1544 words - 7 pages

Mathematics/P2

2

DoE/Feb. – March 2011

NSC

PAGE

MARKS: 150

TIME: 3 hours

This question paper consists of 9 pages, 5 diagram sheets and 1 information sheet.

INSTRUCTIONS AND INFORMATION

Read the following instructions carefully before answering the questions.

1.

2.

3.

4.

This question paper consists of 12 questions.

Answer ALL the questions.

Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in determining the answers.

Answers only will not necessarily be awarded full marks.

5.

6.

7.

An approved scientific calculator (non-programmable and non-graphical) may be used, unless stated otherwise.

Round your answers off to TWO decimal places if necessary, unless stated otherwise.

Diagrams are NOT necessarily drawn to scale.

8.

FIVE diagram sheets for QUESTION 1.2, QUESTION 2.1, QUESTION 2.2, QUESTION 3.1, QUESTION 8.1 and QUESTION 12.3 are attached at the end of this question paper. Write your centre number and examination number on these sheets in the spaces provided and insert them inside the back cover of your ANSWER BOOK.

9.

10.

11.

An information sheet, with formulae, is included at the end of this question paper.

Number the answers correctly according to the numbering system used in this question paper.

Write legibly and present your work neatly.

QUESTION 1

The table below gives a breakdown of the PSL log standings for the 8 top teams at the end of 2008/2009.

POSITION

TEAM

POINTS

1

SuperSport

55

2

Orlando Pirates

55

3

Kaizer Chiefs

50

4

Free State Stars

47

5

Golden Arrows

x

6

Bidvits Wits

x

7

Ajax Cape Town

x

8

Amazulu

42

[Source: http://www.safa-_psl log]

1.1

If the average points for these 8 teams is 48,375, show that

.

(2)

1.2

Draw a box and whisker diagram of the information given on DIAGRAM SHEET 1.

(4)

[6]

QUESTION 2

The individual masses (in kg) of 25 rugby players are given below:

78 102 88 93 81 90 75 60 76 75

68 90 80 77 81 69 60 83 91 100

80 70 81 64 70

2.1

Complete the following table on DIAGRAM SHEET 1

MASS (kg)

FREQUENCY

CUMULATIVE FREQUENCY

60

70

80

90

100

(4)

2.2

Draw an ogive (cumulative frequency curve) of the above information on the grid provided on diagram sheet 2.

(3)

2.3

Calculate the mean mass of the rugby players.

(2)

2.4

How many rugby players have masses within one standard deviation of the mean? From your calculations, calculate the percentage of the rugby players who have masses within one standard deviation of the mean.

(5)

[14]

QUESTION 3

A group of 12 learners was asked to measure their arm span (from fingertip to fingertip) and their height. The data below was gathered.

Arm span (cm)

156

157

160

161

162

165

170

177

184

188

188

194

Height (cm)

162

160

155

160

170

166

170

176

180

187

192

193

3.1

Represent the data as a scatter plot on the grid provided on DIAGRAM SHEET 3.

(4)

3.2

Draw a line of best fit for this scatter plot.

(2)

3.3

Would you expect a person with below average arm span to be below average in height? Give a reason for your answer.

(2)

[8]

QUESTION 4

In the diagram below

PQR with vertices P(– 1 ; 2), Q(– 2 ; – 2) and R(3 ; 0) is given.

4.1

Calculate the angle that PQ makes with the positive x-axis.

(3)

4.2

Determine the coordinates of M, the midpoint of PR.

(2)

4.3

Determine the perimeter of

PQR to the nearest whole number.

(5)

4.4

Determine an equation of the line parallel to PQ that passes through M.

(3)

[13]

QUESTION 5

5.1

The equation of a circle is

.

5.1.1

Prove that the point (2 ; – 9) is on the circumference of the circle.

(2)

5.1.2

Determine an equation of the tangent to the circle at the point (2 ; – 9).

(7)

5.2

Calculate the length of the tangent AB drawn from the point A(6 ; 4) to the circle with equation

.

(5)

[14]

QUESTION 6

The circle, with centre A and equation

is given in the following diagram. B is a y-intercept of the circle.

6.1

Determine the coordinates of B.

(4)

6.2

Write down the coordinates of C, if C is the reflection of B in the line x = 3.

(2)

6.3

The circle is enlarged through the origin by a factor of

.

Write down the equation of the new circle, centre A/, in the form

.

(2)

6.4

In addition to the circle with centre A and equation

, you are given the circle

with centre B.

6.4.1

Calculate the distance between the centres A and B.

(2)

6.4.2

In how many points do these two circles intersect? Justify your answer.

