GRADE 12

MATHEMATICS PAPER 2

DATE:

July 2008

TIME: 3 HOURS

TOPIC:

MID YEAR EXAM

TOTAL MARK: 150

EXAMINER:

Mrs. M Klein

MODERATOR:

Mrs. P Copeland

Name:________________________Teacher: _______________________

Learning Outcomes

Assessment Standards

Question

Mark

3

Space, Shape & Measurement

11.3.1Surface Area & Volume: Pyramids, Cones & Spheres

Coordinate Geometry:

11.3.3 Distance, Gradient, Mid-point, Equation of line, Inclination.

12.3.3 Equation of a Circle & Tangent to a Circle

Transformation Geometry:

11.3.4 Translations, Reflections, Rotations, Enlargements.

12.3.4 Rotation about the origin through any angle.

Rigid Transformations & Enlargements

Trigonometry:

11.3.5 Special Angles, Identities, Reduction Formulae, Equations, Solving Triangles.

12.3.5 Compound Angle Formulae

11.3.6 Solving 2-D problems

12.3.6 Solving 2–D & 3-D Problems

A: 1; 2

B: 1; 7

A: 3

B: 5

A: 5

B: 2; 3; 6

A: 27

B: 28

A: 14

B: 9

A: 24

B: 25

4

Data Handling

& Probability

Descriptive Statistics:

11.4.1 Univariate Data, Central Tendency ( Mean, Median, Mode) & Dispersion (Range, Quartiles, Variance);

Bivariate Data Scatter Plots & graphs of best fit.

A:4

B: 4

A: 15

B: 12

Total

150

COMMENT

SECTION A:KNOWLEDGE & ROUTINE PROCEDURES80 MARKS

QUESTION 1 15 Marks

ABCD is a parallelogram.

a)

Determine the coordinates of E

(2)

b)

Determine the coordinates of D, the 4th vertex of parallelogram ABCD

(4)

c)

Determine the equation of line BA

(3)

d)

Determine the inclination of line BE correct to 1 decimal place.

(3)

e)

Determine the size of angle correct to 1 decimal place.

(3)

QUESTION 2 16 Marks

The equation of Circle centre O passing through points A and B is

a)

Determine the coordinates of O, the centre of the circle.

(5)

b)

What is the radius of the circle?

(1)

c)

If the coordinates of B are determine the value of b, if

(5)

d)

Determine the equation to the tangent to the circle at B.

(3)

e)

What type of quadrilateral is AOBC? Give a reason for your answer.

(2)

QUESTION 317 Marks

are the vertices of a triangle.

ANSWER THIS QUESTION ON THE ON THE DIAGRAM SHEET PROVIDED.

a)

Draw the reflection of about the y-axis

(3)

b)

is the rotation of through an angle of about the origin. Calculate the vertices of and draw it.

(9)

c)

Reduce by a factor of about the origin to form. Give the coordinates of the vertices of .

(3)

d)

By what factor will the area of the new triangle have decreased?

(2)

QUESTION 415 Marks

ANSWER THIS QUESTION ON THE DIAGRAM SHEET PROVIDED.

The following distribution shows the number of orders for groceries received each week in a particular year at a village shop.

18

18

19

19

20

20

20

21

21

21

21

22

22

22

22

22

22

23

23

23

23

23

23

23

23

23

24

24

24

24

24

24

24

24

24

24

24

24

25

25

25

25

25

25

26

26

26

26

26

27

27

27

The above information is summarised on the table below.

Number of Orders Received

18

19

20

21

22

23

24

25

26

27

Number of Weeks

2

2

3

4

6

9

12

6

5

3

a)

Compile a cumulative frequency table.

(5)

b)

Use your table to find

1)

The mode

2)

The median

3)

The range

4)

The quartiles

(5)

c)

Use the axes provided and draw a cumulative frequency curve

1)

Show the median.

2)

Show the quartiles on the graph.

(5)

QUESTION 517 Marks

a)

Simplify as far as possible:

(4)

b)

Solve for

(4)

c)

and

Angle B passes through the point

With the use of a diagram and without the use of a calculator, determine the value of

(9)

SECTION B: COMPLEX PROCEDURES & PROBLEM SOLVING75 MARKS

QUESTION 116 Marks

A slalom water skier enters the course through gates at A and exits through gates at H. A boat that travels in a straight line from A to H at a constant speed of 58km/hr tows him. He is required to enter the course at A, ski around each of 6 buoys and exit the course at H. The distance of each buoy from the boat path is 11,5m. The length of the course is 259m. The buoys are equally spaced.

a)

Determine the coordinates of each buoy

(3)

b)

How long does it take the boat to travel through the course? Give your answer in seconds correct to two decimal places.

