GRADE 12 EXAMINATIONJULY 2012

Mathematics: Paper 3

EXAMINER: Combined Paper MODERATORS: JE; RN; SS; METIME: 2 Hours TOTAL: 100

PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY

1. This question paper consists of 7 questions and an Information Sheet.2. Read the questions carefully.3. Answer all the questions on the question paper.4. You may use an approved, non-programmable, and non-graphical calculator, unless otherwise

stated.5. Round off your answers to TWO decimal places where necessary.6. All the necessary working details must be clearly shown.7. It is in your own interest to write legibly and to present your work neatly.8. Diagrams are not drawn to scale.

NAME_________________________________________________________________________________

Mark Allocation (for educator’s use only)

Q1 Q2 Q3 Q4 Q5 Q6 Q7 TOTAL

Marks Earned

Total Marks

7 29 9 3 20 13 19 100

Grade 12 Mathematics Paper 3 Page 2 of 16

QUESTION 1

1.1) Determine the recursive formula for T k+1 of the sequence 3 ;6 ;12;24 ;…

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1.2) A recursive formula T k+1=a .T k+4 .T k−1 generates the following

sequence, b ;5 ;22 ;64 ;… Determine the value of a and b .

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Grade 12 Mathematics Paper 3 Page 3 of 16

QUESTION 2

All answers involving factorials must be calculated, e.g. 4 !=24

2.1) Using the letters in the word “MATHEMATICALLY”, determine:

2.1.1) The number of 14-letter ‘words’ that can be formed.

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2.1.2) The probability that the new word will start and end on the letter M.

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2.2) If P (A )=15 and P (B )=3

7 find:

2.2.1) P(A∪B) if A and B are mutually exclusive events.

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Grade 12 Mathematics Paper 3 Page 4 of 16

2.2.2) Given that P( A∪B )=P( A )+P(B )−P( A∩B) ;

find P (A∪B ); if A and B are independent events.

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2.3) When South Africa hosted the Soccer World Cup in 2010, a group of 200 fans

were interviewed about Algeria, Bafana-Bafana and Cameroon. They were

asked whether these teams would progress to the second round of the

tournament. An analysis of the data indicated the following:

28 indicated that Algeria and Bafana-Bafana will progress. 42 indicated that only Bafana-Bafana will progress, but not Algeria or Cameroon. 64 indicated that only Cameroon will progress, but not Algeria or Bafana-Bafana. 14 indicated that only Algeria and Cameroon will progress. 98 indicated that Algeria or Bafana will progress, but not Cameroon. 122 indicated that Bafana-Bafana or Cameroon will progress, but not Algeria. 5 indicated that not one of the teams will progress. 3 indicated that all three teams will progress.

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A B

C

Grade 12 Mathematics Paper 3 Page 5 of 16

2.3.1 Draw a Venn-diagram to represent the information.

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2.3.2 From the fans opinions determine the probability that all three teams will progress to

the second round.

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2.3.3 How many fans felt that Bafana-Bafana will progress to the second

round, irrespective of what happens to Algeria or Cameroon?

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Grade 12 Mathematics Paper 3 Page 6 of 16

2.3.4 What is the probability, based on the fans predictions, that only one

team will progress to the second round?

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Grade 12 Mathematics Paper 3 Page 7 of 16

A study was done to determine the effects of sleep deprivation on students’ ability to solve mathematical problems. A total of 13 people were involved in the study. After the sleep deprivation period, the number of errors made for the same problem given to each participant was recorded IN THE TABLE and IN THE SCATTER PLOT below.

16 18 20 22 24 26 280

5

10

15

20

25

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Number of hours without sleep (x)

Num

ber o

f err

ors (

y)

Number of hours without

sleep ( x ) 18 18 19 20 20 21 24 24 26 25 26 27 27

Number of errors( y )

8 6 7 8 9 13 12 14 20 21 21 20 23

Grade 12 Mathematics Paper 3 Page 8 of 16

3.1) Determine the correlation coefficient (to 3dp) for the function illustrated on the previous page.

Interpret this result.

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3.2) Determine, with the use of your calculator, the linear regression equation of

the line of best fit.

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3.3) Use (3.2) to determine how many errors a student would make after 23

hours of sleep deprivation.

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Grade 12 Mathematics Paper 3 Page 9 of 16

QUESTION 4

A sample group of 220 Grade 12 learners were questioned on how many hours they spent on studying in a week before an upcoming Mathematical test. The results are summarized in the table below:

Hours Frequency10 216 1220 2325 6030 7735 3842 8

Determine the standard deviation of the data. (Round off your answer to

three decimal places)

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D

BT

E

A

C

12

2

2

2

1

1

1OX

3

45

2

1

23

1

S

Grade 12 Mathematics Paper 3 Page 10 of 16

QUESTION 5

A, B, C, D, and E, are points on the circumference of a circle with centre O. TBS is a tangent to

the circle at B. DB and AC intersect at X. A1= 600 and C2 = 400

Find the sizes of the following angles and state reasons:

5.1 D1= _________________________________________________________________________

5.2 O2= _________________________________________________________________________

5.3 B2+B3+B4= _________________________________________________________________

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60°

40°

Grade 12 Mathematics Paper 3 Page 11 of 16

5.4 C1= __________________________________________________________________________

5.5 B1= __________________________________________________________________________

5.6 B4= _________________________________________________________________________

5.7 E = __________________________________________________________________________

5.8 B2= _________________________________________________________________________

5.9 X3= _________________________________________________________________________

5.10 X1= __________________________________________________________________________

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AD

F

CB

Grade 12 Mathematics Paper 3 Page 12 of 16

QUESTION 6

Prove that:

6.1.1) ∆ DCF lll ∆BAF

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6.1.2) Show that DCCF =BAAF

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AD E

F

CB

Grade 12 Mathematics Paper 3 Page 13 of 16

6.1.3 If it is further given that EC is a tangent to the circle and BD≪CE .

Prove that ∆ EFC lll ∆ DFB

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6.1.4 Show that DEEF=DCC F

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2

12

312

1 21

Grade 12 Mathematics Paper 3 Page 14 of 16

QUESTION 7

In the diagram PQ and RS are two chords of the circle such that PQ // RS . The tangent to the circle at Q meets RS produced at T and the tangent at S meets QT at V. PS and QR intersect at W. QS and PR are drawn.

7.1 Prove that V 1=2 R1

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Grade 12 Mathematics Paper 3 Page 15 of 16

7.2 Prove that QVSW is a cyclic quadrilateral.

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7.3 Prove that P1+T=R1+ R2

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[19marks]

Total: 100

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Grade 12 Mathematics Paper 3 Page 16 of 16

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