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ST. DAVID’S MARIST INANDA

GRADE 12 MATHEMATICS

PRELIMINARY EXAMINATION

PAPER 2

SEPTEMBER 2014

EXAMINER: Mrs C. Kennedy MARKS: 150

MODERATOR: Mrs L. Nagy TIME: 3 hours

NAME:______

HIGHLIGHT YOUR TEACHER’S NAME:

C. KENNEDY / L. NAGY / L. RIDER / S. RICHARD

INSTRUCTIONS:

This paper consists of 27 pages and a separate Formula sheet. Please check that your paper is complete.

Please answer all questions on the Question Paper and read each question carefully.

You may use an approved non-programmable, non-graphics calculator unless otherwise stated.

Answers must be rounded off to ONE decimal place unless otherwise stated.

It is in your interest to show all necessary working details.

Work neatly. Do NOT answer in pencil.

Diagrams are not drawn to scale.

SECTION A / Q1 / Q2 / Q3 / Q4 / Q5 / Q6 / Q7 / TOTAL
MARKS / 16 / 8 / 10 / 15 / 9 / 7 / 7 / 72
SECTION B / Q8 / Q9 / Q10 / Q11 / Q12 / Q13 / Q14 / Q15 / TOTAL
MARKS / 10 / 16 / 7 / 4 / 13 / 19 / 4 / 5 / 78

SECTION A

QUESTION 1

has vertices A( 3 ; 1 ) , B( -3 ; -2 ) and C( -7 ; 6 ). CAT is a straight line that

intersects the x axis at T.

a)Determine the gradient of BC.(1)

b)Prove that is a right angle.(3)

c)Calculate the size of , correct to the nearest degree.(4)

d)Determine:

i)the co-ordinates of M, the midpoint of AC.(2)

ii)the equation of the circle passing through A, B and C if AC is the

diameter.(3)

iii)the equation of the tangent to the circle found in (ii), at C.(3)

[16]

QUESTION 2

The equation of a straight line RS is .

The equation of a straight line AB is.

Determine the value of pin each case, if :

a)(3)

b)AB cuts RS in the point (-3; 4) (2)

c)the angle of inclination of AB is equal to(3)

[8]

QUESTION 3

The graphs of and are drawn below.

a)Write down:

i) the amplitude of h if .(1)

ii) the period of j if .(2)

b)For which value(s) of x is:

i)(2)

ii)?(2)

iii)(3)

[10]

QUESTION 4 DO NOT USE A CALCULATOR IN THIS QUESTION

a)Evaluate the following. Show all working.

i)(4)

ii)(4)

b)i)Simplify the following expression to a single trig ratio:

(3)

ii)Hence, determine the general solution for:

(4)

[15]

QUESTION 5

Twelve St David’s boys trained to run the 100 m sprint at the Inter-House Athletics. Some

of them took their training more seriously than others. The following table and the scatter

plotshows the days that each athlete trained and the time taken to run the event.

The time taken, in seconds, is rounded to one decimal place.

a)Identify any outlier(s) in the data.(1)

b)Use your calculator to determine the equation of the least squares regression

line. Give your answer correct to 4 decimal digits.(4)

c)Estimate the time taken to run the 100 m sprint for an athlete training for

45days.(2)

d)Calculate the value of r, the correlation coefficient for the data, correct to 4

decimal digits.(1)

e)Comment on the strength of the relationship between the variables.(1)

[9]

QUESTION 6

The vinyl record is a type of gramophone record, most popular from the 1950s to the 1990s, that was most commonly used for mass-produced recordings of music. Below is a Pink Floyd's Dark Side of the Moon record. Only 1000 of this particular record were made.

