Grade 12
September Examination 2012
Mathematics: Paper 3
Time: 2 hoursMarks: 100
Instructions and Information:
Read the following instructions carefully before answering the questions.
1.This question paper consists of 9 questions. Answer ALL the questions.
2.Clearly show ALL calculations, diagrams, graphs etc that you have used in determining your answers. All working should be shown in its proper place.
3.An approved scientific calculator (non-programmable and non-graphical) may be used, unless stated otherwise.
4.If necessary, answers should be rounded off to TWO decimal places, unless stated otherwise.
5.Diagrams are not necessarily drawn to scale.
6.Reasons must be given for all geometry statements
7.For your convenience, a diagram sheet is included for Questions 4, 7, 8 and 9. This should be stapled to the front of your script at the end of the examination.
8.A formula sheet is provided.
9.Number your answers according to the numbering system used in this question paper.
10.It is in your own interest to write legibly and to present your work neatly.
Question 1
1.1Determine the recursive formula for of the sequence 3; 7; 11; 15…(2)
1.2Determine the sum of the first four terms of:
(4)
[6]
Question 2
2.1A teacher draws up the following histogram to represent her class marks in percentages for the term.
2.1.1The teacher has created the impression that the results show a normal distribution. Explain fully how this been done. (2)
2.1.2What is the true situation with respect to the spread of the marks of the class? (1)
2.1.3If the pass mark is 40%, what percentage of the class did not achieve a pass mark for the term? (3)
2.2The following statements are all false. Explain why each is false.
2.2.1If data in a sample is not normally distributed, an increase in sample size will result in a normal distribution. (2)
2.2.2The variance of a sample is always larger than the standard deviation.(2)
2.2.3In a normal distribution, at least 80% of the data lies within one standard deviation of the mean. (2)
[12]
Question 3
3.1In the diagram alongside, two normal distribution curves,labelled A and B, are drawn on the same set of axes.
Below are three sets of means and standard deviations , labelled P, Q and R.
Match curves A and B to P, Q or R. Write only the letters of the curves and the corresponding letter of your answer.
P
Q
R(2)
3.2In a recent year, students entering a local university had a mean entrance examination score of , with a standard deviation of . The distribution was known to be normally distributed.
One student achieved a score of 955, and this was exactly one standard deviation below the mean. Another student achieved a score of 1495 and this was exactly two standard deviations above the mean.
3.2.1Find the mean () and the standard deviation () of the students’ scores.(3)
3.2.2Approximately what percentage of the students’ scores fall between 955 and 1495? (2)
[7]
Question 4
The following table shows the number of orders received after a company called MathsHelp sent out a number of advertising pamphlets.
Number of Pamphlets / Number of Orders600 / 350
1000 / 550
500 / 300
700 / 300
600 / 350
100 / 200
500 / 350
800 / 450
300 / 250
900 / 500
4.1Using the grid on the diagram sheet, draw a scatter plot of the data.(3)
4.2Which function would best fit the data: linear, quadratic or exponential?(1)
4.3Determine, using your calculator, the equation of a linear regression line.(3)
4.4Calculate, using your answer to Question 4.3, the possible number of orders if 200 pamphlets are sent out. (2)
4.5Determine, using your calculator, the correlation coefficient for the data (to 2 decimal places) and comment on your answer. (4)
[13]
Question 5
In Mitchell High School, there are 150 pupils in Grade 12. The Venn diagram given alongside shows the number of pupils taking Mathematics (M), Life Sciences (L) and Geography (G).
50 pupils take Life Sciences and 74 pupils take Mathematics.
5.1Determine the value of and y.(4)
5.2Determine the following:
5.2.1How many pupils take Life Sciences and Geography?(1)
5.2.2(2)
5.3Determine the probability that if a Grade 12 pupil is randomly selected, the pupil does not take Life Sciences but does take Mathematics and Geography. (2)
5.4Determine, showing all your calculations, whether the event that a pupil takes Geography is independent of the event that a pupil takes Mathematics. (4)
[13]
Question 6
6.1In a school, pupils are required to have a four-character password for photocopying. The first character should be a consonant, the second should be a vowel (excluding I and O) and the last two characters should be numerical digits (excluding 0). The numerical digits may be repeated.
6.1.1How many possible passwords are there?(2)
6.1.2Wendy wants her password to start with W. How many possible passwords are there for her? (2)
6.2Given the word “NECESSITY”
6.2.1In how many different ways can the letters of the word be arranged?(2)
6.2.2In how many different ways can the letters be arranged so that they start and end with S? (2)
[8]
Question 7
Refer to the figure alongside.
PMN has V the midpoint of PM.
RT // VN and
MT intersects VN at Q
Giving reasons, determine the following:
7.1(4)
7.2(1)
7.3(2)
7.4(3)
[10]
Question 8
In the diagram alongside, AB is a diameter of the circle ABCF.
AD is a tangent at A.
Let B
8.1. Why is ?(1)
8.2.Prove that (4)
8.3.Prove that CDEF is a cyclic quadrilateral.(3)
8.4.Prove that /// (3)
8.5.If BF = 5cm, BC = 4cm, EF =3cm and FC = 2cm, calculate the length of :
8.5.1.ED(3)
8.5.2.DC(3)
[17]
Question 9
9.1The figure alongside shows a circle with centre O and chord AB. OP is drawn so that OP AB. Use the copy of the diagram on the diagram sheet, or redraw the diagram onto your answer sheet to prove the theorem which states that P is the midpoint of AB.
(5)
9.2Refer to the diagram alongside.
PQ is a chord of a circle with centre O.
PO is produced to S such that PS SQ.
OM PQ.
O and Q are joined.
Giving reasons
9.2.1Prove that PSQ /// PMO(3)
9.2.2Prove that (2)
9.2.3Hence, or otherwise, prove that (4)
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T O T A L M A R K S : 1 0 0
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INFORMATION SHEET: MATHEMATICS
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In ABC:
P(A or B) = P(A) + P(B) – P(A and B)
Page 1 of 11August 2012