Download - Year 12 Mathematics Worksheets
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= Year 12 = Algebra of functions = Worksheet 1
1. For ( )x
xxf1
+= , show that (i) ( )ufu
f =
1,
(ii) ( ) ( ) ( )vfufv
ufuvf =
+ , for { }0\, Rvu ∈ .
2. For ( ) xexf −= 1 , show that (i) ( ) ( ) ( ) ( )ufufufuf −+=− ,
(ii) ( ) ( ) ( ) ( ) ( )vfufvfufvuf −+=+ , for Rvu ∈, .
3. For ( ) xxf −= 1 , show that (i)( ) ( )
( )yf
yfxf
y
xf
−
−=
1 ,
(ii) ( ) ( ) ( ) ( ) ( )yfxfyfxfxyf −+= , for +∈ Ryx, .
4. For ( ) xxf elog1+= , show that (i) ( ) ( ) ( ) 1−+= yfxfxyf ,
(ii) ( ) ( ) 1+−=
yfxf
y
xf , for +
∈ Ryx, .
5. Refer to Q4. Show that (i) ( ) ( )xfy
xfxyf 2=
+ ,
(ii) ( ) ( )[ ]yfxfx
yf
y
xf −=
−
2 .
6. For ( ) 21 xxf += , show that
(i) ( ) ( ) ( ) ( )[ ]12 −+=−++ yfxfyxfyxf ,
(ii) ( ) ( ) xyyxfyxf 4=−−+ ,
for Ryx ∈, .
7. For ( )x
xxf1
+= , show that for +∈ Ryx, ,
( ) ( ) ( ) ( )[ ] ( ) ( )[ ]yfxfyfxfyfxf +−=− .22 .
8. For ( ) xxeexf
−+= , show that
( ) ( ) ( ) ( )[ ] ( ) ( )[ ]yfxfyfxfyfxf +−=− .22 .
9. For ( ) xxeexf
−−= , show that (i) ( )[ ] ( ) ( )xfxfxf 33
3−= ,
(ii) ( )[ ] ( ) ( ) ( )xfxfxfxf 103555
+−= .
10. For ( ) xexf
−−= 1 , show that
( )[ ] ( ) ( ) ( )xfxfxfxf 32333
+−= .
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= Year 12 = Algebra of functions = Worksheet 2
1. Given ( ] Rf →∞− 0,: , ( ) xxxf 22−= , find the rule of 1−f .
2. Given ( )3
11
xxf −= , find the rule of 1−f .
3. Given ( ] Rf →∞− 0,: , ( ) ( )1log 2+= xxf e , find the rule of
1−f .
4. Given [ ] Rg →− 0,: π , ( ) ( )ttg 2sin= , find the rule of 1−g .
5. Given [ ] Rg →− 0,: π , ( ) ( )ttg 2sin= , find the rule of 1−g .
6. Find the rule of the inverse function of ( ) ( )θθ −−−=
121 eh .
7. Express ( ) 1124122 23−+−= xxxxf in the
form ( ) cbxa ++3
. Hence find the rule of 1−f .
8. Given ( ) yxyxyx sincoscossinsin +=+ , express
( ) tttg cossin += in the form ( )bta +sin , where Ra ∈ and
∈
2,0π
b . Hence find the rule of 1−g .
9. Given ( ) tt eetf 22−= and [ )∞∈ ,0t , find the rule of 1−f .
10. Given 21
2
2++=
yyx , where +
∈ Ry , express y in terms
of x.
Numerical, algebraic and worded answers
1. ( ) 111+−=
− xxf 2. ( ) ( ) 3
11 1
−−−= xxf 3. ( ) 11
−−=− xexf 4. ( ) ttg
11sin
2
1 −−=
5. ( ) ttg11
sin2
1 −−−= 6. ( )
θθ
−−=
−
1
1log1
1
eh 7. ( ) ( ) 5223
+−= xxf , ( ) 22
5 3
1
1+
−=
− xxf
8. ( )
+=
4sin2
πttg , ( )
42sin 11 π
−
=
−− ttg 9. ( ) ( )11log1
++=− ttf e 10. ( )4
2
1−±= xxy
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= Year 12 = Algebra of functions = Worksheet 3
Given the rules ( )xf and ( )xg , find the rules ( )( )xgf and
( )( )xfg , state the domain and range in each case.
1. ( ) xxf = ; ( ) 2xxg = .
2. ( ) xxf sin= where [ ]π2,0∈x ; ( ) xxg = .
3. ( ) 12+= xxf ; ( ) xxg elog= .
4. ( ) xxf = ; ( ) xxg cos= .
5. ( ) xxf tan= where
∈
2,0π
x ; ( ) 12+= xxg .
6. ( ) xexf = ; ( ) 542
−−= xxxg .
7. ( )1
1
+=
xxf ; ( ) 2
xxg = .
8. ( ) xxf sin= ; ( ) xexg −= 1 .
9. ( )x
xf1
= ; ( ) xxg cos= where [ ]π2,0∈x .
10. ( )2
11
xxf += ; ( )
1
1
−−=
xxg .
Numerical, algebraic and worded answers. 5. ( )( ) =xgf ( )1tan 2+x ,
−1
2,0
π, [ )∞,1tan ; ( )( ) =xfg 1tan 2
+x ,
2,0π
, [ )∞,1 .
3. ( )( ) =xgf ( ) 1log2
+xe , +R , [ )∞,1 ; ( )( ) =xfg ( )1log 2
+xe , R, [ )∞,0 . 4. ( )( ) =xgf xcos , R, [ ]1,0 ; ( )( ) =xfg xcos , R, [ ]1,1− .
1. ( )( ) =xgf x , R, [ )∞,0 ; ( )( ) =xfg x , [ )∞,0 , [ )∞,0 . 2. ( )( ) =xgf xsin , [ )∞,0 , [ ]1,1− ; ( )( ) =xfg xsin , [ ]π,0 , [ ]1,0 .
6. ( )( ) =xgf54
2−− xx
e , R, ( )∞,0 ; ( )( ) =xfg 542−−
xxee , R, [ )∞− ,9 . 7. ( )( ) =xgf
1
12
+x, R, ( ]1,0 ; ( )( )
( )21
1
+=
xxfg , { }1\ −R , +
R
8. ( )( ) ( )xexgf −= 1sin , R, [ ]1,1− ; ( )( ) x
exfgsin1−= , R, [ ]1
1,1−
−− ee . 10. ( )( ) xxgf = , ( )∞,1 , ( )∞,1 ; ( )( ) xxfg −= , { }0\R , { }0\R .
9. ( )( )x
xgfcos
1= ,
∪
∪
π
ππππ2,
2
3
2
3,
22,0 , ( ] [ )∞∪−∞− ,11, ; ( )( )
xxfg
1cos= , { }0\R , [ ]1,1− .
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= Year 12 = Algebra of functions = Worksheet 4
1. Find a, b, c and d such that
( ) ( )( )( )dxcxbxaxxxxP +++=−−−= 617412 23 .
2. Refer to ( )xP in Q1. Find a, b, c and d such that
( )( )( )dxcxbxaxP +++=
−
2
1.
3. Find the equation of the relation formed after the relation
( ) ( ) 44222
=−++ yx undergoes the following transformations
in the order as shown. Reflection in the y-axis, 4 units down,
2 units left, vertical dilation by factor21 , horizontal dilation by
factor21 .
4. Refer to Q3. Now carry out the transformations in reverse
order. Find the equation of the relation formed.
5. Find the coordinates of the intersection of 11 ++= xy
and xy 2= .
6. The two functions in Q5 undergo the same transformations
as in Q3. Find the coordinates of the intersection of the
transformed functions.
7. If 21
=+x
x , find the value of (i)2
2 1
xx + , (ii)
xx
1+ .
8. Given 3
8loglog =− yx xy , find the positive value of
y
x
e
e
log
log.
9. Use the result in Q8 to solve 3
8loglog =− yx xy and
016 =− yx simultaneously.
10. Given ( )xx
fxf21
3 =
+ , show that ( ) ( )xfxf −=− .
Numerical, algebraic and worded answers. 2. 12=a , 2−=b ,6
1=c , 0=d or any permutation of b, c and d. 5.
3
8,
9
16 8. 3
6.
−−
3
2,
9
17 1. 12=a ,
2
3−=b ,
3
2=c ,
2
1=d or any permutation of b, c and d. 3. 122
=+ yx . 9. 64=x , 4=y .
7(i) 2 (ii) 2. 4. ( ) ( ) 12322
=+++ yx .
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= Year 12 = Algebra of functions = Worksheet 5
A. ( ) ( )xfxf =− B. ( ) ( )xfxf −=− C. ( ) ( )xkfkxf = , k is a real constant D. ( ) ( )xfkxf =+ for some real k
E. ( )( ) xxff = F. ( ) ( ) ( )yfxfyxf +=+ G. ( ) ( ) ( )yfxfyxf =+ H.( ) ( )
22
yfxfyxf
+=
+
I. ( ) ( ) ( )yfxfxyf = J. ( ) ( ) ( )yfxfxyf += K.( )( )yf
xf
y
xf =
L. ( ) ( )yfxf
y
xf −=
M. ( )
( )( )yf
xfyxf =−
From the above functional equations, select (one or more) those that are satisfied by the following solution functions. Show
working.
1. ( ) 1cos +
=
b
xaxf .
2. ( ) ( )xxf 2sin3 −−= .
3. ( ) cxf = , where c is a real constant.
4. ( ) xxf −= .
5. ( ) xxf =
6. ( ) 4xxf =
7. ( ) axexf =
8. ( ) cmxxf +=
9. ( ) 2log xxf e=
Numerical, algebraic and worded answers.
1. A
, D
2. B
, D
3. A
, D
4. C
, E
, F
, H
5. I
, K
6. A
, I,
K
7. G
, M
8. H
9. A
, J,
L
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= Year 12 fm = Matrices (Sim. linear equations) = Worksheet 1
1. Use inverse matrix method to solve the following system of
simultaneous linear equations.
