competency test markers mathematics p2 · 2019-09-23 · competency test markers mathematics p2 may...

COMPETENCY TEST

MARKERS

MATHEMATICS P2 MAY 2015

MARKS: 100

TIME: 2 hours

This paper consists of 23 pages.

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

Kindly print

NAME:

(first name) (surname)

INSTITUTION / SCHOOL:

Section A

Section B

Total

Mathematics P2 2 WCED Competency test 2015 Round 1

Copyright reserved Please turn over

SECTION A – 80 MARKS

Instructions:

Provide full solutions to Questions 1 to 8 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 21 19 16 11 9 5 2

1.1 Calculate the equation of the least squares regression line. (3) 1.2 Calculate the length of the candle after 23 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 In the diagram alongside, is the centre of the circle with equation

2 11694

2.1 Write down the coordinates of . (2) 2.2 Write down the length of the radius of the circle. (1) The chord of the circle is parallel to the axis. It lies below the axis and the length of

is 12 units. 2.3 Determine the coordinates of and . (5) 2.4 Determine the size of , correct to ONE decimal place. (2) The tangents to the circle at the points and meet at the point . 2.5 Calculate the length of , correct to TWO decimal places. (3)

[13]

12

P

A B

N

O

C

y

x



SOLUTION



Question 3 Prove that the equation of the tangent to the circle at the point ; is

[5]

SOLUTION



Question 4 4.1 Show that sin 5 sin 3 2 sin 4 cos (3) 4.2 Hence, or otherwise, solve for if sin 5 sin 3 0 and 0° 180° (5)

[8] SOLUTION



Question 5

and are two parallel chords of a circle with centre and radius .

, 4 and 2 . 5.1 If ∆ ∆ , calculate the size of . (6) 5.2 Calculate the value of if 24. Leave your answer in surd form. (4)

[10]

B

EP

D

A

C



SOLUTION



Question 6 The depth, metres, of water at the entrance to a harbour after hours after midnight on a particular day, is given by 8 4 sin 30 with 0 24 6.1 Determine the maximum depth and the minimum depth of the water. (2) 6.2 Draw the graph of on the system of axes provided. (3) 6.3 Calculate the values of for which 7. Give answers correct to ONE decimal

place. (6) [11]



SOLUTION



Question 7

is a square with the length of a side equal to units. is a point on such that units and with on .

7.1 Prove: 7.1.1 ∆ ∆ (4)

7.1.2 ∆ (4)

7.2 If the area of ∆ is four times the area of ∆ , calculate in terms of . (4)

[12]

P

QL M

NK



SOLUTION



Question 8

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: 8.1 ∥ (3) 8.2 (3) 8.3 (2) 8.4 . (6)

[14]

4 3

21

3 2

1

43 2

1

32

1

21

GB

E C F

D

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 The axis is a tangent to the circle, with centre M, at the point 3; 0 . The point 1; 2 lies on the circle. Determine the equation of the circle. Memorandum 3 1 2 ∴ 16 4 4 ∴ 4 20 ∴ 5 ∴ 3 5 25

[5] You will be assessed as follows: 2 marks for ticks correctly placed 1 mark for mistake(s) identified 1 mark for CA marking 1 mark for reasonable total marks allocated Total: 5 marks LEARNER ANSWER

Midpoint ;

2; 1

∴ 1 2 2 ∴ 2 3 ∴ 2 3 3 ∴ 3 ∴ 3 3 9

T( -1 ; 2 )

M( 3 ; y )

K( 3 ; 0 )



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: 2 marks for ticks correctly placed 1 mark for mistake(s) identified 1 mark for CA marking 1 mark for reasonable total marks allocated 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 In the diagram below, is the centre of the circle. is a diameter of the circle. is a tangent to the circle at .

and are two points on the circle. 32° and 56° Calculate, with reasons, the values of: 3.1 (2) 3.2 (3) 3.3 (4) Memorandum 3.1 34° (’s in a semi circle) (2) 3.2 66° ∴ 66° (tan chord theorem) (3) 3.3 58° ( sum of ) 122° (opp ’s cyclic quad) (4)

[9]

32

565

4

3

2

1

321

G

E

H

D

O

C

B

F



You will be assessed as follows: 2 marks for ticks correctly placed 1 mark for mistake(s) identified 1 mark for CA marking 1 mark for reasonable total marks allocated Total: 5 marks LEARNER ANSWER 3.1 44° (’s in a semi circle) 3.2 76° ∴ 76° (tan chord theorem) 3.3 48° ( sum of ) 132° (opp ’s cyclic quad)



Question 4 In ∆ below, lies on such that 2 .

lies on such that : 3: 8.

is drawn parallel to .

4.1 Write down the value of ∆

∆ (2)

4.2 Calculate the value of the ratio . Show all working to support your answer. (5)

[7] Memorandum

4.1 ∆

∆ (common vertex )

∴ ∆

∆ (2)

4.2 ( ∥

( ∥ )

units 2 units 5 units

∴ (5)

[7]

B

R

D

Q

A

C

P



You will be assessed as follows: 2 marks for ticks correctly placed 1 mark for mistake(s) identified 1 mark for CA marking 1 mark for reasonable total marks allocated Total: 5 marks LEARNER ANSWER

4.1 ∆

∆

∴ ∆

∆

4.2 ( ∥

( ∥ )

2 units units 8 units

∴

competency test markers mathematics p2 · 2019-09-23 · competency test markers mathematics p2 may...

Documents