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hsc maths and physics

 Sydney - Baulkham Hills

Shining light of calculus solves Natures darkest mysteries

Sydney tutor in Calculus, Engineering, GAMSAT, Maths, Physics, HSC Mathematics all levels

We travel to these locations
Epping Castle Hill Carlingford Eastwood Cherrybrook Pymble North Rock The Ponds Dural Kellyville

One to one tuition in the comfort and convenience of students home at $ 60 / hour.
During first lesson I ask questions and assess quality of answers and time taken to answer questions.

Show how to correctly answer questions from high school teacher or other sources eg textbooks.

Explain fundamental concepts in detail and using examples show how fundamental concepts are used to answer specific questions from school or tuition.Emphasise the importance of showing all steps legibly and in logical clear order.

Set homework based only on topics and examples discussed during tuition

Find out date of next exam and topics and prepare appropriately by focusing tuition on future examinable topics as outlined in exam notification sheet.

Working with children check approval from NSW government.



Parents and students

One to one tuition is conveniently held at students own home at a mutually agreed time and day (weekends and weekdays are available)so parents avoid spending the time and hassle of driving and delivering student to and from tuition held at coaching centre...often after hard day at work.

For year 7 - 12 students fee based on $60/ hour at students home or public library

Advance payments not required.

No contracts to sign and therefore parents not locked into paying for unsatisfactory tuition for months in advance

Students should prepare a list of questions and concepts causing difficulty to be discussed during
tuition...this will enable me to determine the students academic level and plan an appropriate program of learning.

Tuition is more effective in terms of learning and obtaining maximium improvement at minimum cost if done on a one to one basis rather then tutoring several students of differing abilities and different ages and school years at same time in a group.Student does not have to wait for question to be answered.

One and half hours tuition for a group of three students at a time implies that each student receives approximately 30 minutes of one to one tuition .This is in general insufficient to cover background and understand the scope and depth of various topics.My group size consists of one student only...this has obvious advantages.

My tuition is personal as possible as there is only one person in the group and the style of tuition is tailored to suit the learning style of the student ,to explain how to obtain correct answer to school and exam problems and other relevent questions.

As there are no other students present , the student need not feel embarassed asking questions. By the way if a students asks many questions there is no increase in fees...if a students asks very few questions there is no decrease in fees. Students often feel inhibited and embarassed about asking my view the only bad question is the
one which is not asked.

My emphasis is to explain the fundamental concepts in mathematics and physics in simple terms and ideas and whenever reasonably possible use existing fundamental laws to deduce more laws equations and rules.When students understand they begin to learn and enjoy the subject. It is difficult if not impossible to enjoy a subject which is only partially understood.It is more interesting and challenging to derive an equation rather then receive it without background explanation and derivations.

I have also prepared some Mathematics / Physics experiments and there is excellent agreement between measured and predicted value.

Experiments in Mathematics help students bridge the gap between theory and practise and better able to understand the more abstract theories eg Simpsons rule...integration and Simpsons and Trapezoidal rule to find area...Newtons law of cooling...maximum and minimum turning points.

For physics the following experiments are available:
potential and kinetic energy...parabolic motion...period of normal pendulum and conical pendulum.measurement of earths gravity...conservation of momentum for elastic and inelastic experiments
. ..Galileos experiment. to calculate
radius and mass of earth using three simple measurements and Newtons Law of Universal Gravitation, Lenzs Law . A good quality accurate experiment is a very effective learning tool and a means to better understand fundamental concepts.

Please note that Year 11 students will follow the new revised Physics syllabus starting this year (2018)
which is far more difficult and of higher standard then the previous syllabus. It is far more appropriate in terms of scope depth and choice of subject matter.for students wishing to study Engineering/ Science at University. .

Homework is given at the end of each tuition session and is based on what has been taught in tuition and at school.
All steps needed to find solution should be written legibly in clear logical order.
Students should study written examples and explanations given in tuition before attempting homework which should be attempted as soon as possible after tuiition This should preferably be completed no later then 4 days after tuition ( while concepts taught in tuition are still fresh in the mind of student)
Students should not spend too much time correcting a solution if the answer is wrong as this can be frustrating and demoralizing The attempted incorrect solution should not be rubbed out.. It is better to let me find the source of error which in many cases is a very simple mistake.