13
73
=+
=−
yx
yx
2. Use inverse matrix method to solve the following system of
simultaneous linear equations.
2
1
4
3
5
2
132
=+
=−
yx
xy
3. Use CAS/calculator to solve for x and y.
=
− 9.7
1.5
7.29.3
3.173.0
y
x
4. Use CAS/calculator to solve for a, p and t.
=
−
−
21
52
103
43
54
65
21
32
43
21
72
51
t
p
a
5. Use CAS/calculator to solve the following system of
simultaneous linear equations.
345
5.12
22.0
132
−=+−
=+−
=−+
=−
yzx
wxy
zyw
zx
6. A system of simultaneous linear equations can be
represented by the following matrix equation.
=
− 4
1
3
2
y
x
ba
ba. The solutions to the equations are 2=x
and 1−=y . Find the values of a and b, and state the
simultaneous equations.
7. Write the system of simultaneous equations in matrix form.
5735
2753
=+
−=−
byax
byax
Find a and b when 3== yx .
8. Consider the simultaneous equations in Q7.
5735
2753
=+
−=−
byax
byax
Use matrix method to find a and b when ax = and by = .
9. Write the system of simultaneous equations in matrix form.
1433
922
1152
−=−+
−=−+−
=+−
rzqypx
rzqypx
rzqypx
Find p, q and r when 1=x , 1−=y and 2=z .
Numerical, algebraic and worded answers.
1. x = 1, y = -2
2. x = -10/9, y = 34/27
3. x = 3.4143, y = 2.0058
4. a = 0.9915, p = -0.4676, t = -0.0638
5. w = 2.5171, x = -0.2051, y = -0.6111, z = -0.4701
6. a = ½, b = 1, x + y = 1, ½ x – 3y = 4
7.
−=
−
57
27
35
53
y
x
ba
ba, a = 2, b = 3
8. a = ±√6, b = ±3
9.
−
−=
−
−−
−
14
9
11
33
22
52
z
y
x
rqp
rqp
rqp
, p = -1, q = 3, r =1
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= Year 12 mm = Matrices = Worksheet 2
1. Use inverse matrix method to solve the simultaneous
equations.
223
445
12
523
−=−+−
−=+−
−=−+−
−=−+−
dcba
dba
dca
dcba
2. Find a, b, c, d and e such that
−
=
−
−
−−−
5
5
3
1
1
1
2
1
1
1
110
0
01
01
00
bd
dceb
eca
edb
bca
3. Find a value for each of x, y and z such that the simultaneous
equations are satisfied.
255
032
4.02.0
=−+−
=++−
=−+
zyx
zyx
zyx
4. Consider
=
−
by
xa 5
12
3. Find a and b such that (i) no x
and y values, and (ii) infinite number of x and y values, will
satisfy the matrix equation.
5. Consider the simultaneous equations ( ) 1051 =+− yxm and
( ) mymx =−+ 33 . Find the values of m such that the equations
(i) have infinitely many solutions, and (ii) have no solutions.
6. Consider
32
15
6422
−=−+
=++
=+−−
zybx
zayx
zyx
Find the values of a and b such that the simultaneous equations
have infinitely many solutions.
7. For the matrix equation
−=
−+
−−
py
x
p
p
3
12
62
11,
find p such that the equation (i) has infinitely many solutions,
(ii) has no solutions and (iii) has a unique solution for each of x
and y.
8. Find a, b and c such that the matrix equation
−=
− 4
1
5
21
52
11
z
y
x
c
b
a
has a unique solution for each of x, y
and z.
Numerical, algebraic and worded answers.
1. a = -2, b = -1, c = 1, d = 2 4. (i) a = -3/2, b ≠ -10/3 (ii) a = -3/2, b = -10/3 7. (i) p = 1 (ii) p = -4 (iii) p ∈ R\{-4,1}
2. a = -1, b = 1, c = -1, d = 1, e = -1 5. (i) m = 6 (ii) m = -2 8. a, b, c ∈ R
3. x = 2/9, y = 4/9, z = 0, or other values 6. a ∈ R, b = 1
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= Year 12 mm = Matrices (Transition) = Worksheet 3
Questions 1, 2, 3 and 4 are related
1. In an isolated country town each household does the weekly
shopping at either Centre A or Centre B. A transition diagram is
shown below. Complete the equivalent transition matrix.
15%
A • •B 10%
this week
B
A
BA
__
__ next week
2. This week 65% of the households shop at Centre A. (i) What
is the percentage of the households expected to be shopping at
Centre B two weeks latter? (ii) In the long term, what is the
percentage of households expected to be shopping at Centre B?
3. What was the percentage of the households shopping at
Centre B two weeks ago?
4. When was Centre B first opened for business?
5. Suppose you wear a fresh pair of socks one day, there is a
60% chance you wear the same pair the next day. If you do not
wear a fresh pair one day, there is a 10% chance you do not
wear a fresh pair the next day. (i) Complete the transition
matrix below. (ii) Find the chance you wear a fresh pair on the
fifth day, given you do not wear a fresh pair on the first day.
(iii) Find the chance you wear a fresh pair in the long term.
_6.0
__
6. Suppose you wear a fresh pair of socks one day, there is a
60% chance you wear the same pair the next day. If you do not
wear a fresh pair one day, there is a 10% chance you do not
wear a fresh pair the next day. On the nth
day the chance you
wear a fresh pair differs from the chance in the long term by
less than 1%, given you do not wear a fresh pair on the first
day. Find n.
7. Suppose you wear a fresh pair of socks one day, there is a
p% chance you wear the same pair the next day. If you do not
wear a fresh pair one day, there is a 10% chance you do not
wear a fresh pair the next day. Find p if the chance you wear a
fresh pair in the long term is 80%.
8. Suppose you wear a fresh pair of socks one day, there is a
40% chance you wear the same pair the next day. If you do not
wear a fresh pair one day, there is a q% chance you do not wear
a fresh pair the next day. Find q such that there is a 30% chance
you wear a fresh pair on the third day, given you do not wear a
fresh pair on the first day.
Numerical, algebraic and worded answers.
1.
90.015.0
10.085.0 3. 15.6% 5. (i)
1.06.0
9.04.0 (ii) 56.25% (iii) 60% 7. 22.5
2. (i) 45.9% (ii) 60% 4. 3 weeks ago 6. 7≥n 8. 78.31
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= Year 12 fm = Matrices (Transition) = Worksheet 4
Questions 1, 2, 3 and 4 are related
1. The following transition matrix can be used to predict the
population in each of 4 towns A, B C and D in subsequent
years. In 2010 the populations of A, B C and D are 10000,
20000, 30000 and 40000 respectively.
DCBA
96.0001.002.0
097.001.002.0
001.098.001.0
04.002.0095.0
D
C
B
A
Find the steady state population of D.
2. Eventually what percentage of the population in each town
will remain in each town?
3. Predict the populations of the 4 towns in 2020. What were
the populations in 2009?
4. In which year did A have its first resident?
5. Complete the transition matrix corresponding to the
transition diagram.
DCBA
D
C
B
A
____
__25.0_
____
____
6. Refer to Q5.
The state
111
444
333
222
becomes
d
c
b
a
after 2 transitions.
Write down the matrix equation to represent the transitions.
Find a, b, c and d.
7. Refer to Q5 and Q6.
Find the steady state.
8. (i) Draw a transition diagram for the transition matrix.
(ii) Find the steady state, given the initial state is
d
c
b
a
.
DCBA
D
C
B
A
3.02.01.04.0
2.01.04.03.0
1.04.03.02.0
4.03.02.01.0
Numerical, algebraic and worded answers.
1. 20000 3. (i) 21763, 20417, 26765, 31055
(ii) 8145, 20013, 30554, 41289
2. 26.67%, 26.67%, 26.67%, 26.67% 4. 2006
25%
75% 60%
• A B •
15%
25%
10%
•C D •
90% 20% 80%
5.
80.010.000
20.090.025.00
0060.025.0
0015.075.0
6.
157,545,207,201
111
444
333
222
80.010.000
20.090.025.00
0060.025.0
0015.075.02
====
=
dcba
d
c
b
a
7.
370
740
0
0
8(ii) ( )( )( )( )
+++
+++
+++
+++
dcba
dcba
dcba
dcba
25.0
25.0
25.0
25.0
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= Year 12 mm = Matrices (Transformations) = Worksheet 5
1.
2.
3.
4.
5. The transformation T is defined by
−+
−=
6
1
30
021
y
x
y
xT . Find the image of the curve
( ) 2222
+−= xy .
6. Refer to Q5. Find 1−T .
7. ( )xf is transformed to ( )xf 232
11 −+ under T defined by
+
=
f
e
y
x
dc
ba
y
xT .
Find the values of a, b, c, d, e and f.
8. Refer to Q7. Find 1−T .
Numerical, algebraic and worded answers.
Write a matrix to transform
the small triangle to the
large triangle.
1.
21
31
0
0 2.
−
10
02 5. y = 24x2 6.
+
−=
−
2
2
0
02
31
1
y
x
y
xT 7. a = -2, b = 0, c = 0, d = 2, e = 3, f = -2 8.
+
−=
−
10
023
21
21
1
y
x
y
xT
Sketch the resultant shape
under the transformation T
defined by
−
=
6
4
20
032
y
x
y
xT
Write a matrix to transform the large
rectangle to the small rectangle.
Sketch the resultant shape
under the transformation T
defined by
−+
−=
1
1
30
02
y
x
y
xT
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= Year 12 = Geometry&Trigonometry = Worksheet 1
1. Find length AB in cm (round to 2 decimal places).
B
1 cm
A
2 cm 3 cm C
2. Find the length of the body diagonal of the rectangular solid.
12 cm
3 cm
4 cm
3. The following solid is a right pyramid with a square base and
height of 2 cm. Find θ °.
θ ° 2 cm
2 cm
4. Find length AB in cm.
B
1 cm
2 cm
A
5. The area enclosed by the triangle is 35 cm2. Find the shortest
distance between the two parallel lines.
