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S T S T I T H I A N S C O L L E G E

D e p a r t m e n t s o f M a t h e m a t i c s

PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY:

1. This question paper consists of 11 questions and 24 pages.

2. A separate green information (formula) sheet will be provided to you.

3. Answer ALL the questions on the question paper in the spaces provided.

4. Clearly show ALL the calculations you have used to determine your answers.

5. Answers only may not necessarily be awarded full marks.

6. You may use an approved scientific calculator (non-programmable and non-

graphical) unless specified otherwise.

7. Please note that diagrams are not drawn to scale.

8. It is in your own interest to write legibly and to present your work neatly.

9. PLEASE FILL IN YOUR NAME AND CIRCLE YOUR TEACHER’S NAME ON

THE BACK PAGE.

GRADE 12

PRELIMINARY (MID-YEAR) EXAMINATION – PAPER 2DATE:

16th July 2018TIME:

3 hours

TOPICS: Statistics, Geometry, Measurement

and Trigonometry

TOTAL MARKS:150

EXAMINER:Mr. P. Statham (GC)

MODERATOR:Mr. R. Rhodes-Houghton (External)

SECTION A

QUESTION 1: [10]

The table below gives the rainfall, in millimetres, recorded for a farming region in South Africa over several years, and the corresponding yield of maize, in tons, obtained over the same period:

Year Rainfall (mm) Yield (tons)1 120 362 115 323 50 244 80 195 75 266 115 277 60 298 135 34

1.1. Calculate the mean and standard deviation of the rainfall recorded over the8-year period (rounded off to two decimal places): (2)

1.2. If rainfall of 30 mm was recorded during the 9th year, without performing any calculations, explain in detail what effect this would have on the standard deviation and why: (2)

1.3. Determine the equation of the least squares regression line for the rainfall (x)

vs. maize yield (y) in the form : (2)(Round off your values of A and B to three decimal places)

1.4. Determine the coefficient of linear regression (r) (rounded off to four decimal places) and explain whether the amount of rainfall received can be used as a good predictor of the expected maize yield: (2)

1.5. It is often stated that predictions using ‘extrapolation’ can be unreliable. From the data supplied in the table, what rainfall values would lead to predictions regarding the maize yield that would be based on extrapolation? (2)

QUESTION 2: [17]

In the diagram below, , and are the three vertices of

. is the midpoint of line AC and lies on the y-axis. with E on AC as shown:

2.1. Show that and : (3)

2.2. Determine the equation of line BE: (4)

x

y

A(-2;3)

B(3;2)

C(k;-5)

M(0;h)

E O

2.3. Hence, determine the coordinates of E: (5)

2.4. Determine the area of : (5)

QUESTION 3: [12]3.

If , without using a calculator, express each of the following in terms of m:

3.1. : (3)

3.2. : (3)

3.3. : (2)

3.4. : (4)

QUESTION 4: [10]4.

In the figure below, the circle with centre O, has points A, B, C, D and E on its

circumference. Reflex angle and . as shown:

Determine the size of the following angles (WITH REASONS):

4.1. : (4)

4.2. : (2)

4.3. : (4)

E

21

D

C

A

B

QUESTION 5: [10]5.

A survey conducted on the monthly petrol consumption, in litres, for a group of 38 St Stithians matric parents, yielded the following results:

Consumption (litres) Frequency Cumulative Frequency

57127322

5.1. Complete the cumulative frequency column in the table above: (2)

5.2. Draw the cumulative frequency curve (ogive) on the grid below: (4)Label your axes appropriately.

5.3. From your ogive, determine an estimate for the median monthly petrol consumption: (2)Show clearly on your ogive where your reading is taken from.

5.4. Determine an estimate for the mean monthly petrol consumption from the results in the table (rounded off to two decimal places): (2)

QUESTION 6: [18]6.

6.1. The graphs of and are sketched below for

:

6.1.1. Write down the values of a and b: (2)

6.1.2. Determine the values of c and d: (2)

6.1.3. Write down the range of : (2)

6.1.4. From the graphs, determine the value of : (2)

6.1.5. For what approximate values of x is : (2)

6.1.6. For what values of x is : (3)

6.2. Solve the equation for (rounded off to one decimal place): (5)

SECTION B

QUESTION 7: [20]7.

7.1. Two circles, and touch each other externally. Find the value of k: (6)

7.2. In the figure below, and are the endpoints of diameter AB of the circle with centre M. BC is a tangent to the circle at B as shown:

7.2.1. Determine the equation of the circle in the form : (5)

7.2.2. Show that the coordinates of C, the point where the tangent at B intersects the

x-axis is : (5)

C

M

x

y

O D

B

A

7.2.3. Determine the size of rounded to one decimal place: (4)

QUESTION 8: [15]8.

8.1. Evaluate without using a calculator: (3)

8.2. Prove that : (5)

8.3. In the diagram below, a tall tree is roughly represented by a right circular cone. The centre of the circular base of the cone is 2 metres above a point B, on the horizontal ground. C and D are points in the same horizontal plane as B, so that

metres, metres, and . The angle of elevation of the top of the tree (A) from point C is as shown:

8.3.1. Calculate the height of the tree AB rounded to one decimal place: (5)

8.3.2. If the radius of the circular base is 4 metres, find the volume of the cone representing the foliage of the tree rounded to one decimal place: (2)

Volume of cone:

QUESTION 9: [17]9.

In the figure below, and are right-angled triangles with lengths

, and . S is a point on RP and T is a point on PQ such that

as shown:


Q

PS

T

R

9.2. Prove : (3)

9.3. Hence or otherwise, show that : (4)

9.4. Hence or otherwise, calculate the value of : (6)

QUESTION 10: [7]10.

The figure below shows . B lies on PR, such that and A lies on

PQ such that . BC is drawn parallel to AR as shown:

10.1. Write down the value of the ratio : (2)

10.2. Calculate the value of the ratio : (5)

QUESTION 11: [14]11.

In the figure below, CD is a tangent to the circle ABDEF at D. Chord AB is produced to

C, and . Chord BE cuts chord AD at H and chord DF at G as shown:




11.4. Hence, prove that is isosceles: (3)

END OF EXAMINATION

MARK RECORD SHEETFOR OFFICIAL USE ONLY

NAME OF LEARNER:

NAME OF TEACHER: SMITH STATHAM MARAIS NONYANE

Statistics AnalyticalGeometry

EuclideanGeometry &

MeasurementTrigonometry

1 / 10

2 / 17

3 / 12

4 / 10

5 / 10

6 / 18

7 / 20

8 / 15

9 / 17

10 / 7

11 / 14

TOTAL / 20 / 37 / 48 / 45

TOTAL / 150 %

jeppe high school for boys - st stithians collegemaths.stithian.com/new caps 2018 prelim...

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