JEPPE HIGH SCHOOL FOR BOYS

ST STITHIANS COLLEGE

Departments of Mathematics

GRADE 12

PRELIMINARY (MID-YEAR) EXAMINATION – PAPER 2

DATE:

16th July 2018

TIME:

3 hours

TOPICS:

Statistics, Geometry, Measurement and Trigonometry

TOTAL MARKS:

150

EXAMINER:

Mr. P. Statham (GC)

MODERATOR:

Mr. R. Rhodes-Houghton (External)

MEMORANDUM

SECTION A

QUESTION 1: [10]

1.

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

36

2

115

32

3

50

24

4

80

19

5

75

26

6

115

27

7

60

29

8

135

34

1.1. Calculate the mean and standard deviation of the rainfall recorded over the

8-year period (rounded off to two decimal places):(2)

√a correct mean

√a correct std. deviation

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)

30 mm of rainfall is very low and would lie further away from the rest of the rainfall data.

Therefore, the standard deviation value would increase (be higher).

√a correct reason

√a correct “increase”

1.3.

Determine the equation of the line of best fit (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)

√a correct A

√a correct B

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)

The regression is moderate, and therefore rainfall received IS UNLIKELY to be a good predictor of the expected maize yield

√a correct r

√a correct explanation

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)

or

√a correct lower value

√a correct upper value

QUESTION 2: [17]

2.

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:

E

O

M(0;h)

C(k;-5)

B(3;2)

A(-2;3)

y

x

2.1.

Show that and :(3)

√m subn into midpoint formula

√a correct h

√a correct k

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

√a correct mAC

√ca correct mBE

√m subn into line formula

√ca correct equation for BE

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

√a correct equation for AC

√m make equations equal to each other

√ca correct xE coordinate

√m subn back to find yE

√ca correct E coordinates

2.4.

Determine the area of :(5)

√m distance formula

√a correct distance for AC

√a correct distance for BE

√m sub into area formula

√ca correct area

QUESTION 3: [12]

3.

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

√a correct angle using

reduction formula

1

m

3.1.

:(3)

√a correct x by Pythag.

√ca correct cos value

3.2.

:(3)

√m use double angle formula

√ca correct subn in formula

√ca correct sin value

3.3.

:(2)

√a correct angle by reduction

√ca correct tan value

3.4.

:(4)

√a correct angle by reduction

√m use double angle formula

√ca correct subn in formula

√ca correct sin value

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:

B

A

C

D

1

2

E

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

4.1.

:(4)

√a correct angle O1

√a correct reason

√a correct angle A

√a correct reason

4.2.

:(2)

√a correct reason

√a correct angle C1+2

4.3.

:(4)

√a √a correct reasons (x2)

√a correct angle C1

√a correct angle C2

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

5

5

7

12

12

24

7

31

3

34

2

36

2

38

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

√a √a correct values (x2)

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.

√m show correct median

√a correct fuel consumption

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

√m use interval midpoints

√a correct fuel consumption

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)

√a correct a

√a correct b

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

√a correct c

√a correct d

6.1.3.

Write down the range of :(2)

√a √a correct range (x2)

6.1.4.

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

√a correct graph values

√a correct value

6.1.5.

For what approximate values of x is :(2)

√a √a correct x values (x2)

6.1.6.

For what values of x is :(3)

√a √a √a correct x values (x3)

6.2.

Solve the equation for

(rounded off to one decimal place):(5)

√m make tan, √a correct value

√a correct reference angle

√ca correct equation

√ca correct angle θ

SECTION B

QUESTION 7: [20]

7.

7.1.

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

√m complete the square

√a correct equation

√m use distance formula

√ca correct centres distance

√m equal to radii of circles

√ca correct k

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:

A

B

D

O

y

x

M

C

7.2.1.

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

√m use distance formula

√a correct distance for AB

√m use midpoint formula

√a correct centre MAB

√ca correct equation

7.2.2.

Show that the coordinates of C, the point where the tangent at B intersects the x-axis is :(5)

√a correct gradient

√ca correct gradient for AB

√m subn into line formula

√ca correct tangent equation

√ca correct x intercept

7.2.3.

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

√m find angle A using mAB

√a correct angle A

√m use isos triangle

√ca correct angle AMD

QUESTION 8: [15]

8.

8.1.

Evaluate without using a calculator:(3)

√m factorise out negative sign

√a correct compound angle

√a correct value (no ca)

8.2.

Prove that :(5)

√m change double angles

√a correct trinomial

√m factorise denominator

√m factorise numerator

√a cancel to get correct

final fraction (no ca)

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)

√m use cosine rule

√a correct subn in cosine rule

√ca correct length of BC

√m correct trig ratio

√ca correct length of AB

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:

√a correct subn in formula

√a correct V (no ca)

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:

R

T

S

P

Q

9.1.

Prove that :(4)

√a correct angle and reason

√a correct angles and reason

√a correct angles and reason

√a correct similarity

9.2.

Prove :(3)

√a correct ratio

√a correct subn of PS

√a correct equation (no ca)

9.3.

Hence or otherwise, show that :(4)

√m use Pythag.

√a correct PQ in terms of x

√a correct subn in equation

√a correct ST (no ca)

9.4.

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

√m use area rule

√a correct sides and angles

√a correct subn in numerator

√a correct subn in denom.

√a correct ratio

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)

√a area prop. to base

(same height)

√a correct ratio

10.2.

Calculate the value of the ratio :(5)

√a correct ratio for PC:CA

√a correct ratios for PC, CA

√a correct ratio for DQ:BQ

√a correct subn in ratio

√a correct ratio

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:

11.1.

Prove that :(4)

√a correct angles and reason

√a correct angles and reason

√a correct angles and reason

√a correct similarity

11.2.

Prove that :(3)

√a correct angles and reason

√a correct angles and reason

√a correct conclusion

11.3.

Prove that :(4)

√a correct angles and reason

√a correct angles and reason

√a correct angles and reason

√a correct conclusion

11.4.

Hence, prove that is isosceles:(3)

√a correct angles and reason

√a correct angles and reason

√a correct reason

END OF EXAMINATION

11

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