Kingsmead College

Grade 12 Preliminary Examination 2014

MATHEMATICS II

Name: _____________________________

Time allowed: hours Total:150 marks

INSTRUCTIONS:

1.Write your name in the space provided.

2.This paper consists of 21 pages. Please check that this paper is complete.

3.Answer all questions in the spaces provided.

4.Non–programmable calculators may be used, but you must show all your working clearly.

5. Give your answers correct to 1 decimal place unless directed otherwise.

6.Write neatly in pen.

7.Diagrams are not drawn to scale.

8.There are extra blank pages at the end of the examination paper if needed.

Q1

Q2

Q3

Q4

Q5

Q6

Q7

Q8

Q9

Q10

Total

26

14

21

14

11

21

13

9

9

12

150

SECTION A

Question 1: [26 marks]

a)In the diagram below, points and are the vertices of in a Cartesian plane.

D

AC cuts the x-axis at D.

i)Calculate the coordinates of D (4)

ii)Calculate the value of if and (4)

iii)If , calculate the size of , correct to one decimal place. (3)

b)A circle with centre passes through the origin and the point .

The tangents at O and N meet at P.

Determine:

i)the equation of the circle (3)

ii)the value of (3)

iii)the equation of OP (2)

iv)the coordinates of P (5)

v)the specific type of quadrilateral represented by POMN (2)

Question 2: [14 marks]

NO CALCULATOR ALLOWED FOR THIS QUESTION:

a)Given: , evaluate

(5)

b)If , express the following in terms of :

i)(3)

ii)(3)

c)Given: . Find the general solution.(3)

Question 3: [21 marks]

a)Complete each of the following statements by filling in the missing words.

i) The angle at the centre of a circle is ……………………….the angle at the circumference.

ii) The tangent to a circle is……………………………. to the radius at the point of contact.(2)

b)In the diagram below, O is the centre of the circle KTUV. PKR is a tangent to the circle

at K. and

Calculate, with reasons, the size of the following angles:

i) (2)

ii) (2)

iii) (2)

iv) (2)

v) (2)

c)

(4)

O is the centre of the circle and ST is a tangent to the circle at S. .

Prove that

( hint : let )

d)A rocket consists of a right circular cylinder of height 20 metres surmounted by a cone whose height and diameter are equal and whose radius is the same as that of the cylindrical section. What should the radius be if the total volume is to be m3?(5)

Vol of Cone

Question 4: [14 marks]

Street vendors in the centre of Johannesburg sell cups of hot chocolate and packets of cooked

chips during lunch hour every day. They kept records of the number of cups of hot chocolate and

the number of packets of chips sold and the temperature at lunchtime on eight days.

Temperature

12

13

14

16

18

22

24

26

Cups of Hot chocolate

300

280

250

210

190

170

150

130

Packets of chips

280

160

250

120

294

206

238

240

a)The correlation coefficient of temperature and cups of hot chocolate sold is . Interpret this correlation coefficient in realistic terms.(2)

b)The correlation coefficient of temperature and packets of chips sold is .

Interpret this correlation coefficient in realistic terms.(2)

c)Determine the least squares regression lines for

i) cups of hot chocolate and temperature (2)

ii)packets of chips and temperature.(2)

d)Hence, estimate the following:

i)the number of cups of hot chocolate sold when the temperature is (1)

ii)the number of packets of chips sold when the temperature is (1)

e)Comment on the reliability of the above predictions.(4)

Total: Section A: 75 marks

Section B

Question 5: [11 marks]

a) On the axes below, sketch the graph of .

Label all intercepts with the axes and the co-ordinates of the turning points. (5)

b)Sketched below are the graphs and

i)Find the values of and .(3)

ii)For which values of is (1)

iii)Write down the new equation of if the y axis is moved to the right,

in simplified from?(2)

Question 6: [21 marks]

a)Without the use of a calculator, evaluate:

(7)

b)i)Prove the identity:

(4)

ii)For which values of is the above identity undefined for .(3)

c)Determine the minimum value of (2)

d)Solve for and :

(5)

Question 7: [13 marks]

RT is a tangent to the circle at T. Lines MN and NT are produced to R and S such that

Let .

a)Prove that RMTS is a cyclic quadrilateral. (5)

b)Prove that /// (5)

c)Hence, find the length of RT if units and units (3)

Question 8: [9 marks]

a) The speeds of 50 motorists were recorded on the N3 between Durban and Johannesburg. The speed limit on this particular stretch of road is120 km/h. The ogive curve, showing the relationship between the speeds of the cars versus the cumulative frequency, is shown below.

i)What percentage of motorists were speeding?(1)

ii)Write down the median speed.(1)

iii)Draw a box and whisker plot using the information from the Ogive.(5)

iv)Comment on the symmetry of the data given(2)

Question 9: [9 marks]

ABC is a straight line with units. B is the centre of the circle with radius of 2 units. P is a point on the circle. and

a)Express in terms of a single trigonometric ratio of (2)

b)Determine the size of in terms of and (1)

c)Hence, find in terms of .(2)

d)Hence show that (4)

Question 10: [12 marks]

The diagram below shows a representation of the chain of a bicycle attached to two circular cogs as represented in the Cartesian plane.

The equations of the circles are given by and .

RPQ is a common tangent to both circles at P and Q respectively, with R a point on the negative x-axis. T is the centre of the larger circle.

a)Write down the coordinates of the centre and length of the radius of the larger circle. (2)

b)Prove that /// (3)

c)Hence find the coordinates of R (4)

d)If , find the length of PQ, leaving answer in surd form (3)

Total: Section B: 75 marks

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