A World of Education…an Education for the World!
GRADE 12 – ADVANCED PROGRAMME MATHEMATICS
EXAMINER: Mr I.L. Atteridge
DATE: 15 July 2011
MODERATOR: Ms B. Maganbhai
TOTAL: 300 marks
TIME: 3 hours
CANDIDATE’S NAME:..................................................................................
CANDIDATE’S MATHEMATICS TEACHER:......................................................
INSTRUCTIONS TO CANDIDATES:
1. Answer all questions in the answer booklet provided.
2. All written work must be done using blue or black ink. Diagrams and graphs must be
drawn neatly using pencil.
3. No correction fluids or tape may be used
4. Scientific, non-programmable calculators may be used unless otherwise stated.
5. Round off to TWO decimal places unless otherwise stated.
6. It is in your own interests to work neatly and to show all necessary steps in calculations.
7. 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 10
SECTION A – ALGEBRA
QUESTION 1
a)
It is given that: 𝑓(𝑥) = 𝑥 4 − 2𝑥 3 + 𝑥 2 − 8𝑥 − 12. If (𝑥 − 3) and (𝑥 + 1)
are factors of 𝑓(𝑥), solve for 𝑓(𝑥) = 0 for 𝑥 ∈ ℂ .
b)
(10)
Solve the following equations for 𝑥 ∈ ℝ:
24
i
𝑥 2 − 𝑥 = 14 − 𝑥(𝑥−1)
(6)
ii
ln|3 − 𝑥| = 1
(3)
iii
iv
√𝑥+3
𝑥 2 −9
≥0
(5)
ln(𝑒 2𝑥−20 ) = 𝑥
(5)
[29]
QUESTION 2
a)
Write the nth term of the following sequence: 1 × 2; 2 × 3; 3 × 4; …
b)
Use mathematical induction to prove the following:
1
1 × 2 + 2 × 3 + 3 × 4 + ⋯ + 𝑛(𝑛 + 1) = 3 𝑛(𝑛 + 1)(𝑛 + 2) for all 𝑛 ∈ ℕ0
(2)
(14)
[16]
QUESTION 3
a)
The function 𝑓 is defined as follows: 𝑓(𝑥) = {
i.
b)
𝑎𝑥 2 + 1 𝑖𝑓 𝑥 ≥ 2
3𝑥 − 2 𝑖𝑓 𝑥 < 2
Determine the value of 𝑎 if the function is continuous for all real
value of 𝑥.
(5)
3
(6)
ii.
If = 4 , prove that 𝑓(𝑥) is differentiable at 𝑥 = 2
i.
On the same system of axes, make neat sketch graphs of the functions
4
defined by: 𝑝(𝑥) = 2|𝑥 + 3| − 4 and 𝑞(𝑥) = | | clearly showing
𝑥
all turning points, intercepts with the axes etc.
ii.
(8)
Hence, using your sketch or by any other means, find the exact solution to:
4
2|𝑥 + 3| − 4 ≥ |𝑥|
(12)
[31]
AP Mathematics
Grade 12
15 July 2011
Page 2 of 10
QUESTION 4
a)
Prove the identity: sec 𝑥 ∙ cosec 𝑥 − cot 𝑥 = tan 𝑥
(6)
b)
The diagram shows the cross-section of a wooden log, of radius 50cm, floating in water.
10cm
50cm
𝜃
50cm
i.
Show that 𝜃 = 1,287 radians.
(4)
ii.
What area of the cross-section of the log is above the water-line?
(8)
[18]
Oh, dear… Surds and you are not friends.
AP Mathematics
Grade 12
15 July 2011
Page 3 of 10
SECTION B - CALCULUS
QUESTION 5
a)
Differentiate the following with respect to 𝑥. You need not simplify your answers but all
exponents must be positive.
i.
ii.
b)
4
(4)
𝑎(𝑥) = √2𝑥 2 + 𝜋
𝑏(𝑥) =
𝑥 4 +1
(5)
2𝑥 3 +3𝑥 −2
1
The volume the cone shown below is given by 𝑉 = 3 𝜋𝑎2 cos2 𝜃 sin 𝜃
𝜋
where 0 < 𝜃 < 2 and 𝑎 is a constant (𝑎 ≠ 0).
Find the maximum value of 𝑉 in terms of 𝑎. Give your answer in radians
correct to two decimal places.
(14)
𝑎
𝜃
2
c)
Determine from first principles 𝑓 ′ (𝑥) where 𝑓(𝑥) =
d)
i.
Find 𝑑𝑥 if 𝑥 3 𝑦 + 𝑥𝑦 3 = 2
ii.
