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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
Question 12 (15 marks) - Start on the appropriate page in your answer booklet
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
Question 13 (15 marks) - Start on the appropriate page in your answer booklet
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
Question 14 (15 marks) - Start on the appropriate page in your answer booklet
a) Simplifyfully
cos cos 120deg cos 120deg
3
b) In the diagram is the diameter of the circle centre and is a tangent at The line intersects the circle at and at The tangent to the circle at intersects
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
45degdeg
1
AB
C
D
M
10
13a
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13 b
(i) If express in terms of and
Hence express in terms of and
2 2
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You may ask for extra writing paper if you need more space to answer question 11
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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
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
Question 12 (15 marks) - Start on the appropriate page in your answer booklet
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
Question 13 (15 marks) - Start on the appropriate page in your answer booklet
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
Question 14 (15 marks) - Start on the appropriate page in your answer booklet
a) Simplifyfully
cos cos 120deg cos 120deg
3
b) In the diagram is the diameter of the circle centre and is a tangent at The line intersects the circle at and at The tangent to the circle at intersects
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
45degdeg
1
AB
C
D
M
10
13a
For which x- values on the curve is the curve increasing 2
13 b
(i) If express in terms of and
Hence express in terms of and
2 2
(ii)
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Q13 -page 1shy- Q 13 page 2 shy
BOS____________Question 13
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Question 14 BOS
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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
Question 12 (15 marks) - Start on the appropriate page in your answer booklet
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
Question 13 (15 marks) - Start on the appropriate page in your answer booklet
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
Question 14 (15 marks) - Start on the appropriate page in your answer booklet
a) Simplifyfully
cos cos 120deg cos 120deg
3
b) In the diagram is the diameter of the circle centre and is a tangent at The line intersects the circle at and at The tangent to the circle at intersects
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
45degdeg
1
AB
C
D
M
10
13a
For which x- values on the curve is the curve increasing 2
13 b
(i) If express in terms of and
Hence express in terms of and
2 2
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I
-QI2middot page 1shy- Q12middot page l shy
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--
Q13 -page 1shy- Q 13 page 2 shy
BOS____________Question 13
Ct) Lj ~V l- tJO ~ ~ cJo = 7
A) Q middot c~S - 6 U U r-
A
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- QI4 page 1shy- QI4middotpage2shy
Question 14 BOS
Lca~4
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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
Question 12 (15 marks) - Start on the appropriate page in your answer booklet
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
Question 13 (15 marks) - Start on the appropriate page in your answer booklet
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
Question 14 (15 marks) - Start on the appropriate page in your answer booklet
a) Simplifyfully
