COMPETENCY TEST MARKERS MATHEMATICS P2 · 2019. 9. 23. · Mathematics P2 1 WCED Competency test 2015 Round 2 COMPETENCY TEST MARKERS MATHEMATICS P2 ROUND 2 – AUGUST 2015 This paper

Mathematics P2 1 WCED Competency test 2015 Round 2

COMPETENCY TEST

MARKERS

MATHEMATICS P2 ROUND 2 – AUGUST 2015

This paper consists of 27 pages.

This competency test consists of Section A (80 MARKS) and Section B (20 MARKS) Answer all questions.

Kindly print

NAME:

(first name) (surname)

IDENTITY NUMBER

__________________________________________________

INSTITUTION / SCHOOL:

Section A

Section B

Total

MARKS: 100

TIME: 2 hours


Copyright reserved Please turn over

SECTION A – 80 MARKS

Instructions:

Provide full solutions to Questions 1 to 10 in the spaces provided in the question paper.

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



Question 1 Round off answers correct to two decimal places. A man buys a box of large candles to use during load shedding. Each candle is 30 cm in length. He selects one candle at random. He lights the candle and, after it has burned continuously for hours, he records its length, cm, to the nearest centimetre. The results are shown in the table below.

hours 5 10 15 20 25 30 35 40 45 cm 27 25 22 18 16 12 9 6 2

1.1 Calculate the equation of the least squares regression line. (3) 1.2 Calculate the length of the candle after 28 hours. Give your answer to the nearest

centimetre. (2) 1.3 It is claimed by the candle manufacturer that the total length of time that such candles

are likely to burn for is more than 50 hours. Comment on this claim, giving a numerical justification for your answer. (2)

[7] SOLUTION



Question 2 The shape shown in the diagram is part of a circle. The centre of the circle is F( 8 ; 4 ) and AD and BC are tangents at A and B respectively. A is the point ( 3 ; 4 ) and B is the point ( 11; 8 ). A piece of wire is stretched from D to A, around part of the circumference of the circle to B and then to C. 2.1 Find the equation of the circle. (3) 2.2 Find the equation of the tangent line, BC, in the form 0 (4) 2.3 Find the length of the wire. (6)

[13]

x

y

CD

B ( 11 ; 8 )

F ( 8 ; 4)A ( 3 ; 4 )



SOLUTION



Question 3 A circle passes through the points 6; 2 and 6; 10 . The line 2 is a tangent to the circle. Determine the equation of the circle.

[6] SOLUTION



Question 4 Prove that sin . sin sin sin sin sin

[5] SOLUTION



Question 5 The diagram shows a square ABCD of side cm, with a point P within the square, such that PC = 6 cm, PB = 2 cm and AP = 2√5 cm. Let PBC =

5.1 Using the cosine rule in PBC, show that cos . (2)

5.2 In PBA, show that sin . (3) 5.3 Hence, or otherwise, show that 56 640 0 (3) 5.4 Find the value(s) of , and hence the length of AB. (3)

[ 11 ]

2

x

x

6

2 5P

D C

BA



SOLUTION



Question 6 A trigonometric function sin is given. It complies with the following conditions: The period is 120° The range is ∈ 6; 2 The co-ordinates of a maximum point is 90°; 2 Determine the values of , and .

[4] SOLUTION



Question 7 The lengths of the sides of a right-angled triangle form a geometric progression with 1. Determine the sizes of the other two angles of the right-angled triangle.

[8] SOLUTION



Question 8 In the diagram, BC and CAE are tangents to circle DAB. BDE is a straight line. BD = BA. Prove that DA BC

[5]

32

12

1

21

A

CB

D

E



SOLUTION



Question 9

is a cyclic quadrilateral and and produced meet the tangent to the circle at and respectively. is the point of contact of the tangent. bisects . Prove that: 9.1 ∥ (3) 9.2 (3) 9.3 (2) 9.4 . (6)

[14]

4 3

21

3 21

43 2

1

32

1

21

GB

E C F

D

A



SOLUTION



Question 10 In ABC, D is a point on AB such that AD : DB = 5 : 3. DE BC with E on AC. 10.1 Determine the value of area ∆ADE

area trapezium DBCE (3)

10.2 For what ratio AD : DB is area ADE = area DBCE? (4)

[7]

ED

CB

A



SOLUTION



BLANK PAGE



SECTION B – 20 MARKS Instructions:

The following questions and solutions are taken from a mock mathematics examination.

Below each question a memorandum with mark allocation is provided. Answers from a candidate are provided in each case. Mark each question WITH A RED PEN and indicate each mark with a tick. Indicate the tick at the appropriate step where the mark is given and circle any

mistakes that you deduct marks for. Indicate with CA (consistent accuracy) next to the tick if you are marking with the

candidate’s mistake. Indicate the total mark allocated for the question.



