king david high school linksfield

KING DAVID HIGH SCHOOL LINKSFIELD

PRELIMINARY EXAMINATION

AUGUST 2020

MATHEMATICS: PAPER II

Time: 3 hours Total: 150 marks

Name:_______________________________ Teacher:_________________ PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This question paper consists of 30 pages and an information sheet of 2 pages. Please check that your paper is complete. 2. Read the questions carefully. 3. Answer all the questions on the question paper and hand it in at the end of the examination. 4. Four blank pages have been included at the end of the exam paper. If you run out of space, use these pages. Make sure that you indicate this when you make use of this space and number these answers exactly as they are in the paper. 5. Diagrams are not necessarily drawn to scale. 6. You may use an approved non-programmable and non-graphical calculator, unless stated otherwise. Ensure that your calculator is in DEGREE mode. 7. Clearly show ALL calculations, diagrams, etc. that you use in determining your answers. Answers only will NOT necessarily be given full marks. 8. Round off to one decimal place unless stated otherwise. 9. It is in your best interest to write legibly and present your work neatly. 10. DO NOT USE TIPPEX.

Q1 (9) Q2 (12) Q3 (12) Q4 (10) Q5 (10) Q6 (10) Q7 (9) Total

Q8 (8) Q9 (14) Q10 (7) Q11 (14) Q12 (14) Q13 (5) Q14 (8) Q15 (8) %

PRELIMINARY EXAMINATION: MATHEMATICS PAPER II of 30 SECTION A

QUESTION 1

The vertices of ABC are ( )2; 5A , ( )4; 0B − and ( ); 5C a − . 1

1; 22

M

−

is the

midpoint of BC and AD BC⊥ .

(a) Determine the value of a. (1)

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(b) Determine the gradient of BC and hence, the equation of AD. (4)

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PRELIMINARY EXAMINATION: MATHEMATICS PAPER II of 30 (c) Line AQ is parallel to BC and has the equation 2 3 4y qx+ = .

(1) Show that the equation can be written as 3

22

qy x

−= + . (2)

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(2) Hence, calculate the value of q. (2)

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[9]

PRELIMINARY EXAMINATION: MATHEMATICS PAPER II of 30 QUESTION 2

(a) If sin47 k = , determine, in terms of k:

(1) ( )sin 133− (3)

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(2) sin77 (4)

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(b) Prove that: sin2 sin

tancos2 cos 1

x xx

x x

+=

+ + (5)

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[12]


(a) Use the diagram below to prove the theorem that states the following:

If DE ST in RST then RD RE

DS ET= . (6)

Construction: ________________

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Proof:

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PRELIMINARY EXAMINATION: MATHEMATICS PAPER II of 30 (b) BC is the diameter of the larger circle and DE is the diameter of the

smaller circle. M is the centre of the smaller circle.

APC is a tangent to the smaller circle at P.

72 , 36PC mm AP mm= = and 45BM mm= .

(1) Prove that MP BA . (3)

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(2) Calculate giving reasons the length of MC. (2)

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PRELIMINARY EXAMINATION: MATHEMATICS PAPER II of 30 (3) Calculate the radius of the larger circle. (1)

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[12]


In the sketch ABCD is a cyclic quadrilateral. AD and BC produced meet at E.

CA bisects ˆ ,BAD CD AE⊥ and ˆ 76BAD = .

Calculate, giving reasons, the size of:

(a) 1A (1)

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(b) 1B (2)

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(c) 2B (2)

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PRELIMINARY EXAMINATION: MATHEMATICS PAPER II of 30

(d) ˆBCD (2)

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(e) E (3)

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[10]


A cardiologist wanted to test the relationship between resting heart rate and the peak heart rate during exercise. Heart rate is measured in beats per minute (bpm). The following set of data was generated from 12 study participants after they had run on a treadmill at 10 km/h for 10 minutes.

Resting Heart Rate

(x) 48 56 90 65 75 78 80 72 82 68 62 76

Peak Heart Rate

(y) 128 165 180 161 151 161 155 154 175 158 158 158

The scatter plot is drawn below.

(a) Use your calculator to determine the equation of the line of best fit. (2)

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(b) Calculate r, the correlation coefficient. (1)

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PRELIMINARY EXAMINATION: MATHEMATICS PAPER II of 30 (c) Estimate what the heart rate of a person with a resting heart rate of 60 bpm

will be after exercise. (2)

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(d) What can you conclude regarding the relationship between resting heart

rate and the heart rate after exercise? Explain your answer briefly. (3)

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(e) Explain whether it is possible to determine the resting heart rate if the peak

heart rate is 170 bpm. (2)

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[10]


(a) In the figure, AB represents a thin metal rod, 130 cm in length, which is

propped up in the corner of a room. OAL is vertical and B is on the

horizontal KOM. ˆ 90KOL = and BC OK⊥ . If ˆ 36,9BOC = and

B is 30 cm from OK.

