MATHEMATICS TRIAL EXAMINATION PAPER II

TIME: 3 HOURS FRIDAY 7 SEPTEMBER 2012 MARKS: 150

PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY:

Answer ALL the questions in the spaces provided.

This paper consists of 5 sections: A, B, C, D and E. Please check that your question paper is

complete.

Hand in each section separately.

Write your Examination Number, name and teacher’s name at the top of each section.

Show all necessary working.

Calculators may be used unless otherwise specified.

A separate DATA SHEET is provided.

Work neatly and legibly.

ANSWERS ONLY NO MARKS

Page 2 of 22

SECTION A (30 Marks)Examination Number: ______________________________Name: ______________________ TEACHER’S NAME: _________________

(Pike; Owen; Lubbe; Botha; Cocks)

PLEASE TURN OVER

Page 3 of 22

QUESTION 1

Refer to the figure. In the figure, A(4 ; 5), B (2 ; -1) and C (10 ; 3) are given. BA is produced to D and DC⊥BC. The circle, with centre A, passes through the midpoint E of BA.

a) Find the coordinates of E, the midpoint of BA. (2)

b) Determine the equation of the circle, centre A, which passes through point E.

(3)

c) Determine the equation of the line BA.

(3)

PLEASE TURN OVER

x

y

B (2 ; -1)

C (10 ; 3)

D

A (4 ; 5)

F

E

Page 4 of 22

d) Determine the equation of the line DC.

(4)

e) Show, by calculation, that the coordinates of D are (6 ; 11).

(3)

f) Find the length of the tangent DF from D to the circle. (Leave your answer in surd form).

(5)

[20]

PLEASE TURN OVER

Page 5 of 22

QUESTION 2

In the sketch below the shape A was created originally and then several transformations carried out.

a) Describe the following transformations using mapping notation i.e. ( x ; y )→

(i) A to B (2)

(ii) B to D (2)

b) Describe the transformation from A to C in words. (3)

c) (i) Find area of Shape C. (1)

(ii) Hence determine the area of Shape E if Shape E is the enlargement (through the origin)

of Shape C by a factor of 25 (2)

[10]

−4 −2 2 4 6

−4

−2

2

4

x

y

AB

C

D

E

Page 6 of 22

[SECTION A: TOTAL 30 MARKS]

x

y

M

Page 7 of 22

SECTION B (31 Marks)Examination Number: ______________________________Name: ______________________ TEACHER’S NAME: _________________

(Pike ;Owen ;Lubbe ;Botha ;Cocks)

QUESTION 3

The length of the radius of the circle with equation x ²−2 x+ y2+4 y=k is 5 units.

Write down the coordinates of the centre, M, of the circle and the value of k .

[6]

PLEASE TURN OVER

Page 8 of 22

QUESTION 4

In the diagram: ( x−3 )2+( y−4 )2=25 is the equation of the circle with centre M, and cuts the y axis at the origin and at A.AB is a tangent to the circle.

a) Find the equation of the tangent AB. (6)

[6]

PLEASE TURN OVER

y

x

M (3 ; 4)

A

B

Page 9 of 22

QUESTION 5

60 boys and 60 girls took part in an activity to determine how far they could jog without stopping.The five number summary for girls was 895; 1250; 1580; 1960 and 2275.The five number summary for boys was 925; 1250; 1740; 2275 and 2460.

a) Draw a box-and-whisker graph for the girls.

(2)

b) How many girls ran further than 1250 m? (1)

c) How many boys ran between 1250 m and 2275 m? (1)

d) Four girls ran a distance shorter than any boy. Calculate the percentage of children who ran between 925 m and 2275 m. (2)

[6]

QUESTION 6

A large company employs 9 salespersons. The commission that each salesperson earned (in rands) in a certain month is shown below.

3 900 5 700 7 300 10 600 13 000 13 600 15 100 15 800 17 100

6.1 Calculate the mean of the above data. (2)

6.2 Calculate the standard deviation for the data. (2)

6.3 The company rates the sales staff according the amount of commission earned. A salesperson whose commission is more than one standard deviation above the mean receives a rating of ‘good’. How many salespersons will receive a rating of ‘good’ for that month? (2)

[6]

PLEASE TURN OVER

800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 23000x

f

Page 10 of 22

QUESTION 7

The ogive gives information on the weekly income of workers in a sample.

a) Complete the cumulative frequency distribution table from the ogive.

Weekly income (R) f cf0 ¿ 1000 250

1000 ¿ 2000 220

2000 ¿ 3000 480

3000 ¿ 4000

4000 ¿ 5000 495

5000 ¿ 6000

(6)

b) Use the ogive to determine the number of workers in the sample who earn R1 500 or more per week. (1)

[7]

[SECTION B: TOTAL 31 MARKS]

1000 2000 3000 4000 5000 6000

100

200

300

400

500

x

f

Weekly income of workers

Cumulative Frequency

−180 −120 −60 60 120 180

−1

1

x

y

−90 −60 −30 30 60 90x

y

15

1

−360 −270 −180 −90 90 180 270 360

−1

1

x

y

−360 −180 180 360

−1

1

x

y

Page 11 of 22

SECTION C (31 Marks)Examination Number: ______________________________Name: ______________________ TEACHER’S NAME: _________________

(Pike ;Owen ;Lubbe ;Botha ;Cocks)

QUESTION 8

Give an equation for each of the following graphs.

a)

(2)

b)

(2)

c)

(2)

d)

(2)

[8]

−180 −90 90 180

−2

−1

1

2

x

y

Page 12 of 22

QUESTION 9

Sketched above is f ( x )=2sin x ; x∈[−180 ° ;180° ].

a) On the same system of axes above, draw a neat sketch of g ( x )=tan x ; x∈[−180° ;180 ° ]. Label 4 exact coordinates on the graph, g(x ), other than the x-intercepts (without the use of a calculator).

