Download Paper

GRADE 12 EXAMINATIONS
SEPTEMBER 2012
MATHEMATICS PAPER 3
“There is salvation in no-one else [apart from Jesus Christ] for there is no other Name
under heaven given among men by which we must be saved.”
Acts 4: 12
Time: 2 hours
Marks: 100
________________________________________________________________________________________________________
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1.
This paper consists of 9 questions; answer all the questions. Also find
attached two formulae pages and 2 diagram pages.
2.
You may freely use an approved, non-programmable, and non-graphics
calculator. Answers must be rounded off to 2 decimal places.
3.
Show all your working details, write legibly and present your work neatly.
All writing must be in black or blue pen.
4.
Diagrams are not necessarily drawn to scale.
5.
Cell phones and other electronic devices may not be brought into the
examinations room, not even to be used as calculators.
6.
Use your time sagaciously: spend no more than half an hour on every
25 marks.
________________________________________________________________________________________________________
GRADE 12 FET: MATHEMATICS PAPER 3
SEPTEMBER 2012
Page 2 of 11
SECTION A
QUESTION 1
1.1
1.2
Consider the recursive formula: Tn + 1 = 2.Tn + 1, T1 = 1
1.1.1 Write down the first five terms of the sequence.
(2)
1.1.2 Write this recursive formula as an explicit algebraic formula.
(2)
Consider the given quadratic sequence of numbers: 2; 5; 10; 17
Give a recursive formula that will generate this sequence.
(4)
_______________________________________________________________________[8]____
QUESTION 2
Two universities administer NBT’s for prospective first-year mathematics students.
Jorik wrote the NBT at University A and scored 78%.
Raka wrote at University B and scored 60%.
The normal distribution graph below represents the results for University A.
65
90
University B revealed the following statistics: Mean (𝑥̅ ) = 49, Standard deviation (σ) = 5
2.1
Calculate the standard deviation (correct to one decimal place) for University A's
entrance examination results.
(2)
2.2
Determine which of the two students performed better in comparison with the
other students who wrote the test at their respective universities. Motivate your
answer with relevant calculations.
(3)
[5]
ELKANAH HOUSE HIGH SCHOOL
PLEASE TURN OVER
GRADE 12 FET: MATHEMATICS PAPER 3
SEPTEMBER 2012
Page 3 of 11
QUESTION 3
In this question round all answers off to one decimal place.
A group of students attended a course in Statistics on Saturdays over a period of 10 months.
The number of Saturdays on which a student was absent was recorded against the final mark
the student obtained. The information is shown in both the table below and the scatter plot.
Number of
Saturdays absent
Final Mark for
examination
0
1
2
2
3
3
5
6
7
96
91
78
83
75
62
70
68
56
3.1
Calculate the equation of the least squares regression line.
(4)
3.2
Calculate the correlation coefficient.
(2)
3.3
Use the correlation coefficient and comment on the trend of the data.
(2) R
3.4
Calculate the predicted final mark of a student who was absent four Saturdays. (2)
3.5
Calculate the values of the lower and upper quartiles.
3.6
An outlier is defined as a value that is 1,5 times the interquartile range away from
the nearest quartile. Determine if any of the final examination marks are outliers. (4)
(4)
[18]
ELKANAH HOUSE HIGH SCHOOL
PLEASE TURN OVER
GRADE 12 FET: MATHEMATICS PAPER 3
SEPTEMBER 2012
Page 4 of 11
QUESTION 4
A and B are two events in a sample space. Suppose P(A) = 0,4; P(A or B) = 0,6 and P(B) = k.
4.1
For what value of k (correct to one decimal) are A and B mutually exclusive?
(2)
4.2
For what value of k (correct to two decimals) are A and B independent?
(4)
[6]
QUESTION 5
The probability that it will rain on a given day is 63%. A child has a 12% chance of falling in
dry weather and is three times as likely to fall in wet weather.
5.1
Draw a tree diagram to represent all outcomes of the above information.
(4)
5.2
What is the probability (correct to two decimals) that a child will fall in dry
weather?