cos cos 120deg cos 120deg
3
b) In the diagram is the diameter of the circle centre and is a tangent at The line intersects the circle at and at The tangent to the circle at intersects
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
45degdeg
1
AB
C
D
M
10
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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
Question 12 (15 marks) - Start on the appropriate page in your answer booklet
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
Question 13 (15 marks) - Start on the appropriate page in your answer booklet
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
Question 14 (15 marks) - Start on the appropriate page in your answer booklet
a) Simplifyfully
cos cos 120deg cos 120deg
3
b) In the diagram is the diameter of the circle centre and is a tangent at The line intersects the circle at and at The tangent to the circle at intersects
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
45degdeg
1
AB
C
D
M
10
13a
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13 b
(i) If express in terms of and
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2 2
(ii)
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I
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--
Q13 -page 1shy- Q 13 page 2 shy
BOS____________Question 13
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A) Q middot c~S - 6 U U r-
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raquo ~ 10 ol Cl-1 - 4C2x21 x+ 17 17 J t -) U 02 88 () If I 1~ --
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- QI4 page 1shy- QI4middotpage2shy
Question 14 BOS
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3 =-shy
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f
-L to Ilt)ards bullbull
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d ~ E - = rsid~ onoosJe eQv4 LoS Cld -RAvci ) 1
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Question 12 (15 marks) - Start on the appropriate page in your answer booklet
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
Question 13 (15 marks) - Start on the appropriate page in your answer booklet
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
Question 14 (15 marks) - Start on the appropriate page in your answer booklet
a) Simplifyfully
cos cos 120deg cos 120deg
3
b) In the diagram is the diameter of the circle centre and is a tangent at The line intersects the circle at and at The tangent to the circle at intersects
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
45degdeg
1
AB
C
D
M
10
13a
For which x- values on the curve is the curve increasing 2
13 b
(i) If express in terms of and
Hence express in terms of and
2 2
(ii)
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I 12
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8- 67~2~1 (j) -
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f 13 110 s~ - e- 67O~3(
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3
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~A 3=F~ (f)
~s-~
You may ask for extra writing paper if you need more space to answer question 11
I
-QI2middot page 1shy- Q12middot page l shy
~ __ _ Iz bull
- -r 8L 3 CL - 2 l shy 2 bull2
J -
1+ I ltr 1+
--2 ltrV L +16
- e b 1shy
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Xi-x- I (i) oV+- +Oo~+-~ --l bull
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You may ask for extra writing paper if you need more space to answer question 12
--
Q13 -page 1shy- Q 13 page 2 shy
BOS____________Question 13
Ct) Lj ~V l- tJO ~ ~ cJo = 7
A) Q middot c~S - 6 U U r-
A
v~ 7T p If (T) as~~
I ]T I ~
dZ~ 1- zC ~lt fJ)~~lrt= 720 ()
raquo ~ 10 ol Cl-1 - 4C2x21 x+ 17 17 J t -) U 02 88 () If I 1~ --
iV Iyen
b pound(1) 4-X c) ~-+--PQ M= ~I-~ I
x -X--lt-lID alL reol X xiplusmn) fTl J =af)-ocy
x 2cfl - ~ay (Qu [(lL A-~
I
