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of 18GRADE 12: MATHEMATICSPRELIM EXAM 2016 PAPER II

MATHEMATICS DEPARTMENT

Grade 12: Mathematics

Paper II

Preliminary Examination 2016

Time: 3 hours 150 marks

NAME:__________________________________________________________

PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY

1. This question paper consists of 18 pages and 13 questions. A separate formula sheet will be

handed out. Please check that your paper is complete.

2. Read the questions carefully.

3. Answer all the questions ON THE QUESTION PAPER. Additional space is provided at the

end of the paper.

4. Number your answers exactly as the questions are numbered.

5. Round off your answers to two decimal digits where necessary.

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

7. Approved non-programmable calculators may be used unless otherwise stated.

8. It is in your own best interest to present your work neatly.

BRIDGE HOUSE SCHOOL PLEASE TURN OVER

of 18GRADE 12: MATHEMATICSPRELIM EXAM 2016 PAPER IISECTION A

Question 1

In the diagram below, A (−2 ;6 ) , B(8 ;−4) and C (−6 ;k ) are three points in the Cartesian

plane with point M the midpoint of AB and AB=BC.

M

C(-6;k)

B(8;-4)

A(-2;6)

x

y

1.1. Determine the coordinates of M . (2)

1.2. Determine the value(s) of k . (6)

1.3. Hence, determine the equation of the line CM if k=−6. (5)


of 18GRADE 12: MATHEMATICSPRELIM EXAM 2016 PAPER II1.4. Determine the value of :

1.4.1. θ if k=−6 (2)

1.4.2. α (2)

1.4.3. C A B (2)

[19]

Question 2

2.1. Simplify without the use of a calculator: (6)

−cos225 ° sin60 °cos765°

+sin (−210 °)tan 210°

2.2. Solve for x if sin ( x−20 ° )=tan 140° for x∈(0 ° ;360) (4)

[10]


of 18GRADE 12: MATHEMATICSPRELIM EXAM 2016 PAPER IIQuestion 3

O is the centre of the circle through A , B ,C and D. AB=BC and A OC=2 x .

2x

12

1

2

321

O

D

C

B

A

Express the following in terms of x, giving reasons for all statements:

3.1. A2 (2)

3.2. A BC (3)

3.3. D (3)

[8]



Use the sketch given below to prove the opposite angles of a cyclic quadrilateral are

supplementary.

x

OD

CB

A

[6]



In the diagram below, AC is a diameter of circle ABC. C2=C3 , AC⊥ FG, AD∥CG ,

BC=6 units and AC=10 units .

G

21

321

32

1

F

C

D

B

A

5.1. Prove that ∆ CAD is isosceles. (3)

5.2. Prove that BCAC

=BGAG (3)



5.3. Prove that ∆ AFG∨¿∨∆ ABC (3)

5.4. Determine the length of AB (2)

5.5. Determine the length of AG (2)

5.6. Determine the length of AF (2)

[15]



Fifteen people took part in a board game tournament. The points scored by each person are

given below.

33 17 20 34 24 52 31 32 18 33 22 36 43 52 28

6.1. Determine the median of the data set. (2)

6.2. Determine the inter quartile range for the data. (3)

6.3. Draw a box and whisker diagram to represent the data. (3)

6.4. Use the box and whisker diagram to comment on the points scored by the

participants of the tournament. (2)

[10]


of 18GRADE 12: MATHEMATICSPRELIM EXAM 2016 PAPER IISECTION B

Question 7

7.1. Use the sketch given to prove tanθ × 1sin θ

−cosθ= 8 pp2−4

(6)

B

A

O

p+2

p-2

y

x

7.2. If cosθ+2cosθ−2

=rs

7.2.1. Prove that cosθ=2 s+2 rr−s (4)



7.2.2. Hence determine the solution for θ if cosθ+2cosθ−2

= 3−5 and θ∈[0° ;540 ° ] (5)

[15]

Question 8

The graph of f ( x )=a tan x for x∈[−180 °;180 ° ] is given below.

8.1. Sketch the graph of g ( x )=sin(x+45° ) on the axis provided above. (3)

8.2. Determine the value of a. (1)

8.3. The graph of h ( x )=cos bx+c a period of 180 ° and a y-intercept at (0 ;3). Determine

the values of b and c. (2)

[6]



The diagram shows points B, C, D and E on the ground such that B, D and C lie in a straight

line. AP is a flagpole. The angle of elevation of A from E is x and from C it is y. The length

of DC is r and AB=AE.

r

y

x

E

D CB

A

9.1. Show thatAE= r tan ysin x (4)

9.2. Calculate the distance of BE if E D C=120° and DE=32 m. (4)

[8]



Given below is the circle with centre A given by the equation (x+2)2+(x−6)2=r2 and the

circle with centre C (a ;b) that lies in quadrant 2. DE is a common tangent to circles A and C

at the point B(−3 ;5). Circle F has the equation x2+ y2=−1+2 x+6 y .

F

E

D

A

B(-3;5)

C(a;b)

x

y

10.1. Write down the coordinates of A. (2)

10.2. Determine the radius of circle A. (2)

10.3. Determine the equation of the tangent DE. (4)



10.4. If ABBC

=13 , determine the coordinates of C. (8)

10.5. Write down the coordinates of F. (4)

[20]



In ∆ ACE, A C E :C A E : A E C=16 :15:5. B and F are the midpoints of AC and AE. ABC is

the diameter of the circle with centre B and G is a point on the circle.

3

12

1

2

1 2

12

1 2

2

1G

B

A

F

DC

E

11.1. Giving reasons for all statements, determine the size of:

11.1.1. C A E (3)

11.1.2. E (1)

11.1.3. F1 (3)

11.2. Show that AG=GD (2)


of 18GRADE 12: MATHEMATICSPRELIM EXAM 2016 PAPER II11.3. Prove that ∆ ACG ≡ ∆ DCG (4)

11.4. Hence, show that CD=2 AB. (2)

[15]

Question 12

In the diagram below, ABCD is a square with sides 7cm long. FC=x ,BF⊥ EF and

∆ BCF∨¿∨∆ FDE.

x

1

2

1

F

E D

CB

A

Show that the area of ∆≝¿ x ¿¿

[4]

Question 13



x 10 20 30 40 50 60 70 80 90 100

y 2 4 3 5 5 6 8 7 9 10

13.1. Dra

w a scatter plot of the data above. (3)

13.2. Determine the equation for the line of best fit for the data. (3)

13.3. Draw the line of best fit on your scatter plot. (2)

13.4. Determine the correlation coefficient of the data. (2)

13.5. “The corresponding y-value for an x-value of 35 is 4,13.” Do you agree with this

statement and if so, is the value reliable? (4)

[14]

TOTAL FOR THIS PAPER: [150]


of 18GRADE 12: MATHEMATICSPRELIM EXAM 2016 PAPER II__________________________________________________________________________________

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BRIDGE HOUSE SCHOOL

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