ST MARY’S DSG, KLOOF
GRADE: 12
AUGUST 2011
ADVANCED PROGRAMME MATHEMATICS
TIME: 3 HOURS
TOTAL: 300 MARKS
MRS J. KINSEY
NAME:______________________________________________
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY
1.
This question paper consists of 10 pages and an insert of 1page containing formula sheets.
2.
Please check that your question paper is complete.
3.
Detach APPENDIX A and hand it in with your answer sheet.
4.
This question paper consists of two modules,
MODULE 1: CALCULUS AND ALGEBRA (200 MARKS)
MODULE 2: STATISTICS (100 MARKS)
5.
Non-programmable and non-graphical calculators may be used, unless otherwise indicated.
6.
All necessary calculations must be clearly shown and writing should be legible.
7.
Diagrams have not been drawn to scale.
8.
Write all your answers in the separate Answer Book provided.
9.
Write all your answers to 2 decimal places, unless otherwise indicated.
Page 2 of 10
Grade 12 AP Maths Trials Examination August 2011
MODULE 1
CALCULUS AND ALGEBRA
QUESTION 1
Prove by induction that n3 + 2n is divisible by 3 for all 𝑛 𝜖 Natural Numbers.
(8)
[8]
QUESTION 2
a) Solve for 𝑥:
b)
(6)
1)
𝑒 2−𝑥 = 4
2)
(𝑙𝑛 𝑥)2 = 𝑙𝑛 ( 𝑥 )
𝑒2
(6)
In a biology experiment the growth of a culture of cells is investigated. The
number of cells, N, at the time t (in hours) is given by the formula:
N=
108
1+4 𝑒 −0,1𝑡
1)
How many cells were there initially?
2)
After how many hours will there be 5 x 107 cells?
3)
How many cells will there be after a long time i.e. as t  ∞
4)
What is the average rate of increase in the number of cells between t=0
and t = 10 hours?
5)
What is the instantaneous rate of increase at time t = 10?
𝑑
Hint: 𝑑𝑡 (𝑒 𝑡 ) = 𝑒 𝑡
(10)
[22]
QUESTION 3
Given that 𝑥 = 1 − 𝑖√3 and 𝑦 = 4 + 𝑖
Find the value of 𝑘 so that 𝑘𝑦 + (𝑥̅ )2 is purely imaginary
(and 𝑥̅ is the complex conjugate of 𝑥.)
[8]
Page 3 of 10
Grade 12 AP Maths Trials Examination August 2011
QUESTION 4
Find values for the constants a and b such that the following function is differentiable on
(0;3)
2𝑥 + 𝑎
𝑓(𝑥) = { 2
𝑏𝑥 + 4
0≤𝑥≤1
1<𝑥≤3
[10]
QUESTION 5
a)
Use the axes provided on APPENDIX A to sketch graphs of the functions
4
defined by 𝑓(𝑥) = 2|𝑥 + 3| − 4 and 𝑔(𝑥) = | |
𝑥
(8)
Clearly indicating all turning points, intercepts with axes etc.
b)
Hence, using your sketch or otherwise, find the EXACT solution (s) to
4
(12)
2|𝑥 + 3|−4 ≥| 𝑥|
[20]
QUESTION 6
Consider the rational function
𝑓(𝑥) =
𝑥 2 −4𝑥+4
𝑥 2 +𝑥−6
a)
Determine the equations of any asymptotes that the graph may have.
b)
Determine the values of 𝑥 for which 𝑓 ′ (𝑥) > 0
c)
Use the grid on APPENDIX A to sketch the graph of f, indicating clearly on
your graph the x and y intercepts as well as any asymptotes the graph may
have.
(8)
(10)
(8)
[26]
Page 4 of 10
Grade 12 AP Maths Trials Examination August 2011
QUESTION 7
a)
Determine from first principles, the derivative of 𝑓 (𝑥) =
b)
Show that 𝐷𝑥[𝑐𝑜𝑠 4 𝑥 − sin4 𝑥] = −2 sin 2𝑥
c)
Determine
d)
If 𝑦 = 𝐴𝑥 2 , write down expressions for:
1)
2)
7.
e)
𝑑𝑦
𝑑𝑥
1
√𝑥
(8)
(6)
if 𝑦 = √𝑥 sec 𝑥
(5)
𝑑𝑦
𝑑𝑥
𝑑2𝑦
(2)
𝑑𝑥 2
Hence show that
𝑑2𝑦
𝑥
2)
𝑥 2 𝑑𝑥 2 = 2𝑦
𝑑𝑥 2
=
𝑑𝑦
1)
𝑑𝑥
𝑑2 𝑦
(6)
[27]
QUESTION 8
A particle P moves in a circular trajectory, anticlockwise with a radius of 2m. It started at A.
P
2
𝜃
A
a)
Give the distance S, travelled when it reaches P, in terms of 𝜃.
b)
Calculate 𝜃 when it reaches its maximum speed, , if the speed of the particle as a
function of the angle 𝜃 is given by the equation 𝑣( 𝜃) =
c)
𝜃
𝜃2 +1
Calculate how far it has travelled when it reaches its maximum speed.