U-~ _ -X-- (T) shoAd b~ -to I J I 4 - (-JC) ~Jolt~
4-l = (x - t ~Sl (gt()amp0 a 0If0 ~ oJ c ~ -4~X =- 1xJ oco -l=v c~I () Ugt-c~~ u _ Apound)0 = f+ltr-(x _ ~
o c 1- ---r r 1 --Ul-) (1- == -
-Ashy
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1 ()lJ - fLL Y
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II~ gt-l y ~ u - I -Wl ) 1 - 2~rY] It YJ I IX-I _V-1 I~ tJtJ41 J ihcu~ ( ~ lt-) (j J
JtNh rirw Ashy [ v ~vb 1 (D - to)I ~
iV - 4- tt 0 - Ja Pev (j) x x JIAot- -4 - 1 ~ -q~-=r-~-- oV=-2t~ 1L~o _ - ~
Ji-~ t w) x - ct p ( rH-q - =- 0 (0-r q + b q of 2-)) - U 1 1
_~ x ~ ((pttt) - r~----T--1)lt--_--- =___ -x~1Kgt ~ -I t[i)
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V5 75-fa Lj
- QI4 page 1shy- QI4middotpage2shy
Question 14 BOS
Lca~4
=I -+shy
- I + J +
3 =-shy
-+shy
f
-L to Ilt)ards bullbull
~M
Ovv-S~
L- F E D 90middot _ e-
~ A-E
EF
L~ e E = r pound e
D c
=shy cto- e
i
f
t
I
i
~
A A
- r-Dr= FE)
d ~ E - = rsid~ onoosJe eQv4 LoS Cld -RAvci ) 1
bv1shy Bi=-CEshy ( OVOV~ 0 ~o~)-shy ~
gp - F D
c i x = 4-a ~ T (l4t at j ~ x--J J
A LL-
Z -( (J) ~t$jilSk~ -rz == l Ashy I ~(~~
m 1 --shy
at Zut in-C I --shy
nDrd l-1 - -I 0 m (Jl-XLjU ltJ shy
(i)~ll~Ao _ J- - -1- x uti ltJ
J 1 - Id] - - x -rJ-~-i-V 3 x +--i 4 l~t -t- ot
uabX v
( uvt 1- at 3 ( 2 et +clte 0 )D
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r ( - ~- ~f
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L ~V~ 6
-
7
Question 13 (15 marks) - Start on the appropriate page in your answer booklet
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
Question 14 (15 marks) - Start on the appropriate page in your answer booklet
a) Simplifyfully
cos cos 120deg cos 120deg
3
b) In the diagram is the diameter of the circle centre and is a tangent at The line intersects the circle at and at The tangent to the circle at intersects
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
45degdeg
1
AB
C
D
M
10
13a
For which x- values on the curve is the curve increasing 2
13 b
(i) If express in terms of and
Hence express in terms of and
2 2
(ii)
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I
-QI2middot page 1shy- Q12middot page l shy
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--
Q13 -page 1shy- Q 13 page 2 shy
BOS____________Question 13
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- QI4 page 1shy- QI4middotpage2shy
Question 14 BOS
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Question 14 (15 marks) - Start on the appropriate page in your answer booklet
a) Simplifyfully
cos cos 120deg cos 120deg
3
b) In the diagram is the diameter of the circle centre and is a tangent at The line intersects the circle at and at The tangent to the circle at intersects
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
45degdeg
1
AB
C
D
M
10
13a
For which x- values on the curve is the curve increasing 2
13 b
(i) If express in terms of and
Hence express in terms of and
2 2
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You may ask for extra writing paper if you need more space to answer question 11
I
-QI2middot page 1shy- Q12middot page l shy
~ __ _ Iz bull
- -r 8L 3 CL - 2 l shy 2 bull2
J -
1+ I ltr 1+
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You may ask for extra writing paper if you need more space to answer question 12
--
Q13 -page 1shy- Q 13 page 2 shy
BOS____________Question 13
Ct) Lj ~V l- tJO ~ ~ cJo = 7
A) Q middot c~S - 6 U U r-
A
v~ 7T p If (T) as~~
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dZ~ 1- zC ~lt fJ)~~lrt= 720 ()
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iV Iyen
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x -X--lt-lID alL reol X xiplusmn) fTl J =af)-ocy
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I
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V5 75-fa Lj
- QI4 page 1shy- QI4middotpage2shy
