**GRADE12–ADVANCED PROGRAMME MATHEMATICS**

**EXAMINER: Mr I.L. AtteridgeDATE**:15 July 2011

**MODERATOR: Ms B. MaganbhaiTOTAL**:300marks

TIME: hours

**CANDIDATE’S NAME:...... **

**CANDIDATE’S MATHEMATICS TEACHER:...... **

**INSTRUCTIONS TO CANDIDATES:**

- Answer all questions in the answer booklet provided.
- All written work must be done using blue or black ink. Diagrams and graphs must be drawn neatly using pencil.
- No correction fluids or tape may be used
- Scientific, non-programmable calculators may be used unless otherwise stated.
- Round off to TWO decimal places unless otherwise stated.
- It is in your own interests to work neatly and to show all necessary steps in calculations.
- Insert your question paper into the back of your answer booklet when handing in.

**THIS EXAMINATION CONSISTS OF 10 PAGES**

AP Mathematics

Grade 12

15 July 2011

Page 1 of 12

**SECTION A – ALGEBRA**

QUESTION 1

a)It is given that: . If and

are factors of , solve for for .(10)

b)Solve the following equations for :

i(6)

ii(3)

iii(5)

iv(5)

[29]

QUESTION 2

a)Write the nth term of the following sequence: (2)

b)Use mathematical induction to prove the following:

for all (14)

[16]

QUESTION 3

a)The function is defined as follows:

i.Determine the value of if the function is continuous for all real

value of (5)

ii.If , prove that is differentiable at (6)

b)i.On the same system of axes, make neatsketch graphs of the functions

defined by: and clearly showing

all turning points, intercepts with the axes etc.(8)

ii.Hence, using your sketch or by any other means, find the exact solution to:

(12)

[31]

QUESTION 4

a)Prove the identity: (6)

b)The diagram shows the cross-section of a wooden log, of radius 50cm, floating in water.

i.Show that radians.(4)

ii. What area of the cross-section of the log is above the water-line?(8)

[18]

**SECTION B - CALCULUS**

QUESTION 5

a)Differentiate the following with respect to . You need not simplify your answers but all exponents must be positive.

i.(4)

ii.(5)

b)The volume the cone shown below is given by

where and is a constant .

Find themaximum value of in terms of . Give your answer in radians

correct to two decimal places.(14)

c)Determine from first principles where (8)

d)i.Find if (7)

ii.Hence, find the equation of the tangent to the curve at the

point(3)

[41]

QUESTION 6

Determine the following without the use of a calculator:

a)(6)

b)(9)

c)(6)

[21]

QUESTION 7

Determine the area enclosed between the graphs of and without using a calculator.

[10]

QUESTION 8

a)Show that if then (6)

b)Hence, use Newton’s Method,correct to five decimal places, to calculate the value

close to of:(8)

[14]

QUESTION 9

Find the area under the curve between and

using strips of equal width, the Riemann sum, and letting .[20]

**SECTION C – MATRICES AND GRAPH THEORY**

QUESTION 10

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.

a)List the vertices that have odd order.(2)

b)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 at least twice. (14)

c)State the length of your shortest route. (4)

d)There is a 6,4km long minor road that is not shown on the network between

B and D. Decide whether or not it is sensible to include BD as part of the main

grading route. Give reasons for your answer. (6)

[26]

QUESTION 11

Use the grids in your answer booklet to answer the following question.

The graph below represents the time it takes to travel between towns in the central Free State. The time is dependent on the distance between the towns and the quality of the roads.

Determine the quickest route, and state the minimum time taken, between:

a)Bothaville to Dealesville (The number of routes is restricted. Refer to your answer booklet.) (10)

b)Kimberley and Welkom(14)

[24]

QUESTION 12

Triangle A is shown on the grid.

a)Triangle A is mapped onto B by a reflection in the -axis. Determine a matrix that gives the resultant co-ordinates. (4)

b)Triangle A is mapped onto C by a reflection in the line Determine a matrix that gives the resultant co-ordinates. (4)

c)Triangle A is mapped onto D by a stretch of scale factor , invariant line the -axis. Determine a matrix that gives the resultant co-ordinates AND draw and label D on the grid. (6)

[14]

QUESTION 13

The point on the Cartesian is mapped onto the point by the transformation described by the matrix .

a)Represent this information as a matrix equation.(2)

b)Solve for and by first setting up a system of equations.(9)

c)Hence, describe fully the transformation that has taken place.(4)

[15]

QUESTION 14

In the diagram, is rotated anticlockwise about the origin and mapped onto. The point A has coordinates and the point M has coordinates

(a)Write down the gradients of and respectively.(2)

(b)Hence, show that the angle of rotation is , correct to one decimal place.(5)

(c)Hence, find the matrix which represents the rotation fromto .(8)

(d)If the coordinates of are find the coordinates of .(6)

[21]

AP Mathematics

Grade 12

15 July 2011

Page 1 of 12