MATHEMATICSPAPER 2GRADE 12JUNE 2015

TIME: 3 HOURSMARKS: 150

RONDEBOSCH BOYS’ HIGH SCHOOL

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Examiner: P GhignoneModerator: T Edwards

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This question paper consists of 14 pages (including an information sheet.

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Page 1 of 15

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 in the ANSWER BOOK provided. Additional space is available at the back of the answer book. Number your work carefully.

3. Use the DIAGRAMS in the ANSWER BOOK for any markings you make.

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

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

6. You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise.

7. If necessary, round off answers to TWO decimal places, unless stated otherwise.

8. Diagrams are NOT necessarily drawn to scale.

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

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

11. Write neatly and legibly.

Page 2 of 15

QUESTION 1

Mr Jacobs owns a laundromat and charges his clients by the weight of their laundry. He kept a record of the weights (in kg) brought in by 22 clients.

0,5 0,8 1,5 1,9 2,1 2,8 2,8 3,3 3,4 3,4 3,64,0 4,5 5,4 6,2 6,4 6,5 7,7 7,8 9,2 9,8 13,0

1.1 Determine the mean weight of the laundry. (2)1.2 Calculate the standard deviation of the data. (2)1.3 How many clients have laundry weights that are within one standard

deviation of the mean? (2)1.4 He calibrates his scale and discovers that the scale was recording a weight

0,5 kg greater than the actual weight. Determine the actual mean and standard deviation. (2)

[8]

Page 3 of 15

QUESTION 2

Further to his investigations, Mr Jacobs collects data for the whole week, and prepares an ogive (cumulative frequency curve) to summarise his data.

2.1 Use the table in the ANSWER BOOK and complete the table below. (3)

Weight (in kg) Cumulative Frequency Frequency0≤ x<22≤x<44 ≤x<66≤ x<88≤ x<1010≤x<1212≤x<14

2.2 Use the frequency data to determine an approximate mean weight of laundry for the week. (3)

2.3 The data set has a median laundry weight of 5,6 kg. Use the ogive to estimate the lower quartile and upper quartile. Show evidence on the ogive. (2)

2.4 A client brings in a load of 13,6 kg. Determine (by calculation) whether this weight is an outlier. (3)

2.5 Mr Jacobs uses a heavy duty machine to wash loads greater than 11 kg. Determine the number of clients who will have their laundry washed in the heavy duty machine. (2)

[13]

Page 4 of 15

Amount of Laundry (kg)

Cum

ulat

ive

Freq

uenc

y

O

S (-5 ; 3) R

Q (3 ; 5)P

y

x

y

x

C(9;-3)

A(-2;-5)

B(3;-1)

QUESTION 3

3.1 Given the points A (−2 ;−5 ) , B (3;−1 ) and C (9 ;−3 ) .

3.1.1 Determine the length of BC. Leave your answer in surd form. (2)3.1.2 Determine the coordinates of M, the midpoint of AB. (2)3.1.3 Find the equation of the line parallel to AB, passing through C. (4)3.1.4 Calculate the size ofθ . (5)

3.2 PQRS is a rectangle with Q(3 ;5), S(−5 ;3) and SR parallel to the x-axis.

3.2.1 Write down the coordinates of R. (2)3.2.2 Show that OQ is perpendicular to OS. (3)3.2.3 Explain why Q, R, O, and S lie on the same circle and give the

coordinates of the centre of this circle. (5) [23]

Page 5 of 15

y

x

A (x ; y )

QUESTION 4

4.1 The equation of a circle is x2+ y2−2 x+4 y−4=0.

4.1.1 Determine the coordinates of A, the centre of the circle and the length of the radius, r. (5)

4.1.2 Calculate the value of p if N (1; p ) with p>0 is a point on the circle.(1)

4.1.3 Determine the equation of the tangent to the circle at N. (2)

4.2 A second circle, centre B, with equation ( x−4 )2+ y2=k2 cuts the circle given in (4.1) twice. Determine the values of k for which point A will be inside the circle B. (4)

[12]

Page 6 of 15

QUESTION 5

5.1 Given: sin 2 A=√158 for 0 ° ≤2 A ≤90°

5.1.1 Determine the value of cos2 Awithout the use of a calculator. (3) 5.1.2 Hence, find the value of cos A without the use of a calculator. Leave

your answer in surd form. (4)

5.2 Given cos ( A−B )=cos A .cosB+sin A . sinB

5.2.1 Derive the formula for cos ( A+B ) . (3)5.2.2 If A+B=90 °, show that:

cos2 ( A−B )=4cos2B sin 2B (4)[14]

QUESTION 6

6.1 Simplify, without the use of a calculator:sin 140° . sin120 °sin 110° .sin 340 ° (6)

6.2 Given: sin x+sin 2 x1+cos x+cos2x

=tan x

6.2.1 Prove the identity. (4)

6.2.2 Determine for which value(s) of x the identity is invalid if x∈[0 ° ;360 ° ]. (3)

[13]

Page 7 of 15

y

P

T

R

Q

y

QUESTION 7

7.1 Given g ( x )=2cos ( x−30 ° )

7.1.1 Make a neat sketch of g ( x ) for the interval [−90 °;270 ° ] on the grid provided. (3)

7.1.2 A straight line is now drawn through the turning points of the graph. Determine the equation of the straight line. (4)

7.2 Determine the general solution for: 32 sin θ−10.3sin θ+9=0 (6)

[13]

QUESTION 8

A vertical post, PT, has its foot in the same horizontal plane as points Q and R.TQ=TR= y and PQ=PR . The angle of elevation from Q to P is θ and P RQ=β.