Question 14 BOS
Lca~4
=I -+shy
- I + J +
3 =-shy
-+shy
f
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~M
Ovv-S~
L- F E D 90middot _ e-
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f
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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
45degdeg
1
AB
C
D
M
10
13a
For which x- values on the curve is the curve increasing 2
13 b
(i) If express in terms of and
Hence express in terms of and
2 2
(ii)
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I
-QI2middot page 1shy- Q12middot page l shy
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J -
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--
Q13 -page 1shy- Q 13 page 2 shy
BOS____________Question 13
Ct) Lj ~V l- tJO ~ ~ cJo = 7
A) Q middot c~S - 6 U U r-
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dZ~ 1- zC ~lt fJ)~~lrt= 720 ()
raquo ~ 10 ol Cl-1 - 4C2x21 x+ 17 17 J t -) U 02 88 () If I 1~ --
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x -X--lt-lID alL reol X xiplusmn) fTl J =af)-ocy
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4-l = (x - t ~Sl (gt()amp0 a 0If0 ~ oJ c ~ -4~X =- 1xJ oco -l=v c~I () Ugt-c~~ u _ Apound)0 = f+ltr-(x _ ~
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- QI4 page 1shy- QI4middotpage2shy
Question 14 BOS
Lca~4
=I -+shy
- I + J +
3 =-shy
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f
-L to Ilt)ards bullbull
~M
Ovv-S~
L- F E D 90middot _ e-
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EF
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f
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d ~ E - = rsid~ onoosJe eQv4 LoS Cld -RAvci ) 1
bv1shy Bi=-CEshy ( OVOV~ 0 ~o~)-shy ~
gp - F D
c i x = 4-a ~ T (l4t at j ~ x--J J
A LL-
Z -( (J) ~t$jilSk~ -rz == l Ashy I ~(~~
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at Zut in-C I --shy
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(i)~ll~Ao _ J- - -1- x uti ltJ
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-
10
13a
For which x- values on the curve is the curve increasing 2
13 b
(i) If express in terms of and
Hence express in terms of and
2 2
(ii)
I
I
b rli
1 ~
J ~ll
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-QI1-pageJ shy -middotQIJ - pag 3shy
Sl-c9J e- +- 11 SJ1 e- - 12 () =e L 30 bull ~dj S lt-e) amp +-125 e-- - R ~ (8- Jj J
I 12
~ r- ~r- R = J 14a-t-2T f-lh~-
ILh-1 shy
13 ~- b72~ I til
~
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13 ~ (e-61~ 13 o e 17 lt3 292 3 7
()-hZJ~ Q 3 bDE=- YJ a il r~ e I
e- 1=67 22- fi1 opound -
-t Y) Q -1-poundIOJ g
5 - c1- ~ t 12 lt 2 plusmn ~ t 0 tcL 12
t -r ~ 1+ t--
s -Of- t-l4t =- 3 +3t (D - 18t~-lY-ampt~ -t)
~HV- I~ i +4- =-D
(~ x -2- =-D shy
2-
~
pound X J-1iWJ ~ == gt-f (D ~ - 32 bull q
8- 67~2~1 (j) -
I
t R- -h irD
LHS - s- ceo 181) + IL iii 1t06
-s shy
f 13 110 s~ - e- 67O~3(
~ DIVIDE
lEI - -lDl - 110f ~ - Lt --J
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12- ([) =- -
c ~ l1n (4 +1) =x ~e= ~ ~A -r~e taN rAj))
J
I - ~ f iCltYa I~
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I t~it CD 3x- ICt~A- -= ~ ~ A + I
3
3t - I = ~~ 4- +- ~ CY- 4 32-1
~A 3=F~ (f)
~s-~
You may ask for extra writing paper if you need more space to answer question 11
I
-QI2middot page 1shy- Q12middot page l shy
~ __ _ Iz bull
- -r 8L 3 CL - 2 l shy 2 bull2
J -
1+ I ltr 1+
--2 ltrV L +16
- e b 1shy
S2S I1C
u CD ~~waf
A- )( kl
=4shy- 0 Oft~
LAPc= LA L
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L Ab
E
b1 L f~c shy L f Pc L 42ad lJ
-lIJ(--r eY nA1 L
Qshy L
IJ JI Ji) C-o rre d S htltoQcl
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j gtIV~- +lt~h s~o
-lt
7shyL
L ~
l-i-1 I-lgtlt--~ x- -I-tii l X-
l b3X- - I k -Xshy
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Xi-x- I (i) oV+- +Oo~+-~ --l bull
