RBHS
Grade 12
Ex: SC
A P Mathematics
September 2012
3 hours
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.
___________________________________________________________
Page 1 of 20
Section A – Algebra and Calculus
Question 1
Given 𝑓(𝑥) = (𝑥 − 1) ln(𝑥 − 1) for 𝑥 > 1 and 𝑔(𝑥) = 𝑒 𝑥 + 1
1.1 Show that 𝑓° 𝑔(𝑥) = 𝑥. 𝑒 𝑥
1.2
Hence solve for 𝑥 if 𝑓° 𝑔(𝑥) = 2𝑥
(4)
(6)
/10/
Question 2
2.1 Prove using Maths Induction that 72𝑛−1 + 32𝑛 is divisible by 8 for
all 𝑛 ∈ ℕ
(12)
2.2
Use Riemann sums to calculate the exact value of the area
bounded by the graph of 𝑓(𝑥) = 6 − 𝑥 2 and the 𝑥-axis, between
the lines 𝑥 = 0 and 𝑥 = 2.
(12)
/24/
Page 2 of 20
Question 3
3.1 Given 𝑓(𝑥) = |2𝑥 + 1|
3.1.1 Solve for 𝑥 if |2𝑥 + 1| = 3
3.2
(4)
3.1.2
Hence solve for 𝑥 if (2𝑥 + 1)2 − |2𝑥 + 1| − 6 = 0
(5)
3.1.3
For what values of 𝑥 will (2𝑥 + 1)2 − |2𝑥 + 1| > 6?
(5)
The graph of the function 𝑔(𝑥) = |𝑥| ln(1 − 𝑥), 𝑥 < 𝑎, is shown
below.
3.2.1
Write down the value of 𝑎.
3.2.2
Sketch 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) 𝑦 = |𝑓(𝑥)|
(4)
3.2.2(b) 𝑦 = 𝑓(|𝑥|)
3.2.2(c) 𝑦 =
1
(2)
(4)
(4)
𝑓(𝑥)
/28/
Page 3 of 20
Question 4
4.1 Given
𝑥
𝑥 2 −1
Break
4.1.2
Hence, calculate the value of
2
1×3
4.2
𝑥
4.1.1
−
𝑥 2 −1
4
3×5
+
into its partial fractions.
6
5×7
−
8
7×9
+ …−
200
199×201
Given 𝑝(𝑥) = 2𝑥 3 + 𝑎𝑥 2 + 𝑏𝑥 − 15 with a zero at 𝑥 = 2 − 𝑖.
Determine the value of 𝑎 and 𝑏.
(8)
(8)
(8)
/24/
Question 5
The functions 𝑓, 𝑔 and ℎ are defined for all real values of 𝑥 by
𝑓(𝑥) = |𝑥|, 𝑔(𝑥) = 3𝑥 + 5 and ℎ(𝑥) = 𝑔(𝑔(𝑥))
5.1
Solve the equation 𝑔(𝑥 + 2) = 𝑓(−12)
(5)
5.2
Find ℎ−1 (𝑥)
(6)
5.3
Determine the values of 𝑥 for which 𝑥 + 𝑓(𝑥) = 0
(5)
/16/
Question 6
Given that 𝑎 + 𝑏 = 2, find the values of 𝑎 and 𝑏 so that the function
𝜋
𝑎𝑥 + 1 𝑖𝑓 𝑥 <
2
𝑓(𝑥) = {
𝜋
sin 𝑥 + 𝑏 𝑖𝑓 𝑥 ≥
2
is continuous for all values of 𝑥.
/9/
Page 4 of 20
Question 7
7.1 Find, leaving your answers completely unsimplified, the following
derivatives if:
7.1.1 𝑦 = (3𝑥 2 − 5)4
(4)
7.1.2
7.2
(4)
𝑓(𝑥) = 𝑥. sec 𝑥
Find the equation of the tangent to the curve 𝑦 3 = 2𝑥𝑦 − 𝑥 2 at the
point (1; 1).
(10)
/18/
Question 8
8.1 Given 𝑓(𝑥) = 4𝑥 − 4 sin 𝑥 − 3𝜋. Use Newton’s method to solve
𝑓(𝑥) = 0. Use 𝑥 = 2 as an initial value. Give your answer correct
to 3 decimal places.
(4)
8.2
r

