BEAULIEU COLLEGE PRELIMINARY EXAMINATIONS GRADE 12 ... CAPS 2018 Prelim... · at point F. Black Widow is standing at the foot of the building at point B. Scarlet Witch and Captain

BEAULIEU COLLEGE: Grade 12 Mathematics July 2018 Paper 2 Page 1 of 26

BEAULIEU COLLEGE

PRELIMINARY EXAMINATIONS GRADE 12

MATHEMATICS PAPER 2

Time: 3 Hours 150 marks

Date: 23 July 2018

Examiner: Ms Smith Moderator: Mr Ruiz-Mesa

EXAMINATION NUMBER: 1 8 1 0 1 2 0 2 0

TEACHER: Ruiz-Mesa Scholefield Smith

_________________________________________________________________________ PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This paper consists of 26 pages and an Information Sheet of 2 pages (i–ii).

Additional space for working out is provided on pages 25 and 26. Please check that

your paper is complete.

2. Write the last three digits of your examination number in the space provided above

and answer all the questions on the question paper.

3. Please note that diagrams are not necessarily drawn to scale.

4. All necessary working details must be shown. 5. Round your answers off to TWO decimal place unless stated otherwise.

6. Approved non-programmable and non-graphical calculators may be used, unless

otherwise stated.

7. Ensure that your calculator is in DEGREE mode.

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


MARKING GRID

Question

Analytical Geometry

Statistics

Trigonometry

Euclidean Geometry and Measurement

1

/7

2

/15

3

/17

4

/6

5

/6

6

/10

7

/14

8

/8

9

/21

10

/10

11

/20

12

/16

TOTAL

PER TOPIC

/43

/21

/39

/47

TOTAL MARK

/150


SECTION A QUESTION 1

A study is done with twelve students to understand the relationship between their heart

rates (in beats per minute) before and after a brisk walk. The results are shown in the table

below:

Before walk 86 62 52 90 66 80 78 74 64 76 80 68

After walk 98 70 56 110 76 96 86 84 74 94 92 90

(a) Calculate the correlation coefficient and then comment on the strength of the

relationship. (Round off correct to four decimal places.) (3)

(b) If the equation of the least squares regression line is y A Bx , calculate the

values of A and B . (Round off correct to four decimal places.) (2)

(c) Should the regression line in (b) be used to predict the heart rate of a student

after a brisk walk if their heart rate before the walk was 108? Explain your

answer. (2)

[7]


QUESTION 2

In the diagram below, points A 3 ;12 , B 5 ; 2 and D 8 ; 7 are given. AB DC

where C is the y -intercept of DC. The x -intercept of DC is E. BC is joined.

x

y

A (3 ; 12)

B (- 5 ; 2)

C

D (8 ; 7)

EO

(a) Determine the length of AB. (Leave your answer in simplest surd form.) (2)

(b) Determine the gradient of AB. (2)


(c) Determine the equation of line CD. (3)

(d) Determine the coordinates of point E. (2)

(e) Determine the size of ˆOCE . (4)

(f) Determine the area of OCE . (2)

[15]


QUESTION 3 PLEASE ENSURE YOUR CALCULATOR IS IN DEGREE MODE. (a) If cos18 .cos20 sin18 .sin20 p , determine the following in terms of p

without the use of a calculator:

(1) cos38 (1)

(2) sin232 (2)

(3) sin76 (3)


(b) Prove that sin2 sin sincos2 cos cos 1

(5)

(c) Determine the general solution for 4sin 2cos2 2 . (6)

[17]


QUESTION 4 In the diagram, the graphs of the functions sinf x a x and tang x bx are drawn

on the same set of axes for 0 ; 225x .

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

(b) Determine the period of 2g x . (2)

(c) Determine the range of 2f x for 0 ; 225x . (2)

[6]


QUESTION 5

O

P

Q

R S

B

12 3

12

31 2

1

2

50 °

(a) Complete the following sentence:

“The opposite angles of a cyclic quadrilateral are ___________________________” (1)

(b) Determine the size of ˆQSR . (5)

[6]

In the diagram alongside O is

the centre of the circle and

QOB is a diameter.

P, R and S are points on the

circumference of the circle.

2O 50


QUESTION 6

A 8 ; 2 , B and D 0 ; 8 are the vertices of a triangle that lie on the circumference of

circle 2 2 2 2 48x y x y with diameter BD and centre M, as shown in the diagram

below:

x

y

A (- 8 ; 2)

B

D (0 ; 8)

M

(a) Determine the coordinates of point M, the centre of the circle. (4)


(b) Determine the coordinates of point B. (2)

(c) Determine the equation of the tangent to the circle at point A. (4)

[10]


Cum

ulat

ive

Freq

uenc

y

Ages of Employees

QUESTION 7

The diagram below is a cumulative frequency diagram for the ages of employees working

at a company.