he 1middot ~vampVll~ ~
e) POIle C~ 2e + ~ e-SiVlJamp - J
)ltS -= ~ Ur-g - I + ~Sl ~-lsln~amp- tf)~
- C-9l A- + ~~I ~ I
- l (u) 9- + ~i amp -J () J-
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You may ask for extra writing paper if you need more space to answer question 12
--
Q13 -page 1shy- Q 13 page 2 shy
BOS____________Question 13
Ct) Lj ~V l- tJO ~ ~ cJo = 7
A) Q middot c~S - 6 U U r-
A
v~ 7T p If (T) as~~
I ]T I ~
dZ~ 1- zC ~lt fJ)~~lrt= 720 ()
raquo ~ 10 ol Cl-1 - 4C2x21 x+ 17 17 J t -) U 02 88 () If I 1~ --
iV Iyen
b pound(1) 4-X c) ~-+--PQ M= ~I-~ I
x -X--lt-lID alL reol X xiplusmn) fTl J =af)-ocy
x 2cfl - ~ay (Qu [(lL A-~
I
U-~ _ -X-- (T) shoAd b~ -to I J I 4 - (-JC) ~Jolt~
4-l = (x - t ~Sl (gt()amp0 a 0If0 ~ oJ c ~ -4~X =- 1xJ oco -l=v c~I () Ugt-c~~ u _ Apound)0 = f+ltr-(x _ ~
o c 1- ---r r 1 --Ul-) (1- == -
-Ashy
~-~~ P-tltyx - JGll1 _2Prcv (0(IC _l-4-X)I Xc-lt- 0 f(J) c 2 L
1 ()lJ - fLL Y
- t(D coc-loJol~ 4 -t x ao v f11-lYhr P+tj x- - Gl p~M =(- shyo-x) ~ cI ~D 1gt rtlv -4- 0 pt lU ltS
II~ gt-l y ~ u - I -Wl ) 1 - 2~rY] It YJ I IX-I _V-1 I~ tJtJ41 J ihcu~ ( ~ lt-) (j J
JtNh rirw Ashy [ v ~vb 1 (D - to)I ~
iV - 4- tt 0 - Ja Pev (j) x x JIAot- -4 - 1 ~ -q~-=r-~-- oV=-2t~ 1L~o _ - ~
Ji-~ t w) x - ct p ( rH-q - =- 0 (0-r q + b q of 2-)) - U 1 1
_~ x ~ ((pttt) - r~----T--1)lt--_--- =___ -x~1Kgt ~ -I t[i)
r---P~ ~(hl~ -2 + 2) _ _ - 0 c-v rv~ CnnfOcvivJ b a-o x 0 _ 2 shy- 1
t- =-~
V5 75-fa Lj
- QI4 page 1shy- QI4middotpage2shy
Question 14 BOS
Lca~4
=I -+shy
- I + J +
3 =-shy
-+shy
f
-L to Ilt)ards bullbull
~M
Ovv-S~
L- F E D 90middot _ e-
~ A-E
EF
L~ e E = r pound e
D c
=shy cto- e
i
f
t
I
i
~
A A
- r-Dr= FE)
d ~ E - = rsid~ onoosJe eQv4 LoS Cld -RAvci ) 1
bv1shy Bi=-CEshy ( OVOV~ 0 ~o~)-shy ~
gp - F D
c i x = 4-a ~ T (l4t at j ~ x--J J
A LL-
Z -( (J) ~t$jilSk~ -rz == l Ashy I ~(~~
m 1 --shy
at Zut in-C I --shy
nDrd l-1 - -I 0 m (Jl-XLjU ltJ shy
(i)~ll~Ao _ J- - -1- x uti ltJ
J 1 - Id] - - x -rJ-~-i-V 3 x +--i 4 l~t -t- ot
uabX v
( uvt 1- at 3 ( 2 et +clte 0 )D
J U (0 -~ nJ Y II -
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r ( - ~- ~f
k (I) ~ -shy I 2-
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--
-QI1-pageJ shy -middotQIJ - pag 3shy
Sl-c9J e- +- 11 SJ1 e- - 12 () =e L 30 bull ~dj S lt-e) amp +-125 e-- - R ~ (8- Jj J
I 12
~ r- ~r- R = J 14a-t-2T f-lh~-
ILh-1 shy
13 ~- b72~ I til
~
~k~~C
~ I
13 ~ (e-61~ 13 o e 17 lt3 292 3 7
()-hZJ~ Q 3 bDE=- YJ a il r~ e I
e- 1=67 22- fi1 opound -
-t Y) Q -1-poundIOJ g
5 - c1- ~ t 12 lt 2 plusmn ~ t 0 tcL 12
t -r ~ 1+ t--
s -Of- t-l4t =- 3 +3t (D - 18t~-lY-ampt~ -t)
~HV- I~ i +4- =-D
(~ x -2- =-D shy
2-
~
pound X J-1iWJ ~ == gt-f (D ~ - 32 bull q
8- 67~2~1 (j) -
I
t R- -h irD
LHS - s- ceo 181) + IL iii 1t06
-s shy
f 13 110 s~ - e- 67O~3(
~ DIVIDE
lEI - -lDl - 110f ~ - Lt --J
V a 2
12- ([) =- -
c ~ l1n (4 +1) =x ~e= ~ ~A -r~e taN rAj))
J
I - ~ f iCltYa I~
~4-t3X -shy
I t~it CD 3x- ICt~A- -= ~ ~ A + I
3
3t - I = ~~ 4- +- ~ CY- 4 32-1
~A 3=F~ (f)
~s-~
You may ask for extra writing paper if you need more space to answer question 11
I
-QI2middot page 1shy- Q12middot page l shy
~ __ _ Iz bull
- -r 8L 3 CL - 2 l shy 2 bull2
J -
1+ I ltr 1+
--2 ltrV L +16
- e b 1shy
S2S I1C
u CD ~~waf
A- )( kl
=4shy- 0 Oft~
LAPc= LA L
-~J
L Ab
E
b1 L f~c shy L f Pc L 42ad lJ
-lIJ(--r eY nA1 L
Qshy L
IJ JI Ji) C-o rre d S htltoQcl
J - ~ L K 11amp1 -4 (i) ~xs Iab~~ uJ - i r
j gtIV~- +lt~h s~o
-lt
7shyL
L ~
l-i-1 I-lgtlt--~ x- -I-tii l X-
l b3X- - I k -Xshy
I x~ - = I d) C-c--~+ vo-lueL
Xi-x- I (i) oV+- +Oo~+-~ --l bull
he 1middot ~vampVll~ ~
e) POIle C~ 2e + ~ e-SiVlJamp - J
)ltS -= ~ Ur-g - I + ~Sl ~-lsln~amp- tf)~
- C-9l A- + ~~I ~ I
- l (u) 9- + ~i amp -J () J-
- 1
- lU-t5
- ~ ~ TTl
I
I
At) K J
- - - - - -
You may ask for extra writing paper if you need more space to answer question 12
--
