ST. DAVID’S MARIST INANDA - St Stithians College 11 Papers/St Davids Marist Inanda... · Page 4...

ST. DAVID’S MARIST INANDA

MATHEMATICS

NOVEMBER EXAMINATION

GRADE 11

PAPER 2

24 NOVEMBER 2016

EXAMINER: MRS C KENNEDY MARKS: 125

MODERATOR: MRS S RICHARD TIME: 2½ Hours

NAME:__________________________________________

HIGHLIGHT YOUR TEACHER’S NAME:

Mrs Kennedy Mrs Nagy Mrs Black Mrs Richard Mr Biller

INSTRUCTIONS:

This paper consists of 24 pages and a separate Formula sheet. Please check that your paper is complete.

Please answer all questions on the Question Paper and read each question carefully.

You may use an approved non-programmable, non-graphics calculator unless otherwise stated.

Answers must be rounded off to ONE decimal place unless otherwise stated.

It is in your interest to show all necessary working details.

Work neatly. Do NOT answer in pencil.

Diagrams are not drawn to scale.

QUESTION Q1

[16]

Q2

[13]

Q3

[4]

Q4

[24]

Q5

[11]

Q6

[6]

Q7

[9]

Q8

[24]

Q9

[10]

10

[8]

TOTAL

[125]

LEARNER”S

MARKS

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Grade 11 November Paper 2 2016

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QUESTION 1 [16 marks]

CALCULATORS MAY NOT BE USED IN THIS QUESTION

a) i) Show that:

cos 1

tansin 1 cos

(5)

ii) Hence, evaluate the following:

cos 60tan 60

sin 60 1

(2)

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b) If sin .cosA A p , express the following in terms of p:

cos( ).cos( ) sin ( ).cos ( )2 2180 A 90 A A 180 A (5)

c) If 𝜃 , 2𝜃 and 3𝜃 are the angles of a triangle evaluate, without the use of a calculator:

cos2𝜃 + cos22𝜃 + cos23𝜃 (4)

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a) If sin cos4 and ,180 360 determine, without a calculator, the value of

sin( ).cos2 180 .

Leave your answer in surd form, if necessary. (6)

b) If 1

sin(x 22 )4

and (x 22 ) [90 ;270 ] , determine the value of cos2x .

(3)

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c) Determine the general solution of:

22sin 3sin 0 (4)

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−135 −90 −45 45 90 135

−4

−3

−2

−1

1

2

3

4

x

y

f

g


Given the graphs of ( ) sin ( ) tanf x 2 x and g x 2x where 135 x 135

Determine:

a) the period of ( )x2

f . (1)

b) one value of x for which ( ) ( )g x 2 f x , for the given domain. (1)

c) the equation of ONE asymptote of ( ),g x 25 where 135 135x . (1)

d) the maximum value of ( )2 f x 1 . (1)

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a) Two straight lines are drawn with equations: 2y x and 5 3 30y x .

i) Determine the coordinates of A, B and C. (5)

ii) Calculate the area of .ABC (2)

A

B C

y

x

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iii) Calculate the size of and . (5)

iv) Write down the coordinates of D, such that ABCD is a paralleogram. (4)

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x

y

B

A

P

b) The line with a gradient of −2 passing through the point 𝑃(3𝑡 ; 2𝑡) intersects

the x-axis at A and the y-axis at B.

i) Determine the equation of the line in terms of t. (3)

ii) The line through P perpendicular to AB intersects the x-axis at C.

Show that the midpoint of line PC has co-ordinates ( ; )t t . (5)

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a) Complete the following statements, using either MEAN, MEDIAN or MODE:

i) The ________________ is more sensitive to extreme outlier data values. (1)

ii) When data is negatively skewed, the ___________ > _____________ . (2)

iii) In a group of data exactly 50% of the data lies above the _____________ . (1)

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b) Refer to the following statements below labelled A to C and match the statements to

their most expected histogram labelled D to F. (3)

A.

The age at death in

developed countries (First

world countries)

B.

The salaries of a major

corporate company.

C.

The results of a test where

some of the students had

been taught and others had

not been taught the section

of work covered.

D.

E.

F.

Fill in the answers in this block:

A→

B→

C→

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0

10

20

30

40

50

60

70

80

90

0 20 40 60 80 100 120

C

U

M

U

L

A

T

I

V

E

F

R

E

Q

U

E

N

C

Y

RANDS

c) In the diagram below, the cumulative frequency curve shows the marked prices of 80

items that are for sale.

If the items are sold at half price,

i) redraw, on the given set of axes, the cumulative frequency curve for the

new prices. (2)

ii) what effect would this have on the standard deviation of the marked prices?

Explain. (2)

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The heights of several plants (in cm) were measured at a certain stage after planting, and the

following data was recorded:

x = days after

planting 14 20 8 15 18 11 14

h = height (cm) 6 11 3 8 10 4

The record of the last height has been lost, but we do know that the regression line

had equation

h = 0,72x – 3,31.

a) Estimate, to the nearest centimetre, what the last recorded height was. (2)

b) Calculate the correlation coefficient for the data relating to the first 6 plants correct to

four decimal places. (i.e. ignoring the last column). (2)

c) Sometime later, another plant’s height 35 days after planting was found to be 30 cm.

Comment on how surprising (or not) this is in the light of your previous results. (2)

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In the diagram below, ABCD and EFGH represent vertical buildings on a slanting ground that

has a constant inclination of 8º.

If DC = 31m and CF = 20m,

a) Write down the size of ˆDCF . (2)

b) Show that DF = 34,5m. (3)

c) Determine the value of , the angle of depression of F from D. (4)

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a) In the diagram:

The tangents at E and G meet at F, 2ˆ 40K and EG GH .

Determine the size of:

i) 2H (2)

ii) F . (4)

1

K E

H G

F

2 3

2

1 1

2

3

2 1

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THIS IS A BLANK PAGE

PLEASE TURN OVER

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O

2

1

65

4

32

2

1

1

Q

R

T

C

A

B

N

P

b) In the diagram, two circles touch each other externally at A.

BAC is a common tangent to both circles at A. PAQ is a double chord.

QR is the diameter of the larger circle. PT is a tangent to the smaller circle.

QRT and NAR are straight lines.

Let 1P x .

i) Name, stating reasons, THREE other angles each equal to x. (4)

ii) Why is PN RQ ? (1)

iii) Prove that:

(1) PN is a diameter of the smaller circle. (4)

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(2) APTR is a cyclic quadrilateral. (4)

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C

S

B

T

A

O

c) A, B and C are points on the circle having centre O. S and T are points on AC

and AB respectively such that OS AC and .OT AB AB = 40 and AC = 48.

If OT = x and 7

,15

OS OT calculate the radius, OA, of the circle. (5)

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O

H

TA

M


a) Use the diagram below to prove the theorem that states that the opposite angles

of a cyclic quadrilateral are supplementary. (5)

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D

E

A

CB

b) In the diagram below, two circles with centres at B and C intersect at A and E

respectively. AB is produced to a point D.

BA and BE are tangents to the smaller circle at A and E respectively.

Prove that ˆˆDBE ACE . (5)

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In the diagrams alongside, arm OA is 10cm long and

rotates clockwise about O.

The connecting rod AB is 30 cm long and point B moves

on a fixed line through O.

In the diagram below, ˆ ˆ,AOB 50 BAO is obtuse and , .OB 35 4 cm

a) How far does B move when angle ˆAOB changes from 50 to 120 ? (6)

b) How far does B move when angle ˆAOB changes from 0 to180 ? (2)

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TOTAL: 125

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