mathematics: paper ii matric preliminary …

MATHEMATICS: PAPER II

MATRIC

PRELIMINARY EXAMINATIONS SEPTEMBER 2021

Marks: 150 Examiner: A Abatzidis Time: 3 Hours Moderator: D Garofoli Reading Time: 10 Min

EXAMINATION NUMBER:

PLEASE READ THESE INSTRUCTIONS CAREFULLY:

1. This question paper consists of 28 pages and a separate Information Sheet.

Please check that your paper is complete.

2. Additional space is given on pages 29 and 30. If this space is used, indicate clearly

which question is being answered.

3. Read the questions carefully.

4. Answer ALL the questions on the question paper. Ensure that you have written your

examination number in the space provided above.

5. Diagrams are not necessarily drawn to scale.

6. All the necessary working details must be clearly shown.

7. Answers only will not necessarily be awarded full marks.

8. Approved non-programmable and non-graphical calculators may be used unless otherwise

stated. Ensure that your calculator is in DEGREE mode.

9. Give answers correct to ONE decimal digit, where necessary.

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

______________________________________________________________________________

MATRIC MATHEMATICS PAPER II: SEPTEMBER 2021 of 30

SECTION A QUESTION 1 A (6; – 4), B (8 ; 2), C (3; a) and D (b ; c) are points on the Cartesian plane. (a) Determine the value of a if A, B and C are collinear. (3) (b) Determine the value of b and c if B is the midpoint of AD. (3) (c) Find the angle of inclination of AB. (2) [8]


QUESTION 2 Tassy measured the heights of several plants (in cm) at different stages after planting, and she

recorded the following data:

x = days after

planting 14 20 8 15 18 11 13

h = height (cm) 6 11 3 8 10 4

Tassy lost the record of the last height, but she does know that the equation of the regression

line is:

h = 0,72x – 3,31

(a) Estimate, to the nearest centimetre, what the last recorded height was. (2)

(b) Calculate the correlation coefficient for the data relating to the first 6 plants correct to

FOUR decimal digits. (i.e. ignoring the last column). (2)

(c) Some time later, Tassy found another plant’s height 25 days after planting to be 20 cm.

Comment on how surprised (or not) Tassy would be in light of her previous results. (2)

[6]


QUESTION 3 Refer to the figure: A circle has equation

2 22 6 15x x y y− + + = (a) Find the coordinates of the centre, A, and the radius of the circle. (4) (b) Determine the co-ordinates of B and C, the x-intercepts of the circle. (4)

A

D (5 ; – 6)

x

y

O B C


(c) Determine the equation of the tangent to the circle at D (5 ; – 6), given A (1 ; – 3). (4) (d) Determine the area of ΔABC. (3)

(e) Given that another circle ( ) ( )2 2

1 1x y b− + − = touches this circle with centre A at one

point only, give one possible value of b. (2)

[17]


QUESTION 4 (a) Refer to the following statements below labelled A to C and match the statements to

their most expected histogram labelled D to F. (3)

A.

The age at death in

developed countries

(First world countries).

B.

The salaries of all

employees at a major

corporate company.

C.

The results of a test

where some of the

students had been taught

and others had not been

taught the section of work

covered.

D.

E.

F.

Fill in the answers in this block:

A →

B →

C →


(b) The diagram below shows a cumulative frequency curve for the lengths of telephone calls

from a home landline during the first 6 months of the year.

Time (minutes)

State whether each of the following statements are True or False: (4)

(1) The distribution of these times is skewed.

(2) The majority of the calls last longer than 6 minutes.

(3) The majority of the calls last between 5 and 10 minutes.

(4) The majority of the calls are shorter than the mean length.

5 10 15 20 25 30

20

40

60

80

100

120

140

160

x

f

Cum

ula

tive

freq

uen

cy


(c) A group of athletes frequently run round a cross-country course in training. The box and

whisker plots below represent the times, in minutes, taken by athletes A, B, C and D to

complete the course a certain number of times.

Time (minutes)

(1) Compare the times taken by athletes C and D. Include variability and

skewness in the analysis. (4)

26 27 28 29 30 31 32 33 34 35 360

x

f

A

B

C

D


(2) Which of the athletes A or B would you pair up to race and possibly win against

(i) C ? Justify your choice. (2)

(ii) D ? Justify your choice. (2)

(d) The following information summarises the year marks for a class of 20 students.

( )20 20

2

1 1

1560 1220i i

i i

x x x= =

− = =

Determine the standard deviation for the class. (2)

[17]


QUESTION 5

(a) Given: sin20op =

Express each of the following in terms of p, showing all working:

(1) sin200o (1)

(2) sin50o (4)

(b) Given that θ ; 2θ , and 3θ are the angles of a triangle evaluate, without the use of a

calculator: 2 2 2cos cos 2 cos 3 + + (4)


(c) Evaluate, without the use of a calculator:

sin124 .sin64 sin214 .sin26o o o o+ (5)

[14]


QUESTION 6

The JP Morgan Chase Tower in Houston is a prism with a height of 305 metres. The cross section is a square with an isosceles triangle removed. Refer to the dimensions as shown. Aerial view JP Morgan Chase Tower

50 m

25 m

305 m


Determine: (a) the volume of the tower. (3) (b) the surface area of the tower. (4)

[7]


QUESTION 7 Refer to the figure: BD is a diameter and ABF is a tangent.