5 cm
6. Find the length of the hypotenuse of triangle D.
1 cm
1 cm A
B D
C
7. Find the total surface area (round to nearest cm2) of the solid.
AB = 3 cm and it is perpendicular to the rectangular base.
A
B
1 cm
2 cm
8. Find the volume (in cm3) of the solid shown in Q7.
9. Find the total surface area (round to nearest cm2) of the solid.
PA = PB = PC = 2 cm and they are perpendicular to each
other. P
B
A
C
10. Find the volume (in cm3) of the solid shown in Q9.
11. Find the (i) total surface area and (ii) volume of the
hemispherical solid in terms of π.
2 cm
Numerical, algebraic and worded answers.
1. 5
.59
cm
2. 1
3 c
m
3. 4
5o
4. 5
cm
5. 1
4 c
m
6. 4
cm
7. 1
1 c
m2
8. 2
cm
3
9. 9
cm
2
10.
8/3
cm
3
11.
(i)
12π
cm
2
(ii
) 16π/3
cm
3
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= Year 12 = Geometry & Trigonometry = Worksheet 2
1. Find the area of the quadrilateral. Round the answer to the
nearest cm2.
Q
5 cm
P 3 cm
4 cm
3 cm R
2 cm
S
2. A quadrilateral similar to the one in Q1 has the diagonal PR
increased to 10 cm. Find the (i) perimeter and (ii) area (round
to the nearest cm2) of this larger quadrilateral.
3. The volume of the prism is 5.2 cm3. Find the volume of a
similar prism with the cross-sectional area 4
1of that of the one
shown.
Cross-
section Length
4. Refer to the prism in Q3. Find the volume of a similar prism
when the length measure is doubled.
5. Find the (i) total surface area and (ii) volume of the
triangular prism.
5 cm
7 cm
6 cm 10 cm
6. Refer to the prism in Q5. Find the volume of a similar prism
when all the length measures are doubled.
7. Refer to the prism in Q5. Find the total surface area of a
similar prism when all the length measures are doubled.
8. Refer to the prism in Q5. Find the total surface area of the
prism if the cross-sectional area is quadrupled.
9. The volume of the solid is 210 cm3. Find the height h of a
similar solid with twice the total surface area.
h = 5 cm
10. Refer to the solid in Q9. Find the volume of a similar solid
with twice the total surface area.
11. Refer to the solid in Q9. Find the height h of a similar solid
with twice the volume.
Numerical, algebraic and worded answers.
1. 9
cm
2
2. (
i) 3
2.5
cm
(
ii)
56
cm
2
3. 1
.3 c
m3
4. 1
0.4
cm
3
5. (
i) 2
09
.39
cm
2
(
ii)
147
.97 c
m3
6. 1
17
5.7
6 c
m3
7. 8
37
.58 c
m2
8. 4
77
.58 c
m2
9. 7
.1 c
m a
pp
rox.
10.
40
cm
3
11.
6.3
cm
app
rox
.
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= Year 12 = Geometry & Trigonometry = Worksheet 3
1. Find θ and φ in degrees.
12 cm 5 cm
θ φ
13 cm
2. Find θ and φ in degrees.
11 cm 5 cm
θ φ
13 cm
3. Find the obtuse angle φ in degrees.
12 cm 5 cm
20° φ
4. Evaluate θcos .
3
4
θ
5. Find the exact value of θtan . Diameter
11
12
θ
6. Find the value of θsin .
θ 28°
9 cm 12 cm
7. Find the altitude h of the triangle.
13 8 h
6
8. Find the area of the quadrilateral. The two diagonals are 12.5
cm and 22.8 cm long.
90°
9. Find the length of AB. A
65°
18 cm
B 20 cm
C
10. Refer to the triangle in Q9. Find the area of ∆ABC.
11. Find the total surface area of the composite solid consisting
of a hemisphere and a circular cone.
Diameter = 6 cm.
7 cm
Numerical, algebraic and worded answers.
1. θ
= 2
2.6
°, φ
= 6
7.4
° 2
. θ
= 2
2.1
°, φ
= 5
5.8
° 3
. 1
24
.8°
4. 0
.6
5. (
√23
)/11
6. 0
.626
7. 5
.56
8. 1
42
.5 c
m2
9. 1
9.2
cm
10.
15
6.4
cm
2
11.
66
cm
2
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= Year 12 = Geometry&Trigonometry = Worksheet 4
1. A cylindrical container (radius 12 cm, height 25 cm) is filled
with water to a depth of 23 cm. A spherical solid (radius 6 cm)
is placed in the water and sunk to the bottom. Find the volume
of spilled water.
2. Refer to Q1. What is the volume measure of spilled water if
all the given length measures are doubled?
3. A closed container in the shape of an inverted square-base
pyramid is filled with water to a depth of 5 cm. Find the ratio
of the volume of water to the volume of air in the container.
10 cm
5 cm
4. Refer to Q3. If the area of the base is 100cm2, find the total
surface area of the pyramid.
5. A house is 3 km west and 1 km north of train station A.
Find the location (distance in km, three figure bearing for
direction) of the house from train station A.
6. Refer to Q5. State the location of train station A from the
house.
7. Refer to Q5 and Q6. The same house is 2 km NE of train
station B. Find the location (distance in km, three figure
bearing for direction) of train station B from train station A.
8. Refer to Q5, Q6 and Q7. Find the shortest distance from the
house to the straight rails between station A and station B.
9. Find the area (in hectares) of the triangular region bounded
by straight lines joining the house, station A and station B.
10. The horizontal distance between X and Y is 120 m.
Estimate the average slope from X to Y.
X 0 m
50 m
Y
100 m
150 m
11. Refer to Q 10. Draw the profile of the vertical cross-section
of the hill between X and Y.
150
50
X Y
Numerical, algebraic and worded answers.
1. 0
2. 0
3. 1
: 7
4. 3
23
.6 c
m2
5. 2
km
300
°T
6. 2
km
120
°T
7. 2
.732
km
270
°T
8. 1
km
9. 1
36
.6 h
ecta
res
10.
0.7
5
Y
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= Year 12 = Geometry&Trigonometry = Worksheet 5
1. Point B is NE of O. Point A is N60°W of O. The angle of
elevation of B from O is 55°. The angle of depression of O
from A is 60°. Find (a) the angle of elevation of A from O, and
(b) the angle of depression of O from B.
B
A
200 m
100 m
2. Refer to Q1. Find the horizontal distance of (a) A from O,
and (b) B from O.
3. Refer to Q1. Find the straight line distance between A and B.
4. Refer to Q1. Find the straight line distance from (a) O to A,
and (b) O to B.
5. Refer to Q1. Find the measure of AOB∠ .
6. Refer to the contour map in Q1. Calculate the land area (in
m2 ) enclosed by AOB∆ .
7. Two solid spheres (radius 1 cm) are in contact when they are
placed inside a rectangular box such that each sphere touches
exactly 5 faces of the box. Find the volume (in cm3) of the box.
8. Two solid spheres (radius 1 cm) are in contact when they are
placed inside a rectangular box such that each sphere touches
exactly 4 faces of the box. Find the volume (in cm3) of the box.
9. Refer to Q8. Calculate the volume of air inside the box when
the spheres are in position.
10. Refer to Q8. If the radius of the 2 identical solid spheres
inside the box is greater than 1 cm, calculate the value of the
ratio, volume of air inside the box : total volume of the spheres.
11. Two solid spheres (radius 1 cm) are in contact when they
are placed inside a rectangular box such that each sphere
touches exactly 3 faces of the box. Find the volume (in cm3) of
the box.
Numerical, algebraic and worded answers.
1. (
a) 6
0°
(b)
55°
2. (
a) 5
7.7
m
(b)
70.0
m
3. 1
01
.6 m
4. (
a) 1
15
.5 m
(b
) 122
.1 m
5. 5
0.6
°
6. 1
95
1 m
2
7. 1
6 c
m3
8. 2
3.3
cm
3
9. 1
4.9
cm
3
10.
1.7
83
11.
31
.4 c
m3
O
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= Year 12 = Calculus = Worksheet 1
1. Differentiate ( )
2
2154
x−− with respect to x.
2. Given ( )( )4
215
2
xxf
−−= , find ( )xf ′ .
3. Find ( )[ ]πexdx
dlog23 − .
4. Find the derivative of ( ) xx 31312 ++ .
5. Evaluate ( )1−′f , given ( )
−= 3
1
8
2
1
xxf .
6. Find ( ) ( )
+
+22
2
2xa
xa
xd
d.
7. Differentiate ( )
3
232 x
e−
with respect to x.
8. Given 123 +×= xy , find dx
dy.
9. Find
−
3
12log
x
dx
de .
10. Given ( ) ( )xxf 3log3 10−= , find ( )xf ′ .
11. Differentiate 2log3
2+xee with respect to x.
Numerical, algebraic and worded answers.
1. 2
0(1
− 2
x)3
2. −
16 /
5(1
− 2
x)5
3. 3
log
eπ
4. (
15
/2)(
1 +
3x)3
/2
5. 1
/ 3
(1 −
x)4
/3
6. −
3 /
2(a
+ x
2)5
/2
7. (
−4/3
)e2
(3−x
)
8. (
3lo
ge2
)2x+
1
9. 2
/ (
2x
− 1
)
10.
−3 /
xlo
ge1
0
11.
3√
(x +
2)
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= Year 12 = Calculus = Worksheet 2
1. Find the derivative of ( )
3
12cos
x−.
2. Differentiate
°
πx
sin with respect to x.
3. Find ( )xf ′ , given ( )
−= 1
2tan
1 xxf
ππ
.
4. Find ( )[ ]1tan +kxdx
d , where k is a constant.
5. Find dx
dy, where
−−
= 1
3
2sin
2cos2
xxy .
6. Differentiate ( )1cos2 +x with respect to x.
7. Find ( )[ ]21cos +x
dx
d.
8. Find dx
dy, where 1tan += xy .