All homework should be written into an A4 sized notebooks . These must be kept as a record of topics covered and the
scope and depth of coverage.A summary book will be developed by the student which will incude formulas examples and an index of topics.
A seprate smaller note book containing index and summary of topics as written by student immeditely after successfully answering homework questions.These contain formula and examples and a list of common mistakes to avoid.It also contains a section of common mistakes to be avoided.

My experience is based on 20 years tuition in following subjects:

Mathematics Years 7-10 all levels

2U General Advanced Years 11-12
2U Advanced Years 11-12

Mathematics ( Ext 1 and Ext 2 ) Years 11-12

Engineering Studies Years 11-12

Physics 2 U Years 11-12

International Baccalaureate Years 11-12 Mathematics (All levels)

International Baccalaureate Years 11- 12 Physics

Gamsat Physics

UMAT Physics

Ten years teaching High School Mathematics (all levels)

Science Years 7-10

Physics (years 11-12) in High School.

Marking of HSC Physics examinations.


The following questions are both unusual
and more difficult then those found in a
normal textbook.


1) Find the three smallest consecutive integers such that

Smallest is divisible by 9
Middle integer is divisible by 7
Largest integer is divisible 23

2) Referring to Q (1) and the divisibility tests what are the largest three consecutive integers less the 1000?

3) In a box there are 8 red and 8 green marbles.
a) What is the smallest number of marbles you
can withdraw (without replacement) to ensure
you have a marble of same colour?

b) if in the box there were 8 red,8 green and 8 yellow
what is answer for (a) ?

4)Given that

(a +b)/c = m and (b+c)/a = n

express (a+c)/b in terms of m and n.

5) Consider a full cylinderical tank of height h and
base circle radius r. Its volume is V.

Three pumps A,B and C extract water.

Pump A extracts water at (V/20) litres per hour
if the suface of water lies between 3h/4 and h.

Pump B extracts water at (V/ 15) litres per hour
if the eater surface lies between h/2 and h.

Pump C extracts water at (V/10) litres per houe
if the water surface lies between 0 and h.

What is time taken for pumps to empty the
full tank of water?

6) Without using calculator find exact value of

1/ ( 1^0.5 + 2^0.5) + 1/ (2^0.5 + 3^0.5)
+ ......1/(20^0.5 + 21^ 0.5)

7) Find the smallest and largest 3 digit number

on division by 6 leaves a remainder of 5
on division by 13 leaves a remainder of 8

8) Solve
1/a + 1/(ab) + 1/(abc) = 5/26

Where a and b and c are integers in ascending
order and not necessarily different in value.

9) Express as powers of prime numbers

eg 1! 2!3!4! = 2 x (3 × 2) × (2× 3 × 4)
= 2^3 × 3 ^2 × 4
= 2^5 × 3^2

10) Given that 29w + 30y + 31z = 366
and that w, y and z are positive integers
in ascending order, find values of w,y and z.

11) Find all real numbers a,b and c such that

a is less then b is less then c

a + b + c = 5

a^2 + b^2 + c ^2 = 15

abc = 1

12) Solve

(4m +5)^0.5 - (3m + 16)^0.5
= (7m-13)^0.5 - ( 6m-2)^0.5

13) The original price of a cd is reduced 15%.
One week later the price is reduced by 10%
off the last price.
Two weeks later it is reduced by 5% off the
last price.
If the final selling price is $7 what is the
original selling price?
Considering the original selling price and the final
selling price what is the percentage discount?

14) The original price of a cd is discounted by
a certain percentage to form a new price.
One week later this new price is further
discounted by twice the certain percentage to
form a final selling price which equals
0.72 of the original price.
What is the original percentage discount?

15) Bob and Jack are both initially stationary on a circular racing track at N 20 degrees East.
Bobs running speed is 1.2 times that of Jacks running
What is the bearing
a) at the first time they meet?
b) the second time they meet?

16) Factorise

( a^2 - b^2)(a^2 - 2ab + b^2)

a^4 + b^4 - 6(ab)^2

a^2 - 4ab + 4b^2 - (c^2 - 6c + 9)

(3ab + 4b -2)^ 2 - ( 5ab +2b - 3)^2

17) In a box there are 5 marbles numbered 3,4,5,6,7

Two consecutive draws are made with replacement
Find the probability of drawing
a) one ball and second ball having smaller number
b) absolute difference in ball numbers equalling 3
c) both balls labelled with prime numbers.
d) sum of numbers on each ball being less then 9

If the first ball is not replaced what are the
answers for (a) (b) (c) (d) ?

