2015 year 11 yearly examination p 3u... · 2020. 9. 28. · exam consists of 9 pages. this paper...

BAULKHAM HILLS HIGH SCHOOL

2015 YEAR 11

YEARLY EXAMINATION

Mathematics Extension 1

General Instructions Reading time ndash 5 minutes

Working time ndash 2 hours

Write using black or blue pen

Board-approved calculators may be used

In Questions 11ndash14 show relevant mathematical reasoning andor calculations

Marks may be deducted for careless or badly arranged work

Total marks ndash 70 Exam consists of 9 pages This paper consists of TWO sections Section 1 ndash Page 2-4 (10 marks) Attempt Question 1-10 Allow about 15 minutes for this section Section II ndash Pages 5-9 (60 marks) Attempt questions 11-14 Allow about 1 hours and 45 minutes for

this section

2

Section 1 ndashMultiple Choice (10 marks) Attempt all questions

Answer the following in the booklet provided

1 is a diameter of the circle

If ang 40deg then ang is

(A) 40deg

(B) 50deg

(C) 60deg

(D) not enough information

2 What is another expression for cos

(A) cos cos sin sin

(B) cos cos sin sin

(C) sin cos cos sin

(D) sin cos cos sin

3 What is the acute angle to the nearest degree between the lines 1 3 and 4 6 5 0

(A) 15deg

(B) 38deg

(C) 52deg

(D) 75deg

4 is equal to

(A)

(B)

(C)

(D)

A B

P

3

5 Let tan where 0deg 180deg

Which of the following gives the correct expression for 2 sin 2 cos

(A)

(B)

(C)

(D)

6 The coordinates of a point that divides the interval 27 to 120 externally in the ratio 43 is

(A) 44 28

(B) 44 28

(C) 54 21

(D) 5421

7 If cos and 0deg 180deg then tan is equal to

(A) or3

(B) or3

(C) 2

(D) 2

8 The expression sin radic3 cos can be written in the form 2 sin The value of is

(A) 30deg

(B) 30deg

(C) 60deg

(D) 60deg

4

9 The line is a tangent to the circle at and is a secant meeting the circle at and

Given that 6 9 and what is the value of

(A) 2

(B) 3

(C) 12

(D) 18

10 A point moves in the -plane such that tan cot is its parametric presentation with parameter where is any real number The locus of is then a

(A) Parabola

(B) Circle

(C) Hyperbola

(D) Straight line

End of Section I

AT 6

x

C

B

9

Not to scale

5

Section II ndash Extended Response All necessary working should be shown in every question

Question 11 (15 marks) - Start on the appropriate page in your answer booklet

Marks

a) Solve 2 3

21

3

b) Theacuteanglebetweenthelines 3 and is45degi Showthat

33 1

1

ii Hencefindallthevaluesof

12

c) Inthediagram isparallelto and isthecentreofthecircleang 80degandang 110deg

CopythediagramintoyourbookletFindthevalueofang givingreasons

3

d) Solve5 cos 12 sin 13 0overthedomain0 360deg

3

e) How many distinct permutations of the letters of the word are possible in a straight line when the word begins and ends with the letter

1

f) If tan and tan

express tan in terms of

2

End of Question 11

QP

S R T

O

Not to scale

6


Marks

a) Simplify 4 8 2

2

2

b) From a group of 7 girls and 6 boys 3 girls and 2 boys are chosen

(i) How many different groups of 5 are possible

(ii) If the group of 5 stand in a line how many arrangements are possible if the boys stand together

2

2

c) In the diagram two circles intersect at and and are straight lines

Draw the diagram into your answer booklet

(i) Explain why ang ang

(ii) Hence or otherwise show that is a cyclic quadrilateral

1

2

d) (i) Sketchthegraphof |1 2 |

(ii) Henceorotherwisesolve|1 2 |

2

2

e) Prove that cos 2 tan sin 2 1

2

End of Question 12

B

A

E

C

D

P K

Not to Scale

7


Marks

a) 4 men 2 women and a child sit at a round table

In how many ways can these 7 people be arranged

(i) Without restrictions

(ii) If the child is seated between the two men

1

1

b) Consider the function

(i) State the domain of the function

(ii) Show that the function is an odd function

(iii) Show that the function is increasing throughout its domain

(iv) Evaluate

(v) Using the above information sketch the function showing any essential features