Assume the record to be a circle with centre O. Radius r is drawn through the vertices of

. BO and CO are joined. OP is drawn perpendicular to chord BC. BC is k units

in length.

a)Give a reason why .(1)

b)Express in terms of giving reasons.(2)

c)Determine the length of the diameter in terms ofk and a trigonometric function of . (4)

[7]

QUESTION 7 ( REASONS MUST BE GIVEN IN THIS QUESTION )

In the figure JKLM is a quadrilateral with . A circle through J and M cuts

the quadrilateral JKLM at X and Y. O is the centre of the circle and .

a)Calculate the size of .(4)

b)Hence, prove that XYLK is a cyclic quadrilateral.(3)

[7]

SUB-TOTAL: 72

SECTION B

QUESTION 8

a)If where , determine, using a diagram,

an expression forin terms of p.(4)

b)i)Prove the identity:

(4)

ii)Hence, evaluate:

(2)

[10]

QUESTION 9

a)Given:

Determine the value of k if the length of the radius of the circle is 10 units.(4)

b)The image alongside is an aerial photograph of

circles made in fields by flattened crops.

The diagram below is a representation of the above crop circles on the Cartesian plane.

Equal circles with centres A and E touch equalcircles with centres B and D respectively.

Circles with centres B and D touch each otherat C.A, B, C, D and E lie on the same straight line.

B(0;0) is the centre of the circle passing through C(20;15), which is the centre of

the circle passing through the origin.Perpendicular lines are drawn from C and E

meeting the x axis at Q and R respectively.

i)Determine the equation of the circle with centre B.(2)

ii) Determine the co-ordinates of D.(2)

iii)If the radius of the larger circle is 25 units and the radius of the smaller

circle is 5units:

(1)Determine the value of(2)

(2)Hence, determine the length of QR(2)

(3)Determine the length of ER and hence write down the equation of the

circle with centre E.(4)

[16]

QUESTION 10

a)Building regulations require the provision of lightning protection for thatched roof

structures. In the diagram below, AD represents a vertical lightning mast constructed

on the same horizontal plane as a house.

B and C are two points on the ground which are in line with the front side MN of

the house. B and C lie on the circle, centre D, of lightning protection provided by the lightning mast.

is the angle of elevation from B to the top of the mast at A.

.

i)Express BD, the radius of protection, in terms of p and .(1)

ii)Hence, show that .(4)

iii)If BC = 29,5 metres, p = 21,2 metres and , calculate the

size of .(2)

[7]

QUESTION 11

The riverine rabbit is one of the most endangered animals in

Southern Africa. It is estimated that there are less than

250 riverine rabbits left in the wild. The mass of 60 riverine

rabbitsis summarized by the box and whisker plot below:

Given the frequency table below:

Determine the values for:

i)A(1)

ii) B(1)

iii) C (1)

iv)D(1)

[4]

QUESTION 12(REASONS MUST BE GIVEN IN QUESTIONS 12, 13 & 14)

a)Given the following statement:

A line drawn parallel to one side of a triangle divides the remaining two

sides in proportion.

Given:

Complete the following table: the proof of the above theorem:(4)

STATEMENT / REASON
/ same height
/ same height
but / iii) ......

b)In the diagram below, S and T are the midpoints of RQ and PS respectively.

.

Calculate the numerical value of the following:

i)(3)

ii)(3)

iii)(3)

[13]

QUESTION 13

a)In the diagram below, A, B and C are points on the circle with centre O.

AB produced meets tangent XCY at Y.

.

Express in terms of p.(5)

b)In the diagram below, MATH is a cyclic quadrilateral with MH = TH.

Chords MT and AH intersect at Q. AM is extended to S such that

Prove that:

i)HS is a tangent to circle MATH at H.(4)

ii)(4)

iii) (2)

iv)(4)

[19]

QUESTION 14

A flashlight projects an image of a square ABCD, on a wall 4m away. One side of square ABCD measures 75cm.

How far from the flashlight should the square be placed so that the areaof the image,, is 1? (4)

[4]

QUESTION 15

By multiplying each of the numbers in a particular set A by three and then adding eight, we obtain a new set B.

a)What is the relationship between the means of sets A and B?(2)

b)What is the relationship between the standard deviations of sets A and B? (3)

[5]

SUB-TOTAL: 78

ST DAVID’S MARIST INANDA PAPER II 2014