18) A circular cone has a base circle radius r standing
on horizontal ground
and a perpendicular height h above horizontal
When it is half full of water what is the height
of water surface above ground?

19) Consider a rectangle of lengths 7m by 12m.
The two shorter opposite sides are both
increased in length by the same amount and
the area of rectangle increases by 20 square
What is the change in perimeter of the rectangle?

20) Consider the sequence
What is the value of the 40' th term if squares
and cubes are omitted?
What is the value of 35 th term if odd square
and odd cubic numbers are omitted?

21) Consider a rectangle whose perimeter is 26 metre.
One of the sides is increased by 1 metre...
the other side is increased 9 metre...the new
shape remains rectangular. and the area
of the new rectangle increases by 86 square
What are the dimensions of the original area?

22) A resturant can hire only two waiters at a time.
It can choose from seven different persons.
If the resturant is open every day , what is the
maximum number of days that can pass without
repeating the hire of same pair of waiters?

23) Given that

Log( (m^2 ) × (p^2n)) = 1

Log (m^(2n) × (p^2)) = 1

where m and p are both positive numbers
and n is a positive integer

Prove that Log((m^n) × (p^n)) lies
between zero and 1.

24) Given that

m^2 + n^3 = p^4
Show that m , n and p cannot be prime numbers.
Note that 2 is the only even prime number.

25) Given that m and n are any positive or negative
numbers show that

( 1 + lml) ÷ ( 1 + lnl)
is less then or equal to
( 1 + lm-nl)

Note that lcl means absolute value of c.

26) Consider 4 positive integers.
The second integer is 1.5 times the first integer a.l
The third integer is 6 times the first.
The fourth integer is 8 times the second integer.
What is the minimum average of all four

27) The lengths of a pair of sides of a triangle
are added to give 21,16 and 13 cm.

a) without finding the lengths of each side
find the perimeter of the triangle.

b) find the length of each side of the triangle

c) without using a calculator or trigonometry
find area of triangle in surd form.

28) Consider a quadrilateral whose side
lengths taken 3 at a time and added
give 25 , 27 , 28 and 31.

a) without finding the lengths of any of the sides
what is the perimeter of the quadrilateral?

b) find the lenths of each side of the quadrilateral

c) what is the minimum length of each diagnol?

29) Two clocks A and B are set at 8.0 am.
Clock A uniformly gains 40 seconds every hour.
Clock B uniformly loses 20 seconds every 1.5
At what time will Clock A be 800 seconds
ahead of clock B?

30) The addition of three numbers gives Y E S

Find one solution for the value of each digit Y,E and S.

31) What is first digit on the right for the sum

a) 2^ 17 + 4 ^ 17

b) 3^20 + 7^ 60

c) 5^39 + 8^43 - 7^29

31) Given that the capital letters represent a digit
between 0 and 9
and D is even find the value of the sum.

32) The sum of the digits of the number 2536 is 16.
What is the next largest number whose digits
also add up to 16?

33)Consider the following sequence


Which letter represents the

a) 45 th term
b) 93 rd term

34) Prove that

a) f(t) + f(-t) is even

b) f(w) - f(-w) is odd

Are the following odd or even ? Give reasons.

2f(t) - 3f(-t)

f(t) ÷ f(-t)

f( y) × f(-y)

( g( -t))^2

35) Consider the pattern

a) Find the value of the 256 'th term

b) Find the sum of the first 163 terms

36) Bill spends a total of $4.82 purchasing stamps
at 25 cents each and envelopes at 33 cents
each. How many stamps and how many
envelopes did he buy?

37) Let 499998 × 500002 = w
Find the sum of the digits for w.

38) The addition of three side lengths of a
rectangle give 100 cm.
The addition of a diiferent combination of
three side lengths give 65 cm.
What is perimeter of rectangle?

39) How many 3 digit numbers are divisible by
a) 14 and 38
b) 18 and 126
c) 35 and 40 and 28

40) How many positive integers greater then 100
and less then 400can be expressed as rhe
product of two odd prime numbers?

41) Consider the following number grid.

a 2 m

6 y b

z 18 c

The sum of the 3 numbers when added vertically
horizontally or diagnolly is identical.
Find the value of each pronumeral.

42) Find the sum of the digits of the number

35^2 ×( 625^ 2020)× ( 4^4043)

43) The counting numbers starting from 7
are written out

Find the value of
a) 15 th digit

b) 30th digit

c) 50th digit

d) 109th digit

Find the sum of first 50 digits.

