Herzlia Senior High School

HERZLIA SENIOR HIGH SCHOOL

“If you will it, it is no legend”

MARKS:150

TIME:3 HOURS

This question paper consists of 16 pages, 6 diagram sheets and an information sheet.

INSTRUCTIONS AND INFORMATION

Read the following instructions carefully before answering the questions.

1. This question paper consists of 11 questions.

2. Answer ALL the questions.

3. Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in determining your answers.

4.Answers only will not necessarily be awarded full marks.

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

unless stated otherwise.

6. If necessary, roundoff answers to TWO decimal places, unless stated otherwise.

7. Diagrams are NOT necessarily drawn to scale.

8.6 diagram sheets for answering QUESTION 1.1, QUESTION 1.3, QUESTION 2.1, QUESTION 2.2, QUESTION 8, QUESTION 9, QUESTION 10 and QUESTION 11 are attached at the end of this question paper. Write your name in the space provided and insert the sheets inside the back cover of your ANSWER BOOK.

9.An information sheet, with formulae, is included at the end of the question paper.

10. Number your answers correctly according to the numbering system used in this question paper.

11. Write neatly and legibly.

QUESTION 1

The data below shows the pulse rate of a sample of 12 people when they rest

and then again after 2 minutes of jogging.

1.1Draw a scatter plot of the data given on the grid provided on DIAGRAM

SHEET 1.(3)

1.2Calculate the equation of the least squares line for this data.(4)

1.3Draw the least squares line on the scatter plot.(2)

1.4Calculate the correlation coefficient.(1)

1.5Comment on the correlation of the data.(1)

1.6If Joan’s heart rate after jogging is 86 beats per minute, what is her

resting heart rate, in beats per minute?(1)

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QUESTION 2

The time taken (to the nearest minute) for a certain task to be completed was

recorded on 48 occasions and the following data was obtained:

2.1Complete the cumulative frequency table on DIAGRAM SHEET 2.(2)

2.2Draw an ogive (cumulative frequency curve) for the given data on

DIAGRAM SHEET 2. (3)

2.3Determine, from the ogive, the median, lower quartile and upper

quartile for the data.(3)

2.4Draw a box and whisker diagram of the data (assume minimum = 11

and maximum = 30).(2)

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QUESTION 3

ABCD is a quadrilateral with vertices A(1;6), B(3;0), C(6;1) and D(7;t) in a

Cartesian plane. AD || BC.

3.1Calculate the gradient of BC.(2)

3.2Determine , the angle of inclination of BC.(2)

3.3Show that AB is perpendicular to BC.(2)

3.4Find the value of t.(3)

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QUESTION 4

4.1In the diagram below, the equation of the circle having centre M

is . Tangent PQ touches the circle at point P

and Q is the point (8;3).

4.1.1Show that M has coordinates (2;–5).(4)

4.1.2Find the coordinates of the x-intercepts of the circle.(4)

4.1.3Give the radius of the circle.(1)

4.1.4Find the length of PQ.(4)

4.2In the diagram below, a circle has a diameter with equation .

The tangent at point E on the circle cuts the x-axis at F(12;0).

4.2.1Determine the equation of the tangent at E.(3)

4.2.2Determine the coordinates of E.(3)

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QUESTION 5

5.1Given: and

5.1.1Use a calculator to determine , correct to one decimal place.(2)

5.1.2Hence determine the value of , correct to one

decimal place.(1)

5.2In the diagram below, T(8;k) is a point in the first quadrant.

If , determine without using a calculator:

5.2.1The value of k(2)

5.2.2The value of (leave your answer in surd form)(3)

5.3Simplify the following as far as possible without using a calculator:

5.3.1(4)

5.3.2(5)

5.4By replacing 2A with 3A – A and 4A with 3A + A, prove theidentity . (6)

5.5Determine the general solution of .(7)

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QUESTION 6

In the diagram below, the graphs of and and a third

trigonometric graph are sketched for the interval .

6.1Write down the values of a, b and c.(3)

6.2Use the graphs to solve the following for :

(2)

6.3The equation of the third graph sketched is given by .

Determine the values of p and q.(3)

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QUESTION 7

7.1Prove that in any acute-angled ABC, .(3)

7.2In the figure below, A, B and C are in the same horizontal plane. P is a

point vertically above A and B and C are 20 units apart. The angle of

elevation from B to P is .

7.2.1Prove that .(4)

7.2.2Given that AB = AC, show that .(2)

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QUESTION 8

In the diagram below, O is the centre of the circle. Chord AB is perpendicular to

diameter DC. CM : MD = 4 : 9 and AB = 24 units.

8.1Determine an expression for DC in terms of x if CM = 4x units.(1)

8.2Determine an expression for OM in terms of x.(2)

8.3Hence, or otherwise, calculate the length of the radius.(4)

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QUESTION 9

9.1In the diagram below, O is the centre of the circle. PQ is a tangent to the

circle at A. B and C are points on the circumference of the circle. AB,

AC and BC are joined.

Prove the theorem that states .

(6)

9.2In the diagram below, two circles have a common tangent TAB. PT is

a tangent to the smaller circle. PAQ, QRT and NAR are straight lines.

Let .

9.2.1Name, with reasons, THREE other angles equal to x.(5)

9.2.2Prove that APTR is a cyclic quadrilateral.(5)

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QUESTION 10

10.1Use the diagram below to prove the theorem which states that if VW || YZ

then .

(6)

10.2In PQR below, B lies on PR such that 2PB = BR. A lies on PQ such that

PA : PQ = 3 : 8. BC is drawn parallel to AR.

10.2.1 Calculate (by showing all your working) the value of the ratio .(5)

10.2.2 Write down the value of .(2)

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QUESTION 11

11.1Two circles touch each other at point A. The smaller circle passes through O,

the centre of the larger circle. Point E is on the circumference of the smaller

circle. A, D, B and C are points on the circumference of the larger circle.

OE || CA.

11.1.1 Prove, with reasons, that AE = BE.(2)

11.1.2 Prove that AED ||| CEB.(3)

11.1.3 Hence, or otherwise, show that AE2 = DE.CE.(2)

PLEASE TURN OVER

11.2In the accompanying figure, AB is the diameter of circle ADCB. Chords

AC and BD intersect at E. EP is perpendicular to AB.

11.2.1 Prove that BPE ||| BDA.(3)

11.2.2 Hence show that .(2)

11.2.3 Prove that .(5)

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TOTAL : 150

HERZLIA SENIOR HIGH SCHOOL

GRADE 12 MATHEMATICS PAPER 2

MONDAY 1STSEPTEMBER 2014

DIAGRAM SHEET

(PLEASE PLACE IN YOUR ANSWER BOOK)

NAME:………………………………………………..

DIAGRAM SHEET 1

QUESTIONS 1.1 AND 1.3

DIAGRAM SHEET 2

QUESTION 2.1

QUESTION 2.2

DIAGRAM SHEET 3

QUESTION 8

QUESTION 9.1

DIAGRAM SHEET 4

QUESTION 9.2

QUESTION 10.1

DIAGRAM SHEET 5

QUESTION 10.2

QUESTION 11.1

DIAGRAM SHEET 6

QUESTION 11.2