MARK RECORD SHEETFOR OFFICIAL USE ONLY

NAME OF LEARNER:

QUESTION AIM 3Trigonometry

AIM 4Geometry

AIM 5Statistics

1 /222 /22

3 /9

4 /275 /126 /16

7 /16

8 /9

9 /5

10 /9

TOTALS /34 /107 /18

/150 %

ST STITHIANS GIRLS’ COLLEGE

GRADE 12MATHEMATICS: PAPER 2

24 July 2015

S T S T I T H I A N S C O L L E G E

INSTRUCTIONS:

1. This paper consists of 10 questions. Answer ALL of the questions.

2. This question paper consists of 22 pages.

3. Clearly show ALL calculations you have used to determine the answers.

4. An approved scientific calculator (non-programmable and non-graphical) may be used, unless otherwise specified.

5. If necessary, answers should be rounded off to TWO decimal digits, unless stated otherwise.

6. Diagrams are not necessarily drawn to scale.

7. It is in your own interest to write legibly and to present your work neatly.

2

GRADE 12

MATHEMATICS PRELIM EXAM – PAPER 2

DATE: 24 July 2015 TIME: 180 minutes

TOPICS: Paper 2 TOTAL MARKS: 150

EXAMINER: Cluster N143 MODERATOR: Cluster N143

QUESTION 1 [22]

1.1) M (1 ;b) is the midpoint of the line segment joining A(a ;4) and B (5 ;6 ) .

Find the values of a and b . (3)

1.2) The points C (1 ; ‒2 ) , D(5 ;1) and E(c2;c+1) are collinear.

Find the value(s) of c . (4)

3

1.3) Given points A (−2 ;2 ) ,B(3 ;4 ) and C (4 ; 0) on the Cartesian plane as sketched:

1.3.1) Calculate the size of AC B , rounded off to one decimal digit. (6)

1.3.2) Show that the midpoint, M , of AC is (1 ;1). (2)

xC(4 ; 0)

B(3 ; 4)

A(‒2 ; 2)

4

1.3.3) Determine the equation of the circle which has AC as a diameter.

Give your answer in the form ¿. (3)

1.3.4) Determine by calculation, whether point B lies inside or outside this circle.Give a reason for your answer. (2)

1.3.5) Write down the value of the shortest distance from B to the circle.

(Leave your answer in surd form) (2)

5

6

QUESTION 2 [18]

Refer to the diagram below. Given circle with centre O and equation .

is the centre of the larger circle. A common tangent touches the circles at

B and D respectively.

2.1) B lies on the circumference of the small circle. Determine the value of t. (3)

2.2) C is the midpoint of BD. Determine the coordinates of D. (2)

G (m;0)

C

B

O x

D

•

y

7

2.3) Determine the gradient of DG. (3)

2.4) Show that (3)

2.5) Determine the equation of the circle with centre G. (3)

2.6) Determine the size of angle . (4)

8

QUESTION 3 [22]

3.1) Trigonometric functions f (x) and g(x ) are given below, with x∈ [−90 °;360 ° ]:

3.1.1) Write down the equations of f and g. (2)

3.1.2) Write down the period of f . (1)

3.1.3) Write down the amplitude of g. (1)

3.1.4) Determine the values of x where f ( x ) . g (x)≥0 for x∈[90 ° ;270° ] (4)

g

f

9

3.2) If θ, 2θ and 3θ are the angles of a triangle, evaluate cos2θ+cos2 2θ+cos2 3θ

without the use of a calculator: (4)

3.3) Without the use of a calculator, solve sin 32 °cosx+cos32° sinx=sin 75 ° for x,

where −360 °≤ x≤360 ° : (5)

10

3.4) You are riding the Colossus at Ratanga Junction and notice that consecutive peaks,

T and L , of the ride are in proportion to each other.

You also notice as you are riding, that L OK=SO T=β.

3.4.1) Determine the value of cos ( 90°+β ).

