mathematics paper 2 preliminary examination grade …

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ST DAVID’S MARIST INANDA

MATHEMATICS

PAPER 2 PRELIMINARY EXAMINATION GRADE 12

14 SEPTEMBER 2020

NAME: _____________________________________________________________

PLEASE HIGHLIGHT YOUR TEACHERS NAME:

MRS KENNEDY

MS VAZZANA

MRS NAGY

MR VICENTE

MRS RICHARD

MRS BLACK

INSTRUCTIONS:

✓ This paper consists of 27 pages. Please check that your paper is complete.

✓ A separate two-page information sheet with formulae is included. Please

check that your paper is complete.

✓ Please answer all questions on the Question Paper.

✓ You may use an approved non-programmable, non-graphics calculator unless

otherwise stated.

✓ It is in your interest to show all your working details.

✓ Work neatly. Do NOT answer in pencil.

✓ Diagrams are not drawn to scale.

✓ All Euclidean Geometry MUST be answered with REASONS

SECTION A Q1 [7]

Q2 [9]

Q3 [19]

Q4 [9]

Q5 [9]

Q6 [10]

Q7 [12]

TOTAL [75]

LEARNER’S MARKS

SECTION B Q8 [18]

Q9 [7]

Q10 [19]

Q11 [10]

Q12 [14]

Q13 [7]

TOTAL [75]

LEARNER’S MARKS

EXAMINER: MRS L BLACK MARKS: 150

MODERATOR: MRS C KENNEDY TIME: 3 Hrs

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SECTION A

QUESTION 1 [7 MARKS] (Calculations in this question must be to two decimal

places)

Measurements were collected from 12 people to determine whether people with

longer arms are taller than people with shorter arms.

The scatter graph is drawn below:

a) Use your calculator to determine the equation of the line of best fit. (3)

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b) Marie’s arm span is measured to be 160cm. What would her estimated height

be? (2)

c) The point ( ; )x y with coordinates (165.5;167.667) is labelled P on the scatter

graph. Use this information and the graph to identify and write down using the

corresponding letter from the table (eg A, B, C…) of a candidate having below

average arm-span with above average height. (1)

d) Suppose the correlation (r) between arm span and height is calculated for two

cases:

Case 1: Candidate J is included

Case 2: Candidate J is excluded.

How would the correlation compare:

i) The r values would be the same.

ii) The r value would be closer to 1 for case 1 than for case 2.

iii) The r value would be closer to 1 for case 2 than for case 1.

iv) The r value would be closer to -1 for case 1 than for case 2.

(just write the number of your choice below) (1)

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QUESTION 2 [9 MARKS] (Calculations in this question must be to two decimal

places)

A group of 30 learners each randomly rolled two dice once and the sum of the values

on the uppermost faces of the dice was recorded. The data is shown on the

frequency table below.

Sum of the values on the uppermost

faces

Frequency

2 0

3 3

4 2

5 4

6 4

7 8

8 3

9 2

10 2

11 1

12 1

a) Calculate the mean of the data. (2)

b) Determine the median of the data. (2)

c) Determine the standard deviation of the data. (2)

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d) Determine the number of times that the sum of the recorded values of the

dice is within ONE standard deviation interval from the mean. Show your

calculations. (3)

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QUESTION 3 [19 MARKS]

In the diagram below, A(−3; 4) , B(4; 8), C(5; 0) and D are the vertices of

parallelogram ABCD. BC is extended to E to meet DE which is parallel to the x-

axis.

a) Determine the equation of line BE. (4) b) i) Determine the coordinates of P, where P is the point of intersection of

the diagonals of ABCD. (2)

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ii) Hence, determine the coordinates of D (2)

c) With a reason, prove that ABCD is a rhombus. (3)

d) i) Calculate the size of ˆACB . (4)

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ii) Hence, calculate the area of ∆ABC. (round to the nearest whole number) (4)

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Answer this question WITHOUT using a calculator:

The point ( ;8)P k lies in the first quadrant

such that 17OP = units and ˆTOP = as

show in the diagram alongside.

a) Determine the value of k. (2)

b) Write down the value of cos . (1)

c) If it is further given that 180 + = , determine cos . (2)

d) Hence, determine the value of sin( ) − . (4)

