Preliminary Examination

TRINITYHOUSEHIGH SCHOOL

MATHEMATICS

PRELIMINARY EXAMINATION

PAPER 2

GRADE 12

8 September 2009MARKS: 150

TIME: 3hours

EXAMINER:R. van der Westhuizen

MODERATORS:R. Swiegers, R. Vorster, A. Wright

LEARNING OUTCOMES:LO 2LO 3LO 4

ASSESSMENT STANDARDS:AS 3, 4AS 1, 3 – 6AS 1

LEARNER’S NAME:______CLASS: ____

______

INSTRUCTIONS

%

1. Highlight your teacher’s name above.
2. Show all necessary calculations.
3. Round off to two decimal places, unless specified otherwise.
4. Non-programmable calculators may be used, unless instructed otherwise.
5. Diagrams are not necessarily drawn to scale.
6. This question paper consists of 8 typed pages.
7. An information sheet and an answer sheet are provided.

SECTION / A / B
QUESTION / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / TOTAL
MAXIMUM / 16 / 19 / 14 / 22 / 14 / 12 / 12 / 15 / 14 / 12 / 150
MARK

SECTION A

QUESTION 1

a), and are points in the Cartesian plane.

1)Calculate the length of , leaving answer in simplest surd form. (2)

2)Calculate the coordinates of S, the midpoint of RC. (2)

3)Determine the size of , correct to one decimal place. (3)

4)Determine the equation of the line AR. (3)

5)Determine whether . (2)

b)If and , use a diagram to calculate the value of . (4)

16 marks

QUESTION 2

a)Evaluate the following without using a calculator:

(5)

b)Simplify into a single term:

(6)

Page 1 of 8 TRINITYHOUSE Mathematics 2 - Preliminary 2009

c)Zakumi is the official mascot for the 2010 FIFA World Cup. His name is a composition of ‘ZA’ for ‘South Africa’ and ‘kumi’ which translates into ‘10’ in various languages across Africa. He is a leopard that loves to play football and his main priority is to turn the 2010 World Cup into one joyful and unforgettable party! Assume Zakumi’s front and back to be identical.

1)Write down the rule for the transformation from A to. (1)

2)Write down the rule for the transformation from A to. (2)

3)The lowest point of the 2010 emblem lies at. Write down the coordinates ofif the emblem is rotated anti-clockwise about the origin. (2)

4)The bottom left point of the South African flag is given by and the area of the flag is 1,5 square units. If the flag is enlarged through the origin by a scale factor of 3, write down the coordinates of and the increased area of the flag. (3)

19 marks

Page 2 of 8 TRINITYHOUSE Mathematics 2 - Preliminary 2009

QUESTION 3

a)The equation of the circle in the diagram is . A tangent is drawn at the point .

1)Calculate the value of r. (2)

2)Determine the equation of the tangent at A. (4)

b)The following trigonometric functions are given in the interval :

Write down the following:

1)Period of f. (1)

2)Range of f. (2)

3)The equation of the reflection of f in the X-axis. (2)

4)The coordinates at the maximum turning point of g. (2)

5)The equation of the asymptote of h. (1)

14 marks

QUESTION 4

ANSWER THIS QUESTION ON THE ANSWER SHEET

22 marks

Page 3 of 8 TRINITYHOUSE Mathematics 2 - Preliminary 2009

QUESTION 5

a)Evaluate the following expression without using a calculator:

(5)

b)Prove the following identity:

(4)

c)Determine the general solution of the equation (5)

14 marks

Page 4 of 8 TRINITYHOUSE Mathematics 2 - Preliminary 2009

SECTION B

QUESTION 6

a)A point undergoes the following chain of transformations:

• Reflection in the line
• Rotation about the origin of clockwise
• Enlargement through the origin by a factor of

Express the image of T in terms of x and y. (3)

b)Determine the equation of the line which is a clockwise rotation of the line about the origin. (3)

c)Five data values are represented as follows:

1)Determine the value of x if the mean of the data set is15. (4)

2)Hence determine the median of the data set. (2)

12 marks

QUESTION 7

a)Solve for x if where (4)

b)Prove that (5)

c)If determine in terms of p. (3)

12 marks

Page 5 of 8 TRINITYHOUSE Mathematics 2 - Preliminary 2009

QUESTION 8

a)K is the centre of a circle with equation . is a point outside the circle that lies on the tangent that touches the circle at T.