(a) How many people work at the company? (1)

(b) Use the cumulative frequency diagram to complete the following table: (2)

Ages of Employees Frequency 20 30x 7

30 40x

40 50x 26

50 60x

60 70x 12


(c) Determine an estimate for the mean ages of the employees. (2)

(d) Determine an estimate for the standard deviation of the ages. (2)

(e) Draw a box and whisker plot of the data on the same grid as the cumulative

frequency diagram. (4)

(f) Describe the skewness of the ages of the employees. (1)

(g) The company wants to lower the retirement age of their employees to 60.

They will then follow a policy to employ experienced workers between the

ages of 40 and 50 to replace the retired employees.

How will this influence the standard deviation? Explain. (2)

[14]

TOTAL SECTION A: [75]


x

y

A B

NO

12

SECTION B QUESTION 8 The diagram alongside shows the

sketch of the circle

2 2 1692 14

x y .

N is the centre of the circle and

A and B are points on the

circumference of the circle.

AB x -axis and AB 12 units.

Determine the coordinates of points A and B. (8)

[8]


QUESTION 9

(a) Prove the theorem that states the angle between a tangent and a chord is

equal to the angle in the alternate segment.

A

B

C

D E

O

Given: DE is a tangent to circle centre O at A. B and C are points on the circle.

Required to prove: __________________________________________ (1)

Construction: ______________________________________________ (1)

Proof: (5)


(b) In the diagram alongside, chord BA and

tangent TC of circle ABC are produced

to meet at R.

BC is produced to P with

RC = RP.

AP is not a tangent.

(1) Prove that ACPR is a cyclic quadrilateral. (5)

A

B

CP

R

T

12 3

4

12

3

12

1 2


(2) Prove that CBA RPA . (6)

(3) Prove that CB RARCAC

(3)

[21]


QUESTION 10 In the Cartesian plane below, the following is given:

Two equal circles with centres A and B.

One smaller circle with centre C that touches both circles A and B.

Equation of circle A: 2 27 6 36x y

Equation of circle B: 2 223 6 36x y

The radius of circle C is 4 units.

E is a point on the x -axis.

AB CE and CE intersect AB at D such that AD DB .

x

y

A B

C

D

E

(a) Explain why CE is a vertical line. (2)


(b) Determine the x -coordinate of C. (2)

(c) Determine the length of CD. (4)

(Hint: Use a construction.)

(d) Determine the equation of circle C. Leave your answer in the form

2 2 2x a y b r . (2)

[10]


QUESTION 11

In the diagram below, TBD is a common tangent to both circles at point B.

AKC is a chord of the larger circle and a tangent to the smaller circle at K.

Chords MN and BK intersect at F. Chord PA is produced to D. BMC, BNA and

BFKP are straight lines.

A

B

C

D

F

K

N M

P

T

1 23

123

4

12

312 3

4

12 3

4

12 3

(a) Prove MN CA . (4)


(b) Prove 2 2ˆ ˆM N . (3)

(c) Prove BK BMKP MC

(8)

(d) Prove DAP is a tangent to the circle passing through points A, B and K. (5)

[20]


QUESTION 12 (a) The Avengers are on a mission. Falcon is standing lookout on the top of a building

at point F. Black Widow is standing at the foot of the building at point B. Scarlet

Witch and Captain America are holding the perimeter at points S and C

respectively. Vision is standing on the other end of the building at point V.

34°30°

120°

F

B

SC

V

55 m

The height of the building is 55 m. The angles of elevation from points C and S

to F are 30 and 34 respectively. B, S, C and V are all points in the same

horizontal plane and ˆCBS 120 .

(1) Determine the distance between Scarlet Witch and Black Widow,

i.e. the length of SB. (2)


(2) Determine the distance between Scarlet Witch and Captain America,

i.e. the length of SC. (4)

(3) If BV BC and ˆBVC 80 , determine the distance between Vision and

Captain America, i.e. the length of VC. (3)


Determine the area of DEH . (7)

[16]

TOTAL SECTION B: [75] TOTAL: [150]

(b) In AFC alongside,

AF BE and AC GD .

HD ED , FC 36 units

and ˆGDF 40 .

AB 1BC 2

and AF 12FG 7

.

A

B

CDEF

G

H

40°


ADDITIONAL SPACE FOR WORKING OUT


ADDITIONAL SPACE FOR WORKING OUT

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BEAULIEU COLLEGE PRELIMINARY EXAMINATIONS GRADE 12 ... CAPS 2018 Prelim... · at point F. Black Widow is standing at the foot of the building at point B. Scarlet Witch and Captain