Find the sum of the first 20 even digits.

44) Given M = 12!
How many

a) square numbers are factors of M

b) cubes are factors of M

45) J is the smallest integer which has the following properties
It is simultaneously the sum of
3 consecutive integers
4 consecutive integers
5 consecutive integers

What is value of J?

46) Given that a,b and c are positive integers

a + (c ÷(cb + 1)) = 45/7

47) A hotel has 20 rooms each of which has either
one or two or three beds.
There are a total of 44 beds.
In how many rooms are there one,two or three
How many different solutions are there?

48) Given the following arrangement of numbers


5 7

9 11 13

15 17 19 21

a) sum of first 5 rows
b) sum of first 20 rows

If there are a total of 10 rows find
a) the sum of numbers in the first two columns
b) the sum of numbers in the first 9 rows
c) the sum of numbers in the 10 th row

49) Let J be a two digit number.
How many possible values of J are there
a) the ones digit is at least 4 more then the tens

b) the ones digit is twice the tens digit.

c) the ones digit is less then the tens digit

If J is even what are answers to above?

50) J is an integer such that the sum of two of
its factors is equal to 162 and the difference between the same factors is 82. Find the value of J?

51) The integer 540 is the product of three factors.
A pair of these factors is such that the difference
in squares is 299. What is value of each of these
three factors of 540?

52)The product of 540 and the positive integer J
is a perfect cube.Find the first three smallest
of J.

53) An irregular hexagon has 3 adjacent sides each
equal in length to b and another 3 adjacent sides
each equal in length to c.
The sum of three adjacent sides is equal
to 29 cm.
The sum of another three adjacent sides
is equal to 25 cm.
All side lengths are integers.
Find the values of b and c.

54) A cone is of perpendicular height 7cm. It has a
circular base of radius 5 cm. The top of the
cone is directly above the centre of the circular
A horizontal band Is painted on the outside of the
cone. The top of the band forms a circle 6 cm
above the circular base....the bottom of the
band forms a circle 2 cm above the
circular base.All 3 circles lie in planes parallel
to each other.
What is the area of the painted band?

55) The sum of a fraction and its reciprocal is
equal 73/24
What is the value of each fraction?

56) The four vertices of a square ABCD lie on the
the circumference of a circle of radius 8cm.
The four sides of a larger square EFGH are all
tangent to the same circle.
Find the value of the area of the large and small
square and area bound by one side of the large
square and the arc of the circle.

57) Triangle ABC has a perimeter
of 17 + (109)^(0.5)

AB = 109^(0.5)

Without finding lengths of BC and CA find
a) area of triangle ABC

b) perpendicular height from AB to C

Also find lengths of AC and CB

58) The capital letters represent a positive
integer between 0 and 9.


For the above multiplication find
the value of each letter.

59) Show that
(( a + 1)! - a!) ((a + 1)! + a!) = a! (a + 2)! ÷ (a + 1)

60) The fraction 11/13 is converted to a recurring
decimal. What is the sum of
a) the first 960 digits to the right of the decimal
b) the first 851 digits to the right of the decimal

c) the first 357 even digits to the right of the
decimal point?
61) solve the following inequalities

a) (3m +1)/( 2m - 5) is greater then -2

b) ( 4n- 3)/ (3n^2 + 1) is greater then -3

c) ( 6p - 2)/(4p^2 + 1)

is greater then

(3p - 5)/ (2p^2 +7)

62) Two digit number is 6 more then 7 times
the sum of its digits.
The tens digit is 1 less then 3 times the units
What is value of this two digit number?

63) Rectangle ABCD.
P is the midpoint of BC.
Q is the midpoint of CD.
R lies on BA
area of RPQ ÷ area of ABCD = 0.4
Find value of
a) RB/RA
b) perimeter of RPQ/ perimeter of ABCD

64) Consider the GP 3 9 27 81 etc
What is the last digit for the sum
of the first

a) 100 terms

b) 203 terms

c) 307 terms

65) The inequality
4/9 is less then 13/m is less then 6/11
If m is an integer find its value

66) Given that
2^15 × 125^6 = J
What is the sum of the digits of J?

67) consider a 4 digit number J

sum of the first two digits is 13
sum of second and third digit is 7
sum of third and fourth digit is 6
sum of first and last digit is 12

Without finding actual values of digit
is J divisible by 3. Give reasons.