(2)

[12]

QUESTION 7

The point (x ; 2) is rotated about the origin through an angle of 150( in an anticlockwise direction to give the point (– 3 ; y). Calculate the values of x and y.

[5]

QUESTION 8

In the diagram below

MNP is given with vertices M(– 5 ; 2), N (6 ; 4) and P(2 ; – 4).

MNP is enlarged by a factor of 1,5 to

.

8.1

Draw

on the grid provided on DIAGRAM SHEET 4.

(3)

8.2

Write down the values of:

8.2.1

(2)

8.2.2

(2)

8.3

If the above transformation is applied to

n more times to get the image

, write down the value of

.

(2)

[9]

QUESTION 9

Consider the point A (– 12 ; 6). The point is reflected about the x-axis to A/.

9.1

Write down the coordinates of A/.

(1)

9.2

An alternative transformation from A to A/ is a rotation about the origin through α°, where

. Calculate α.

(6)

[7]

QUESTION 10

10.1

If sin 28° = a and cos 32°= b, determine the following in terms of a and/or b :

10.1.1

(2)

10.1.2

(3)

10.1.3

sin 4°

(4)

10.2

Prove without the use of a calculator, that if sin 28° = a and cos 32° = b, then

.

(4)

10.3

Evaluate each of the following without using a calculator. Show ALL working.

10.3.1

(7)

10.3.2

(4)

10.4

Determine the general solution of:

(7)

10.5

Consider:

10.5.1

For which values of x,

, will this expression be undefined?

(3)

10.5.2

Prove that

for all other values of x.

(5)

[39]

QUESTION 11

The sketch below shows one side of the elevation of a house. Some dimensions (in metres) are indicated on the figure.

Calculate, rounded off to ONE decimal place:

11.1

EC

(3)

11.2

(3)

11.3

Area of

DEC

(2)

11.4

The height EF

(3)

[11]

QUESTION 12

The graph of

is drawn below.

12.1

Write down the period of f.

(1)

12.2

Write down the amplitude of h if

.

(2)

12.3

Draw the graph of

for

on the grid provided on DIAGRAM SHEET 5.

(3)

12.4

Use the graph to determine the number of solutions for

,

.

(1)

12.5

For which values of x is g(x) ( 0?

(2)

12.6

For which values of x is

and

?

(3)

[12]

TOTAL:

150

CENTRE NUMBER:

EXAMINATION NUMBER:

DIAGRAM SHEET 1

QUESTION 1.2

QUESTION 2.1

MASS

(kg)

FREQUENCY

CUMULATIVE FREQUENCY

60

70

80

90

100

CENTRE NUMBER:

EXAMINATION NUMBER:

DIAGRAM SHEET 2

QUESTION 2.2

CENTRE NUMBER:

EXAMINATION NUMBER:

DIAGRAM SHEET 3

QUESTION 3.1

CENTRE NUMBER:

EXAMINATION NUMBER:

DIAGRAM SHEET 4

QUESTION 8.1

CENTRE NUMBER:

EXAMINATION NUMBER:

DIAGRAM SHEET 5

QUESTION 12.3

INFORMATION SHEET: MATHEMATICS

;

;

M

In (ABC:

P(A or B) = P(A) + P(B) – P(A and B)

MATHEMATICS P2

FEBRUARY/MARCH 2011

B

C

A

M(– 5 ; 2)

GRADE 12

NATIONAL

SENIOR CERTIFICATE

A

B

C

E

D

F

G

7,5

3,5

9,4

� EMBED Equation.3 ���

f

P(2 ; – 4)

N(6 ; 4)

N(6 ; 4)

P(2 ; – 4)

M(– 5 ; 2)

Copyright Reserved

Please turn over

Copyright Reserved

Please turn over

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Chart1

0

5

12

19

23

25

Mass (kg)

Cumulative frequency

CUMULATIVE FREQUENCY CURVE

Sheet1

60 0

70 5

80 12

90 19

100 23

110 25

Sheet1

0

0

0

0

0

0

Mass (kg)

Frequency

Cumulative Frequency Curve

Sheet2

Sheet3

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Chart1

162

160

155

160

170

166

170

176

180

187

192

193

Arm span (cm)

Height (cm)

Scatter plot showing arm span vs height

Sheet1

156 162

157 160

160 155

161 160

162 170

165 166

170 170

177 176

184 180

188 187

188 192

194 193

Sheet1

0

0

0

0

0

0

0

0

0

0

0

0

Arm Span (cms)

Height (cms)

Scatter Plot showing Arm Span vs Height

Sheet2

Sheet3

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