(4)

b) (i) Write the equation sin x=sin 2 xas an equation containing only tan x and 2sin x.(Show all working)

(2)

(ii) Determine the general solution for sin x = sin 2 x

(5)

[11]

PLEASE TURN OVER

f

Page 13 of 22

QUESTION 10

Solve for x if: cos3 x=−0,5 and x∈[90 ° ;270° ]

[5]

QUESTION 11

In this question round off all degrees found to the nearest degree.

a) Given: AC=6√2 units ; BC=3√2 units and Area of ∆ ABC=9 units2

Find the value of θ where θ∈[90° ;180 ° ].

(3)

b) The gradient of BC= 950 . Find the gradient of AC (correct to 2 decimal places).

(4)

[7]

C3 2

6 2

A

B

x

y

Page 14 of 22

[SECTION C: TOTAL 31 MARKS]

Page 15 of 22

SECTION D (27 Marks)Examination Number: ______________________________Name: ______________________ TEACHER’S NAME: _________________

(Pike ;Owen ;Lubbe ;Botha ;Cocks)

QUESTION 12

Simplify: sin (180 °+ x ) . cos² (90 °+x )+sin (−x ) . sin ²(90 °−x)

[7]

QUESTION 13 (NO CALCULATOR)

If tan2θ=−247 where 0 °<2θ<180 ° , find, with the aid of a sketch, the value of:

a) cos2θ

(3)

b) cosθ

(2)

[5]

Page 16 of 22

PLEASE TURN OVERQUESTION 14

Prove that:sin 3xsin x

− cos3 xcos x

=2

[4]

QUESTION 15

The equation sin θ+cosθ=sin 24 °+cos 24 ° has 2 solutions in the interval (0 ° ;90 °). By inspection 24 ° in one solution, what is the other solution?

[1]

QUESTION 16

If sin 2 25 = m, calculate without using a calculator, the value of cos20 ° cos70 °+sin 20° sin 70°in terms of m.

[4]

Page 17 of 22

PLEASE TURN OVER

QUESTION 17

∆ ABC has AB=2units, AC=3units. And BC 2 = 13 + √108 Calculate the the size of A.

[3]QUESTION 18

Describe the transformation of the point (a;b):

(a;b )→( a−√3b2

; a√3+b2 )

[3]

3 units

2 units

CB

A

O120

S'

S (-1 ; 4)

x

y

Page 18 of 22

[SECTION D: TOTAL 27 MARKS]SECTION E (31 MARKS)

Examination Number: ______________________________Name: ______________________ TEACHER’S NAME: _________________

(Pike ;Owen ;Lubbe ;Botha ;Cocks)

QUESTION 19

The point S(-1 ; 4) is rotated through an angle of 120º anticlockwise about the origin as shown in the diagram.Determine the coordinates of S’.

[4]

QUESTION 20

Without the use of a calculator, determine the value of cos ² (325 ° )−si n2145 °cos340 ° . tan 200°

10 units

60RQ

P

Page 19 of 22

[8]

PLEASE TURN OVERQUESTION 21

a) Complete: sin ² x+cos ² x=¿____________________________ (1)

b) Hence, determine the value(s) of tan x if 5cos ² x+5sin xcos x=3.

(5)

[6]

QUESTION 22

a) Complete: sin [180 °−(P+Q ) ]=¿¿________________________________ (1)

b) In the figure, PQR is a triangle with Q=60° and PQ=10units.

If sin (P+Q )=58 , calculate the length of PR without using a calculator,

leaving your answer in surd form.

(4)

Page 20 of 22

[5]

PLEASE TURN OVER

Page 21 of 22

QUESTION 23

The ornamental tea-cup in the picture is in the form of a cone;The top of which has been cut off to leave a truncated cone.

The inner diameter of the top of the cup is 8 cm, the diameter of the innerBottom is 3 cm and the depth of the cup is 10 cm.

The volume of the cone [Vol=13πr 2h]

(Leave all answers in terms of π)

In the diagram:DC=n ;EB=m; AB=a;BC=b.

Fact: a

a+b=mn

a) Use the fact given above to prove that x, the height of the part of the cone that has been cut off, is 6 cm.

(3)b) Calculate the volume of the part of the cone that was “cut off”.

(2)

c) Calculate the volume of the cone of which the depth is x + 10 cm.

(2)

d) Calculate the volume of the cup.

(1)

[8]

C

Bm

n

A

E

D

b

a

10 cm

x cm

8 cm

3 cm

Page 22 of 22

[SECTION E: TOTAL 31 MARKS]