9. Find
ydy
d
tan
1.
10. Differentiate ( )1tan +x with respect to x.
11. Find
′4
πf , given ( )
xxf
cos
2= .
Numerical, algebraic and worded answers.
1. 2
/3 s
in[2
(1−x
)/3
]
2. 1
/180
co
s(x°
/π)
3. 1
/2 s
ec2(π
x/2
−1)
4. k
sec
2(k
x+
1)
5. −
sin
(x/2
)−2/3
co
s(2
x/3
−1)
6. −
2 s
in(x
+1
)co
s(x+
1)
7. −
2 (
x+1
)sin
(x+
1)2
8. s
ec2√
(x+
1)
/ 2
√(x
+1
)
9. −
1/s
in2y o
r −
sec2
y/t
an2y
10.
sec2
(x+
1)
/ 2
√tan
(x+
1)
11. 2
1/4
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= Year 12 = Calculus = Worksheet 3
1. Given ( )2
xxf −= , find ( )xf ′ .
2. Find dx
dyfor ( )( )12 −+= xxy .
3. Find 542 −− xx
dx
d.
4. Find dx
dy, where
−=
2tan
πxy and ( )π,0∈x .
5. Differentiate 1
1
+
−
x
x with respect to x.
6. Differentiate 1
1
+
+
x
x with respect to x.
7. Evaluate dx
dy for
−=
xy e
πlog at 1=x .
8. Evaluate ( )1−′ ef for ( ) ( )1
1log
+
+=
x
xxf e .
9. Find ( )xf ′ for ( ) 1log += xxf e .
10. Find ( )
−−1
21
xe
x
dx
d.
11. Differentiate bxe
−cos2 with respect to x.
Numerical, algebraic and worded answers.
1. −
1/2
fo
r x>
0, 1
/2 f
or
x<
0
2. −
(2x+
1)
for
−2<
x<
1,
2
x+
1 f
or
x<−2
or
x>1
3. 2
x−4
4. −
sec2
(x−
π/2
) fo
r 0
<x<
π/2
s
ec2(x
−π/2
) fo
r π/
2<
x<
π 5
. (
3−
x)/
[2(x
+1
)2√
(x−1
)]
6. −
1/[
2(x
+1
)3/2]
7. 1
8. 0
9. −
1/(
x+1
) fo
r x<
−1,
1
/(x+
1)
for
x>−1
10.
(1
−x)(
x−3
)e1
−x
11.
−e
2co
s√(x
−b
) sin√
(x−b
) /
√(x
−b
)
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= Year 12 = Calculus = Worksheet 5
1. The position x (in metres) of a particle moving in a straight
line is given by 1882+−= ttx at time t (in seconds). Find the
(i) average velocity, i.e. average rate of change of x with
respect to t over the interval [ ]5,4 and (ii) instantaneous
velocity, i.e. instantaneous rate of change of x with respect to t ,
at 5=t .
2. The graph shows the temperature T (in °C) of boiling water
decreases when the burner is turned off at 0=t . Estimate
(i) the average rate of change in temperature in the first 30
minutes and (ii) the rate of change in temperature at 30=t min.
T
100
50
0 30 t(min)
3. The volume V (in litres) of water remaining in a tank after
draining for t minutes is given by ( )2
60150000
−=
ttV . Find
the rate at which the water is draining after 30 min.
4. A 4-metre ladder leans against a vertical wall. If the bottom
of the ladder slides away from the wall at 0.3 ms-1
, find the
speed of the top of the ladder sliding down the wall when the
bottom of the ladder is 2 m from the wall.
5. Refer to the ladder in Q4. The sliding ladder makes an angle
θ with the vertical wall at time t. Find the rate of increase of θ
(in °s-1
) when the bottom of the ladder is 2 m from the wall.
6. A spherical balloon is inflated at 80 cm3s
-1. How fast is the
radius r (in cm) increasing when 20=r ?
7. Refer to the balloon in Q6. How fast is the surface area A (in
cm2) increasing when 20=r ?
8. Two cars move away from the intersection of two
perpendicular straight roads. Car A travels at 60 kmh-1
and car
B at 80 kmh-1
. If both cars are at the intersection initially, at
what rate are they moving apart after 6 min?
9. Refer to the two cars in Q8. At what rate are the two cars
moving apart after 6 min if initially car B is at the intersection
and car A is 3 km from the intersection?
10. Refer to the two cars in Q8. If both cars are at the
intersection initially, at what rate are they moving apart when
they are 2 km from each other?
11. The volume of a cube increases at 0.5 cm3s
-1. How fast does
the surface area increase when the length of its edge is 20 cm?
Numerical, algebraic and worded answers.
1. (
i) 9
ms-1
(i
i) 2
ms-1
2. (
i) −
2.5
°C
min
-1
(
ii)
−0
.9 °
Cm
in-1
3. 8
33
.3 L
min
-1
4. 0
.173
2 m
s-1
5. 4
.96
°s-1
6. 0
.016
cm
s-1
7. 8
cm
2s-1
8. 1
00
km
h-1
9. 9
8 k
mh
-1
10. 1
00
km
h-1
11. 0
.1 c
m2s-1
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= Year 12 = Calculus = Worksheet 6
1. Water flows out of a tank at a rate of ( ) ( )( )2512 −+= tttr
litres per minute at time 0≥t (min). Find the time when the
flow is the quickest.
2. Find the area of the largest rectangle that can fit inside the following triangle. 3 4
3. Find the area of the largest square that can fit inside the following triangle. 3 4
4. Find the radius of a 1-litre cylindrical can, which will minimise the cost of the metal to make it.
5. Find the point on the line 102 =+ yx that is closest to the
point ( )3,6 .
6. A right circular cylinder is placed inside a sphere of radius 5 cm. Find the largest possible volume of the cylinder.
7. A right circular cylinder is placed inside a sphere of radius 5 cm. Find the largest possible surface area of the cylinder.
8. At what production level will the average cost per television be lowest if the cost ($) of producing x televisions each week is
( ) 2001.02.0260 xxxC ++= ?
9. The volume (kL) of water in a pond at day t is given by
( )t
t
tV
e
=2
log2
, where 1≥t . Find the maximum volume of
water in the pond.
10. Find the area of the largest rectangle that has each of its sides touching a vertex of the given rectangle (4 cm by 3 cm). 3 4
11. In terms of p and q, where 0, >qp , find the area of the
smallest right-angle triangle with the point ( )qp, lying on its
hypotenuse. y
• ( )qp,
0 x
Numerical, algebraic and worded answers.
1. 1
min
2
. 3
squ
are
unit
s
3. 1
44
/49 s
quar
e u
nit
s
4. (
50
0/ π
)1/3 c
m
5. (
4, 2
)
6. 5
00
(√3
) π/9
cm
3
7. 2
5(1
+√5
)π c
m2
8. 5
10
9
. 1
/e k
L
10. 4
9/2
cm
2
11. 2
pq
squ
are
un
its
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= Year 12 = Calculus = Worksheet 7
1. Use ( ) ( ) ( )afhafhaf ′+≈+ to estimate ( )01.1f given
( ) 34 5xxxf −= .
2. Use ( ) ( ) ( )afhafhaf ′+≈+ to estimate ( )5.24f and
( )5.25f given ( ) xxf = .
3. Use ( ) ( ) ( )afhafhaf ′+≈+ to estimate ( )745.2f , given
( ) xxf elog= and 718.2≈e .
4. Given 0.3=x is an approximate solution to the equation
023 23 =−+− xxx , use ( ) ( ) ( )afhafhaf ′+≈+ to find a
better approximation of the solution.
5. Given 3 xy = , use xdx
dyy ∆≈∆ to find the % change in y
when x changes from 125 to 126.
6. Given xey = , use xdx
dyy ∆≈∆ to find the % change in y
when x increases by 0.01.
7. Use ‘left’ rectangles of unit width to estimate the area under
the graph of xey = between 0=x and 3=x .
8. Use ‘right’ rectangles of unit width to estimate the area
under the graph of xey = between 0=x and 3=x . Find the
average of the left and right-rectangles estimates.
9. Use ‘left’ rectangles of 6
π in width to estimate the area
under the graph of xy sin= between 0=x and 2
π=x .
10. Use ‘right’ rectangles of 6
π in width to estimate the area
under the graph of xy sin= between 0=x and 2
π=x . Find
the average of the left and right-rectangles estimates.
11. Use ‘right’ rectangles of 10 units in width to estimate the
area bounded by the curve xy 10log10
1= , the x-axis and the
line 20=x .
Numerical, algebraic and worded answers.
1. −
4.1
1
2. 4
.95
, 5.0
5
3. 1
.010
4. 2
.9
5. 0
.27
%
6. 1
%
7. 1
+e+
e2 s
q u
nit
s
8. e
+e
2+
e3 s
q u
nit
s
(
1+
2e+
2e
2+
e3)/
2 s
q u
nit
s
9. (
1+
√3) π
/12
sq
unit
s
10.
(3+
√3)π
/12
sq
un
its
(2+
√3)π
/12
sq
un
its
11. 2
.3 s
q u
nit
s
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= Year 12 = Calculus = Worksheet 8
Q1 to 6. Deduce the graph of gradient function ( )xf ′ from the given graph of function ( )xf .
1. y
•
0 x
•
2. y
0 x
3. y
•
0 x
4. y
0 x
5. y
0 x
6. y
0 x
Q7 to 12. Deduce the graph of the original function ( )xf from the given graph of ( )xf ′ or an anti-derivative function ( )xF .
7. ( )xf ′
•
0 x
•
8. ( )xF
0 x
9. ( )xf ′
0 x
10. ( )xF
0 x
11. ( )xf ′
0 x
12. ( )xF
0 x
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= Year 12 = Calculus = Worksheet 9
1. ( ) xxxF 2cos3sin −= is an anti-derivative of ( )xf on
6,0π
. Find the exact value of ( )dxxf∫6
0
π
.
2. ( ) xxxF elog= is an anti-derivative of ( )xf on [ ]1,2 −− .