68) Express as an exact fraction

If the first digit is located immediately to the
right of the decimal point
what is the value of the 100th digit?

69) Only one of the following statements are true
The wallet contains at least $75
The wallet contains at least $60
The wallet contains at least $48

What is maximum amount of money
in wallet . Assume an integral number of dollars

70) Given that

a! ÷ b! = 2730

Find the values of a and b.

71) By cosidering the area of a regular pentagon
or otherwise show that

cos 36 = 0.25 (1 + 5^0.5)

Hence find exact value of
Sin72 tan36 cos18 sin81

Using only a compass, sharp pencil,
straight edge only show how to construct
a)an angle of 36 degrees

b) regular pentagon

72) Given that

4/71 is less then a/b is less then 16/ 105
and a and b are positive integers.

Find one solution to the inequality.

73) Rectangle has side lengths a and b where
b is larger then a.

Using a sharp pencil, good quality compass,
straight edge show how to construct
a square of area ab.

74) the vertices of a rectangle lie on the circumference
of a circle of radius 5 cm.
The perimeter of this rectangle is 28 cm
Find the value the side lengths of the rectangle
and its area.

75)Thirteen consecutive positive integers are each
divided by 13. What is the sum of the remainder?

76)Five consecutive positive integers are divided
by 7. What is the maximum and minimum sum of
the remainder?

77) Given that
45 - 4a = 4b + 19
i) average of a and b
ii) the value of (a +b)^3 + (a + b) ^2

78) A dice has 6 faces numbered 1 to 6.
Three of these dice are rolled...the sum
of these numbers shown on the uppermost
faces is added.
List all the possible sums.
How many different possible sums are possible?

79) Allen ,Bob and Clive are wearing ties which are
green,purple and red in colour but not
necessarily in that order.
Allen is not wearing tje purple tie.
Bob says to Clive " l like your red tie"
What colour tie is Bob wearing?

80) If a three digit number is divided by 7 or 11
the remainder is 3 in each case.
What is the smallest and largest such three
digit number?

81) A six sided dice has the following numbers
on its faces
0 0 3 4 5 7
Four such dice are thrown simultaneously .
The sum of numbers appearing on uppermost
side is added.

a) List all possible sums
b) What are number of different sums?

82) Commencing with the number 50 the positive
integers are listed in increasing order but the
digit 8 is omitted .

What is the value of the 10th , 35th, 87th

83) The sum of three positive integers A,Band C is 64.
C is 11 more then A.
B differs from one of the other numbers by 2
and the other number by 9.
Find the value of all three numbers.

84) Consider five consecutive multiples of a given
The average of the first two multiples is 76.
The average of the last two multiples is 100.
What are the values of each multiple?

85)Find the greatest prime factor of
5! + 7!

5! + 7! + 9!

86) Consider the numbers between 21 and 99.
How many such numbers are there such
the difference in digits is equal to
c) 7

87) Consider the consecutive integers starting
with 75 and ending with 162.

What is the probability that the number chosen

a) contains the digit 7 at least once
b) contains the digit 8 at least twice
c) does not contain the digit 3
d)does not contain an odd digit
e) is an odd number and divisible by 7
f) none of the digts are 5 or 8
g) must contain two even digits only

88) The two prime numbers A and B
such that 5A + 9B = 712
Find two values of A and two corresponding
values of B.

89) A retailer has access to a supplier who
is always able to supply brass house
The retailer has available for sale only the digits
0 , 7 and 8.

Assume that a house number cannot start with
How many three digit house numbers
can be formed?
How many four digit house numbers
can be formed?
How many house numbers containing
up to four digits can be formed?

90) An odd number lies between 303 and 385.
The sum of its digits is five times the tens digit.
What is the number?

91) How many three digit numbers between
209 and 580 have decreasing value of
digits ( when reading from left to right )
eg 531 and 310
but not 513 or 301)

92) A tape 4 cm wide is used to completely
cover the outside of a rectangular box

12 cm by 10 cm by 8cm.
What was length of tape used if no overlap

93) Consider a sector ABC of radius r cm subtending
an angle of 60 degrees at A.
Sector ADE also subtends an angle of 60
degrees at A and has a radius which is one cm
less then r.

The area of a portion of the annulus DBCE is 13
square cm .The point D lies on AB and E lies
on AC.

Sketch large good quality diagram

What is the value of r?