ˆ 2ABE x= and AB = EB. Determine the following angles in terms of x , giving reasons:

(a) 1C

(b) ˆECD

(c) 1E

(d) D

(e) 2E

[7]

76 marks

E

D

B

A

C

F

2x

1

2

2

2x

1


SECTION B QUESTION 8

(a) Prove:

( ) ( )( ) ( )

2 2cos 45 sin 452sin

cos30 cos 30 sin30 sin 30

o o

o o o o

A AA

A A

− − −=

+ + + (5)

(b) Determine the general solution: 33tan 2 tan2x x= (6)


(c) Given: 2 1 cos2tan

1 cos2

PP

P

−=

+

(1) Prove the identity. (5) (2) For which values of P will the identity be undefined? (3)

[19]


QUESTION 9 (a) Marianne is riding with the cable car from Station A to Station B, the highest point at the

top of Mount Glacier. The ride takes 16 minutes with the cable car travelling at an average speed of 2 metres per second. Assuming the cable car moves in a straight line forming a 25o angle of elevation from Station A, find the height of Mount Glacier correct to the nearest metre. (4)

25o

B

Cable •

A

Mount Glacier


(b) Dave is constructing a portable basketball net for an indoor court. The structure is made from a metal tubing support, with dimensions as shown in the diagram (not drawn to scale).

B

C

92o

H

78o

72o

A

1,3 m


Determine the length of tubing that Dave needs for AB. (6)

[10]


N

M

P

RQ

QUESTION 10 Refer to the figure: P is the centre of the circle with a radius of 73 units. M is the midpoint of chord QR. N is a point on PR such that PN = 40 units. MN⊥ PR.

(a) Give a reason why PM⊥ QR. (1) (b) Determine, giving reasons, the length of MR, correct to the nearest whole number. (4) [5]


QUESTION 11

(a) Given: Δ XYZ and Δ LMN such that ˆ ˆ ˆ ˆ ˆ ˆ; andX L Y M Z N= = = .

Using the given diagram, prove the theorem which states that if two triangles are equiangular, then their sides are in proportion.

i.e. Prove that XY XZ

LM LN=

Construction: (1) Proof: (5)

Z

X L

Y M N


(b) In the diagram, O is the centre of the circle with diameter AOB. The tangent through C intersects AD produced at F.

OD⊥AC and CF⊥AF

Prove that:

(1) Δ FCD ||| Δ CAB (4)

1

E

D

F

B

O

C

A

1

1

1

2

2

2

2

3

3 1

2

4


(2) FC.CB = 2 FD.AE (4)

(3) 1 2ˆ ˆA A= (3)

[17]


QUESTION 12 So far, all the Analytical Geometry that we have done has been two-dimensional, in the co-ordinate plane. Solids in space are generally three-dimensional. Co-ordinates in the plane can be extended to co-ordinates in space. For this we need three co-ordinate axes that are mutually perpendicular to each other and pass through the origin O. These are usually the x, y and z-axes as shown in the sketch. This gives three co-ordinate planes.

• The (horizontal) xy-plane where z = 0.

• The (vertical) yz-plane where x = 0.

• The (vertical) xz-plane where y = 0.

❖ A point P in this system will have three co-ordinates ( ); ;P x y z .

❖ The distance between two points ( ); ;A A AA x y z and ( ); ;B B BB x y z becomes

( ) ( ) ( )2 2 2

A B A B A BAB x x y y z z= − + − + −

❖ And the midpoint M between A and B becomes

; ;2 2 2

A B A B A Bx x y y z zM

+ + +


(a) Determine the equation of the sphere with centre ( ); ;C h k l and radius r.

Note: A sphere is the set of all points ( ); ;P x y z such that the distance PC = r. (2)

(b) Determine the centre and the radius of the following sphere:

2 2 2 4 2 6 2 0x y z x y z+ + + + − − = (3)


45 90 135 180 225 270 315 360

−0.5

−0.25

0.25

0.5

0.75

1

1.25

1.5

x

y

[5] QUESTION 13 Deborah is approximating the phases of the moon by using trigonometric functions as models.

The function f has equation ( ) sinf x a x b= + and this can be used to estimate the dates for the

moon phases for August 2021 in the southern hemisphere.

August 2021

90o 180o 270o 360o

(a) Give the value of a and b. (2)

f g


(b) The model assumes a 28-day moon cycle, and on 1 August 2021 Deborah saw the last quarter (half moon, waning). On what date in August did Deborah see the new moon? (2)

(c) State the equation of g in the form ( ) ( )cosg x m x p q= − + . (3)

(d) The graph of g shows the moon phases for September 2021.

On what date in September can Deborah expect to see the new moon? (2)

[9]


S

P

T

RQ

QUESTION 14 Refer to the figure: In ΔPTR: PT ǀǀ QS, RS = 5 units, ST = x units,

RQ = 3x + 1 units, PQ = 6 units

(a) Calculate the value of x, with reasons. (4)

(b) Assuming x = 3 and area ΔQRS = p units2, find the area of QSTP in terms of p. (5)

[9]

74 marks

TOTAL: 150 marks

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ADDITIONAL SPACE

REMEMBER TO CLEARLY INDICATE AT THE QUESTION THAT YOU USED THE ADDITIONAL SPACE TO ENSURE THAT ALL ANSWERS ARE MARKED.


ADDITIONAL SPACE

mathematics: paper ii matric preliminary …

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