Find the exact value of ( )dxxf∫−
−
1
2
.
3. ( ) xexxF −= 2 is an anti-derivative of ( )xf on [ ]1,2 −− . Find
the exact value of ( )dxxf∫−
−
2
1
.
4. Find the indefinite integral of ( ) 323
−− x .
5. Evaluate dxx∫
−
− −
1
31
2.
6. Find ( ) dxx∫−
−1
23 .
7. Given ( )
−=
−222
xx
eexf , find ( )dxxf∫ .
8. Evaluate dxx
dxx
∫∫
+
− ππ
002
sin2
sin .
9. Evaluate dxxxdx
d∫
+−
21
0
21
1.
10. Given ( ) ππ
=∫ dxxg
2
0
, evaluate ( ) dxxgx
∫
−
2
0
22
cos
π
π.
11. Evaluate dxx
x∫
+
+3
1
3
1
1.
Numerical, algebraic and worded answers.
1. 3
/2
2. l
og
e4
3. 4
e2 −
e
4. 1
/[4
(3−2
x)2
] +
c
5. l
og
e4
6. 2
√(3
x−2
)/3
+ c
7. 4
(ex/2+
e−x/2)
+ c
8. 4
9. 1
/3
10. 0
11. 2
0/3
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= Year 12 = Calculus = Worksheet 10
1. Given 2
log1log
x
x
x
x
dx
d ee −=
, evaluate dx
x
xe
e∫
−
1
2
log1.
2. Show that ( )( ) ( ) ( )xxxdx
d2tan2sec212sec =+ . Hence find
( ) ( )( )dxxx∫ 2tan2sec .
3. Find the derivative of ( )x2cos . Hence evaluate
( ) ( )dxxx∫6
0
2tan2sin
π
.
4. Find ( )xf , given ( ) ( )xxf 2sec2=′ and 18
=
πf .
5. Find
+ 2
1 xdx
d and hence dx
x
xt
∫+0
21
.
6. Find ( )xf ′ , given ( ) xxxf sin= . Hence find dxxx∫ cos .
7. Show that ( ) xenx
dx
d+ ( ) xenx 1++= , where n is an integer.
Hence find ( ) dxenxx
∫ + .
8. Find the area of the region bounded by the x-axis and the
curve ( )( )12 −+= xxy .
9. Find the area of the region bounded by the x-axis and the
curve ( )( )( )2123 −−−= xxxy .
10. Find the area of the region bounded by ( ) ( )2221 −+= xxy
and 16=y .
11. Find the area of the region bounded by ( )( )211 −+= xxy
and 15 += xy .
Numerical, algebraic and worded answers.
1. 1
/e
2. ½
sec
(2x)
+ c
3. −
√(s
in(2
x)t
an(2
x))
1
−1/√
2
4. ½
(ta
n(2
x)+
1)
5. x
/√(1
+x
2)
√
(1+
t2)
− 1
6. x
cosx
+ s
inx
x
sinx +
co
sx +
c
7. (
x+
n−1
)ex +
c
8. 4
.5 s
q u
nit
s
9. 0
.5 s
q u
nit
s
10. 6
2.5
sq u
nit
s
11. 2
53
/12
sq
unit
s
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= Year 12 = Calculus II = Worksheet 1
1. Find ( )( )22sec x
dx
d.
2. Given ( ) ( )12cos 2 += tectf , find ( )tf ′ .
3. Find the derivative of ( )1cot 22 +x with respect to x.
4. Is ( ) ( )xdx
dx
dx
d 22sectan = ? Why?
5. Find ( )( )21sin x
dx
d −.
6. Given ( )2
2
1 1cos
= −
ttf , find ( )tf ′ .
7. Find the derivative of
xar
1tan2 2 with respect to x.
8. Find the coordinates of the point of inflection in the graph of
( ) 115tan5 1 −+= − xy .
9. Find the x-coordinate of the point(s) of inflection in the
graph of xexy −= 2 .
10. Find the turning point(s) and/or point(s) of inflection in the
graph of2
102 ++
=xx
y .
11. Given dxx
y ∫ −+
−=
24
2, find
dx
dy.
Numerical, algebraic and worded answers.
1. 4
xta
n(x
2)s
ec2(x
2)
2. −
2co
t(√
(2t+
1))
cose
c2(√
(2t+
1))
/ √
(2t+
1)
3. −
4xc
ot(
x2+
1)c
ose
c2(x
2+
1)
4. Y
es, ta
n2x a
nd
sec
2x d
iffe
r by
a c
on
stan
t.
5. 2
x /
√(1
−x
4)
6. 4
cos−
1(1
/t2)
/ [t
√(t
4−
1)]
7. −
2ta
n(1
/√x)
/ [(
1+
x)√
x]
8. (
−1/5
, −
1)
9. 2
−√2
, 2+
√2
10.
T.P
.(−1
/2, 40
/7)
I.P
. (−
(3+
√21
)/6, 3
0/7
)
(−
(3−√
21
)/6
, 30/7
)
11.
−2/(
4+
x−
2)
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= Year 12 = Calculus II = Worksheet 2
1. Given 3=dr
dp and ( ) 11sin4 1 +−= − pq , find
dr
dq when
1=q .
2. Given θθ
2−=d
dr and θ
θ=
d
dR, find
dr
dR when
2
πθ = .
3. If 2
4
4
xdx
dy
+= and 2=
dt
dy, find
dx
dt in terms of x.
4. Evaluate dt
dx when 1log −= xy e , 1−=
dt
dy and 0=x
5. The volume of water in a container is given by 3
3
1hV π= m
3
when the depth is h m. Water is drained from the container at a
constant rate of 2
π m
3s
-1. Find the rate of decrease in the depth
of water when 2
1=h .
6. The profile of a skate ramp is given by
−= −
15
cos31 x
y .
Find dt
dx when 2−=
dt
dy at 2=x . (Length in m, time in s)
7. Given ( )
15
1 2
2
=−−
yx
, find dx
dy at 6=x .
8. Refer to Q7. Find dt
dy at 6=x when 2−=
dt
dx.
9. Given ( ) 113 2 ++=+ yxyx , find dy
dx in terms of x and y.
10. Use calculus to find the coordinates of the points where the
graph of ( )1244 22 −−=+ yxyx has a vertical or horizontal
tangent line.
11. Refer to Q10. Find the exact coordinates of the points
where the graph of ( )1244 22 −−=+ yxyx has a gradient of 1.
Numerical, algebraic and worded answers.
1. 1
2
2. −
1/2
3. 2
/ (
4+
x2)
4. 1
5. 1
ms−
1
6. 8
/3 m
s−1
7. ±
1/2
8. ±
1
9. (
6xy+
6y−
1)
/ (1
−3y2
)
10.
Ho
ri:
(1,0
), (
1,−
4)
Ver
t: (
0,−
2),
(2
,−2
)
11.
(1+
1/√
5, −2
−4/√
5)
(1−1
/√5
, −2
+4
/√5
)
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= Year 12 = Calculus II = Worksheet 3
1. Find dx
x∫
− 223
2.
2. Find dx
xx∫
−−
22
1.
3. Evaluate dtt
∫
+
1
3/1
231
2.
4. Find dxxx
∫ ++ 122
12
.
5. Evaluate dxx
∫−
−
2
1
2.
6. Find dxx
x
∫
+
−
1
1
7. Find
+
−
2
1
1
1tan
xdx
d. Hence find dx
xx
xx
∫
++
+
22
224
35
.
8. Evaluate dxxxx
xx
∫
−+−
+−4
3
23
2
8126
34 in exact form.
9. Find dxxx
xx
∫
++
+
4224
3
.
10. Find dxx
x
∫
tan
sec2
.
11. Evaluate dxxx
e
e e
∫
2
log
1.
Numerical, algebraic and worded answers.
1. 2
/3 s
in−
1(x
/√2
) +
c
2. c
os−
1(1
−x)
+ c
3. π
/(3
√3
)
4. t
an−
1(2
x+
1)
+ c
5. l
og
e4
6. l
og
e(1+
x)2
− x
+ c
7. −
2x /
(x
4+
2x
2+
2)
x
2/2
+ t
an−
1[1
/(1
+x
2)]
+ c
8. l
og
e2 −
3/8
9. ¼
log
e(x
4+
2x
2+
4)
+ c
10.
log
e|ta
nx| +
c
11.
log
e2
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= Year 12 = Calculus II = Worksheet 4
1. Find dxxx∫
− 2
1 .
2. Anti-differentiate ( ) ( )xx 2cos2sin 32 .
3. Given ( )2
1 x
xxf
−=′ , find ( )xf .
4. Given ( )
=
2tan
2sec
24 xxxg , find ( )dxxg∫ .
5. Find dxx
x∫
−1
2
.
6. Anti-differentiate ( ) ( )xx sin12sin − .
7. Find ( )dxkx∫cot .
8. Evaluate ( )dxx∫ −π2
0
cos1 .
9. Given ( ) ( )nxxf 2cos=′ , find ( )xf .
10. Find ( )dxx∫4
sin .
11. Evaluate dxxx
∫
π
0
22
2cos
2sin .
Numerical, algebraic and worded answers.
1. −
(1
−x2)3
/2 /3
+ c
2. s
in3(2
x)
/6 −
sin
5(2
x)
/10 +
c
3. −
√(1
−x2)
+ c
4. 2
tan
3(x
/2)
/3 +
2ta
n5(x
/2)
/5 +
c
5. −
2√
(1−
x)
+ 4
(1−
x)3
/2/3
−
2(1
−x)5
/2/5
+ c
6. 4
(1−
sinx)5
/2/5
− 4
(1−
sinx)3
/2/3
+ c
7. l
og
e|si
n(k
x)|
/ k
+ c
8. 4
√2
9. x
/2 +
sin
(2n
x)
/(4n
) +
c
10. 3
x/8 −
sin
(2x)
/4 +
sin
(4x)
/32
+ c
11.