(2)
(10)
(2)
[14]
Grade 12 AP Maths Trials Examination August 2011
Page 5 of 10
QUESTION 9
A rough sketch of the graph 𝑦 = 𝑥 3 − 6𝑥 2 + 9𝑥 − 1 is shown. There are 𝑥 intercepts at the
point A; B and C and stationary points at (1;3) and (3;-1) as indicated.
Use the Newton Raphson formula to find the 𝑥 intercept at Point B correct to 6
decimal places.
[10]
QUESTION 10
a)
b)
Determine the following integrals
1)
∫ 𝑥√3 − 𝑥 𝑑𝑥
(6)
2)
∫ 𝑥 cos(2𝑥) 𝑑𝑥
(5)
3)
∫ sin(𝑥 + 1) sin(3𝑥 − 2)𝑑𝑥
(8)
Find the area under the curve f(x) = -x2 + 2x + 4 between x = 0 and x = 3 using n
strips of equal width, the Riemann Sum and then letting n → ∞
(16)
[35]
Page 6 of 10
Grade 12 AP Maths Trials Examination August 2011
QUESTION 11
A rectangular wooden beam is to be cut from a uniform circular log of radius r metres (see
diagram). A detail of the cross section is also given.
x
r
b
a)
b)
The stiffness of the beam(s) is given by the equation 𝑠 = 𝑘𝑏𝑥 3 , where 𝑘 is a constant
and 𝑏 and 𝑥 are the breadth and length of the beam respectively (see sketch).
1)
Find b in terms of r and x
2)
Show that 𝑠 = √(4𝑟 2 − 𝑥 2 ). 𝑥 3
(8)
Hence calculate the value of 𝑥 of the stiffest beam that can be cut from a log of
radius 0,2m (i.e. maximise).
Give the answer correct to 2 decimal places
(It is not necessary to show that this answer will give a maximum).
(12)
[20]
Grade 12 AP Maths Trials Examination August 2011
Page 7 of 10
MODULE 2
QUESTION 1
The letters of the word PARALLELOGRAM are arranged to form words with the same number
of letters.
a)
b)
How many new words can be made if the words don’t necessarily have to make
sense?
What is the probability that a word formed from the letters of the word
PARALLELOGRAM start and ends with the same letter?
(6)
(12)
[18]
QUESTION 2
A small packet of Jelly Tots contains 5 pink, 7 yellow, and 3 green sweets. Only these 3
colours are represented in the packet. If I remove 3 jelly tots, one at a time, WITHOUT
replacement, calculate:
a)
The probability that all three are pink.
(8)
b)
The probability that I will get one of each colour.
(8)
[16]
QUESTION 3
A coffee machine is regulated so that it dispenses an average of 200ml of coffee per cup. If
the amount of coffee is normally distributed, with a standard deviation of 6ml
a)
What is the probability that a cup contains between 197ml and 203ml?
b)
How many cups are expected to overflow if 210ml cups are used for the next 100
drinks?
(8)
(10)
[18]
QUESTION 4
If 𝑃(𝑋 = 𝑥) =
2𝑥+1
𝑘
where 𝑥 ∈ {2; 3; 4; 5}
a)
Find the value of 𝑘.
(7)
b)
The mean of any discrete probability function is defined to be 𝑥̅ = ∑ 𝑥. 𝑃(𝑋 = 𝑥)
find the mean.
(3)
[10]
Page 8 of 10
Grade 12 AP Maths Trials Examination August 2011
QUESTION 5
The probability density function of a continuous random variable is given by
𝑓(𝑥) = 3𝑘𝑥 2 + 𝑘
0≤𝑥≤2
a)
Evaluate 𝑃(𝑋 = 1)
(1)
b)
Find the value of 𝑘.
(5)
c)
Find 𝑃(1 ≤ 𝑋 < 2)
(4)
[10]
QUESTION 6
Consider a sample of 100 women with a mean weight of 61,236kg and a standard deviation of
5,433kg. Find a 92% confidence interval for the mean weight of the women.
[8]
QUESTION 7
In a sample of 400 shops it was discovered that 136 of them sold television sets below the list
prices that had been recommended by the manufacturers. Calculate the 95% confidence
limits for this estimate.
[8]
QUESTION 8
A group of 10 students obtained the following marks for the Preliminary examinations in
Mathematics for Paper 1 and Paper 2.
Student
A
B
C
D
E
F
G
H
I
J
Paper 1
42
84
50
42
33
50
69
81
50
35
Paper 2
31
83
42
60
28
63
59
92
73
40
a)
b)
Using your calculator, find r (the correlation coefficient) to 2 dp and also the
equation of the best-fit regression line.
(8)
Using your equation from (a), find the expected mark for a student on Paper 2 if
he/she obtains 75% for Paper 1.
(4)
[12]
Page 9 of 10
Grade 12 AP Maths Trials Examination August 2011
Name_______________________________
Appendix A
Module 1
Question 5
4
𝑓(𝑥) = 2|𝑥 + 3| − 4 and 𝑔(𝑥) = | |
𝑥
Grade 12 AP Maths Trials Examination August 2011
Question 6
𝑥 2 − 4𝑥 + 4
𝑓(𝑥) = 2
𝑥 +𝑥−6
Page 10 of 10