(2)
What is the probability (correct to two decimals) that a child will not fall on
any given day?
(4)
5.3
[10]
QUESTION 6
The seven numbers 0, 1, 2, 3, 4, 5 and 6 are used to make three-digit codes.
6.1
How many unique codes are possible if digits can be repeated?
(2)
6.2
How many unique codes are possible if the digits cannot be repeated?
(2)
6.3
In the case where digits may be repeated, how many codes are numbers that
are greater than 300 and exactly divisible by 5?
(4)
[8]
ELKANAH HOUSE HIGH SCHOOL
PLEASE TURN OVER
GRADE 12 FET: MATHEMATICS PAPER 3
SEPTEMBER 2012
Page 5 of 11
SECTION B
QUESTION 7
ABCD is a parallelogram with diagonals intersecting at F. FE is drawn parallel to CD.
AC is produced to P such that PC = 2AC and AD is produced to Q such that DQ = 2AD.
B
A
F
C
P
E
D
Q
7.1
Show that E is the midpoint of AD.
(3)
7.2
Prove PQ || FE.
(3)
7.3
Calculate the ratio of area ΔDEF : area ΔAPQ.
(5)
_________________________________________________________________________________________________[11]
ELKANAH HOUSE HIGH SCHOOL
PLEASE TURN OVER
GRADE 12 FET: MATHEMATICS PAPER 3
SEPTEMBER 2012
Page 6 of 11
QUESTION 8
In the diagram below, O is the centre of the circle. AB is a diameter of the circle.
Chord CS produced meets chord EB produced at D. Chord EC is parallel to
chord BS. CO and AC are joined. Let CÔB = 2x
C
1
3
2
A
1
O
S
2
2x
E
1
2
3
B
D
8.1
Find, with reasons, six angles with a magnitude of x.
(9)
8.2
Show that ΔDBS is isosceles.
(2)
8.3
Show that Ĉ1 = Ĉ3.
(2)
8.4
Show that BOCD is a cyclic quadrilateral.
(3)
_________________________________________________________________________________________________[16]_
ELKANAH HOUSE HIGH SCHOOL
PLEASE TURN OVER
GRADE 12 FET: MATHEMATICS PAPER 3
SEPTEMBER 2012
Page 7 of 11
QUESTION 9
In the diagram below PDA is a tangent to the circle ACBT at A. CTP and BTD are straight lines.
AC || BD. Let Â1 = x.
A
3
2 1
1
C
1
2
1
2
3
T
4
D
2
P
1 2
B
9.1
Prove that ΔABC ||| ΔADT.
(6)
9.2
Prove that PT is a tangent to the circle ADT at T.
(5)
9.3
Prove that ΔAPT ||| ΔTPD.
(3)
9.4
If 3AD = 2AP, show that AP2 = 3PT2 .
(4)
_________________________________________________________________________________________________[18]_
ELKANAH HOUSE HIGH SCHOOL
PLEASE TURN OVER
GRADE 12 FET: MATHEMATICS PAPER 3
SEPTEMBER 2012
Page 8 of 11
DIAGRAM SHEET
QUESTION 7
B
A
F
C
P
E
D
Q
QUESTION 8
C
1
3
2
A
1
O
E
ELKANAH HOUSE HIGH SCHOOL
S
2
2x
B
D
PLEASE TURN OVER
GRADE 12 FET: MATHEMATICS PAPER 3
SEPTEMBER 2012
Page 9 of 11
QUESTION 9
A
3
2 1
1
C
1
2
1
2
3
T
4
D
2
P
1 2
B
ELKANAH HOUSE HIGH SCHOOL
PLEASE TURN OVER
GRADE 12 FET: MATHEMATICS PAPER 3
SEPTEMBER 2012
Page 10 of 11
MATHEMATICS
INFORMATION SHEET
PAGE 1
𝑥=
−𝑏 ± √𝑏 2 − 4𝑎𝑐
2𝑎
𝐴 = 𝑃(1 ± 𝑖. 𝑛)
𝐴 = 𝑃(1 ± 𝑖)𝑛
𝑥[(1 + 𝑖)𝑛 − 1]
𝐹=
𝑖
𝑃=
𝑥[1 − (1 + 𝑖)−𝑛 ]
𝑖
𝑛
∑1 = 𝑛
𝑛
𝑖=1
∑𝑖 =
𝑛
𝑛
∑(𝑎 + (𝑖 − 1)𝑑) = (2𝑎 + (𝑛 − 1)𝑑)
2
𝑖=1
𝑛(𝑛 + 1)
2
𝑖=1
∞
𝑛
𝑛
∑ 𝑎𝑟
𝑖=1
𝑖−1
𝑎(𝑟 − 1)
=
; 𝑟≠1
𝑟−1
𝑇𝑛 = 𝑎𝑛2 + 𝑏𝑛 + 𝑐
(𝑛 − 1)(𝑛 − 2)
𝑇𝑛 = 𝑇1 + (𝑛 − 1)𝑓 +
𝑠
2
∑ 𝑎𝑟 𝑖−1 =
𝑖=1
𝑎
; −1 < 𝑟 < 1
1−𝑟
Where 𝑓 is the first term of the first difference and
𝑠 is the first term of the second difference.
