ST BENEDICT’S COLLEGE

SUBJECT Mathematics Paper 2 DATE 24 July 2015GRADE 12 MARKS 150EXAMINER Mrs Serafino MODERATOR Gr 12 EducatorsNAME DURATION 3 HoursCLASS

QUESTION NO. DESCRIPTION MAXIMUM MARK ACTUAL MARK

1 ; 2 Statistics 26

3 Trigonometry 10

4 Analytical Geometry 9

5 Trig Functions 15

6 Analytical geometry 18

7 ; 8 Euclidean Geometry 10

9 Trigonometry 17

10 Analytical Geometry 13

11 ; 12 Euclidean Geometry 24

13 Trigonometry 8

150

TEACHER’S SIGNATUREPARENT’S SIGNATURE

INSTRUCTIONS:1. This paper consists of 13 questions and 19 pages.

2. Read the questions carefully.

3. Answer all questions.

4. You may use an approved non-programmable and non-graphical calculator, unless otherwise stated.

5. Round off your answers to two decimal digits where necessary.

6. All necessary working details must be shown. Answers only, without the relevant calculations will not be given marks.

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

Grade 12 2 of 23 Mathematics Paper 2

SECTION A

QUESTION 1 12 MARKSBelow is a set of data recording the grams of fat to the total calories in sandwiches.

SANDWICH TOTAL FAT (g)

TOTAL CALORIES

Hamburger 9 260

Cheeseburger 13 320

Quarter Pounder 21 420

Quarter Pounder with Cheese

30 530

Big Don Burger 31 560

Gill Sandwich Special 31 550

Gill Special with Avo 34 590

Crispy Chicken 25 500

Fish Fillet 28 560

Grilled Chicken 20 440

Grilled Chicken Light 5 300

a) Calculate the line of best fit that models the data. 2

b) Using the equation of the line of best fit, calculate the how many calories a sandwich with 32 g of fat will contain. 2

Grade 12 3 of 23 Mathematics Paper 2

0 5 10 15 20 25 30 35 400

100

200

300

400

500

600

700

Total Fat

Tota

l Cal

orie

s

c) Calculate the correlation coefficient and comment on the relationship between fat and calories. 4

d) How many grams of fat will a sandwich with 723 calories contain? 2

e) Is the answer in question (d) reliable? Motivate your answer. 2

Grade 12 4 of 23 Mathematics Paper 2

QUESTION 2 14 MARKSThe cumulative frequency curve below shows the mass of 280 students.

a) How many students have a mass of less than 73 kilograms? 1

b) Find the number of students that have a mass between 80 and 85 kilogram. 1

c) Complete the following frequency table. 2

Mass (m ¿ 60≤m<65 65≤m<70 70≤m<75 75≤m<80 80≤m<85 85≤m<90

Frequency 20 10

d) Hence, calculate the estimated mean of the students’ mass. 2

e) Write down the standard deviation of the mass of the students. 1

Grade 12 5 of 23 Mathematics Paper 2

Mass (kg)

(70 ;75)

(75 ;200)

f) Using the given Cumulative Frequency polygon, write down the five number summary for the given data. 3

g) Draw a box and whisker diagram of the given set of data. 3

h) After a relaxing Matric Vac, all the students have gained 2 kilograms of mass each. 1

Which of the following statements would be correct for the given data set?

1) The standard deviation of the data set changes and the mean stays the same.

2) The standard deviation as well as the mean changes.

3) The standard deviation stays the same and the mean changes.

Grade 12 6 of 23 Mathematics Paper 2

QUESTION 3 10 MARKSa) If 2sin2β−2cos2β=√3 and β∈ [0 °;90 ° ] ; find the value of: cos2 β 4

b) Evaluate the following without the use of a calculator: 6

sin 252 °tan 234 ° .cos306 ° . sin 324 °

Grade 12 7 of 23 Mathematics Paper 2

−10 −5 5 10

−5

−4

−3

−2

−1

1

2

3

4

5

x

y

QUESTION 4 9 MARKSIn the diagram below, PQRS is a rectangle with vertices P (−4 ; 0 ) , Q (4 ; a ) ,R (6 ;0 )∧S. Q lies in the first quadrant.

a) Using analytical methods, show that a=4. 4

b) Determine the equation of the straight line through the points S and R. 2

c) Determine the equation of the circle that has diameter PR. Give the equation of the circle in the form (x−a)2+( y−b)2=r2. 3

Grade 12 8 of 23 Mathematics Paper 2

P

Q

R

S

QUESTION 5 15 MARKSGiven: f ( x )=1+sin x and g ( x )=cos2 x

a) Calculate the points of intersection of the graphs of f and g for x∈ [0° ;360 ° ] 8

b) Draw sketch graphs of f and g for x∈ [0° ;360 ° ] 4

90 180 270 360

−2

−1

1

2

3

x

y

c) For which values of x will f (x)≤g (x) 3

Grade 12 9 of 23 Mathematics Paper 2

QUESTION 6 18 MARKS

In the diagram above, A, B and C are the vertices of a triangle. AC is extended to cut the x-axis at D. The diagram is not drawn according to scale.

