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Redhill High School 2019 Gr 12 Prelim Paper 2 1 of 24

REDHILL HIGH SCHOOL

GRADE 12 PRELIMS

MATHEMATICS PAPER 2

20 SEPTEMBER 2019

_________________________________________________________________________________

Examiner: V van Rooy Moderator: G Evans & L Friedman Time: 3 Hours Marks: 150 NAME: ________________________ Teacher: GE / VvR / LF _________________________________________________________________________________

PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY

1. This question paper consists of 24 pages and 13 questions. Please check that

your question paper is complete.

2. Read the questions carefully.

3. Answer ALL the questions on the question paper.

4. Diagrams are not necessarily drawn to scale.

5. You may use an approved non-programmable and non-graphical calculator,

unless otherwise stated.

6. Ensure that your calculator is in DEGREE mode.

7. All the necessary working details must be clearly shown. Answers only will not

necessarily be awarded full marks.

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

9. Round off to two decimal places unless otherwise stated. _________________________________________________________________________

SECTION A

Q1 Q2 Q3 Q4 Q5 Q6 Q7 SUBTOTAL

20 17 8 7 6 15 9 82

SECTION B

Q8 Q9 Q10 Q11 Q12 Q13 SUBTOTAL

15 17 9 12 9 6 68

TOTAL /150


Question 1 [20 marks]

a) ( ) ( ) ( )A 4;3 , B 6;13 , C 4;11− and D are the vertices of a parallelogram.

The diagonals intersect at M.

(1) Determine the coordinates of D. (4)

A( 4 ; 3 )

B( 6 ; 13 )

C( −4 ; 11 )

D X

Y

M


(2) Use analytical methods to show that ˆAMB 90= . (3)

(3) Determine the size of ˆABM correct to one decimal place. (4)


b) A circle is given in the diagram with equation 2 2x y 10+ = . The midpoint of

chord AB is point ( )D 2; 1− − .

(1) Determine the equation of line AB. (4)

A

B

X

Y

D( −2 ; −1 )


(2) Hence, determine the coordinates for points A and B. (5)



Answer the following questions WITHOUT using a calculator.

a) If sin a = and 90 180 , determine the value for tan in terms of a.

(4)

b) Simplify, ( ) ( )

21 cos 2A

cos A .cos 90 A

−

− + completely, given that 0 A 90 . (5)


c) Determine the value of sin3x.cosy cos3x.siny+ , if 3x y 270+ = . (2)

d) Consider ( ) ( )g x 4cos x 30= − +

(1) Write down the maximum value of ( )g x . (1)

(2) Determine the range of ( )g x 1+ . (2)

(3) The graph of g is shifted 60° to the left and then reflected about the

x-axis to form a new graph h. Determine the equation of h in its simplest

form. (3)



The following identity is given: 1 sin 2A sin A cos A

cos 2A cos 2A cos A sin A

++ =

−

a) State the values of A for which the identity is undefined. (3)

b) Hence, prove the identity. (5)



In the diagram, the graphs of ( )f x asinx= and ( )g x tanbx= are drawn on the

same system of axes for the interval 0 x 225 .

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

b) Write down the period of ( )f 3x . (2)

c) Determine the values of x in the interval 90 x 225 for which

( ) ( )f x .g x 0 (3)

−180 −135 −90 −45 45 90 135 180 225 270 315 360

−5

−4

−3

−2

−1

1

2

3

4

5

x

y

Y

X

f

g



In the diagram DE FG BC, AD 36 cm, DF 24 cm, AE 48 cm= = = and

DE GC 40 cm= =

Determine, giving reasons, the length of:

a) EG (3)

b) BC (3)

A

B

C

D

E

F

G

36

48

40

40

24



a) In the circle with centre O, OT QP, OS PR, OT 8 units, PQ 30 units⊥ ⊥ = =

and PR 23 units= . Determine x if it is given that OS x= . (5)

O

P

Q

R

S

T 8

30

23


b) In the diagram below the two circles intersect at D and B. AB is a straight

line and cuts the circle through B, C and D in E. BC and AC are also

straight lines. Given that 2ˆFC FD and F 180 2x= = −

(1) Determine, giving reasons, the size of the following in terms of x:

(i) ˆBED (5)

(ii) A (2)

F

1

1

2

A

D 1

C 2 2

E 1

2

B

4 3


(2) Hence, prove that ED BC (3)