Q13 -page 1shy- Q 13 page 2 shy
BOS____________Question 13
Ct) Lj ~V l- tJO ~ ~ cJo = 7
A) Q middot c~S - 6 U U r-
A
v~ 7T p If (T) as~~
I ]T I ~
dZ~ 1- zC ~lt fJ)~~lrt= 720 ()
raquo ~ 10 ol Cl-1 - 4C2x21 x+ 17 17 J t -) U 02 88 () If I 1~ --
iV Iyen
b pound(1) 4-X c) ~-+--PQ M= ~I-~ I
x -X--lt-lID alL reol X xiplusmn) fTl J =af)-ocy
x 2cfl - ~ay (Qu [(lL A-~
I
U-~ _ -X-- (T) shoAd b~ -to I J I 4 - (-JC) ~Jolt~
4-l = (x - t ~Sl (gt()amp0 a 0If0 ~ oJ c ~ -4~X =- 1xJ oco -l=v c~I () Ugt-c~~ u _ Apound)0 = f+ltr-(x _ ~
o c 1- ---r r 1 --Ul-) (1- == -
-Ashy
~-~~ P-tltyx - JGll1 _2Prcv (0(IC _l-4-X)I Xc-lt- 0 f(J) c 2 L
1 ()lJ - fLL Y
- t(D coc-loJol~ 4 -t x ao v f11-lYhr P+tj x- - Gl p~M =(- shyo-x) ~ cI ~D 1gt rtlv -4- 0 pt lU ltS
II~ gt-l y ~ u - I -Wl ) 1 - 2~rY] It YJ I IX-I _V-1 I~ tJtJ41 J ihcu~ ( ~ lt-) (j J
JtNh rirw Ashy [ v ~vb 1 (D - to)I ~
iV - 4- tt 0 - Ja Pev (j) x x JIAot- -4 - 1 ~ -q~-=r-~-- oV=-2t~ 1L~o _ - ~
Ji-~ t w) x - ct p ( rH-q - =- 0 (0-r q + b q of 2-)) - U 1 1
_~ x ~ ((pttt) - r~----T--1)lt--_--- =___ -x~1Kgt ~ -I t[i)
r---P~ ~(hl~ -2 + 2) _ _ - 0 c-v rv~ CnnfOcvivJ b a-o x 0 _ 2 shy- 1
t- =-~
V5 75-fa Lj
- QI4 page 1shy- QI4middotpage2shy
Question 14 BOS
Lca~4
=I -+shy
- I + J +
3 =-shy
-+shy
f
-L to Ilt)ards bullbull
~M
Ovv-S~
L- F E D 90middot _ e-
~ A-E
EF
L~ e E = r pound e
D c
=shy cto- e
i
f
t
I
i
~
A A
- r-Dr= FE)
d ~ E - = rsid~ onoosJe eQv4 LoS Cld -RAvci ) 1
bv1shy Bi=-CEshy ( OVOV~ 0 ~o~)-shy ~
gp - F D
c i x = 4-a ~ T (l4t at j ~ x--J J
A LL-
Z -( (J) ~t$jilSk~ -rz == l Ashy I ~(~~
m 1 --shy
at Zut in-C I --shy
nDrd l-1 - -I 0 m (Jl-XLjU ltJ shy
(i)~ll~Ao _ J- - -1- x uti ltJ
J 1 - Id] - - x -rJ-~-i-V 3 x +--i 4 l~t -t- ot
uabX v
( uvt 1- at 3 ( 2 et +clte 0 )D
J U (0 -~ nJ Y II -
- o1c+a t ~ccl- ---J
~n+-l TY-= Jrr -~ +-u -4
cJ - T -- J
Jli + ~ (dj 2A -at) I
- J f- - 4- tC
1 - Jctplusmn -r-l Ip R ~)( Jb t (( -1shy1tN I - c -plusmn -CL L~ -tc k -shy D - cd =shy-lt) ~
- In plusmn- +shy 0 middottr -shy ~
- 0 t - Jtt 1 cD en e-I 001 ( l i n A~ ~ ~
Ix o-t ~Jt+ -~ =shy -
r ( - ~- ~f
k (I) ~ -shy I 2-
CJb-e
1
t4 ) (l(OVL B r F D
A eEf eLpound ~D 29
~
eIll--u~ L
L ~V~ 6
-
--
-QI1-pageJ shy -middotQIJ - pag 3shy
Sl-c9J e- +- 11 SJ1 e- - 12 () =e L 30 bull ~dj S lt-e) amp +-125 e-- - R ~ (8- Jj J
I 12
~ r- ~r- R = J 14a-t-2T f-lh~-
ILh-1 shy
13 ~- b72~ I til
~
~k~~C
~ I
13 ~ (e-61~ 13 o e 17 lt3 292 3 7
()-hZJ~ Q 3 bDE=- YJ a il r~ e I
e- 1=67 22- fi1 opound -
-t Y) Q -1-poundIOJ g
5 - c1- ~ t 12 lt 2 plusmn ~ t 0 tcL 12
t -r ~ 1+ t--
s -Of- t-l4t =- 3 +3t (D - 18t~-lY-ampt~ -t)
~HV- I~ i +4- =-D
(~ x -2- =-D shy
2-
~
pound X J-1iWJ ~ == gt-f (D ~ - 32 bull q
8- 67~2~1 (j) -
I
t R- -h irD
LHS - s- ceo 181) + IL iii 1t06
-s shy
f 13 110 s~ - e- 67O~3(
~ DIVIDE
lEI - -lDl - 110f ~ - Lt --J
V a 2
12- ([) =- -
c ~ l1n (4 +1) =x ~e= ~ ~A -r~e taN rAj))
J
I - ~ f iCltYa I~
~4-t3X -shy
I t~it CD 3x- ICt~A- -= ~ ~ A + I
3
3t - I = ~~ 4- +- ~ CY- 4 32-1
~A 3=F~ (f)
~s-~
You may ask for extra writing paper if you need more space to answer question 11
I
-QI2middot page 1shy- Q12middot page l shy
~ __ _ Iz bull
- -r 8L 3 CL - 2 l shy 2 bull2
J -
1+ I ltr 1+
--2 ltrV L +16
- e b 1shy
S2S I1C
u CD ~~waf
A- )( kl
=4shy- 0 Oft~
LAPc= LA L
-~J
L Ab
E
b1 L f~c shy L f Pc L 42ad lJ
-lIJ(--r eY nA1 L
Qshy L
IJ JI Ji) C-o rre d S htltoQcl
J - ~ L K 11amp1 -4 (i) ~xs Iab~~ uJ - i r
j gtIV~- +lt~h s~o
-lt
7shyL
L ~
l-i-1 I-lgtlt--~ x- -I-tii l X-
l b3X- - I k -Xshy
I x~ - = I d) C-c--~+ vo-lueL
Xi-x- I (i) oV+- +Oo~+-~ --l bull
he 1middot ~vampVll~ ~
e) POIle C~ 2e + ~ e-SiVlJamp - J
)ltS -= ~ Ur-g - I + ~Sl ~-lsln~amp- tf)~
- C-9l A- + ~~I ~ I
- l (u) 9- + ~i amp -J () J-
- 1
- lU-t5
- ~ ~ TTl
I
I
At) K J
- - - - - -
You may ask for extra writing paper if you need more space to answer question 12
--
Q13 -page 1shy- Q 13 page 2 shy