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a) Draw the graphs of ( ) tan 1f x x= + and ( ) cos2g x x= for [ 180;180]x − on the

axes below. Clearly show all intercepts, turning points and asymptotes. (4)

b) Write down the period of 1

4g x

. (1)

c) If ( ) cos(2 20 )h x x= − + , describe fully in words, the transformation

from g to h. (2)

−180 −135 −90 −45 45 90 135 180

−3

−2

−1

1

2

3

x

y

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d) For which values of x, where 0x , will '( ) ( ) 0f x g x (2)

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Two circles intersect at D and E. Chord RE of the smaller circle is a tangent to the

larger circle at E. N is a point on the small circle.

EN and RD are produced to meet the bigger circle at A. RN, ED and DN are drawn.

V is a point on the larger circle and AV and EV are drawn.

a) Prove that 1 1 2ˆ ˆ ˆD E E= + (4)

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b) Prove that |||EDR AER . (3)

c) If 2 .AV DR AR= and 3ER cm= determine the length of AV. (3)

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Refer to the given diagram, P and Q are the centres of the two circles. From A a

straight line is drawn to cut circle P at B and C, and circle Q at D and E. AGF is a

common tangent to circle P at G and to circle Q at F. GCDF is a cyclic quadrilateral.

Prove:

a) 1 1ˆP Q= (6)

3

3

4

1

4

3

3

2

E

21 2

2

2

1

1

1

1

1

1

P

D

Q

F

C

G

AB

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b) AG AC

GF CE= (2)

c) AB AC

BD CE= (4)

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SECTION B


The diagrams below show a representation of the chain of a bicycle attached to two

circular cogs as represented in the Cartesian plane.

The equations of the circles given by 2 2 1x y+ = and 2 2( 6) 9x y− + = . RPQ is a

common tangent to both circles at P and Q, with R a point on the negative x-axis.

T is the centre of the larger circle.

a) Write down the coordinates of the centre and length of the radius of the larger

circle. (2)

b) i) Prove that |||RPO RQT . (3)

y

xT

Q

O

P

R

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ii) Hence, determine the coordinates of R, showing all working. (4)

c) If ( 3;0)R − , Determine:

i) The length of PQ, leaving your answer in surd form. (5)

ii) The equation of the tangent RPQ. (leave in surd form) (4)

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Given two circles 2 2 2 6 2x y mx y+ + − = and 2 2 2( 5) ( )x y n p− + + = .

a) Determine m and n if the circles are concentric. i.e. they have the same

centre. (4)

b) Determine two values of p if it is further given that the radii of the circles differ

by 2 units. (3)

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******THIS PAGE IS BLANK FOR EXTRA WORKING SPACE***********

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a) i) Prove that 2sin (2cos 1)(cos 1)

tan2 2sincos2

x x xx x

x

− ++ =

(6)

ii) Hence calculate the values of x if:

1. tan2 2sin 0x x+ = for [0;360]x (6)

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2. tan2 2sinx x+ is undefined for [0;360]x (3)

b) Prove, without the use of a calculator, that if sin28 a = and cos32 b = ,then

2 2 11 1

2b a a b− − − = (4)

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In the diagram, P, Q and R are three points in the same horizontal plane and SR is a

vertical tower of height h meters. The angle of elevation of S from Q is , ˆPRQ = ,

ˆ 30RQP = and 6PQ meters= .

a) Express QR in terms of h and . (2)

b) Express ˆQPR in terms of . (1)

P

Q R

S

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c) Hence show that 3(1 3 tan )h = + . (7)

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In the figure below. ADC is a tangent at D to the circle with centre O.

ˆ 90 .AOB = AO cuts the circle at E. EDB is a straight line.

a) If 1O = , express 1C and 1D in terms of . (6)

α

43

1

2

2

2

2

1

1

1

A

BC

DE

O

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b) If 2CB units= and 4OD units= , find the length of OA.

Leave your answer in simplified surd form if necessary. (8)

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The time shown on the clock is exactly twelve minutes past ten or 10 12h .

The clock is now represented by the figure below:

a) Determine the size of B in terms of x, giving reasons. (3)

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b) Hence, find the size of B in degrees. Show all working. (4)

TOTAL: 150

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