3
8
area of circle
A chord of a circle which subtends an angle of 𝜃 at the centre cuts
off a segment equal in area to
3
8
of the whole circle.
8.2.1
Show that 4𝜃 − 4 sin 𝜃 = 3𝜋
(4)
8.2.2
Using 8.1 and 8.2.1, solve for 𝜃.
(2)
/10/
Page 5 of 20
Question 9
9.1 Find
9.2
𝜋
4
9.1.1
∫0 (sin 𝑥 + cos 𝑥)2 𝑑𝑥 (show working)
(8)
9.1.2
∫ (1+sin 𝑥)2 𝑑𝑥
(8)
9.1.3
∫ 𝑥 sin(2𝑥) 𝑑𝑥
(10)
cos 𝑥
A normal to the graph of 𝑦 = √𝑥 has the equation 𝑦 = −4𝑥 + 𝑎
where 𝑎 ∈ ℝ.
9.2.1
Show that 𝑎 = 18.
(8)
9.2.2
Find the area of the shaded region.
(6)
/40/
Page 6 of 20
Question 10
The loop 𝑦 2 = 𝑥(5 − 𝑥)2 is shown.
The shaded region is rotated about the 𝑥-axis. Find the volume of the
solid formed.
/8/
Question 11
𝑥 2 −7𝑥+14
Given the graph of 𝑓(𝑥) = (𝑥−1)(𝑥−3). James has done some calculations
and discovered the following:
 there are no real 𝑥-intercepts
 the 𝑦-intercept is
14
3
 the vertical asymptotes are at 𝑥 = 1 and 𝑥 = 3
 there is no oblique asymptote
11.1 Calculate the 𝑥-values at the turning points.
(9)
11.2 Find lim 𝑓(𝑥) and hence write down the equation of the horizontal
𝑥→∞
asymptote.
(4)
/13/
Page 7 of 20
Section B – Matrices and Graph Theory
Question 12
The trapezium T has vertices with co-ordinates (1; 1); (1; 3); (4; 3) and
(3; 1). Shapes A, B, C, D and E are images of T under different
transformations. These transformations are illustrated in the diagram
below.
12.1 Describe in detail the following transformations in words:
12.1.1 T  E
12.1.2 T  B
(3)
(3)
12.2 Quote the matrix that would effect the following transformation:
12.2.1 T  A
(2)
12.2.2 T  C
(2)
12.2.3 T  D
(2)
/12/
Page 8 of 20
Question 13
ABC has co-ordinates A (1; −1); B (1; 1) and C (4; 1)
13.1 Give the co-ordinates of C’, the image of C, if ABC is reflected in
the line 𝑦 = 3𝑥.
(6)
13.2 Give the co-ordinates of A’’, the image of A, if ABC is rotated
130° anti-clockwise about the origin. [i.e. use the original coordinates]
(4)
/10/
Question 14
1 −2 0
14.1 Find the inverse of matrix 𝑃 if 𝑃 = ( 3
1 5)
−1 2 3
(10)
14.2 Solve the following equations simultaneously, using Gaussian
reduction. Be sure to show relevant working in the process of
obtaining solutions.
3𝑥 − 𝑦 + 𝑧 = 2
2𝑥 + 𝑦 − 4𝑧 = −13
𝑥 − 𝑦 − 2𝑧 = −2
Page 9 of 20
(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.
A
14
10
D
4
G
B
8
5
17
15
13
12
E
11
11
7
H
16
9
C
6
F
14
I
15.1 Use 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)
15.2 State the length of you minimum spanning tree.
(2)
15.3 Draw your minimum spanning tree.
(4)
/14/
Page 10 of 20
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.
B
2,5
9
E
7,5
C
1,5
D
9
H
6
6
10,5
10,5
3
4,5
A
G
3
6
J
4,5
F
7,5
3
3
I
16.1 Use 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)
16.2 Write down the corresponding route.
(2)
16.3 A new subway is constructed connecting C to G directly. The time
taken to walk along this subway is 𝑥 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 𝑥.
(6)
/20/
Page 11 of 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.
105
195
90
60
30
C
75
B
D
30
A
120
G
90
F
30
15
E
165
90
120
H
The total length of the streets is 1215 metres.
17.1 Find 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)
17.2 For an optimal Chinese postman route, state:
17.2.1 the number of times that the vertex F would occur.
17.2.2 the number of times that the vertex H would occur.
Page 12 of 20
(2)
(2)
/14/
Question 18
18.1 The complete graph 𝐾𝑛 has every one of its 𝑛 vertices connected
to each of the other vertices by a single edge.
18.1.1 Find the total number of edges in the graph 𝐾5 .
(2)
18.1.2 State the number of edges in a minimum spanning tree for
the graph 𝐾5 .
(2)
18.1.3 State the number of edges in a Hamiltonian circuit for the
graph 𝐾5 .
(2)
18.2 A 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
Page 13 of 20
BLANK PAGE
Page 14 of 20
INFORMATION SHEET
General Formulae
x=
n
n
i 
1 n
i 1
n
i 2 
i 1
x0
x0
 x if
x 
 x if
– b ± b 2 – 4ac
2a
i 1
n(n  1) n 2 n


