Matric Trials Paper 2 MEMO - St Stithians Collegemaths.stithian.com/New CAPS 2016 Prelim Papers/Hilton... · 2016-10-19 · CORE MATHEMATICS PII Page 5 of 24 (ii) Determine the inter-quartile

CORE MATHEMATICS PII Page 1 of 24

HILTON COLLEGE

TRIAL EXAMINATION AUGUST 2016

CORE MATHEMATICS PAPER 2

Time: 3 hours 150 marks

PLEASE READ THE FOLLOWING GENERAL INSTRUCTIONS CAREFULLY. 1. This question paper consists of 24 pages. There is also a separate yellow information sheet.

Please check that your paper is complete. 2. Read the questions carefully. 3. This question paper consists of 13 questions. Answer all questions. 4. You may use an approved non-programmable and non-graphical calculator, unless a specific

question prohibits the use of a calculator. 5. Round off your answers to one decimal digit where necessary, unless otherwise stated. 6. All necessary working details must be shown. 7. It is in your own interest to write legibly and to present your work neatly. 8. Please note that the diagrams are NOT necessarily drawn to scale. 9. Please ensure that your calculator is in DEGREE mode. Please do not turn over this page until you are asked to do so

EXAMINATION NUMBER:

M E M O

I pledge that I have neither given nor received help with this assessment.

Date: ______________________________ Signed: ______________________________


SECTION A

QUESTION 1 (a) Give the equation of the circle shown below: (2)

2 22 2 4x y

(b) Give the equation of the line which has an angle of inclination of 45o and and passes through the point 0; 3 . (2)

3y x (c) “The longer diagonal of a kite bisects the shorter diagonal at 90o” Use the above statement to find the equation of the longer diagonal. (4)

3;5

3

1

31

5 33

4

14

3

mid pt of AB

gradient of AB

gradient of longer diagonal

c

c

y x

http://www.clipartkid.com/kite-clipart-black-and-white-clipart-panda-free-clipart-images-wge8V1-clipart/

8

a radius a centre

a slope a intercept

a mid-pt

a slope

m -‘ve reciprocal

m substitution

ca


QUESTION 2 Consider the sketch of Guy cycling up a hill. The equation of the straight line representing the

road is 3

xy and the equation of the circle representing the rear wheel is

2 28 4 1004x x y y

(a) Determine the angle the road makes with the horizontal. (2)

1

tan

1tan 30

3

m

http://cliparting.com/free-bike-clip-art-11887/

(b) Assuming that units are in cm, determine the diameter of the rear wheel, in cm. (4)

2 2

2 2

2 2

8 4 1004

4 2 1004 16 4

4 2 1024

32

64

x x y y

x y

x y

radius is

diameter is cm

6

m link to gradient

ca

m completing the square

a

ca

ca


QUESTION 3 Determine the possible value(s) of k if the point 3;D k is a distance of 10 units from the point

9;3A .

2 2

2

2

3 9 3 10

36 3 100

3 64

3 8

5

k

k

k

k

k ll or

QUESTION 4 (a) The Grade 12 Core Maths Marks for Term 2 are depicted in the cumulative frequency curve shown below:

(i) How many matrics do Core Maths? (1) 78

5

m distance formula a

m solving for k

ca ca

a

k Q3Q1


(ii) Determine the inter-quartile range of the marks. Show by means of (3) dotted lines where you have read off any values you have used in your calculation.

3

1

74

52

74 52 22

Q

Q

IQR

(iii) What percentage of pupils achieved a distinction (80% or more)? (2)

16

100 20.5%78

(iv) Give a value for k if 40% of candidates achieved a mark of less than k? (2)

40% ~ 31

56%

candidates

k

m for showing on graph

m for subtracting Q1 from Q3

a for both correct

a m for % calculation

a

ca


(b) Ten Grade 11 boys achieved the following marks in Science and Mathematics in the June 2016 Examinations.)

Science (x)

Maths (y)

90 93

74 61

49 60

87 88

77 82

72 62

77 90

62 43

64 77

89 80

(i) Determine the equation of the line of best fit, the least squares regression line. Give both parameters to 3 decimal places. (2)

0.886 7.988y x

(ii) Use your answer to (b) (i) to predict the Maths mark for a boy who achieves a mark of 80% for Science. (2)

78.9%

(iii) Calculate, to 2 decimal places, the correlation coefficient and comment on what it means for the relationship between the Science and Mathematics marks given. (3)

0.71r There is a reasonably strong, positive relationship between performance in Maths and performance in Science.