A survey was conducted at a local supermarket relating the distance that shoppers

lived from the store to the average number of times they shopped at the store in a

week. The results are shown in the table below:

Distance from the store in

km 1 2 3 4 5 7 8 10

Average number of times

shopped per week 12 10 7 7 6 2 3 2

0

1

2

3

4

5

6

7

8

9

10

11

12

13

0 1 2 3 4 5 6 7 8 9 10

Scatter plot

Distance from store in km

Av

era

ge n

um

be

r o

f ti

me

s p

er

we

ek


a) Use the scatter plot to comment on the strength of the relationship

between the distance a shopper lived from the store and the average

number of times she/he stopped at the store in a week. (2)

b) Calculate the correlation coefficient of the data. (1)

c) Calculate the equation of the least squares regression line of the data. (2)

d) Use your answer in (c) to estimate the average number of times that a

shopper living 6 km from the supermarket will visit the store in a week. (2)

e) Sketch the least squares regression line on the scatter plot. (2)


SECTION B


a) It is given that sinp 3cosp 1+ = .

(1) Show that the equation can be written as ( )1

sin 60 p2

+ = (3)

(2) Hence, determine the general solution of sinp 3cosp 1+ = (4)


b) A, B and C are three points on the circumference of a semi-circle in a

horizontal plane. AC is the diameter of the semi-circle. EC is a slanted

pole in the vertical plane through C, forming an angle of with the

vertical. The angle of elevation of E at A is . ˆBC x and ACB= = .

(1) Show that x

ACcos

=

(2)

(2) Determine an expression for ˆAEC in terms of and . (3)

(3) Prove that ( )

x.sinEC

cos .cos

=

− (3)

A

B

C

D E

x



a) Determine the value(s) of k if the circles 2 2x y 2x 2y 2+ − − = and

2 2x y 8x 10y k+ − − = touch each other externally. (8)


b) A circle with centre ( )P m;n touches the y-axis at point Q and passes

through the point ( )R 1; 1− − . Point P lies on the line 2x y 4 0+ + = .

(A possible diagram is shown)

(1) Show that n 2m 4= − − . (2)

(2) Write down the coordinates for Q, in

terms of n. (1)

(3) Determine the coordinates of point P. (6)

Q

x

y

P(m;n)

R(-1;-1)

Q



The box and whisker diagram below shows the marks (out of 80) obtained in a

History test by a class of nine learners.

32 62

20 30 40 50 60 70 80

a) Comment on the skewness of the data. (1)

b) Write down the range of the marks obtained. (1)

c) If the learners had to obtain 32 marks to pass the test, estimate the

percentage of the class that failed the test. (1)

d) In ascending order, the second mark is 28, the third mark is 36 and the

sixth mark 69. The seventh and eight marks are the same. The average

mark for this test is 54.

28 36 69

Fill in the marks of the remaining learners in ascending order. (6)



PQ is a diameter of the circle through points P, W, R and Q. SP is a tangent to the

circle at point P. Let 1P x= .

a) Why is ˆPRQ 90= ? (1)

b) Prove that 1ˆP S= . (2)

Q

R

P

S

T

W

1

1

1 1 1

2 2

2

2 2

3


c) Prove that SRWT is a cyclic quadrilateral. (2)

d) Prove that QWR ||| QST (3)

e) If QW 5 cm, TW 3 cm, QR 4 cm and WR 2 cm= = = = . Determine the

lengths of:

(1) TS (2)

(2) SR (2)



ABCD is a square with side length of 9 cm. BE EF⊥ .

a) Prove that BAE ||| EDF (4)

E A

B C

D

F

1

1

1

2


b) If AE x cm= , show that the area of ( )

2x 9 x

EDF18

− = (5)



The diagram represents an aquarium in the shape of a hemispherical dome having

radius 25 metres. Within the dome is a solid right pyramid with a rectangular base.

The centre of the rectangle is also the centre of the hemisphere.

a) The top view of the hemispherical dome is represented in the diagram below.

ON p= units and 1N = . Show that RN 2p.cos= . (3)

b) If the pyramid has a square base and p 10= metres, determine, to the

nearest whole number, the volume of water required to completely fill the

aquarium. (3)

(3

sphere

4V r

3= & pyramid

1V area of base height

3= )

R

O

N

1

h

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