94) ABC is a sector AB and AC are joined to form a cone ,vertex at A
and circular base( lying in horizontal plane) of perimeter arc length CB = w is formed.

What is radius of circular base and perpendicular
height cone?

95) Hollow cone having a perpendicular height h
and base circle of radius r is filled to 2/3 of its
maximum volume with water.
Hollow cylinder having two circular parallel
ends (both lying in vertical plane) and each of
radius 1.5 r.

7 the water contained in the cone is poured
into the cylinder.

What is the height of the water level in the cylinder
in terms of h?

96)Consider the following long multiplication.

* * 5
1 4 6
4 * * 0
* * 0 0 *
* * * * *
* * * 8 * *

Find the value of the product as shown on last line.

97) Given that the square of a three digit number
gives a five digit number as shown

M A M × M A M = M A D A M

where the capital letters represent integers
from 0 to 9

Find the value of each capital letter.

98) Circular cone of height h and radius ; circular
base lies in horizontal plane. Liquid is added
so that it is 3/4 full.
What is height of horizontal liquid line above
circular base in terms of h and r?
What is wetted area of inside of cone?

The same cone which is 3/4 full of liquid is now tipped upside down...the vertex is directly below the
circle centre and circle lies in horizontal plane.

What is height of horizontal liquid line directly above
vertex of cone?

What is wetted area of inside of cone?

99) A circular cylinder of radius r and perpendicular
height h is filled to 2/3 of its maximum volume
with liquid.
It is then turned through 90 degrees so that
the circular ends are both in the vertical plane.
What is the height of the suface of liquid above
the lowest point on the circular end.

100) Consider a rectangular prism BCDEFGHI.

BCDE and FGHI both lie in the vertical plane.

Base DEHI lies in horizontal plane.

DE , CD and BF are of lengths k, m and n

Prism is 1/4 full of liquid.

Prism is rotated 30 degrees anticlockwise
about axis EH.

What is new height of water surface above EH

101) Consider a rectangle ABCD of side lengths
8 cm = AB and 12 cm = AD

E is a point on AD such that
2 x EC = EB

Find the perimeter of triangles EAB and EDC.

102) The product of the squares of two numbers
is 925444
Find the numbers.

The product of the squares of two consecutive
odd numbers is 54686025
One odd number is 4 greater then the other
odd number.
Find the value of these numbers.

103) Consider two circle of areas B and C of differing radii w and y respectively are constructed.
Circle centres are shown.

Using a compass sharp pencil and straight edge
( not a ruler) only
show how to construct a circle whose area is B + C.

104) M and N are positive integers

M^ 0.5 and N^ 0.5 are both surds

Given that
M^0.5 + N ^ 0.5 = P

where P is a positive rational number.
Show that P does not exist.

105) Consider two pizzas of differing radii 8cm
and 12 cm are shared among 13 people.
The pizzas are divided into 13 sectors
of identical area.
How can this be done?

106) consider a quadrilateral ABCD.
Using only a compass, sharp pencil,
straight edge only divide quadrilateral into
a) 5 quadrilaterals of equal area
b) 3 quadrilaterals whose areas are in
ratio of 1: 2: 4

107) Two rectangular prism shaped cakes are of
identical height.

One cake measures 80 cm by 75 cm.
The other cake measures 40cm by 30cm.

The cakes are to be shared equally among
18 guests (same volume of slice) Each guest is
to receive only one slice of cake.

How can this be done?

108) Consider 4 concentric circles. The innermost circle and each of the three annuli each have the same area.
The outermost circle has an area of 36pi
Find the radii of all concentric circles.

109) Given any three points A,B and C and using
only a compass,sharp pencil and straight
edge only ,construct a circle passing through all
three points A,B and C.

Explain why the procedure used is valid.

Given any three points E,F and G .
The three straight lines EF,FG and GE are
tangent to a circle.
Construct this circle using sharp pencil,
compass and straight edge.
Explain why the procedure used is valid.

110) Show that the largest value of a three digit
number minus the smallest value of a three
digit number (having the same digits) is divisible
by 99.
All the digits are different.
Eg 732 - 237 = 495 = 99 x 5

111) Given that
a lies between zero and one inclusively
b lies between zero and one inclusively

Show that
a/ (1 + b) + b/ (1 + a)
lies between zero and one inclusively.

112) Givem a trapezium ABCD find the value of
cosA + cosB + cosC + cosD
where A,B,C and D refer to the internal
angles of the trapezium.

