RONDEBOSCH BOYS’ HIGH SCHOOL - Prelim Websitemaths.stithian.com/New CAPS 2016 Prelim Papers/Rondebosch... · Web view7.You may use an approved scientific calculator (non-programmable

MARKS: 150 EXAMINER: P GHIGNONEE DU TOITS CARLETTI

TIME: 3 HOURS MODERATOR: T EDWARDS

This question paper consists of 16 pages including 1 formula sheet

RONDEBOSCH BOYS’ HIGH SCHOOL

SENIOR CERTIFICATE

MATHEMATICS (PAPER 2)12 SEPTEMBER EXAMINATION - 2016

GRADE 12

Mathematics/P2 RBHS September 2016

INSTRUCTIONS AND INFORMATION

Read the following instructions carefully before answering the questions.

1. This question paper consists of 13 questions.

2. Answer ALL the questions in the SPECIAL ANSWER BOOK provided.

3. Clearly show ALL calculations, diagrams, graphs et cetera that you used to determine the answers.

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

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

6. Diagrams are NOT necessarily drawn to scale.

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

8. An INFORMATION SHEET with formulae is included at the end of the question paper.

9. Write neatly and legibly.

Page 2 of 18


QUESTION 1

The graph below shows the monthly maximum temperatures in a certain city.

1.1 Calculate the mean monthly maximum temperature. (3)

1.2 Calculate the standard deviation of the monthly maximum temperature. (2)

1.3 It is predicted that one hundred years from now, global warming is likely to increase the city’s monthly maximum temperature by 5° C in December, January and February. It will also result in an increase of 1° C in the other months of the year.

1.3.1 By how much does the mean increase? (2)

1.3.2 Without calculating the new standard deviation, describe the effect that the predicted increases in temperature will have on the standard deviation. Make reference to the range in your answer. (2)

[9]

Page 3 of 18


QUESTION 2

2.1 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.

Time (in minutes) Frequency11≤ t<15 615≤ t<19 719≤ t<23 1523≤ t<27 1827 ≤ t<31 2

2.1.1 Complete the cumulative frequency table. (1)

2.1.2 Draw an ogive for the given data. (4)

2.1.3 Determine the following from the ogive:

a) the median (1)b) lower quartile (1)

2.1.4 How many people took longer than 25 minutes to complete task? (2)

2.2 Below is the scatter plot representing a set of data:

Choose the letter that corresponds to the most correct answer.

2.2.1 The equation of the least squares regression line is: A y=58,28 x−4,48 B y=4,48 x+58,28C y=58,28−4,48 x (1)

2.2.2 This plot could be representing the relationship between:A temperature and thirstB training and fitnessC late-coming and success (1)

[11]

Page 4 of 18

O R x

y

T

S

63,43º

P(-5:0)


QUESTION 3

In the diagram below, P is the point(−5 ;0). The inclination of the line PT is 63,43°. S is the midpoint and the y-intercept of PT . R is a point on the x-axis such that PO :∨¿2 :3.

3.1 Determine:

3.1.1 The gradient of PT , correct to the nearest integer value. (2)

3.1.2 The equation of PT in the form y = mx + c. (2)

3.1.3 The distance PS in simplified surd form. (2)

3.1.4 The coordinates of T . (2)

3.2 Determine the coordinates of R. (2)

3.3 Calculate the area of ∆ PTR . (4)

[14]

Page 5 of 18

y

x

20

O

C(-1;26)

A

R

B(-1;1)


QUESTION 4

In the figure the circle has a centre B(−1 ;1) . CAR is a tangent at Awith C (−1 ;26) . C B A=R=θ . CA=20 units.

Calculate:

4.1 the length of the radius BA. (3)

4.2 the equation of the circle. (3)

4.3 the equation of CA, show working. (4)

4.4 the equation of AB, given that the equation of CA is y=43

x+27 13 . (3)

4.5 the coordinates of A. (3)

[16]

Page 6 of 18

x

y

B

O


QUESTION 5

5.1 In the diagram below, two circles are drawn. Circle O touches circle centre B externally with B in the fourth quadrant.

The equation of circle O with centre the origin is given by x2+ y2 = 45. The equation of the circle centre B is given by (x−2 p)2 + ( y+ p)2 = 20. Determine the value of p. (5)

5.2 Prove that the radius of the circle with equation shown below can never exceed √13 for any value of θ.

x2+ y2+4 x cosθ+8 y sin θ+3=0 (5)

[10]

Page 7 of 18


QUESTION 6

6.1 Given 13 sin 2 A=12 ,where 90 ° ≤2 A ≤270 °.

Without the use of a calculator, use a sketch in the correct quadrant to determine the following: Label the relevant angle(s).