𝑓(𝑥 + ℎ) − 𝑓(𝑥)
ℎ→0
ℎ
𝑓 ′ (𝑥) = lim
𝑑 = √(𝑥2 − 𝑥1 ) 2 + (𝑦2 − 𝑦1 ) 2
𝑀[
𝑥1 + 𝑥2 𝑦1 + 𝑦2
;
]
2
2
𝑚=
𝑦2 − 𝑦1
𝑥2 − 𝑥1
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )
𝑚 = tan 𝜃
(𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 = 𝑟 2
ELKANAH HOUSE HIGH SCHOOL
PLEASE TURN OVER
GRADE 12 FET: MATHEMATICS PAPER 3
SEPTEMBER 2012
Page 11 of 11
MATHEMATICS
INFORMATION SHEET
PAGE 2
𝐼𝑛 ∆𝐴𝐵𝐶:
𝑎
𝑏
𝑐
=
=
sin 𝐴 sin 𝐵 sin 𝐶
𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐. cos 𝐴
1
𝑎𝑟𝑒𝑎 𝑜𝑓 ∆𝐴𝐵𝐶 = 𝑎𝑏. sin 𝐶
2
sin(𝛼 + 𝛽) = sin 𝛼. cos 𝛽 + cos 𝛼. sin 𝛽
sin(𝛼 − 𝛽) = sin 𝛼. cos 𝛽 − cos 𝛼. sin 𝛽
cos(𝛼 + 𝛽) = cos 𝛼. cos 𝛽 − sin 𝛼. sin 𝛽
cos(𝛼 − 𝛽) = cos 𝛼. cos 𝛽 + sin 𝛼. sin 𝛽
cos2 𝛼 − sin2 𝛼
cos 2𝛼 = { 1 − 2 sin2 𝛼
2 cos 2 𝛼 − 1
sin 2𝛼 = 2 sin 𝛼. cos 𝛼
(𝑥, 𝑦) = ((𝑥𝐴 cos 𝛼 − 𝑦𝐴 sin 𝛼) ; ((𝑦𝐴 cos 𝛼 + 𝑥𝐴 sin 𝛼))
𝑥̅ =
∑𝑥
𝑛
𝑣𝑎𝑟 =
𝑥̅ =
∑𝑛𝑖=1(𝑥𝑖 − 𝑥̅ )2
𝑛−1
∑ 𝑓. 𝑥
𝑛
𝑣𝑎𝑟 =
∑𝑛𝑖=1(𝑥𝑖 − 𝑥̅ )2
𝑛
∑𝑛𝑖=1(𝑥𝑖 − 𝑥̅ )2
√
𝑠. 𝑑 =
𝑛
𝑃(𝐴) =
𝑛(𝐴)
𝑛(𝑠)
ELKANAH HOUSE HIGH SCHOOL
𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 𝑎𝑛𝑑 𝐵)
PLEASE TURN OVER