Page | 1

Please read the following instructions carefully:

1. All answers will be completed on this question paper. A separate data sheet is provided. Please check that all pages are printed.

2. Please write your name in the given space and read all questions carefully.

3. Diagrams are not necessarily drawn to scale.

4. You may use an approved non-programmable and non- graphical calculator unless otherwise stated. Ensure your calculator is in degree mode.

5. Round off your answers to one decimal digit where necessary, unless otherwise stated.

6. All necessary working details and geometrical reasons must be clearly given to gain the allocated marks.

7. Present your work neatly and write legibly.

Name :

TREVERTON COLLEGE TRIAL EXAMINATION 2017

SUBJECT: MATHS PAPER II GRADE: 12

TOTAL: 150 MARKS TIME: 3 HOURS

EXAMINER: MRS BEBB MODERATOR: MR BALIE

Page | 2SECTION A

QUESTION 1

AB is the line 2 x− y+6=0 and CD is the linex−2 y=4 .

a) Write down the co-ordinates of A, B, C and D, the intercepts with the axes.

(6)

b) Write down the equation of the line that is perpendicular to AB and passes through A

(2)

B D

C

A

y

Ox

Page | 3

c) Prove ΔOAB is equiangular to ΔODC and therefore similar.

(7)

d) Prove that ABCD is a cyclic quadrilateral

(2)

[17]

QUESTION 2

In the given diagram O (0 ; 0) is the centre of the circle. L (x ; y) and N (12 ; 5) are two points on the circle. LON is a straight line. The point M (t ; -1) lies on the tangent to the circle at N.

a) Write down the equation of the circle (2)

.

b) Determine the value of t

L

N

O M (t;-1)

Page | 4

(6)

c) Another circle is defined by the equation x2+6 x+ y2+4 y=4 .

Rewrite this equation in the form ( x−a )2+( y−b )2=r2 and determine the

value of a+b+r

(4)

[12]

QUESTION 3

In the above diagram Δ ABC has D and E on line BC so that AD ll TE

BD is 10 cm and DC = 15 cm and AT:TC = 2:1

a) Show that D is the midpoint of BE.

(3)

A

B C

D E

T

10cm

Page | 5

(b) Determine the value of

Area ΔBTEArea ΔTEC

(2)

[5]

QUESTION 4

Using the given diagram of circle ABC with O as the centre,

prove the theorem which states

[4]

A

B

C

O

Page | 6QUESTION 5

O is the centre of circle ABCDE with , AB = DC and .

Calculate the size of:

a)

(2)

b)

(2)

c)

(6)

[10]

B

O

12

C1

3

4

2

12

3

A

D

E

Page | 7QUESTION 6

a) Without the use of a calculator determine:

1)

(3)

2)cos 2 A if cos A=−12

13and A∈ [0 ° ;180° ]

(3)

b) Simplify without the use of a calculator:

1−cos2 xsin 2 x

(6)

Page | 8c) Give the general solution for x if: sin2 x = 2cos x + 2

(5)

[17]

QUESTION 7

In the given diagram, the box and whisker plot depicts the number of push ups done by 36 students in one minute. The interquartile range and the range are 15 and 25 respectively.

a) Determine a and b

(2)

b) The sports coach puts the pupils through a training program and then re-tests them and plots the new data on a cumulative frequency curve to see if anyone had improved.

a 29 3917 b

Page | 9

1) Draw a box and whisker plot of the new data

(3)

2) Would you say this data is skewed? Why?

(2)

3) Which part of the distribution indicates the best improvement?

(1)

4) Comment on the effectiveness of the training program based on

the data, giving reasons for your conclusions

(3)

[11]

Number of Push ups

Cumulative Frequency

5

10

20

250

30

35

15

40

10

45

505

15 20 25

(49;36)

30 35 40 45 505

Page | 10SECTION B

QUESTION 8

An athlete’s ability to take and use oxygen effectively is

called his / her VO2-max.

Twelve athletes with pre-recorded VO2-max readings ran for one hour. The

distances (in kilometres) that they each covered are represented in the table:

VO2-max reading 20 55 30 25 40 30 50 40 35 30 50 40

Distance run (km) 8 18 13 10 11 12 16 14 13 9 15 12

a) Draw a scatter plot of the data. (Place VO2-max on the horizontal axis)

(4)

b) Use the correlation coefficient to describe the correlation between the two sets of data.

Page | 11

(2)

c) Determine the equation of the least squares regression line (line of best fit) and draw it on the graph.

(5)

d) Use the method of interpolation to predict the distance run if an athlete has a

VO2-max value of 26.

(2)

[13]

QUESTION 9

Page | 12

In the figure, the graphs of f ( x )=2 cos(x+a ) and g( x )=1+sin bxare given forx∈ [−180° ;180 ° ]

a) Determine the values of a and b using the graphs.

(2)

b) Determine the values of x for which

g ( x )f ( x )

≤0

(3)

c) The y axis is translated 30 ° to the right. Determine the new equation of f

in the form y=c sin( x+d )with reference to the new set of axes.

(2) [7]

QUESTION 10

a) Prove that: sin 3 x − sin x = 2cos2 x sin x

Page | 13

(6)

b) A rectangular roof which is 12m by 9m slopes at an angle ofWrite an expression to determine the slope of its diagonal (A DB ) interms of

(5)

[11]

12 m A

B

C

α

α.

D

E

9 m

α

Page | 14QUESTION 11

In the given diagram line AE is the diameter of the circle with centre O and a radius of

1 unit. .

. Determine an expression for:

a) p in terms of x

(2)

b) AB in terms of x

(3)

c) Hence determine BE in terms of p

(3)

[8]

A

D

E

O

B

2

C

1

x

p

Page | 15

x

y

P

MA(2 ; 3)

BDO

QUESTION 12

A (2 ; 3), the midpoint of radius OP, lies on the circumference of the smaller circle with diameter OA. Write down:

a) The equation of the larger circle.

(3)

b) The equation of the smaller circle.

(3)

c) Determine the equation of the tangent to the large circle at P.

(4)

d) Hence, determine the equation of a tangent to the large circle which is parallel to the tangent in (c).

(2)

(2)

[12]

x

y

P

M

A(2 ; 3)

O BD

Page | 16QUESTION 13

21

321

3

2

1

2

1T

P

SR

Q

TP and TS are tangents to the circle. Q is a point on PR such that Q1 = P1 .

Prove the following:

a) TQ // SR.

(4)

b) TQ bisects S QP .

(5)

Page | 17 [9]

QUESTION 14

In the above diagram AOB is the diameter of the semi-circle, centre O, MO NB, ON

and MB intersect at K and .

a) Prove that MB bisects

`

(4)

b) Express in terms of x

(2)

c) Calculate the value of

A B

1

O

NM

K

23

1

2

1

12

2

3

Page | 18(3)

[9]

QUESTION 15

F

21

2 1

321

321

D

A

C

B

E

In the diagram, BC is the diameter of the circle BCDE. BD bisects A BC , and A = E3

ProveAD2 = DF . DB

[5]

Page | 19

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