(5)

c)

What is the minimum distance the skier has to travel in order enter the course at A, pass around all six buoys and exit at H

(6)

d)

Assuming the skier is in the course for the same length of time as the boat, what is the skier’s average speed?

(2)

QUESTION 2 10 Marks

Prove the identity:

a)

(5)

b)

For what values of is the identity undefined?

(5)

QUESTION 35 Marks

The following question was given to candidates to answer:

Determine, without the use of a calculator, the value of sin66º in terms of t.

Joyce gave the following solution:

Line 1

Line 2

Line 3

Unfortunately, Joyce's answer is incorrect.

a)

Explain why Joyce's answer is incorrect.

(1)

b)

Give a correct solution to Joyce's problem.

(4)

QUESTION 412 Marks

a)

Fifty shoppers were asked what percentage of their income they spend on groceries.

Six answered that they spend between 10% and 19%, inclusive. The full set of responses is given in the table below.

PERCENTAGE

FREQUENCY (f)

10 - 19

6

20 - 29

14

30 - 39

16

40 - 49

11

50 - 59

3

Calculate the mean percentage of family income allocated to groceries.

ANSWER THIS QUESTION ON THE DIAGRAM SHEET PROVIDED.

(5)

b)

The marks of 8 learners in a test for which the maximum mark is 10, were:

7, 4, 9, 4, 9, 5, 4, 6.

Calculate the standard deviation of this data.

(7)

QUESTION 59 Marks

In the diagram above:

Triangle A'B'C is the reflection of triangle ABC about the line x = 4; and

Triangle A''B''C'' is the reflection of triangle A'B'C' about the line x = 9

This combination of transformations is called a composition of transformations

a)

Determine (with justification) the values of p and q: the co-ordinates of C'

(3)

b)

Determine (with justification) the values of r and s: the co-ordinates of B''

(3)

c)

Describe (with justification) a single transformation that would have the same result as the composition of transformations: reflection about the line x = 4 followed by reflection about the line x = 9, i.e. a single transformation that could be used to transform triangle ABC into triangle A''B''C''

(2)

d)

By considering the point A(1;3) discuss whether or not the composition of transformations: reflection about the line x = 9 followed by reflection about the line x = 4 will give the same result as the composition above.

(1)

QUESTION 610 Marks

a)

In the figure, TC is a vertical tower. Two wires leading from the top of the tower are staked at position A and position B so that A; B and C are all in the same horizontal plane. The angles TAB and TBA are measured and found to be β and θ respectively. The distance between the stakes is x metres. If the angle of elevation from stake A to the top of the tower is α, calculate the height of the tower in terms of x; β; θ and α.

(5)

b)

A container is lifted off a boat at point D by crane AG. The arm of the crane AB is rotated through an angle of to AC. The container is then lowered onto the quay from E to F. The angle of elevation of D from F is. The length of AB is r metres.

1)

Prove that

(3)

2)

Determine the distance the container was lowered onto the quay. (The length of EF.)

(2)

QUESTION 713 Marks

A dartboard consists of three concentric circles with diameters dividing it into 12 equal sectors as shown. The radii of the circles are 10cm, 8cm, and 6cm respectively. The diameter AB is parallel to the x-axis. Diagonals CD and EF have equations

and respectively.

a)

Determine the co-ordinates of the centre P of the dartboard. (Leave your answer in surd form if necessary).

(5)

b)

Hence find the equation of the second circle

(2)

c)

Write down the equation of the diameter AB

(1)

d)

Scores for a particular sector are shown on the outer ring of the dartboard. If a dart lands on the inner ring of that sector, the score remains unchanged. If a dart lands between the first and second rings of that sector, the score is doubled, and if a dart lands in the outer ring of that sector, the score is halved. For example, the dart shown in the figure scores 40 points.

1)

In which ring is the point R (8; 6) located?

(2)

2)

In which sector is the point R (8; 6) located?

(2)

3)

Determine the score obtained by a dart landing at R

(1)

MATHEMATICS PAPER 2

ANSWER SHEET

NAME: ____________________________

SECTION A: QUESTION 1

b)

c)

d)

SECTION A:QUESTION 4

a) Cumulative Frequency Table

NUMBER OF ORDERS (x)

NUMBER OF WEEKS (f )

CUMULATIVE FREQUENCY

18

19

20

21

22

23

24

25

26

27

b)

SECTION B:QUESTION 4

a)

PERCENTAGE

FREQUENCY (f)

MIDPOINT OF INTERVAL (x)

TOTAL

10 - 19

6

20 - 29

14

30 - 39

16

40 - 49

11

50 - 59

3

TOTAL

TOTAL

b)

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