π /8
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= Year 12 = Calculus II = Worksheet 5
1. Anti-differentiate 2
23
2
xx
x
−−.
2. Find dxxx
x∫ +−
−2
23
1.
3. Anti-differentiate 2
2
1
x−.
4. Find dxx
∫ + 22
1.
5. Evaluate ( )
dxx
∫− +
1
1
22
1. 6. Find
( )dx
x
x∫ + 2
2.
7. Given ( )34
32
2
+−
+=′
xx
xxxf , find ( )xf .
8. Given ( )34
32
2
++
+=′
xx
xxxf , find ( )xf .
9. Given ( )( )2
11
1
−−=
xxf , find ( )dxxf∫ .
10. Given ( )( )2
11
1
−+=
xxf , find ( )dxxf∫ .
11. Find ( )
dxx
x∫ −+ 2
2
11.
Numerical, algebraic and worded answers.
1. −
(3/4
)lo
ge|3
−2x−
x2| +
c
2. (
1/2
)log
e|3
−2
x+x
2| +
c
3. (
√2 /
4)l
og
e|(√
2+
x)/
(√2
−x)|
+ c
4. (
1/√
2)t
an−
1(x
/√2
) +
c
5. 2
/3
6. l
og
e|2+
x| +
2/(
2+
x)
+ c
7. x
+ l
og
e(|x
−3
|9/|x−
1|2
) +
c
8. x
− l
og
e|x+
1| +
c
9. (
1/2
)log
e|x/
(2−
x)|
+ c
10.
tan
−1(x
−1
) +
c
11.
x +
lo
ge|x
2 −
2x+
2| +
c
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= Year 12 = Calculus II = Worksheet 6
1. The graph of ( )dxxfy ∫= is shown below. Sketch ( )xfy = .
y
0 x
2. The graph of ( )xfy = is shown below. Sketch ( )dxxfy ∫= .
y
0 x
3. The graph of ( )dxxfy ∫= is shown below. Sketch ( )xfy = .
y
0 x
4. The graph of ( )xfy = is shown below. Sketch ( )dxxfy ∫= .
y
0 x
5. Evaluate dxx
∫−
−
1
1
1
2sin without using graphics calculator.
6. Evaluate dxx∫−
−
−
1
1
1
2cos
πwithout using graphics calculator.
7. Evaluate ( ) ( ) dxxx∫−
+++
5
5
22 12sin10
312cos3.0 without
using graphics calculator.
8. Evaluate dxxx
∫−
−
1
1
22
3tan
2
3
3sec
2
3without using
graphics calculator.
9. Evaluate dxx∫ −
5.1
0
23
3 without using graphics calculator.
10. Evaluate ( )( )dxxixi
∫−
+−
2
2 22
2.
11. Evaluate dxx
x∫
−
π
π
sin.
Numerical, algebraic and worded answers.
5. 0
6. 0
7. 3
8. 3
9. π
/√3
10.
π/√2
11.
≈3.7
04
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= Year 12 = Calculus II = Worksheet 7
1. Find the area of the region bounded by the curvex
xy
tan= ,
the x-axis, 2
π−=x and
2
π=x .
2. Find the exact area of the region bounded by the curve
( )1log −= xye
, the y-axis, 0=y and 1=y .
3. Find the exact area of the region bounded by the curve
2cot
xy = , the x-axis,
2
π=x and π=x .
4. Find the exact area of the region bounded by the curve
( )1cos 1−=
−xy , the y-axis, 0=y and π=y .
5. Find the area of the region bounded by the curves
xy1
sin3
2 −= and
2
3sin
xy = .
6. Find the exact area of the region bounded by the curves
212 xy −−= and 212 xy −= .
7. Find the exact volume of the 3D shape formed by rotating
the curve 212 xy −= about the x-axis.
8. Find the exact volume of the 3D shape formed by rotating
the curves 212 xy −−= and 212 xy −= about the y-axis for
[ ]1,0∈x .
9. Given the curve2
1
1
x
y
−= , where
2
30 ≤≤ x , find the
exact volume of the 3D shape formed by rotating it about the y-
axis.
10. Given the curve ( )21−= xy , where 30 ≤≤ x , find the exact
volume of the 3D shape formed by rotating it about the y-axis.
11. Find the volume of the 3D shape formed by rotating the
curve xxy sin= about the x-axis for [ ]2,0∈x .
Numerical, algebraic and worded answers.
1. ≈
2.1
78
2. e
3. l
og
e2
4. π
5. ≈
0.2
39
6. 2
π
7. 1
6π
/3
8. 8
π /
3
9. π
/2
10. 4
5π
/2
11.
≈7
.29
6
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= Year 12 = Calculus II = Worksheet 8
1. Verify that 2
12
2++= xey
x is a solution to
022 =+− xydx
dy.
2. Find the value(s) of constant k such that kxkxy cossin −= is
a solution to 02
2
=+ ydx
yd.
3. Find the constants a and b such that xxey = is a solution to
02
2
=++ bydx
dya
dx
yd.
4. Verify that ( )1−+=axax
ea
bAey satisfies bay
dx
dy+= ,
where a and b are positive constants.
5. Verify that ( )bxey ax sin= satisfies the equation
( ) 02 22
2
2
=++− ybadx
dya
dx
yd.
6. Verify that ( ) ( )xxy ee logcoslogsin += satisfies the equation
02
22
=++ ydx
dyx
dx
ydx .
7. Solve 291
3
xdx
dy
+= , given 0
3
1=
−y .
8. Solve 0log =− tdt
dxt e , given 1=x when 2log =te .
9. Solve ttdt
xdcos32
2
2
+= , given 2=x and 3=dt
dx when 0=t .
10. Use technology to evaluate y when 1=x , given
( )2sin x
dx
dy= where 2=y when 0=x .
11. Use technology to evaluate V when 2=t , given
( ) 11log ++= tdt
dVe where 5=V when 1=t .
Numerical, algebraic and worded answers.
2. ±
1
3. a
= −
2, b
= 1
7. t
an−
1(3
x)
+ π
/4
8. (
log
et)2
/2
9. t
4/1
2 +
3t
− 3
cost
+ 5
10.
≈2
.31
11.
≈6
.91
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= Year 12 = Calculus II = Worksheet 9
1. Find the general solution to ( ) 11 =−dx
dyy .
2. Find the general solution to ydx
dy−=1 .
3. Solve 2
21 ydx
dy−= for y, given
2
1=y when
22
π=x .
4. Find the solution to 2
11
ydx
dy+= , where 3−=y when
2=x .
5. Find the general solution to 0222 =−+− yy
dx
dy.
6. Find the general solution to ( ) 1122 =+−
dx
dyyy .
7. Find the general solution to yydx
dyelog= .
8. Given y
y
dx
dy −=
1 and 1=y when 0=x . Find x when
0=y .
9. Use Euler’s method with step size of 0.1 to find the
approximate solution to yxdx
dy+= at 3.0=x , given ( ) 10 =y .
10. Use Euler’s method with step size of 0.1 to find the
approximate solution to 22
yxdx
dy+= at 2.0=x if ( ) 10 =y .
11. Use Euler’s method with step size of 0.1 to find the
approximate solution to x
edx
dyx = at 2.1=x , given ( ) 21 =y .
Numerical, algebraic and worded answers.
1. y
= 1
± √
(2x+
c)
2. y
= 1
± k
ex
3. y
= 1
/√2 s
in(x
√2),
x
∈[−
π/(2
√2
), π
/(2
√2)]
4. y
= −
√(x
2 −
1)
5. y
= t
an(x
+c)
+ 1
6. y
= (
3x+
c)1
/3 +
1
7. e
^(±
ex −
c)
8. −
4/3
9. ≈
1.3
62
10.
≈1.2
22
11.
≈2.5
45
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= Year 12 = Calculus II = Worksheet 10
1. The rate of change of V with respect to t is inversely proportional to 1+t . Initially 100=V and 80=V when 5=t .
Set up a differential equation for V, solve it to find V when 9=t .
2. A population grows at a rate proportional to its size. If the initial population is 10000 and it doubles every unit of time. Find the population after (i) 2 (ii) 3 (iii) 2.73 units of time.
3. The rate of decay of a radioactive substance is directly proportional to the remaining mass m of the substance. The time taken for a half of the substance remaining in the sample is 3.2 hours. Find the proportion of the substance remaining in the sample after another two hours.
4. The gradient of the tangent to a curve ( )xfy = is partly
proportional to x and partly to x
1. The curve passes through
the origin, ( )2,1 and ( )11,4 . Find y when 9=x .
5. The surface temperature T of an object changes in time t at a rate proportional to the difference between the temperature of the object and the temperature To of the surrounding medium.
If the temperature of the object drops by 10°C in 5 minutes. Find the drop in temperature in the next 5 minutes, given the
surrounding temperature is constant 20°C and the initial
temperature is 80°C.
6. The acceleration a of a particle moving in a straight line is directly proportional to the square of its speed v. It has an
initial speed of 80 ms−1. Five seconds later the speed is 56 ms−1.
Find the time when the speed is 10 ms−1.
7. A thermometer is taken from a house at 21°C to the outside.
One minute later it reads 27°C, another minute later it reads
30°C. Find the temperature outside the house.
8. A person borrows $10000 at 10.95% interest compounded daily. Set up a differential equation for the amount owing at time t days. Find the amount $A owing a year later.
9. A tank contains 2000 L of salt solution with a concentration of 0.3 kg of salt per litre. Pure water runs into the tank at 50 L per minute and the well mixed solution runs out at the same rate. Find the amount of salt in the tank after 5 minutes.
10. Refer to Q9. Instead of pure water, a solution with a concentration of 0.2 kg of salt per litre runs into the tank. Find the amount of salt in the tank after 5 minutes. Find the concentration of salt in the tank eventually.
11. Refer to Q9. Instead of running out at the same rate, the well mixed solution runs out at 40 L per minute. Use Euler’s method (step size of 1 minute) to find the approximate amount of salt in the tank after 5 minutes.