BOS____________Question 13
Ct) Lj ~V l- tJO ~ ~ cJo = 7
A) Q middot c~S - 6 U U r-
A
v~ 7T p If (T) as~~
I ]T I ~
dZ~ 1- zC ~lt fJ)~~lrt= 720 ()
raquo ~ 10 ol Cl-1 - 4C2x21 x+ 17 17 J t -) U 02 88 () If I 1~ --
iV Iyen
b pound(1) 4-X c) ~-+--PQ M= ~I-~ I
x -X--lt-lID alL reol X xiplusmn) fTl J =af)-ocy
x 2cfl - ~ay (Qu [(lL A-~
I
U-~ _ -X-- (T) shoAd b~ -to I J I 4 - (-JC) ~Jolt~
4-l = (x - t ~Sl (gt()amp0 a 0If0 ~ oJ c ~ -4~X =- 1xJ oco -l=v c~I () Ugt-c~~ u _ Apound)0 = f+ltr-(x _ ~
o c 1- ---r r 1 --Ul-) (1- == -
-Ashy
~-~~ P-tltyx - JGll1 _2Prcv (0(IC _l-4-X)I Xc-lt- 0 f(J) c 2 L
1 ()lJ - fLL Y
- t(D coc-loJol~ 4 -t x ao v f11-lYhr P+tj x- - Gl p~M =(- shyo-x) ~ cI ~D 1gt rtlv -4- 0 pt lU ltS
II~ gt-l y ~ u - I -Wl ) 1 - 2~rY] It YJ I IX-I _V-1 I~ tJtJ41 J ihcu~ ( ~ lt-) (j J
JtNh rirw Ashy [ v ~vb 1 (D - to)I ~
iV - 4- tt 0 - Ja Pev (j) x x JIAot- -4 - 1 ~ -q~-=r-~-- oV=-2t~ 1L~o _ - ~
Ji-~ t w) x - ct p ( rH-q - =- 0 (0-r q + b q of 2-)) - U 1 1
_~ x ~ ((pttt) - r~----T--1)lt--_--- =___ -x~1Kgt ~ -I t[i)
r---P~ ~(hl~ -2 + 2) _ _ - 0 c-v rv~ CnnfOcvivJ b a-o x 0 _ 2 shy- 1
t- =-~
V5 75-fa Lj
- QI4 page 1shy- QI4middotpage2shy
Question 14 BOS
Lca~4
=I -+shy
- I + J +
3 =-shy
-+shy
f
-L to Ilt)ards bullbull
~M
Ovv-S~
L- F E D 90middot _ e-
~ A-E
EF
L~ e E = r pound e
D c
=shy cto- e
i
f
t
I
i
~
A A
- r-Dr= FE)
d ~ E - = rsid~ onoosJe eQv4 LoS Cld -RAvci ) 1
bv1shy Bi=-CEshy ( OVOV~ 0 ~o~)-shy ~
gp - F D
c i x = 4-a ~ T (l4t at j ~ x--J J
A LL-
Z -( (J) ~t$jilSk~ -rz == l Ashy I ~(~~
m 1 --shy
at Zut in-C I --shy
nDrd l-1 - -I 0 m (Jl-XLjU ltJ shy
(i)~ll~Ao _ J- - -1- x uti ltJ
J 1 - Id] - - x -rJ-~-i-V 3 x +--i 4 l~t -t- ot
uabX v
( uvt 1- at 3 ( 2 et +clte 0 )D
J U (0 -~ nJ Y II -
- o1c+a t ~ccl- ---J
~n+-l TY-= Jrr -~ +-u -4
cJ - T -- J
Jli + ~ (dj 2A -at) I
- J f- - 4- tC
1 - Jctplusmn -r-l Ip R ~)( Jb t (( -1shy1tN I - c -plusmn -CL L~ -tc k -shy D - cd =shy-lt) ~
- In plusmn- +shy 0 middottr -shy ~
- 0 t - Jtt 1 cD en e-I 001 ( l i n A~ ~ ~
Ix o-t ~Jt+ -~ =shy -
r ( - ~- ~f
k (I) ~ -shy I 2-
CJb-e
1
t4 ) (l(OVL B r F D
A eEf eLpound ~D 29
~
eIll--u~ L
L ~V~ 6
-
-QI2middot page 1shy- Q12middot page l shy
~ __ _ Iz bull
- -r 8L 3 CL - 2 l shy 2 bull2
J -
1+ I ltr 1+
--2 ltrV L +16
- e b 1shy
S2S I1C
u CD ~~waf
A- )( kl
=4shy- 0 Oft~
LAPc= LA L
-~J
L Ab
E
b1 L f~c shy L f Pc L 42ad lJ
-lIJ(--r eY nA1 L
Qshy L
IJ JI Ji) C-o rre d S htltoQcl
J - ~ L K 11amp1 -4 (i) ~xs Iab~~ uJ - i r
j gtIV~- +lt~h s~o
-lt
7shyL
L ~
l-i-1 I-lgtlt--~ x- -I-tii l X-
l b3X- - I k -Xshy
I x~ - = I d) C-c--~+ vo-lueL
Xi-x- I (i) oV+- +Oo~+-~ --l bull
he 1middot ~vampVll~ ~
e) POIle C~ 2e + ~ e-SiVlJamp - J
)ltS -= ~ Ur-g - I + ~Sl ~-lsln~amp- tf)~
- C-9l A- + ~~I ~ I
- l (u) 9- + ~i amp -J () J-
- 1
- lU-t5
- ~ ~ TTl
I
I
At) K J
- - - - - -
You may ask for extra writing paper if you need more space to answer question 12
--
Q13 -page 1shy- Q 13 page 2 shy
BOS____________Question 13
Ct) Lj ~V l- tJO ~ ~ cJo = 7
A) Q middot c~S - 6 U U r-
A
v~ 7T p If (T) as~~
I ]T I ~
dZ~ 1- zC ~lt fJ)~~lrt= 720 ()
raquo ~ 10 ol Cl-1 - 4C2x21 x+ 17 17 J t -) U 02 88 () If I 1~ --
iV Iyen
b pound(1) 4-X c) ~-+--PQ M= ~I-~ I
x -X--lt-lID alL reol X xiplusmn) fTl J =af)-ocy
x 2cfl - ~ay (Qu [(lL A-~
I
U-~ _ -X-- (T) shoAd b~ -to I J I 4 - (-JC) ~Jolt~
4-l = (x - t ~Sl (gt()amp0 a 0If0 ~ oJ c ~ -4~X =- 1xJ oco -l=v c~I () Ugt-c~~ u _ Apound)0 = f+ltr-(x _ ~
o c 1- ---r r 1 --Ul-) (1- == -
-Ashy
~-~~ P-tltyx - JGll1 _2Prcv (0(IC _l-4-X)I Xc-lt- 0 f(J) c 2 L
1 ()lJ - fLL Y
- t(D coc-loJol~ 4 -t x ao v f11-lYhr P+tj x- - Gl p~M =(- shyo-x) ~ cI ~D 1gt rtlv -4- 0 pt lU ltS
II~ gt-l y ~ u - I -Wl ) 1 - 2~rY] It YJ I IX-I _V-1 I~ tJtJ41 J ihcu~ ( ~ lt-) (j J
JtNh rirw Ashy [ v ~vb 1 (D - to)I ~
iV - 4- tt 0 - Ja Pev (j) x x JIAot- -4 - 1 ~ -q~-=r-~-- oV=-2t~ 1L~o _ - ~