(Leave your answer in surd form if necessary) (3)

3.4.2) Determine the value of p. (2)

OS

T (p ; 10)

L (10 ; 8)

K

11

QUESTION 4 [9]

Mr Mears is curious to see the distribution of heights of all his History students.

The table below summarises the individual heights (in cm) of 61 History students.

4.1) Complete the table by filling in the unknown values for (a) and (b): (2)

Height Intervals in cm Frequency Cumulative Frequency140≤x<150 0 0150≤x<160 5 5160≤x<170 11 16170≤x<180 (a) 38180≤x<190 13 51190≤x<200 7 (b)200≤x<210 3 61

4.2) Below is an Ogive for the heights of the History students:

150 160 170 180 190 200 2100

10

20

30

40

50

60

70

Cumulative Frequency of Heights

12

Use the Ogive to estimate the values of Q1, Q2 and Q3, and show on the Ogive how you

read off your answers.

4.2.1) Q1 (1)

4.2.2) Q2 (1)

4.2.3) Q3 (1)

4.3) Which height interval(s) contain(s) heights from the 90th percentile. (2)

4.4) Use the table of information to calculate an estimate for the mean of the History

students’ heights. (2)

13

QUESTION 5 [20]

5.1) In the figure O is the centre of the circle and DB=DF . AF ,BEand BF are straight lines, and F=20 °.

Find, with reasons, the magnitude of the following angles:

5.1.1) D2 (3)

5.1.2) A (3)

5.1.2) O2 (2)

5.1.3) C1 (2)

F

C2

1

343 B

21

D 21

23 1

4O

E A

14

15

5.2) In the diagram below, parallelogram KLMN is given. T is not the centre of the circle. L=66o and N1=24o. Determine the size of M 1. (5)

1

4

3

3

3

2

2

2

2

2

1

1

1

1

LP

K

T

M

N

16

5.3) An arch of a bridge is such that it is an arc of a circle and its height is 36m andits span is 96m. (i.e. and ).

Calculate with reasons the radius OD of the arch, i.e. calculate the length of OD.(Hint Let ) (5)

C

• O

BA

D

17

QUESTION 6 [16]

6.1) OABCD is a right pyramid with a square base with sides of length 4cm as shown

in the diagram below. O A B = 50 ° and OA = OB .

6.1.1) Determine the length of OA. (2)

6.1.2) Determine the length OE, the slant height of triangle OAB, where E is the

midpoint of AB. (3)

6.1.3) Show that the perpendicular height is … (2)

AD

BC

O

18

6.1.4) Hence, or otherwise, calculate the volume of the pyramid. (2)

6.2) Given AB ll CD, , , , , and Find the Area of the shaded ΔBCD. (7)

42°

35°

6

9

15

ED

C

B

A

8

19

QUESTION 7 [12]

7.1.1) Prove the identity: (5)

7.1.2) Hence, determine the maximum value of , and the value

of x to give this maximum, where (2)

7.2) Determine the general solution of: cos (25 °−2θ )=sin 4θ (5)

20

QUESTION 9 [9]

In the figure below, ΔABC has D and E on BC, and . and AD ll TE.

9.1) Write down the numerical value of (1)

9.2) Show that D is the midpoint of BE. (2)

9.3) If , calculate the length of TE. (2)

F

B DE

C

T

A

21

9.4) Calculate the numerical value of:

9.4.1) (1)

9.4.2) (3)

22

QUESTION 10 [10]

Refer to the figure below. LM=KL, and LZ is a tangent to the circle at L. XZ ∥ ln

and KM produced meets XZ at Y . KNX is a straight line.

10.1) Prove that YK∨¿ ZL. (4)

3

ZL

3

2 1

2

1

2

214

2 1

1

2 1

Y

KM

N

X

23

10.2) Prove that ∆ XYM∨¿∨∆KYX .(4)

10.3) Prove XZ . XY=KY . LZ (5)

24