2
2 2
nn  12n  1 n 3 n 2 n



6
3
2 6
n 2 n  1
n 4 n3 n 2
i 




4
4
2
4
i 1
2
n
3
z  a  bi
z*  a  bi
n A  n B  n  AB
 A
n A  n B  n  
B
n An  n n A
log a x 
log b x
log b a
Calculus
f '( x)  lim
h 0
n
f ( x  h) – f ( x )
h
 f ' g ( x).g ' ( x) dx 
b
 x n1 
x
dx




 n  1 a
a
b
ba n
Area  lim 
  f  xi 
n  n 
i 1
dy dy dt


dx dt dx
f ( g ( x))  c
 f ( x).g ' ( x)dx  f ( x).g ( x)   g ( x). f ' ( x) dx  c
xr 1
f ( xr )
 xr 
f ' ( xr )
b
V    y 2 dx
a
Page 15 of 20
Function
x
Derivative
nx n 1
n
sin x
cos x
cos x
 sin x
tan x
sec 2 x
cot x
cosec x
 cosec 2 x
sec x. tan x
 cosec x. cot x
f ( g ( x))
f ' ( g ( x)). g ' ( x)
f ( x). g ( x)
g ( x). f ' ( x)  f ( x).g ' ( x)
f ( x)
g ( x)
g ( x). f ' ( x)  f ( x). g ' ( x)
g ( x)2
sec x
Trigonometry
1
A  r 2
2
s  r
In ABC:
a
b
c
=
=
sin A sin B sin C
a 2  b 2  c 2 – 2bc. cos A
Area 
sin 2 A  cos 2 A  1
1
ab.sin C
2
1  tan 2 A  sec 2 A
1  cot 2 A  cosec 2 A
sin  A  B  sin A. cos B  cos A sin B
cos A  B  cos A cos B  sin A sin B
sin 2 A  2 sin A cos A
cos 2 A  cos 2 A  sin 2 A
1
sin( A  B)  sin( A  B)
2
1
sin A. sin B  cos( A  B)  cos( A  B )
2
1
cos A. cos B  cos( A  B)  cos( A  B)
2
sin A. cos B 
Matrix Transformations
 cos 

 sin 
 sin  

cos  
 cos 2

 sin 2
sin 2 

 cos 2 
Page 16 of 20
NAME_______________________
ADVANCED PROGRAMME MATHEMATICS
GRADE 12
SEPTEMBER 2012
RONDEBOSCH BOYS’ HIGH SCHOOL
𝒆𝒊𝝅 + 𝟏 = 𝟎
Question
1
2
3
4
5
6
7
8
9
Max.
marks
10
24
28
24
16
9
18
10
40
Question
10
11
12
13
14
15
16
17
18
Max.
marks
8
13
12
10
20
14
20
14
10
Actual
marks
Actual
marks
TOTAL MARK
300
Page 17 of 20
3.2.2(a)
3.2.2(b)
3.2.2(c)
Page 18 of 20
15.
14
10
13
5
6
12
11
G
11
B
F
9
7
H
E
C
6
D
I
9
G
10,5
9
H
6
1,5
14
3
4,5
10,5
16
3
2,5
7,5
C
E
4
A
15
17
D
16.
B
8
A
6
J
4,5
F
7,5
Page 19 of 20
3
3
I
17.
105
195
90
60
30
C
75
B
D
30
A
120
G
90
F
30
15
E
Page 20 of 20
165
90
120
H