a) Calculate the gradient of:

i) AD 2

ii) BC 2

b) Calculate the size of D CB . 3

Grade 12 10 of 23 Mathematics Paper 2

D

A(3 ;7 )

C(−3 ;4 )

B(1 ;−6 )

M

[G(a ;b )

c) Determine the coordinates of M, the midpoint of BC. 2

d) If G(a;b) is a point such that A, G and M are collinear, show that b=2a+1. 3

e) Hence calculate the value of b if GC ¿√17. 6

Grade 12 11 of 23 Mathematics Paper 2

QUESTION 7 6 MARKSIn the diagram O is the centre of circle CNEL. KE is a tangent at E. N and T are points on KO. KO∥ EL.

Complete the following tables, using appropriate reasons. 3 + 3

Statement Reason

a) E3=90 °

∴ T 2=90 °

∴CT=TE

b) E1+ E2=L

But: L=O2

∴O2=E1+ E2 Deductive logic

∴COEK is a cyclic quadrilateral

Grade 12 12 of 23 Mathematics Paper 2

C

O123

T 23

N1 2

1 2 3EK

L

SECTION B

QUESTION 8 4 MARKSThe following statements are given:

AB bisects CD

CD bisects AB

AB and CD bisect each other

AB is perpendicular to CD

AB bisects CD at right angles

CD bisects AB at right angles

AB and CD bisect each other at right angles

In the space provided under each diagram, write down the statement that applies MOST accurately to the given diagram.

______________________________________ ____________________________________

_________________________________ _________________________________

QUESTION 9 17 MARKS

Grade 12 13 of 23 Mathematics Paper 2

A

BC

D

A

B

C

D

×

׿

¿

∞

∞

A

B

C

D

B

A

C

D

∥

∥

a) If tan α=3 and cos α<0, determine without the use of a calculator and with the aid of a sketch, the

value of 2sinα2cos α

2 . 4

b) Prove that: cos (x+45 °)

cos x+cos(90°+x )=√22 4

c) Prove that √−cos2 (90 °−θ )−cosθ cos (−θ) is non-real for all real angles θ . 3

d) You may not use a calculator in this question:

Grade 12 14 of 23 Mathematics Paper 2

Prove that for any angle B: 4sin B cos B cos2B sin 15 °

sin 2B ¿¿¿ 6

Grade 12 15 of 23 Mathematics Paper 2

QUESTION 10 13 MARKSIn the figure below, the line y=−2x+7 cuts the circlex2+ y2−2 x+10 y=39 at points A and B. A second circle, concentric to the first, touches the line AB at H, as shown in the diagram. (Concentric circles have the same centre) Leave the answer to the following questions in surd form, where necessary.

a) Calculate the coordinates of the centre, as well as the radius of the outer circle. 5

b) Determine the coordinates of H. 6

Grade 12 16 of 23 Mathematics Paper 2

[G

P

y

x

A

H

B

c) Assuming point H to be (5 ;−3 ) ; determine the equation of the inner circle. 2

Grade 12 17 of 23 Mathematics Paper 2

QUESTION 11 10 MARKSIn the sketch, O is the centre of circle EFG. EG and EOF are produced to H and J respectively so that HJ⊥EJ. JG is a tangent to the circle at G.

a) Prove that GFJH is a cyclic quadrilateral. 3

b) If H 2=x , determine G2 in terms of x. 4

Grade 12 18 of 23 Mathematics Paper 2

O FE

G

H

J1

21

21

1

2

3

3

42

21

c) Prove that OJ is a diameter of circle OJG. 3

Grade 12 19 of 23 Mathematics Paper 2

QUESTION 12 14 MARKS

In the figure TD is a tangent to circle ABCD. AD ∥ BC. AB and DC produced meet at W. TBS is a straight line.

If B1=B3, prove that:

a) BWTD is a cyclic quadrilateral. 4

b) TBS is a tangent to circle ABCD. 5

Grade 12 20 of 23 Mathematics Paper 2

A B

C

D T

W

S

1 1

1

11

2 2

2

2

2

3

34

5

≪

≪

c) TW ∥ BC. 5

QUESTION 13 8 MARKS

Grade 12 21 of 23 Mathematics Paper 2A

In the figure, ABCDFE is the open cover of a book standing upright. AC and AF are the diagonals of the identical rectangles ABCD and AEFD respectively. AB=m units and CF=n units. Prove that:

a) cosC D F=1−¿¿ n2

2m2 3

b) If cosC A F=b; show that AC=¿ √ n2

2(1−b) 4

c) If m=15 units, n=12 units, calculate the size of C D F. 1

Grade 12 22 of 23 Mathematics Paper 2

MATHS FOR ALL

Bm

n

E

CD

F

Grade 12 23 of 23 Mathematics Paper 2