Hence, find the equation of the tangent to the curve at the
√𝑥+5
𝑑𝑦
point (1; 1)
(8)
(7)
(3)
[41]
AP Mathematics
Grade 12
15 July 2011
Page 4 of 10
QUESTION 6
Determine the following without the use of a calculator:
a)
∫ 𝑥 2 ∙ √𝑥 3 + 2 ∙ 𝑑𝑥
𝜋
2
b)
∫ cos 3𝑥 ∙ 𝑐𝑜𝑠𝑥 ∙ 𝑑𝑥
c)
∫
−
𝜋
2
cos 𝑥
∙ 𝑑𝑥
(sin 𝑥 − 1)2
(6)
(9)
(6)
[21]
QUESTION 7
Determine the area enclosed between the graphs of 𝑓(𝑥) = −𝑥 2 + 9 and 𝑔(𝑥) = 𝑥 + 7 without
using a calculator.
[10]
AP Mathematics
Grade 12
15 July 2011
Page 5 of 10
QUESTION 8
a)
Show that if
𝑎
∫ (8𝑥 3 − 7)𝑑𝑥
= 10 then 2𝑎4 − 7𝑎 − 28 = 0
(6)
2
b)
Hence, use Newton’s Method, correct to five decimal places, to calculate the value
𝑎
close to 2,2 of:
∫ (8𝑥 3 − 7)𝑑𝑥
= 10
2
(8)
[14]
QUESTION 9
Find the area under the curve 𝑓(𝑥) = 2𝑥 − 𝑥 2 + 4 between 𝑥 = 0 and 𝑥 = 3
using 𝑛 strips of equal width, the Riemann sum, and letting 𝑛 → ∞.
[20]
AP Mathematics
Grade 12
15 July 2011
Page 6 of 10
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.
b)
Starting and finishing at A, find a route of minimum length that covers every
(2)
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]
AP Mathematics
Grade 12
15 July 2011
Page 7 of 10
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.
Bothaville
51
Kroonstad
Bloemhof
62
41
25
44
29
Hoopstad
52
47
Welkom
Christiana
33
25
20
38
46
Hertzogville
62
45
Warrenton
55
Bultfontein
48
36
60
35
35
Kimberley
31
Dealesville
82
Winburg
Brandfort
62
Boshof
36
58
70
49
46
Bloemfontein
Determine the quickest route, and state the minimum time taken, between:
a)
b)
Bothaville to Dealesville (The number of routes is restricted. Refer to your answer
booklet.)
(10)
Kimberley and Welkom
(14)
[24]
AP Mathematics
Grade 12
15 July 2011
Page 8 of 10
QUESTION 12
Triangle A is shown on the grid.
a)
Triangle A is mapped onto B by a reflection in the 𝑥-axis. Determine a matrix that gives
the resultant co-ordinates.
b)
Triangle A is mapped onto C by a reflection in the line 𝑦 = 𝑥. Determine a matrix that
gives the resultant co-ordinates.
c)
(4)
(4)
Triangle A is mapped onto D by a stretch of scale factor −2, invariant line the 𝑦-axis.
Determine a matrix that gives the resultant co-ordinates AND draw and label D on the
grid.
(6)
[14]
QUESTION 13
The point (2; 3) on the Cartesian is mapped onto the point (2; 11) by the transformation
𝑝
described by the matrix (
𝑞
𝑞−4
).
𝑝
a)
Represent this information as a matrix equation.
(2)
b)
Solve for 𝑝 and 𝑞 by first setting up a system of equations.
(9)
c)
Hence, describe fully the transformation that has taken place.
(4)
[15]
AP Mathematics
Grade 12
15 July 2011
Page 9 of 10
QUESTION 14
y
N
A(7; 24)
M(-15; 20)
B
O
x
In the diagram, ∆𝑂𝐴𝐵 is rotated anticlockwise about the origin and mapped onto∆𝑂𝑀𝑁. The
point A has coordinates (7; 24) and the point M has coordinates (−15; 20)
(a)
Write down the gradients of 𝑂𝐴 and 𝑂𝑀respectively.
(2)
(b)
Hence, show that the angle of rotation is 53,1𝑜 , correct to one decimal place. (5)
(c)
Hence, find the matrix which represents the rotation from 𝑂𝐴𝐵 to 𝑂𝑀𝑁.
(8)
(d)
If the coordinates of 𝐵 are (10; 20) find the coordinates of 𝑁.
(6)
[21]
AP Mathematics
Grade 12
15 July 2011
Page 10 of 10