1

2

2

1

2

c) The diagram below shows the parabola 4 and the points 2 and 2

(i) Show that the equation of the chord is 2 2

(ii) The line joining and passes through the point 0 2 Show that 2

(iii) The normals to the parabola 4 at points and intersect at

The coordinates of are ndash 2

(DO NOT PROVE THIS) Find the equation of the locus of

2 1 2

End of Question 13

x

y

Q(2aq aq2)

P(2ap ap2)

x2

= 4ay

8


a) Simplifyfully

cos cos 120deg cos 120deg

3

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at Let ang

(i) Prove ang 90deg

(ii) Prove

2 2

c) In the diagram 2 is a point on the parabola 4

(i) Show that the normal to the parabola at has equation 2

(ii) This normal cuts the and axes at and respectively

Find the ratio of in simplest terms

2 2

A

O

B

E

F

G

D C

Not to scale

x

y

T(2at at2)

x2

= 4ay

Y

XO

Not to Scale

9

d) CD is a vertical pole of height 1 metre that stands with its base on horizontal ground A is a point due South of such that the angle of elevation of from is 45deg

is a point due East of such that the angle of elevation of from is is the midpoint of

(i) Show that cosec

(ii) Find an expression for CM in terms of cosec

2 2

End of Examination

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1

AB

C

D

M

10

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2

Section 1 ndashMultiple Choice (10 marks) Attempt all questions

Answer the following in the booklet provided

1 is a diameter of the circle

If ang 40deg then ang is

(A) 40deg

(B) 50deg

(C) 60deg

(D) not enough information

2 What is another expression for cos

(A) cos cos sin sin

(B) cos cos sin sin

(C) sin cos cos sin

(D) sin cos cos sin

3 What is the acute angle to the nearest degree between the lines 1 3 and 4 6 5 0

(A) 15deg

(B) 38deg

(C) 52deg

(D) 75deg

4 is equal to

(A)

(B)

(C)

(D)

A B

P

3



(A)

(B)

(C)

(D)


(A) 44 28

(B) 44 28

(C) 54 21

(D) 5421


(A) or3

(B) or3

(C) 2

(D) 2


(A) 30deg

(B) 30deg

(C) 60deg

(D) 60deg

4



(A) 2

(B) 3

(C) 12

(D) 18


(A) Parabola

(B) Circle

(C) Hyperbola

(D) Straight line

End of Section I

AT 6

x

C

B

9

Not to scale

5



Marks

a) Solve 2 3

21

3


33 1

1


12



3


3


1

f) If tan and tan


2

End of Question 11

QP

S R T

O

Not to scale

6


Marks

a) Simplify 4 8 2

2

2




2

2





1

2



2

2


2

End of Question 12

B

A

E

C

D

P K

Not to Scale

7


Marks





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1





(iv) Evaluate


1

2

2

1

2







2 1 2

End of Question 13

x

y

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x2

= 4ay

8


a) Simplifyfully


3


at Let ang

(i) Prove ang 90deg

(ii) Prove

2 2





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A

O

B

E

F

G

D C

Not to scale

x

y

T(2at at2)

x2

= 4ay

Y

XO

Not to Scale

9



(i) Show that cosec


2 2

End of Examination

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t- =-~

V5 75-fa Lj


Question 14 BOS

Lca~4

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3 =-shy

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(A)

(B)

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(B) 44 28

(C) 54 21

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(B) or3

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(B) 30deg

(C) 60deg

(D) 60deg

4



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(B) 3

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(B) Circle

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(D) Straight line

End of Section I

AT 6

x

C

B

9

Not to scale

5



Marks

a) Solve 2 3

21

3


33 1

1


12



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3


1

f) If tan and tan


2

End of Question 11

QP

S R T

O

Not to scale

6


Marks

a) Simplify 4 8 2

2

2




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2





1

2



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2


2

End of Question 12

B

A

E

C

D

P K

Not to Scale

7


Marks





1

1





(iv) Evaluate


1

2

2

1

2







2 1 2

End of Question 13

x

y

Q(2aq aq2)