113) Given that
(d + e)/f = 4 and (e + f)/d = 0.8
Find the value of (d + f)/e

114) Solve
(3y + 6)^0.5 + y^0.5 = 14

115) The difference between squares of two
prime numbers is 240.
Find the values of the two prime numbers

116) The sum of two consecutive prime numbers
is 244 and the difference is 18.
Find the value of each prime number.

117) Consider the following

11 × 11 = 121
111 × 111 = 12321
1111 × 1111 = 1234321
Find the value of

a) 11111 × 11111

b) 11111 × 11111

c) 11112 × 11111

d) 111114 × 111115

e) 111109 × 111113

f) 2222 × 3333

g) 8888 × 2222 + 44444

h) 5555 × 2222
118) Find the algebraic expression for the average of
all different 3 digit numbers
produced by rearranging the order of the digits
a,b and c
where a does not equal b does not equal c.

Check your answer if the digits are 1,7 and 9.

119) consider the following number pattern
2 3 4
5 6 7 8 9
10 11 12 13 14 15 16

What is the 4th number in the 9th row.
What is the sum of numbers in 9th row?

120) Consider three classes of English,Maths and

A total of

35 students study English.
41 students study Mathematics .
9 students study both Mathematics and
and Physics.
17 students study only English.
26 students study only Maths.
13 students study only Physics.

What is number of students studying all
of above subjects?
What is number of students studying
both English and Physics?

121) The number 51368 is multiplied

b) 500,000,000,000,005
What is the sum of the digits of the product?

122) Prove that a five digit number is divisible by 11

if the the sum of digits in the first, third fifth
column minus sum of digits in second ,fourth
column is a multiple of 11.

eg 10857

(7 +8 +1) - (5 +0) =11 = 11× 1
Therefore 10857 is divisible by 11

123) Consider a six digit number abcdef
Show that is divisible by 47 if

31a + 36b + 13c + 6d - 37e + f
is divisible by 47

124) Rectangular loop ABCD of dimensions 9m x 6m
is in a horizontal plane at a height of 4m above
horizontal ground.

A chain of length 5m has a tiny ring attatched
at both ends. One ring is free to slide along
the perimeter ABCD; the other ring is attatched
to the neck collar of a pet lizard which roams
at ground level.
Draw suitable diagrams to show this.
Over what ground surface area can lizard graze.

125) Consider the two digit numbers between
20 and 89

How many of these numbers have digits
differing by
a) 3
b) 4
c) at least 8
d) zero

126) The number
85761324975271m is divisible by 11.
What is the value of m?

127) The number 79341a63b
is divisible by 11.
Find 3 values of a and 3 corresponding
values of b.

128) The number 72,186,5a5,86b,476
is divisible by 198
Find one pair of values of a and b.
Do not use calculator.

129) In a bag there are a total of 16 coins and total
value of 395 cents.
The coins are 50 cent , 20 cent and 5 cent.
The number of 5 cent coins is greater then
the number of 50 cent coins.

How many of each type of coin are in the bag?

130) Show that the sum of the squares of five
consecutive integers cannot be a square

131) Consider the sequence
80, 82 ,83, 84 , 85,86,87....
The integers which are perfect squares
and or perfect cubes are omitted.
What is value of

10th term
17th term
24th term
30th term

132) Consider the sequence of consecutive even
2,4,6,... .
Find the sum of the first
a) 18 numbers
b) 30 numbers

Find the sum of the digits of the first
a) 20 numbers
b) 50 numbers

133) The number 109,032 is divisible by factors
What are the next three next which are
divisible by the same factors.

134) The rectangle BCDE has

BC =9 cm CD = 4cm

An arc of radius 6 cm having centre at C
is drawn. The arc intersects BC at F and
ED at G.
Draw a good quality sketch.
Find the area and perimeter of BFGE.

Consider the rectangle BCDE.
Using centre C an arc of radius 10 cm
to meet to meet BE at I and ED at J.
Find the area and perimeter of IJE.

135) The relevent sections of a book
start from page 8. This requires
a total of 189 digits.
What is the last page number of
the relevant section?

136) The two digit number ab is mutiplied
by another two digit number ba
to give 2701.
Find the values of a and b.

137) Consider the three digit numbers
abc and acb

Show that
(abc)^2 - (acb)^2 is divisible by

(i) 9

(ii) 9 (b - c)

138) Consider the original number abcd
where a ,b,c and d are digits between 0 and 9.