Calculate:

(1) the distance from B to the corner O, of the room.

Round your answer, correct to the nearest cm. (2)

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(2) the size of ˆOBA . (3)

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(b) PQR is drawn with 20 4PQ x= − , RQ x= and ˆ 60Q = .

(1) Show that the area of 25 3 3PQR x x = − . (2)

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(2) Determine the value of x for which the area of PQR will be maximum.

(3)

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[10]


In ABC , P is a point on AB. Q and R are points on BC such that AP PB= and BQ QR= . AR cuts PC in T.

(a) Prove RT QP . (2)

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(b) (1) If 5

4

BR

RC= , determine

CT

TP. (3)

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(2) Hence, determine Area CTR

Area CPQ

. (4)

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[9]

SECTION A: 72

PRELIMINARY EXAMINATION: MATHEMATICS PAPER II of 30 SECTION B

QUESTION 8

Given: cos 2 and cos2 7p= = , calculate the value(s) of:

(a) p. (5)

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(b) if 0 ;180 . (3)

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[8]


B is the centre of the circle. CAR is a tangent to the circle at A.

( )1; 26C − .

ˆ ˆ and 20 units.CBA ARQ CA= = =

(a) Determine the length of AB, the radius of the circle. (3) ______________________________________________________________

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(b) Determine the equation of the circle in the form ( ) ( )2 2 2x a y b r− + − = . (2)

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(c) What is the equation of the tangent CR? (4)

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PRELIMINARY EXAMINATION: MATHEMATICS PAPER II of 30 (d) If it is given that the equation of the radius is 4 3 1y x+ = , determine

the coordinates of A. (5) ______________________________________________________________

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[14]


A taxi company has 200 taxi cabs. The cumulative frequency curve below shows the fares in hundreds of Rands taken by the cabs on a particular morning.

(a) Use the curve to estimate:

(indicate on the ogive where you read off your answers)

(1) the median fare taken by the company that morning. (1)

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(2) the percentage of taxis in which the fare collected was between

R4 000 and R8 000. (2)

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PRELIMINARY EXAMINATION: MATHEMATICS PAPER II of 30 (b) The company charges R75 per kilometre travelled. There are no other charges

involved. Use the curve to answer the following questions:

(1) On the morning in questions, 40% of the cabs travel less than

m kilometres. Determine the value of m. (2)

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(2) How many cabs travelled less than 60 km on this morning? (2)

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[7]


(a) A circle touches the X-axis at ( )7; 0A and passes through the point ( )2; 1T −

as shown in the diagram below.

Determine the coordinates of the centre of the circle. (5)

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PRELIMINARY EXAMINATION: MATHEMATICS PAPER II of 30 (b) The diagram shows a circle with centre N and equation

2 2 1494 2

4x x y y− + + = . Chord AB of the circle is parallel to the X-axis and

lies below it. The length of AB is 12 units.

(1) Determine the coordinates of N. (3)

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(2) Determine the radius of the circle. (1)

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(3) Calculate the coordinates of A and B. (5)

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[14]


(a) (1) Show that ( )cos 30 2sinx x+ = can be written as 3 cos 5sinx x= . (4)

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(2) Hence, or otherwise solve for x if ( )cos 30 2sinx x+ = ; 0 ; 360x . (3)

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(b) On the axes below, the graph of ( ) 2sinf x x= has been drawn; 0 ; 360x .

On the same axes sketch the graph of ( ) ( )cos 30g x x= + . (3)


(c) Use the graphs to determine the values of 0 ; 360x for which:

(1) ( ) ( ). 0f x g x . (2)

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(2) 4sin 3 cos sinx x x − . (2)

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[14]


During the moderation process after the mini prelim that was out of 150 marks it was

decided to adjust the raw scores. Two options were discussed.

Option One: Add 7,5 marks to each student’s raw score.

Option Two: Add 10% of the marks they did not get.

For example:

Raw Scores Option 1 Option 2

100 107,5 105

80 87,5 87

(a) How will option one affect the raw median and standard deviation? (2)

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(b) How will option two affect the raw median and standard deviation? (3)

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[5]


In the diagram diameter AOB intersects chord CE at D so that CE ED= .

If and : 3 : 2OA r AB CD= = , show that ( )3 5

units.3

rEB

−= (8)

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[8]


Refer to the figure below.

• D and E are points on BC and AC

respectively of ABC

• AD and BE intersect at F

• BD CE=

• AD BE=

• FDCE is a cyclic quadrilateral.

(a) Prove:

ABD BCE (4)

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(b) Write down an angle equal to 1A . (1)

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(c) Prove that ABC is an equilateral triangle. (3)

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[8]

SECTION B: 77

PRELIMINARY EXAMINATION: MATHEMATICS PAPER II of 30 EXTRA LINES: (Please number carefully if you use this space!)

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PRELIMINARY EXAMINATION: MATHEMATICS PAPER II of 30 ______________________________________________________________

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king david high school linksfield

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