8.1 Express PQ in terms of y and θ. (2)

8.2 Prove that QR=2 y .cos βcosθ (4)

[6]

Page 8 of 15

21

O

S

R

Q

P

QUESTION 9

9.1 In the diagram below, P, Q, R and S are points that lie on the circumference of the circle with centre O.

Given below is the partially completed proof of the theorem that states that P+ R=180°. Complete the proof by filling in the missing sections.

Construction: Join SO and QO.

Statement ReasonO1=… …O2=2 R ….O1+O2=2 P+2 R And O1+O2=… …∴2 P+2 R=360 ° ∴ P+ R=180 °

(5)

Page 9 of 15

58

12

3

12

ZY X

W

V

U

9.2 In the figure below, UVWX is a cyclic quadrilateral, with UW∨¿YZ and tangent YXZ touching the circle at X. U W X=58°.

Determine the values of the following angles, showing all steps and reasons:9.2.1 X1 (2)

9.2.2 X3 (2)

9.2.3 X2 (2)

9.2.4 V 1 (2)

Page 10 of 15

40

12

12

1

2

4 123

2

1

FE

D C

B

A

9.3 In the diagram below, AC is a chord of circle ABCDE.AFD⊥EFB , ∠D1=40° and ∠E=∠B1 .

In the following questions, give a reason for each statement:

9.3.1 Name THREE angles each equal to 50 ° . (4)

9.3.2 Calculate the size of∠DCB. (2)

9.3.3 Prove that EB||DC. (2)

9.3.4 Prove that AC is a diameter of the circle. (3)

[24]

QUESTION 10

In the diagram alongside, Q is a point on MP and T is a point on PN of ∆MNP ,such that QT||MN.S is the midpoint of MN. R is another point on MN such that QR||PS.MQ:QP = 5:2

Page 11 of 15

SR

T

Q

NP

M

Calculate, with reasons, the numerical value of the following:

10.1 PTTN (2)

10.2 RSMN (5)

[7]

Page 12 of 15

12

1

21

243

1 23

3

4

2

1

T

U QP

W

S

QUESTION 11

In the diagram, WPTU is a cyclic quadrilateral with UW=UT .Chords WT and PU intersect at Q. PW extends to S such that SU∨¿WT .

11.1 Complete the statement: The angle between the tangent to a circle and the chord drawn from the point of contact is … (2)

11.2 Prove that:

11.2.1 US is a tangent to circle PWUT at U. (5)

11.2.2 ∆ SPU∨¿∨∆ SUW (4)

11.2.3 SU 2=SP .SW (2)

11.2.4 SU 2 .QU=PU .SW 2 (4)

[17]

Page 13 of 15

INFORMATION SHEET: MATHEMATICS

x=−b±√b2−4ac2a

A=P (1+¿ ) A=P (1−¿ ) A=P (1−i )n A=P (1+i )n

∑i=1

n

1=n ∑i=1

n

i=n(n+1 )

2T n=a+ (n−1 )d Sn=

n2 (2a+(n−1 )d )

T n=a rn−1 Sn=

a ( rn−1 )r−1

; r ≠1 S∞=a

1−r ;−1<r<1

F=x [ (1+i )n−1 ]

iP=

x [1−(1+i )−n ]i

f ' ( x )= limh→0

f ( x+h )−f ( x )h

d=√(x2−x1 )2+( y2− y1 )2 M( x1+x2

2;y1+ y2

2 )y=mx+c y− y1=m (x−x1 ) m=

y2− y1

x2−x1m= tan θ

( x−a )2+ ( y−b )2=r2

In ΔABC:a

sin A= b

sinB= c

sinC a2=b2+c2−2bc .cos A area∆ ABC=12ab . sinC

sin (α+β )=sinα .cos β+cos α . sin β sin (α−β )=sinα .cos β−cos α . sin β

cos (α+β )=cos α .cos β−sin α .sin β cos (α−β )=cosα .cos β+sin α .sin β

cos2α={cos2α−sin 2α1−2sin2α2cos2α−1

sin 2α=2sin α .cosα

( x ; y )→ ( xcosθ− y sinθ ; y cosθ+ xsin θ )

x=∑ fxn σ 2=

∑i=1

n

(xi−x )2

n

P (A )=n (A )n (S )

P (A∨B )=P ( A )+P (B )−P ( A∧B )

Page 14 of 15

y=a+bx b=∑ ( x−x ) ( y− y )

∑ ( x−x )2

Page 15 of 15