P(2ap ap2)

x2

= 4ay

8


a) Simplifyfully


3


at Let ang

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A

O

B

E

F

G

D C

Not to scale

x

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x2

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Y

XO

Not to Scale

9



(i) Show that cosec


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End of Examination

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I



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-Ashy


1 ()lJ - fLL Y








t- =-~

V5 75-fa Lj


Question 14 BOS

Lca~4

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3 =-shy

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f


~M

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EF

L~ e E = r pound e

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A LL-


m 1 --shy

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(B) 3

(C) 12

(D) 18


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(B) Circle

(C) Hyperbola

(D) Straight line

End of Section I

AT 6

x

C

B

9

Not to scale

5



Marks

a) Solve 2 3

21

3


33 1

1


12



3


3


1

f) If tan and tan


2

End of Question 11

QP

S R T

O

Not to scale

6


Marks

a) Simplify 4 8 2

2

2




2

2





1

2



2

2


2

End of Question 12

B

A

E

C

D

P K

Not to Scale

7


Marks





1

1





(iv) Evaluate


1

2

2

1

2







2 1 2

End of Question 13

x

y

Q(2aq aq2)

P(2ap ap2)

x2

= 4ay

8


a) Simplifyfully


3


at Let ang

(i) Prove ang 90deg

(ii) Prove

2 2





2 2

A

O

B

E

F

G

D C

Not to scale

x

y

T(2at at2)

x2

= 4ay

Y

XO

Not to Scale

9



(i) Show that cosec


2 2

End of Examination

45degdeg

1

AB

C

D

M

10

13a


13 b



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I



o c 1- ---r r 1 --Ul-) (1- == -

-Ashy


1 ()lJ - fLL Y








t- =-~

V5 75-fa Lj


Question 14 BOS

Lca~4

=I -+shy

- I + J +

3 =-shy

-+shy

f


~M

Ovv-S~


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EF

L~ e E = r pound e

D c

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f

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c i x = 4-a ~ T (l4t at j ~ x--J J

A LL-


m 1 --shy

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5



Marks

a) Solve 2 3

21

3


33 1

1


12



3


3


1

f) If tan and tan


2

End of Question 11

QP

S R T

O

Not to scale

6


Marks

a) Simplify 4 8 2

2

2




2

2





1

2



2

2


2

End of Question 12

B

A

E

C

D

P K

Not to Scale

7


Marks





1

1





(iv) Evaluate


1

2

2

1

2







2 1 2

End of Question 13

x

y

Q(2aq aq2)

P(2ap ap2)

x2

= 4ay

8


a) Simplifyfully


3


at Let ang

(i) Prove ang 90deg

(ii) Prove

2 2





2 2

A

O

B

E

F

G

D C

Not to scale

x

y

T(2at at2)

x2

= 4ay

Y

XO

Not to Scale

9



(i) Show that cosec


2 2

End of Examination

45degdeg

1

AB

C

D

M

10

13a


13 b



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(ii)

I

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(iv) Evaluate


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End of Question 13

x

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t- =-~

V5 75-fa Lj


Question 14 BOS

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1

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(iv) Evaluate


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1

2







2 1 2

End of Question 13

x

y

Q(2aq aq2)

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= 4ay

8


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3


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B

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-Ashy


1 ()lJ - fLL Y








t- =-~

V5 75-fa Lj


Question 14 BOS

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3 =-shy

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O

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E

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t- =-~

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t- =-~

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t- =-~

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Question 14 BOS

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Question 14 BOS

Lca~4

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=shy cto- e

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m 1 --shy

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uabX v

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-

2015 year 11 yearly examination p 3u... · 2020. 9. 28. · exam consists of 9 pages. this paper...

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