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

+−= .



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



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− .



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 .



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

Copyright itute.com 2010 Free download from www.itute.com Do not reproduce without permission

= 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


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


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 .


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


= 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.


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


= 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

__




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.


p% chance you wear the same pair the next day. If 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%.



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.


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


= 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


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


= 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 .


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


= 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


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


= 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.


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

.


= 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


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



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


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



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.


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


= 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.


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)



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= .


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



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.


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

)



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?


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



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


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



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 .


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



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



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.


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



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 .


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


= 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.


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)



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.


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

)



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.


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



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 .


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



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.


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



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.


5. 0

6. 0

7. 3

8. 3

9. π

/√3

10.

π/√2

11.

≈3.7

04



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 .


2cot

xy = , the x-axis,

2

π=x and π=x .


( )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 .


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



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 .


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



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 .


approximate solution to 22

yxdx

dy+= at 2.0=x if ( ) 10 =y .


approximate solution to x

edx

dyx = at 2.1=x , given ( ) 21 =y .


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



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.


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


= 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 .


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)



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.


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

)



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 .


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)



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 .


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



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


= 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.


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)



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.

•

•

•

•



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.


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



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.


r

+=

ttt

1)( i

−+

tt

1j.


r

+=

2

2 1)(

ttt i

−+

2

2 1

tt j.

9. What is different between the particle motions in Q7 and

Q8?


r ( )tt 2sec2)( = i ( )t2tan− j.


r2

1

2)(

t

tt

+= i

21

3

t+− j.


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



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.


r ( ) tt tan= i t2sec+ j, where 2

0π

<≤ t . Find the magnitude

and direction of its velocity at 4

π=t .


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.


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


= 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



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



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



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.


yxy sin to sketch a

solution curve.


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 .



given by xxxf tan)( . Use

the slope field to sketch the

graph of the anti-derivative

of xf that passes through

the origin.



given by41

1)(

xxf

.

Use the slope field to sketch

the graph of the anti-

derivative of xf that

passes through the origin.



given byx

xxf

)sin()( .

Use the slope field to sketch

the graph of the anti-

derivative of xf that

passes through the origin.



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.


(ii) Estimate the value of a.

3. A slope field for ( )( )byaxky −−=′ is shown, where k, a and b are

constants.


(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

year 12 mathematics worksheets

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