15

a slope a intercept

a m for substitution if shown

a

a a


QUESTION 5 (a) In each case simply give the quadrant(s) in which must lie if: (i) cos 0 tan 0and (1)

IV

(ii) 3 3

sin tan5 4

and (1)

III

(b) Give the equations for each of the following graphs: (i) (2)

2siny x

(ii) (2)

cos 1y x

(iii) (2)

3 tany x

a

a

a sin a amplitude of 2

a cos a reflected and shifted

a tan a amplitude


(c) If sin 20 p then determine the following in terms of p: (i) cos 20 (2)

2

2

cos 20 1 sin 20

1 p

(ii) sin 200 (2)

sin 20

p

(iii) cos 250 (2)

sin 20

p

(iv) cos140 (2)

2

2

cos 40

1 2sin 20

2 1p

m identity / diagram

a

m reduction

a

m reduction

a

m double angle formula

a


(d) Consider the diagram with lengths and angles as marked:

(i) Determine, to one decimal place if necessary, the length of AC. (2)

6sin 30

612

sin 30

AC

AC cm

(ii) Hence, or otherwise, determine, to one decimal place, the length of AD. (3)

2 2 212 13 2 12 13 cos60

157

12.5

AD

AD cm

21

m correct method

ca

m cos rule a

ca


QUESTION 6 (a) Complete the following statements: (i) If a line cuts two sides of a triangle in the same proportions then…. (1)

It is parallel to the third side (ii) The exterior angle of a cyclic quadrilateral is …. (1) Equal to the opposite interior angle

(iii) If two triangles have corresponding sides in the same proportion then …. (1) They are equiangular OR They are similar

(b) In the diagram below CP is a tangent to the circle at C. AB || CD. AE = ED.

ˆPCD 40 and ˆBDC 60

a

a

a

a


Calculate, giving reasons, the sizes of:

(i) 2B (2)

40 tan - chord theorem

(ii) 1B (1)

60 . ' ||alt s on lines

(iii) E (2) 120 . 'opp s of CQ

(iv) 1D (2)

30 'sum of s of isos

(v) 1A (2)

2

1

D 20 .

A 100

opp s of CQ

sum s of

12

a a

a with reason

a a

a a

a a


QUESTION 7 (a) In the diagram below, FE || DC and DE || BC. AF = 5 cm and FD = 2 cm.

Calculate, with reasons, the length of DB to one decimal place. (5)

5

25

27 5

22.8

AF AEprop. int. theorem

FD ECAD AE

but prop. int. theoremDB EC

DBDB cm

a a

m seocnd use of prop. int. theorem

a

ca


(b) In the diagram below A is the centre of the circle. CD = 6 cm and EF = 1 cm. Calculate, giving reasons, the length of the radius. Hint: let the radius be x. (3)

2 2 2

3

1 3 .

5

CE cm from centre

x x Pythag

x

TOTAL FOR SECTION A: 75 MARKS

8

wj

wj

ca


SECTION B QUESTION 8 Consider the diagram:

(a) Determine to 1 decimal place, giving reasons where necessary. (5)

1 1 1tan 1 tan

345 18.4

26.6

153.4 opp s of CQ

(b) If it is further given that BC is a diameter of the circle then find the equation of the circle. (4)

2 22

2 2

1 5 2 4; 2;3

2 2

4 3 5 2 10

2 3 10

centre is

r

x y

9

m a

ca

ca reason

m a

a

ca


QUESTION 9 Consider the diagram below, showing the circle with equation 2 2 6 2 15 0x y x y

Determine, to 1 decimal place, the length of the tangent (AB) to the circle from the point 8; 1A ,

to the point of contact B, giving reasons where necessary.

2 2

2

2 2 2

2 2 2

2

3 1 15 1 9 25

25 5

,

11 2 125

125 25

10

x y

r and r

AB AO r pythagoras radius tangent

but AO

AB

AB units

6

m finding centre and radius a

m with reason

a

a

ca


QUESTION 10 (a) John was asked to prove the tan-chord theorem, viz. that “the angle between a tangent and a chord is equal to any angle subtended by that chord in the alternate segment”. He knows that he needs to prove that 2

ˆ ˆA D and he has

remembered that he needs to construct the diameter and to join FC. He has made the constructions correctly with dotted lines but now he is stuck! Complete the proof for John. (6)

2

1 2

1

1 2

2

ˆ

ˆ ˆ 90

ˆ 90

ˆ ˆ 90

ˆ

ˆ

ˆ ˆ

let A x

A A radius tangent

A x

but C C in semi circle

F x s of

but D x s in same segment

A D

wj

wj

wj

wj

a

m


(b) Consider the diagram below showing parallelogram ABCD with diagonals AC and BD drawn, intersecting at G. EF||BD , CF = 3 cm and FD = 5 cm

(i) Determine the ratio CE

EA giving reasons. (4)

3. int .

