book. You do not need to work out any values – simply show how the shape changes.

3.2.2(a) (4)

3.2.2(b) (4)

3.2.2(c) (4)

/28/

Question 4

4.1Given

4.1.1Break into its partial fractions.(8)

4.1.2Hence, calculate the value of

(8)

4.2Given with a zero at .

Determine the value of and .(8)

/24/

Question 5

The functions , and are defined for all real values of by

, and

5.1Solve the equation (5)

5.2Find (6)

5.3Determine the values of for which (5)

/16/

Question 6

Given that , find the values of and so that the function

is continuous for all values of .

/9/

Question 7

7.1Find, leaving your answers completely unsimplified, the following derivatives if:

7.1.1(4)

7.1.2(4)

7.2Find the equation of the tangent to the curve at the point .(10)

/18/

Question 8

8.1Given . Use Newton’s method to solve . Use as an initial value. Give your answer correct to 3 decimal places.(4)

8.2

( area of circle)

A chord of a circle which subtends an angle of at the centre cuts off a segment equal in area to of the whole circle.

8.2.1Show that (4)

8.2.2Using 8.1 and 8.2.1, solve for .(2)

/10/

Question 9

9.1Find

9.1.1 (show working)(8)

9.1.2(8)

9.1.3(10)

9.2A normal to the graph of has the equation where .

9.2.1Show that .(8)

9.2.2Find the area of the shaded region.(6)

/40/

Question 10

The loop is shown.

The shaded region is rotated about the -axis. Find the volume of the solid formed.

/8/

Question 11

Given the graph of . James has done some calculations and discovered the following:

· there are no real -intercepts

· the -intercept is

· the vertical asymptotes are at and

· there is no oblique asymptote

11.1Calculate the -values at the turning points.(9)

11.2Find and hence write down the equation of the horizontal asymptote.(4)

/13/

Section B – Matrices and Graph Theory

Question 12

The trapezium T has vertices with co-ordinates ; ; and . Shapes A, B, C, D and E are images of T under different transformations. These transformations are illustrated in the diagram below.

12.1Describe in detail the following transformations in words:

12.1.1T E(3)

12.1.2T B(3)

12.2Quote the matrix that would effect the following transformation:

12.2.1T A(2)

12.2.2T C(2)

12.2.3T D(2)

/12/

Question 13

ABC has co-ordinates A ; B and C

13.1Give the co-ordinates of C’, the image of C, if ABC is reflected in the line .

(6)

13.2Give the co-ordinates of A’’, the image of A, if ABC is rotated anti-clockwise about the origin. [i.e. use the original co-ordinates](4)

/10/

Question 14

14.1Find the inverse of matrix if (10)

14.2Solve the following equations simultaneously, using Gaussian reduction. Be sure to show relevant working in the process of obtaining solutions.

(10)

/20/

Question 15

A diagram has been given for you to use in your answer book.

The following network shows the lengths, in kilometres, of roads connecting nine villages, A, B, …, I.

15.1Use Prim’s algorithm starting from E, showing the order in which you select the edges, to find a minimum spanning tree for the network.(8)

15.2State the length of you minimum spanning tree.(2)

15.3Draw your minimum spanning tree.(4)

/14/

Question 16

A diagram has been given for you to use in your answer book.

The network below shows some paths on an estate. The number on each edge represents the time taken, in minutes, to walk along a path.

16.1Use Dijkstra’s algorithm on the network to find the minimum walking time from A to J. You must show evidence that you have used Dijkstra’s algorithm.(12)

16.2Write down the corresponding route.(2)

16.3A new subway is constructed connecting C to G directly. The time taken to walk along this subway is minutes. The minimum time taken to walk from A to G is now reduced, but the minimum time taken to walk from A to J is not reduced. Find the range of possible values for .(6)

/20/

Question 17

A diagram has been given for you to use in your answer book.

Benny delivers newspapers to houses on an estate. The network shows the streets on the estate. The number on each edge shows the length of the street, in metres.

Benny starts from the newsagents located at vertex A, and he must walk along all the streets at least once before returning to the newsagents.

The total length of the streets is 1215 metres.

17.1Find the length of an optimal Chinese postman route around the estate, starting and finishing at A. (You are given that the shortest distance from G to B is 210, and the shortest distance from A to H is 150)(10)

17.2For an optimal Chinese postman route, state:

17.2.1the number of times that the vertex F would occur.(2)

17.2.2the number of times that the vertex H would occur.(2)

/14/

Question 18

18.1The complete graph has every one of its vertices connected to each of the other vertices by a single edge.

18.1.1Find the total number of edges in the graph .(2)

18.1.2State the number of edges in a minimum spanning tree for the graph .(2)

18.1.3State the number of edges in a Hamiltonian circuit for the graph .(2)

18.2A simple graph G has six vertices and nine edges, and G is an Eulerian circuit. Draw a sketch to show a possible graph of G.(4)

/10/

TOTAL MARKS: 300

BLANK PAGE

INFORMATION SHEET

General Formulae

Calculus

FunctionDerivative

Trigonometry

In ABC:

Matrix Transformations

NAME_______________________

ADVANCED PROGRAMME MATHEMATICS

GRADE 12

SEPTEMBER 2012

RONDEBOSCH BOYS’ HIGH SCHOOL

Question

1

2

3

4

5

6

7

8

9

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10

24

28

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16

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11

12

13

14

15

16

17

18

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8

13

12

10

20

14

20

14

10

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TOTAL MARK

300

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