Numerical, algebraic and worded answers.
1. d
V/d
t =
k/t
, ≈
74.3
2
. (
i) 4
000
0 (
ii)
800
00
(
iii)
≈ 6
63
46
3. 0
.324
2
4. 4
5
5. 8
.3°C
6
. 8
1.7
s
7. 3
3°C
8
. d
A/d
t =
(lo
ge1
.00
03
)A
$1
115
7.0
2
9. 5
29
.5 k
g
10. 5
76
.5 k
g, 0
.2 k
g p
er l
itre
1
1. 5
42
.9 k
g
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= Year 12 = Complex numbers = Worksheet 1
1. Express 1− in terms of i.
2. Express i43 + in yix + form.
3. Express i125 + in yix + form.
4. Express i in yix + form.
5. Express i− in yix + form.
6. Express i158 − in yix + form.
7. Express ( )312
1
2
1ii +
− in polar form.
8. Express 1
62
−
−
i
i in polar form.
9. Express
−−
2
32
πθicis in polar form.
10. Simplify
n
cis
12
6
1
62
π, where n is an integer.
11. Simplify (i)
+
−
33
2
62
ππciscis and
(ii)
−
−
33
2
62
ππciscis .
Numerical, algebraic and worded answers.
1. i
or
−i
2. 2
+i
or
−2−
i
3. 3
+2i
or
−3−2
i
4. 1
/√2
+ 1
/√2
i
o
r −1
/√2
−1/√
2 i
5. 1
/√2
− 1
/√2
i
o
r −1
/√2
+1/√
2 i
6. 5
/√2
−3/√
2 i
o
r −5
/√2
+3/√
2 i
7. 2
cis
( π/1
2)
8. 2
cis
( π/1
2)
9. 2
cis
θ 1
0. 4
n
11.
(i)
4/√
3
(ii)
(4
/√3
)cis
(−π/
3)
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= Year 12 = Complex numbers = Worksheet 2
You need a ruler and/or a protractor to do Q1 to 4. Both axes have the same scale.
1. z is shown in the argand diagram. Plot iz , z− and iz− .
Im(z)
Re(z)
• z
2. θcisz 2= is shown in the argand diagram. Plot z , 1−z and
1+z . Im(z)
Re(z)
• z
3. 1z and 2z are shown in the argand diagram.
Plot 21 zz + , 21 zz − and 212 zz + .
Im(z)
• 2z Re(z)
• 1z
4. αcisz 21 = and βcisz =2 are shown in the argand diagram.
Plot 21zz , 12 / zz and 3
2z .
Im(z)
• 2z Re(z)
• 1z
5. Simplify
5
2
1
2
1
− i .
6. Simplify ( )( )6
3
1
3
i
i
+
−.
7. Find the cube roots of 8− in yix + form.
8. Find z such that 82
3
−=z . Express answers in yix + form.
9. Simplify
22
22
−−
+ zzzz.
10. Simplify (i) ( )( )zizizz −+ and (ii) ( )( )zizizz ++ .
11. Given αcisz 21 = and βcisz =2 , find 2
21 zz − in terms of
α and β.
Numerical, algebraic and worded answers.
5. 1
/√2
−(1
/√2
)i
6. 1
7. 1
+i√
3,
−2+
0i,
1−i
√3
8. 4
+0i,
−2
+i2
√3,
−2−i
2√3
9. |z
|2
10.
(i)
2|z
|2
(ii)
2|z
|2i
11. 5
− 4
cos(
α−β
)
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= Year 12 = Complex numbers = Worksheet 3
1. Plot the sixth roots of −1 in the argand diagram below.
Im(z)
Re(z)
2. Plot the cube roots of i and i− in the argand diagram below.
Im(z)
Re(z)
3. Show that 36 1 i±=− .
4. Solve 015 =+z . Express the solutions in polar form.
5. Factorise iz −2 .
6. Factorise iz +2 .
7. Factorise ( ) izziz −−++− 33 23 .
8. Factorise 13 +z .
9. Factorise 164 −z .
10. Factorise 44 +z .
11. Factorise 646 −z .
Numerical, algebraic and worded answers.
3. (
−1)^
(1/6
)
=
((−
1)^
(1/2
))^(1
/3)
=
(±
i)^(1
/3)
4. c
is( π
/5),
cis
(3π/
5),
cis
(π),
c
is(−
π/5
), c
is(−
3π/
5)
5. (
z−1
/√2−
1/√
2 i
) (z
+1
/√2+
1/√
2 i
)
6. (
z+
1/√
2−
1/√
2 i
) (z
−1/√
2+
1/√
2 i
)
7. (
z−
i)(z
+i)
(z−3
−i)
8. (
z+
1)(
z−
1/2
−i√
3 /
2)(
z−1
/2+
i√3
/2
)
9. (
z−2
)(z+
2)(
z−2
i)(z
+2i)
10.
(z−1
−i)(
z−1
+i)
(z+
1−
i)(z
+1
+i)
11.
(z−2
)(z+
2)(
z+
1−
i√3
)×
(z+
1+
i√3
) )(
z−1
−i√3
) )(
z−1
+i√
3)
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= Year 12 = Complex numbers = Worksheet 4
1. Find 3 1 i+ . Write your answers in polar form.
2. Change your answers in Q1 to exact yix + form.
3. Find 8 1− . Write your answers in polar form.
4. Change 8
πcis to exact yix + form.
5. Use the conjugate root theorem and the fundamental theorem
of algebra to explain why dczbzaz +++ 23 has at least one
real root for Rdcba ∈,,, .
6. Show that iz +−1 is a factor of 862 23 +−+ zzz . Find the
other factors.
7. Find the roots of zzz ++ 23 .
8. Solve 06432 23 =−+− zzz .
9. Given iz −−1 is a factor of ( ) qpzzzP ++= 3 , find p and
q R∈ . Hence solve ( ) 0=zP .
10. Consider ibaz += , find a and b such that iz =2 . Hence
solve 14 −=z .
Numerical, algebraic and worded answers.
1. 21/6cis(π/12), 21/6cis(3π/4), 21/6cis(−7π/12)
2. 2−4/3(1+√3) − 2−4/3(1−√3)i, −2−1/3 + i2−1/3, 2−4/3(1−√3) − 2−4/3(1+√3)i
3. cis(−7π/8), cis(−5π/8), cis(−3π/8), cis(−π/8), cis(π/8), cis(3π/8), cis(5π/8), cis(7π/8)
4. √(2+√2) /2 + i /√(4+2√2)
5. Cubic polynomial has 3 roots (FTofA). For real coefficients, either all roots are real, or a pair of complex conjugate roots + 1 real root (CRT).
6. z−1−i, z+4
7. 0, −1/2−i√3 /2, −1/2+i√3 /2
8. z = 3/2, i√2, −i√2
9. p = −2, q = 4, z = −2, 1+i, 1−i
10. a = ±1/√2 and b = ±1/√2, z = 1/√2 + 1/√2 i, −1/√2 − 1/√2 i, 1/√2 − 1/√2 i, −1/√2 + 1/√2 i
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= Year 12 = Complex numbers = Worksheet 5
Sketch in the complex plane the following subsets of C, the set of complex numbers.
1. ( )
<−+3
21:
πizArgz
2. { }izzz −+≥ 2:
3. ( )( )
=−
−
2
1
1Re
1Im:
z
zz
4. { }2: =++ zizz
5. { }2: =−+ zizz
6. { }1)1Re(: +=− zzz
7. { }1)1Re(: +>− zzz
8. { }12)1Re(: +=− zzz
9. { }1)1Re(2: +=− zzz
10. { }zzz arg2: =
11. { }21:3
2
3: <−≤∩
≤< izzArgzzππ
12. { } { }1:)2Im(: ≤+∪=+ izzzzz
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= Year 12 = Vectors = Worksheet 1
1. Vectors a and b are as shown. Construct vectors b + a and
a −−−− b.
a
b
2. Refer to Q1. Describe a vector that is linearly independent of
a and b.
3. Find a vector c that is linearly dependent on vectors p, q and
r.
4. Vector r has a magnitude of 10 and makes angles of 30°, 45°
and 60° respectively with i, j and k. Express r in terms of i, j
and k.
5. Find the magnitude of p = 3i − 4j + 5k, and the exact
values of αcos , βcos and γcos , where α, β and γ are the
angles that p makes with the x, y and z axes respectively.
6. Find the scalar product of the two vectors shown below.
120° 5
5
7. Find the values of c and d so that 2i + 2j – ck is
perpendicular to i + dj + 6k.
8. Find the projection of i + k onto −i + j – 2k, i.e. the scalar
resolute of i + k in the direction of −i + j – 2k.
9. Resolve 10i + 7j − 11k into two components, one is parallel
to 5k and the other perpendicular to it.
10. Resolve 10i + 7j − 11k into two components, one is parallel
to 4i + 2j − 3k and the other perpendicular to it.
11. a, b and c are orthogonal vectors. Express the cosine of the
angle between a + b + c and c in terms of a, b and c.
Numerical, algebraic and worded answers.
2. E
.g. a
vec
tor
that
po
ints
o
ut
of
(or
into
) th
e pag
e.
3. E
.g. c
= 2
p −
q +
0.2
s
4. 5
√3i
+5√2
j +
5k
5. 5
√2, 3
√2 /
10
, −4
√2 /
10
, √2
/2
6. 1
2.5
7. c
∈ R
, d
= 3
c −
1
8. −
√6 /
2
9. −
11
k, 1
0i
+ 7
j
10. 1
2i
+ 6
j −
9k
, −2
i +
j −
2k
11.
|c| /
√(|a
|2 +
|b|2
+ |c
|2)
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= Year 12 = Vectors = Worksheet 2
1. Use vectors to prove that the diagonals of a rhombus are
perpendicular.
2. Use vectors to prove the cosine rule.
3. Use vectors to prove that the angle subtended by the
diameter of a semi-circle is a right angle.
4. If two linearly independent vectors are of equal magnitude,
prove that their sum is perpendicular to their difference.