53 5

8 . sec

3 3

13 13

CE CFprop theorem

EG FDlet CE p then EG p

GA p diagonals of parm bi t one another

CE p

EA p

(ii) Determine Area CEF

Area ABCD

giving reasons. (4)

2

|||

3 9

8 64

sec1.

4

CEF CGD AAA

areas of similar figures are in the ratioCEF

equal to the square of the ratios ofCGD

their sides

diagonals bi t area of parm andbut CGD parm ABCD

CGD DAG equal base and height

9

256

CEF

ABCD

wj

m

a

a

a

wj

a

wj


(c) In the diagram below, AB is a diameter of circle ABC with centre O. Chord BC is produced to D. OD AB and OD cuts AC at E.

Prove, giving reasons: (i) That AOCD is a cyclic quadrilateral. (4)

1 2

3

3 3

ˆ ˆ 90

ˆ 90

ˆ ˆ

C C in semi circle

C s on straight line

C O

AOCD is a cyclic quadrilateral converse s in same segment

(i) 2 1

ˆ ˆC D (3)

1 2

2 2

2 1

ˆˆ

ˆ ˆ .

ˆ ˆ

D A s in same segment

but A C isos radii

C D

wj

wj

a

wj

wj

wj

a


(iii) OCE ||| ODC (4)

2 1

2

1 2 3

ˆ ˆ

ˆ

ˆ ˆˆ

|||

C D proved

O is common

E C C third of

OCE ODC AAA

(iv) 2OE.OD OC (2)

2

|||

.

OC OEs

OD OC

OC OE OD

27

a

a

wj

wj

wj

a


QUESTION 11 The marks of a class of 23 boys have a mean of 73% and a standard deviation of 10. Angus and Gavin have marks of 79% and 67% respectively. Calculate, to 2 decimal places, the standard deviation of the remaining boys if Angus and Gavin leave the class.

232

1

2 2

22 23

212

1

100

2300

22 23

, 36 36

:

2300 36 36 2228

2228

21

ii

ii

Variance initially

x x

Suppose Angus and Gavin are number and then

now x x and x x

Since x is unchanged by their leaving

x x

so the new variance

and the new standa

222810.30

21rd deviation

5

a

a

m

a

ca


QUESTION 12

(a) Determine the general solution of the following equations:

(i) 1

sin 2 cos 20 cos 2 sin 202

(4)

1sin 2 20

230

2 20 210 360 2 20 330 360

115 180 175 180

key angle

k or k with k Z for both

k or k

(ii) sin cos 3 (5)

sin sin 90 3

90 3 360 180 90 3 360

22.5 90 45 180

with k Zk or k

for both

k or k

m

m

a a

m

am

ca ca


(b) In the diagram below, AB is a straight line 1000m long. P represents an object moving along AB. DC is a vertical tower with C, A and B points in the same horizontal plane. The angles of elevation of D from A and B are 20 and respectively.

(i) Find the length of AC rounded to 2 decimal places. (2)

154tan 20

154423.11

tan 20

AC

AC m

(ii) Find the value of to the nearest degree. (6)

2 2 2

2 2

1

ˆ2 . cos

423.11 1000 2 423.11 1000 cos30

667.96

154tan

667.96154

tan 13.0667.96

BC AC AB AC AB BAC

BC

BC m

now

a

ca

m cos rule

a

ca

m trig ratio a

ca


(iii) Let be the angle of elevation of D from P. (5) Determine the maximum value of to one decimal place.

The angle will be greatest when P is closest to C which will happen when CP is perpendicular to AB When this happens:

1

sin 30423.11

423.11 sin 30

211.56

154tan

211.56154

tan211.56

36.1

CP

CP

CP m

now

22

m

m using 90o triangle

a

m

ca


QUESTION 13 A point P moves in the plane in such a way that its distance from the point 2;3A is always equal

to its distance from the line 1y . Two possible positions for P are shown and labelled as P1 and P2 to aid your understanding.

By letting the point P be the point ;P x y , determine, in standard form, the equation

of the function on which P moves while satisfying the condition of being equidistant from the point 2;3A and the line 1y .

2 2 2 2

2 2 2 2

2 2 2

2

2

2 3 1 0

2 3 1 0

4 4 6 9 2 1

4 12 4

34

x y y

x y y

x x y y y y

x x y

xy x

TOTAL FOR SECTION B: 75 MARKS

TOTAL FOR PAPER: 150 MARKS

6

m distance formula m equating distances

m squaring

a

ca

ca

Documents

Matric Trials Paper 2 MEMO - St Stithians Collegemaths.stithian.com/New CAPS 2016 Prelim Papers/Hilton... · 2016-10-19 · CORE MATHEMATICS PII Page 5 of 24 (ii) Determine the inter-quartile