Question 1 In the diagram, ABCD is a parallelogram, with A( 4 ; 1 ) and B( 6 ; 9 ). The diagonals of ABCD intersect at M( 0 ; 3 ). 1.1 Calculate the coordinates of C. (2) 1.2 Determine the equation of the straight line CD. (3) 1.3 Prove that AMB = 90. (3) 1.4 Calculate the size of ABM (rounded off to ONE decimal digit). (5) 1.5 Determine the value of if A, B and P( ; 5 ) are collinear. (3) 1.6 Q lies on BM such that the area of QADC is equal to of the area of ABCD.

Determine the co-ordinates of Q. (2) /18/

x

y

O

D

C

M ( 0 ; 3 )

A ( 4 ; -1 )

B ( 6 ; 9 )



Memorandum 1.1 4; 7 (2) 1.2 5 ∴ 5 ∴ 7 5 4 ∴ 5 27 (3) 1.3 1 1 ∴ 1 ∴ 90° (3) 1.4 5 ∴ 78,7° 1 ∴ 45° ∴ 33,7° (5) 1.5 5 ∴ 4 30 5 ∴ 5 26 ∴ (3) 1.6 2; 5 (2)

/18/ You will be assessed as follows:

2 marks for ticks correctly placed

1 mark for mistake(s) identified

1 mark for CA marking indicated

1 mark for reasonable total marks allocated

Total: 5 marks



LEARNER ANSWER 1.1 4; 5 1.2 5 ∴ 5 ∴ 5 5 4 ∴ 5 25

1.3 1 1 ∴ 90° 1.4 4√2 6√2 √104 ∴ cos √ √ √√ √ ∴ cos 0.83 ∴ 33.7° 1.5 5 ∴ 4 30 5 ∴ 5 34 ∴ 1.6 2; 6



Question 2 Determine the general solution of: tan 2 5 sin 2 Memorandum tan 2 5 sin 2 ∴ 5 sin 2 ∴ sin 2 5 sin 2 cos 2 ∴ sin 2 5 cos 2 1 0 ∴ sin 2 0 or cos 2 ∴ 2 0° 180° or 2 78,46° 360° ∴ 0° 90° or 39,23° 180°; ∈

[7] You will be assessed as follows:



1 mark for CA marking indicated


Total: 5 marks LEARNER ANSWER tan 2 5 sin 2 ∴ 5 sin 2 ∴ sin 2 5 sin 2 cos 2 ∴ 5 cos 2 1 ∴ cos 2 ∴ 2 78,46° 360° ∴ 39,23° 180°



Question 3 If cos 12 ° , determine the value of the following, without using a calculator, in terms of

: 3.1 cos 168° (2) 3.2 tan 102° (3) 3.3 sin 66° (3)

[8] Memorandum 3.1 cos 168° cos 12° (2) 3.2 tan 102° tan 78° √ (3) 3.3 sin 66° cos 24° 2 cos 12° 1 2 1 (3)

[8] You will be assessed as follows:



1 mark for CA marking


Total: 5 marks

1 - a21

a

78

12



LEARNER ANSWER 3.1 cos 168° cos 12° 3.2 tan 102° tan 12° °°

° 3.3 sin 66° sin 60° 6° sin 60° cos 6° cos 60° sin 6° √ cos 12° 1 1 cos 12°

√ 1 1



Question 4 In the diagram below, a tangent EA to the circle meets CD produced at A. CE is produced to B so that AB = AE. It is further given that AED = 35 and CDE = 65. 4.1 Calculate, giving reasons, the size of 4.1.1 C (2) 4.1.2 E1 (2) 4.2 Now prove that ABED is a cyclic quadrilateral. (3) 4.3 Find the size of B1 and hence deduce that AB is a tangent to the circle through B, D

and C. (4) [11]

Memorandum 4.1 4.1.1 35° (tan chord theorem) (2) 4.1.2 65° (ext of DEC) (2) 4.2 65° (’s opp equal sides) ∴ is cyclic (ext = int opp ) (3) 4.3 35° (’s in same seg) ∴ ∴ is a tangent to the circle (conv tan chord theorem) (4)

[11]

32

1

3

2

1

21

21

35

65

D

CE

B

A


You will be assessed as follows:



1 mark for CA marking


Total: 5 marks LEARNER ANSWER 4.1 4.1.1 90° ( in semi circle) ∴ 25° (’s of DEC) 4.1.2 55° (ext of DEC) 4.2 55° (’s opp equal sides) ∴ is not cyclic (ext int opp ) 4.3 35° (’s in same seg) ∴ is not a tangent to the circle ( )

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COMPETENCY TEST MARKERS MATHEMATICS P2 · 2019. 9. 23. · Mathematics P2 1 WCED Competency test 2015 Round 2 COMPETENCY TEST MARKERS MATHEMATICS P2 ROUND 2 – AUGUST 2015 This paper