5. P, Q and R are points of trisection of sides AB, AC and BC
respectively. Use vectors to show that BPQR is a
parallelogram.
A
• Q
P •
C
• R
B
6. ABCD is a parallelogram. M is the midpoint of AB. Use
vectors to show that DM and AC trisect each other.
B C
M
A D
7. Use vectors to show that any two medians of a triangle
trisect each other.
8. Use vectors to show that any two body diagonals of a
parallelepiped bisect each other.
9. Given 10=OA a, 6=OB (a + b) and 15=OC b. Show that
points A, B and C are collinear.
A
B C
O
10. Use vectors to prove that the midpoints of the sides of a
quadrilateral are the vertices of a parallelogram.
•
•
•
•
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= Year 12 = Vectors = Worksheet 3
1. Find the value of γβα 222coscoscos ++ , where α, β and γ
are the angles that a vector makes with the x, y and z axes
respectively.
2. Find a vector perpendicular to 3i - 2j + 4k.
3. Calculate the angle between vector 3i − 4j − 35 k and the x-
y plane.
4. Use a vector method to find the shortest distance from the
point ( )5,2,3 − to the line that passes through ( )1,2,3 and ( )2,0,1 .
5. P and Q are points with position vectors p and q
respectively. If |p| = 4, |q| = 7 and p•q = 20, find PQ .
6. Given points ( )4,3 −A , ( )0,7B and M between A and B, find
the coordinates of M such that MBAM 3= .
7. Given points ( )4,3 −A , ( )0,7B and M on the extension of AB ,
find the coordinates of M such that BMAM 3= .
8. Find a vector perpendicular to i + j and j − k.
9. Find a vector p such that i + j, j − k and the vector p are
linearly dependent.
10. If a and b are linearly independent and d is perpendicular to
both a and b, find a vector c in terms of a and b such that c and
d are also perpendicular.
11. Show that vectors 2i − 3j + 5k, i − j + 2k, i + 2j + k and
i + 7j are linearly dependent.
Numerical, algebraic and worded answers.
1. 1
2. E
.g. 2i
+ j
− k
3. 6
0°
4. 4
5. 5
6. (
6, −
1)
7. (
9, 2
)
8. E
.g. i
− j
− k
9. E
.g. i
+ 2
j −
k
10.
E.g
. c
= 2
a −
b
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= Year 12 = Vectors = Worksheet 4
1. Determine the cartesian equation corresponding to the vector
equation a2
θ= i +
−1
3
θj.
2. In terms of θ, find a vector equation of the locus of point P,
which is the circle as shown. y
A P
θ
0 2 x
3. Determine the cartesian equation corresponding to the vector
equation b ( )1−= p i + ( )21 p− j.
4. In terms of θ, find a vector equation of the locus of point P,
which is the ellipse as shown. y
A 1 P
θ
0 2 x
5. Refer to Q4. If 12
−=t
θ at time t, find a vector equation of
the locus of point P in terms of t.
6. Derive the cartesian equation of the path given by
r1
1)(
+=
tt i − ( )1−t j.
7. Derive the cartesian equation of the path given by
r
+=
ttt
1)( i
−+
tt
1j.
8. Derive the cartesian equation of the path given by
r
+=
2
2 1)(
ttt i
−+
2
2 1
tt j.
9. What is different between the particle motions in Q7 and
Q8?
10. Derive the cartesian equation of the path given by
r ( )tt 2sec2)( = i ( )t2tan− j.
11. Derive the cartesian equation of the path given by
r2
1
2)(
t
tt
+= i
21
3
t+− j.
Numerical, algebraic and worded answers.
1. y
= 2
/3 x
− 1
2. r
= 2
cosθ
i +
2si
nθ j
3. y
= −
x2 +
2x
4. r
= 2
cosθ
i +
sinθ j
5. r
= 2
cos(
t/2 −
1)
i
+
sin
(t/2
−1
) j
6. y
= 2
− 1
/x
7. x
2 −
y2 =
4
8. x
2 −
y2 =
4
9. S
ame
pat
h b
ut
d
iffe
ren
t vel
oci
ty
10.
x2/4
− y
2 =
1
11.
x2/4
+ y
2/9
= 1
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= Year 12 = Vectors = Worksheet 5
1. Find dx
d[ 2x i x− j].
2. Given F x= i 2y− j, where x and y are functions of t, find
dF/dt.
3. Find ∫[2x i x− j] dx.
4. The position of a particle is given by
r ( ) ( )tt 2cos3= i ( )t2sin4− j at time 0≥t . Find the
magnitude and direction of its velocity at 8
3π=t .
5. Refer to the particle in Q4. Find its acceleration and show
that it is towards the centre of the path.
6. The position of a particle is given by
r ( ) tt tan= i t2sec+ j, where 2
0π
<≤ t . Find the magnitude
and direction of its velocity at 4
π=t .
7. The position of a particle is given by
r ( )t ( )ntacos= i + ( )nta sin j + bt k, where 0, >ba .
Describe its motion.
8. The acceleration of a particle moving in a plane is given by
a = −5j. Initially it is at r = 14i, and has a velocity of 7i − 10j.
Find the cartesian equation of its path including domain.
9. The position vectors of particle A and B are rA5
t= i 5
t
e+ j
and rB t= i ( )telog+ j respectively, where 0>t . Find the time
when the two particles are closest.
10. Refer to Q9. Find the closest approach of the two particles.
11. Refer to Q9. Find the closest distance between the paths of
the two particles.
Numerical, algebraic and worded answers.
1. 2
x i
−1
/2√x
j
2. d
x/d
t i
− 2
y dy/d
t j
3. x
3/3
i −
2x
3/2/3
j +
c
4. 5
√2;
−0.6
i +
0.8
j
5. a
= −
4r
6. 2
√5;
√5 /
5 i
+ 2
√5 /
5 j
7. U
pw
ard
cir
cula
r h
elix
,
r
adiu
s a
, p
erio
d 2
π/n
,
a
scen
din
g s
pee
d b
8. y
= −
5/9
8 x
2 +
10,
[
14
, ∞
)
9. t
≈ 1
.12
7
10. 1
.448
11.
√2
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= Year 12 = Slope (direction) fields = Worksheet 1
The slope fields for certain first order differential equations are
shown in A to J. Write next to each one a solution (from the
following list) to the corresponding differential equation.
1.x
y1
2. xy elog 3. xey 4.2
1
xy 5. 3xy
6. xy e log 7. 2xy 8. xy tan 9. xy 2 10. xy 3
A.
B.
C.
D.
E.
F.
G.
H.
I.
J.
Answers.
A5
B4
C1
D2
E7
F6
G8
H3
I10
J9
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= Year 12 = Slope (direction) fields = Worksheet 2
Match each differential equation to the corresponding slope
field (A to J): 1. 22 yxdx
dy 2.
y
x
dx
dy
2 3.
x
y
dx
dy
2
4. y
x
dx
dy
2 5. 22 yx
dx
dy 6. yx
dx
dy 7. xy
dx
dy
8. xdx
dy 9. 2xy
dx
dy 10. yx
dx
dy 2
A.
B.
C.
D.
E.
F.
G.
H.
I.
J.
Answers.
A5
B2
C8
D1
E6
F3
G9
H4
I7
J10
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= Year 12 = Slope (direction) fields = Worksheet 3
Sketch the slope fields of the differential equations (A to K) for
yx, where 2,1,0,1,2 x and 2,1,0,1,2 y .
In each case on the slope field, sketch the graph of the solution
curve passing through the point 1,0 if it exists.
A. 212
1y
dx
dy
B. 312
1x
dx
dy
C. 212
1y
dx
dy
D. 2y
dx
dy
E. y
x
dx
dy 1
F. x
y
y
x
dx
dy
G. 1y
x
dx
dy
H. xydx
dy
I. yxdx
dy 2
J. 22
2
1yx
dx
dy
K. 22
2
1yx
dx
dy
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= Year 12 = Slope (direction) fields = Worksheet 4
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Use the slope field for
122 yxy to sketch
the solution curve that passes
through the origin.
Use the slope field for xeyy to sketch the
solution curves that satisfy
the initial conditions.
(a) 10 y , (b) 10 y .
Use the slope field for 2yxyy to sketch a
solution curve.
Use the slope field for
yxy sin to sketch a
solution curve.
Use the slope field for
xyy 2sin to sketch the
solution curve that passes
through (0,1).
Use the slope field for xey 1 to sketch a
solution curve.
The gradient of each line
segment in the slope field is
given by2
1)(
xxf . Use
the slope field to sketch
three members of the family
of anti-derivatives of xf .
The gradient of each line
segment in the slope field is
given by xxxf tan)( . Use
the slope field to sketch the
graph of the anti-derivative
of xf that passes through
the origin.
The gradient of each line
segment in the slope field is
given by41
1)(
xxf
.
Use the slope field to sketch
the graph of the anti-
derivative of xf that
passes through the origin.
The gradient of each line
segment in the slope field is
given byx
xxf
)sin()( .
Use the slope field to sketch
the graph of the anti-
derivative of xf that
passes through the origin.
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= Year 12 = Slope (direction) fields = Worksheet 5
1. A slope field for ( )yayy −=′ is shown, where a is a constant.
(i) Given ( ) 5.00 =y , estimate the values of ( )5.0y and ( )0.1y .
(ii) Estimate the value of a.
2. A slope field for y
xay
sin
sin=′ is shown, where a is a constant.
(i) Given ( ) 0.10 =y , estimate the values of ( )5.0y and ( )0.1y .
(ii) Estimate the value of a.
3. A slope field for ( )( )byaxky −−=′ is shown, where k, a and b are
constants.
(i) Given ( ) 5.01 =y , estimate the values of ( )5.0y and ( )0.2y .
(ii) Estimate the values of k, a and b.
Answers 3(i) -0.45, 0.83 (ii) 2, 2, 1 2(i) 0.8, 1.55 (ii) 2 1(i) 1.9, 3.6 (ii) 4