1)Determine the coordinates of K. (4)

2)Show that (4)

3)Calculate the value of k such that the length of the tangent PT is a minimum. (2)

b)Write down a possible data set of nine values that would be represented by the box-and-whisker plot below:

(5)

15 marks

Page 6 of 8 TRINITYHOUSE Mathematics 2 - Preliminary 2009

QUESTION 9

a)The Y-axis is a tangent to the circle with centre at the point P.

M lies on the line.

The point lies on the circle.

Calculate the values of a and b showing all necessary working out. (6)

b)The pyramid of Giza is situated near Cairo. It has a height of 140 metres and a squarebase with a side length of 230 metres.

In the sketch F is on the same horizontal plane as ABCDE. The shadow that the pyramid is casting is represented by EF and VO is the perpendicular height of the pyramid. The length of the shadow is 52 metres.

1)Calculate the magnitude of to the nearest degree. (2)

2)Mr Vorster used Pythagoras to calculate the distance between the top of the pyramid and the end of the shadow, i.e. VF, to be 218 metre. Using a different method to calculate the length of VF, determine whether you agree with him or not. (6)

14 marks

Page 7 of 8 TRINITYHOUSE Mathematics 2 - Preliminary 2009

QUESTION 10

a)Most modern soccer balls are stitched from 32 panels of waterproofed leather – 12 regular pentagons and 20 regular hexagons. The 32-panel configuration is the spherical polyhedron corresponding to the icosahedron; it is spherical because the faces bulge due to the pressure of the air inside.

The surface area of each hexagonal panel has been calculated to be 52,68 cm ². The length from the vertex of each pentagon to the centre of the pentagon is 3,8 cm.

Calculate the total amount of leather that is used to make the soccer ball. (6)

b)The perpendicular distance from a point to a line is given by

If the length of the perpendicular from the origin to the line is 5 units, prove that (6)

12 marks

Page 8 of 8 TRINITYHOUSE Mathematics 2 - Preliminary 2009

TRINITYHOUSEHIGH SCHOOL

MATHEMATICS

PRELIMINARY EXAMINATION

PAPER 2

ANSWER SHEET

LEARNER’S NAME:______CLASS: ____

Question 4

a)Fifty students wrote an entrance examination for entry into the faculty of Actuarial Science at UCT. The results were reported as grouped data.

1)Complete the frequency table of the results of the examination on the answer sheet. (2)

Marks / Frequency / Cumulative
frequency / Midpoint of interval
/ 4
/ 11
/ 20
/ 10
/ 5

2)Draw an ogive to represent the data. (4)

3)Use the ogive to determine the approximate inter-quartile range for the entrance examination. (3)

4)Calculate the approximate standard deviation for the data. (3)

b)In 2006 Mr Solomons bought a small business for R 100 000. At the end of each year his financial reports showed the value of the business as follows:

YEAR / 2006 / 2007 / 2008 / 2009 / 2010
Value of business / R 100 000 / R 180 000 / R 350 000 / R 680 000 / R 1 300 000

1)Draw a scatter plot to represent the data. (4)

2)Does a linear, quadratic or exponential function best describe the data? Give a reason for your choice. (2)

3)Describe the correlation that you notice between the points. (2)

4)Assuming that the business maintains the rate of growth, explain how you could use your graph to estimate the value of Mr Solomon’s business at the end of 2011. (2)

22 marks