Ji-~ t w) x - ct p ( rH-q - =- 0 (0-r q + b q of 2-)) - U 1 1
_~ x ~ ((pttt) - r~----T--1)lt--_--- =___ -x~1Kgt ~ -I t[i)
r---P~ ~(hl~ -2 + 2) _ _ - 0 c-v rv~ CnnfOcvivJ b a-o x 0 _ 2 shy- 1
t- =-~
V5 75-fa Lj
- QI4 page 1shy- QI4middotpage2shy
Question 14 BOS
Lca~4
=I -+shy
- I + J +
3 =-shy
-+shy
f
-L to Ilt)ards bullbull
~M
Ovv-S~
L- F E D 90middot _ e-
~ A-E
EF
L~ e E = r pound e
D c
=shy cto- e
i
f
t
I
i
~
A A
- r-Dr= FE)
d ~ E - = rsid~ onoosJe eQv4 LoS Cld -RAvci ) 1
bv1shy Bi=-CEshy ( OVOV~ 0 ~o~)-shy ~
gp - F D
c i x = 4-a ~ T (l4t at j ~ x--J J
A LL-
Z -( (J) ~t$jilSk~ -rz == l Ashy I ~(~~
m 1 --shy
at Zut in-C I --shy
nDrd l-1 - -I 0 m (Jl-XLjU ltJ shy
(i)~ll~Ao _ J- - -1- x uti ltJ
J 1 - Id] - - x -rJ-~-i-V 3 x +--i 4 l~t -t- ot
uabX v
( uvt 1- at 3 ( 2 et +clte 0 )D
J U (0 -~ nJ Y II -
- o1c+a t ~ccl- ---J
~n+-l TY-= Jrr -~ +-u -4
cJ - T -- J
Jli + ~ (dj 2A -at) I
- J f- - 4- tC
1 - Jctplusmn -r-l Ip R ~)( Jb t (( -1shy1tN I - c -plusmn -CL L~ -tc k -shy D - cd =shy-lt) ~
- In plusmn- +shy 0 middottr -shy ~
- 0 t - Jtt 1 cD en e-I 001 ( l i n A~ ~ ~
Ix o-t ~Jt+ -~ =shy -
r ( - ~- ~f
k (I) ~ -shy I 2-
CJb-e
1
t4 ) (l(OVL B r F D
A eEf eLpound ~D 29
~
eIll--u~ L
L ~V~ 6
-
--
Q13 -page 1shy- Q 13 page 2 shy
BOS____________Question 13
Ct) Lj ~V l- tJO ~ ~ cJo = 7
A) Q middot c~S - 6 U U r-
A
v~ 7T p If (T) as~~
I ]T I ~
dZ~ 1- zC ~lt fJ)~~lrt= 720 ()
raquo ~ 10 ol Cl-1 - 4C2x21 x+ 17 17 J t -) U 02 88 () If I 1~ --
iV Iyen
b pound(1) 4-X c) ~-+--PQ M= ~I-~ I
x -X--lt-lID alL reol X xiplusmn) fTl J =af)-ocy
x 2cfl - ~ay (Qu [(lL A-~
I
U-~ _ -X-- (T) shoAd b~ -to I J I 4 - (-JC) ~Jolt~
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o c 1- ---r r 1 --Ul-) (1- == -
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1 ()lJ - fLL Y
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- QI4 page 1shy- QI4middotpage2shy
Question 14 BOS
Lca~4
=I -+shy
- I + J +
3 =-shy
-+shy
f
-L to Ilt)ards bullbull
~M
Ovv-S~
L- F E D 90middot _ e-
~ A-E
EF
L~ e E = r pound e
D c
=shy cto- e
i
f
t
I
i
~
A A
- r-Dr= FE)
d ~ E - = rsid~ onoosJe eQv4 LoS Cld -RAvci ) 1
bv1shy Bi=-CEshy ( OVOV~ 0 ~o~)-shy ~
gp - F D
c i x = 4-a ~ T (l4t at j ~ x--J J
A LL-
Z -( (J) ~t$jilSk~ -rz == l Ashy I ~(~~
m 1 --shy
at Zut in-C I --shy
nDrd l-1 - -I 0 m (Jl-XLjU ltJ shy
(i)~ll~Ao _ J- - -1- x uti ltJ
J 1 - Id] - - x -rJ-~-i-V 3 x +--i 4 l~t -t- ot
uabX v
( uvt 1- at 3 ( 2 et +clte 0 )D
J U (0 -~ nJ Y II -
- o1c+a t ~ccl- ---J
~n+-l TY-= Jrr -~ +-u -4
cJ - T -- J
Jli + ~ (dj 2A -at) I
- J f- - 4- tC
1 - Jctplusmn -r-l Ip R ~)( Jb t (( -1shy1tN I - c -plusmn -CL L~ -tc k -shy D - cd =shy-lt) ~
- In plusmn- +shy 0 middottr -shy ~
- 0 t - Jtt 1 cD en e-I 001 ( l i n A~ ~ ~
Ix o-t ~Jt+ -~ =shy -
r ( - ~- ~f
k (I) ~ -shy I 2-
CJb-e
1
t4 ) (l(OVL B r F D
A eEf eLpound ~D 29
~
eIll--u~ L
L ~V~ 6
-
- QI4 page 1shy- QI4middotpage2shy
Question 14 BOS
Lca~4
=I -+shy
- I + J +
3 =-shy
-+shy
f
-L to Ilt)ards bullbull
~M
Ovv-S~
L- F E D 90middot _ e-
~ A-E
EF
L~ e E = r pound e
D c
=shy cto- e
i
f
t
I
i
~
A A
- r-Dr= FE)
d ~ E - = rsid~ onoosJe eQv4 LoS Cld -RAvci ) 1
bv1shy Bi=-CEshy ( OVOV~ 0 ~o~)-shy ~
gp - F D
c i x = 4-a ~ T (l4t at j ~ x--J J
A LL-
Z -( (J) ~t$jilSk~ -rz == l Ashy I ~(~~
m 1 --shy
at Zut in-C I --shy
nDrd l-1 - -I 0 m (Jl-XLjU ltJ shy
(i)~ll~Ao _ J- - -1- x uti ltJ
J 1 - Id] - - x -rJ-~-i-V 3 x +--i 4 l~t -t- ot
uabX v
( uvt 1- at 3 ( 2 et +clte 0 )D
J U (0 -~ nJ Y II -
- o1c+a t ~ccl- ---J
~n+-l TY-= Jrr -~ +-u -4
cJ - T -- J
Jli + ~ (dj 2A -at) I
- J f- - 4- tC
1 - Jctplusmn -r-l Ip R ~)( Jb t (( -1shy1tN I - c -plusmn -CL L~ -tc k -shy D - cd =shy-lt) ~
- In plusmn- +shy 0 middottr -shy ~
- 0 t - Jtt 1 cD en e-I 001 ( l i n A~ ~ ~
Ix o-t ~Jt+ -~ =shy -
r ( - ~- ~f
k (I) ~ -shy I 2-
CJb-e
1
t4 ) (l(OVL B r F D
A eEf eLpound ~D 29
~
eIll--u~ L
L ~V~ 6
-
-