6.1.1 cos2 A (3)

6.1.2 cos A (3)

6.2 Evaluate the following without the use of a calculator:

cos70 ° . cos10°+cos 20° . cos80 ° (4)

6.3 Given tan B=34 where 0 °<B<90 °.

6.3.1 Prove that 3 cos x+4 sin x=5 sin ( B+x ) . (3)

6.3.2 Hence, or otherwise, find the general solution to:

3 cos x+4 sin x=52 (6)

[19]

Page 8 of 18


QUESTION 7

In the figure, the graphs of f ( x )=2cos (x+a)and g ( x )=1+sin bx are given for

x∈[−180 °;180 ° ].

7.1 Determine the values of a and b using the graphs. (2)

7.2 Determine the values of x for which g ( x )f ( x )

≤ 0. (3)

7.3 The y−axis is translated 30 ° to the right. Determine the new equation of f in the form y=csin( x+d) with reference to the new set of axes. (2)

[7]

Page 9 of 18

15,6

20 m

40 m

9 m

G

F

ED

C B

A


QUESTION 8

8.1 In an acute angled triangle ∆ ABC:

8.1.1 Prove that c2=a2+b2−2 ab . cos C (5)

8.1.2 Deduce that: 1+cos C=(a+b+c)(a+b−c)2 ab

(4)

8.2 In the accompanying figure D , E and F are three vertices of the floor of a rectangular hall. G is a light on the ceiling such that D ,G and F lie in the same vertical plane. The angle of elevation of G from D is 15,6 °, DE=40m, EF=9m and DG=20 m.

Calculate:

8.2.1 the length of GF in metres (3)

8.2.2 the height of the light, G, above the floor. (2)

[14]

Page 10 of 18


Page 11 of 18


QUESTION 9

PA and PB are tangents to circle M with AC=BC. A2=x .

9.1 Complete the table in the answer book.

Statement Reason

B1=x

A1=x

PA=PB

¿ x ∠' s opp=sides

P2=x alt∠' s ;CB∨¿PQ

¿ x corresp ’s; CB∨¿ PQ(5)

9.2 Prove that:9.2.1 ABRP is a cyclic quadrilateral. (2)9.2.2 AP=BQ (3)

9.3 Prove that: BCAR

= ABAQ (2)

[12]

Page 12 of 18

x

2

1

4

32

1

43

2

1

32

12

1

21

S

C

B

Q

RP

A M


QUESTION 10

10.1 In the diagram below, O is the centre of the circle.

Prove the theorem that states: BO C=2 × B A C (4)

10.2 In the diagram, O is the centre of the circle with PQ∨¿∨¿. Q PO=2x

Find the size of P Q R in terms of x. (4)

[8]

Page 13 of 18

CB

A

O

2x

O R

QP

S


QUESTION 11

Given: PQ=30 unitsQR=20 unitsPR=40 unitsQS=10 unitsRS=15 units

11.1 Prove that ∆ PQR∨¿∨∆ RSQ.

(3)

11.2 Prove that QS∨¿PR. (2)

11.3 If PS and QR intersect at X , calculate the length of QX . (5)

11.4 Find: area ∆ PQRarea PQSR (2)

[12]

Page 14 of 18

1510

20

30 40

R

S

Q

P


QUESTION 12

12.1 In the figure, AB∨¿ DC. C T D=90 °.AT=3 units, CT =5 units and CD=x units.

12.1.1 Prove that ∆ ABT∨¿∨∆CDT .(3)12.1.2 Find the area of the circle in terms of x. (5)

12.2

12.2.1 ∆ PQR is inscribed in a circle with Q=R=4 P. Find the size of P in degrees. (3)

12.2.2 P, Q and R form part of a regular polygon inscribed in the same circle. Q and R are adjacent vertices. Using the value of P found in 12.2.1, determine the number of sides of the polygon.

(4)[15]

Page 15 of 18

x

5

3

T

C

DA

B

R

Q

P

D

C

B

A


QUESTION 13

Four circular coins of unequal sizes lie on a table so that each coin touches two, and only two, of the others. Prove that the four points of contact, ABCD are concyclic.

[3]

[TOTAL 150]

Page 16 of 18


INFORMATION SHEET: MATHEMATICS

x=−b±√b2−4ac2a

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

T n=a+ (n−1 ) d Sn=n2 (2a+(n−1 ) d )

T n=a r n−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

sin B= c

sin C a2=b2+c2−2 bc .cos A area ∆ ABC=12

ab . sin C

sin (α +β )=sin α . cos β+cos α . sin β sin (α−β )=sin α .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 2α1−2sin2 α2cos2 α−1

sin 2 α=2sin α . cosα

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 17 of 18


y=a+bx b=∑ ( x−x ) ( y− y )

∑ ( x−x )2

Page 18 of 18

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RONDEBOSCH BOYS’ HIGH SCHOOL - Prelim Websitemaths.stithian.com/New CAPS 2016 Prelim Papers/Rondebosch... · Web view7.You may use an approved scientific calculator (non-programmable