The first and last digits of original
number are swapped to form a new number.

7 is added to the new number whiich is equal
to twice the original number .

Find the value of a,b,c and d.

138) Given that
a + b + c = 14

a^2 + b^2 + c^2 = 78

Find the value of (ab + ac + bc)^3

139) Prove that
the product of four consecutive integers
plus one gives a perfect square

140) Show that n^6 - n^2 is divisible by 5
a) using induction

b) considering the last digit of n^6 and n^2

141) Show that 3^0.5 and 7^0.5 and 11^0.5
cannot be terms (not necessarily consecutive)
of an arithmetic sequence.

142) Show that if
a^0.5 and b^0.5 and c^0.5 are members of an
arithmetic sequence (not necessarily
consecutive then (ac) is a perfect square
not necessarily integral.

148) Given that abc - bca = 288 and a ,b and c are
digits between 0 and 9 and a is even
find values of a , b and c.

If a is odd find values of a , b and c.
If a is even find values of a, b and c.
149) Given that the total price of
8 apples and 11 mandarins is $15.50
11 apples and 8 mandarins is $14.90

find the difference in price between
one apple and one mandarin.



1) How to find the value of e from first principles

Must first understand the concept of e

given that f(x)= B ( power x)

Does there exist a value of B such that

df/dx = B (power x)

(Unchanged by differentiation) and if so find its value

Using the fundamental definition of differentiation

(f(x+h) - f(x))/h= ( B (power (x+h)) - B (power x))/h

B (power x)= B(power x)(B (power h) -1)/h

1 =( B(power h ) -1) / h

h + 1 = B (power h)

Log (h+1) = h Log B (must use base 10 ..why)

B = 10 power(Log(h+1)/h )

Let h equal a very small number eg 0.000001

B = 10 (power( (Log 1.000001)/0.000001)

B = 10 power 0.4329

B= 2.718 (approximately value of e

2) prove that there is only one value of e ( using calculus)

3) given length of each side of a scalene triangle find its area
( without using Herrons formula or trigonometry)

4) Generate Pythagorean triads
These are whole numbers a,b, c such that
axa +bxb= cxc

5×5 + 12×12 = 13× 13

5) Given that m and n are both positive integers

m^3 + n^3 = 854

find the values of m and n using algebra.

6) a rectangular prism has 3 faces of area 7 , 8 and 9 square metres
What is its volume and length of each side
What is the length of each diagnol for each
face of the prism

7) a star has a radius of 42673 4896875 metre

If its radius increases by 3.5metre what is its change
in circmference and surface area.

8) how to balance complex chemical equations using simple algebra ( no guesswork)

How could you show that a given chemical
equation cannot be balanced.

9) show that a (to the power of zero) = 1
You may use the rule
a^m × a^n = a^(m +n)

10) at the end of 6 months the price of a house
Increases by 10%
In another 6 months the price decreases by 10%
How much has the price increased over one year?

11) using a pencil,compass, straight edge only show how to divide a straight line
into any numer of equal lengths 3 ,5,6 , 11 equal lengths

12) using a compass ,straight edge , pencil only show how to construct angles of
60, 45, 90, 30, 15,75 ,150, degrees

13) using a compass ruler pencil construct an exact length of ( square root of 34 ) cm

14)using a compass ,ruler ,sharp pencil construct an area
of ( square root of 35) square centimetres.

15) given a rectangle sand compass pencil straight edge only show how to divide it into 7 smaller equal area rectangles

16) given a triangle ABC ,compass ,sharp pencil, straight edge show how to divide any triangle ABC into
a triangle having

One fifth of area of ABC

One eleventh area of ABC

17) develop the formula for the area of the trapezium given that the lengths of the parallel sides are a and b
and h is the perpendicular distance between the parallel sides.

18) show that the sum of the two lengths of any triangle is larger then third side

19) consider a triangle
AB= 4m-10

Find the value of
(sinA) ÷ ( sinB)

Find value of all internal angles

20) Assuming the sum of positive numbers is positive
and the product of positive numbers is positive
prove that

a) the product of a positive and negative number is

b) the product of two negative numbers is positive

21) Without using calculator find which is larger

( square root of 7) + (square root of 5)


(Square root of 2) + ( square root of 10)

22) a triangle has side lengths

m×m + 1

m×m +7

3m + 1

Find the minimum value of m and minimum
area of triangle.

23) The Chefs Problem

Recipe is as follows
( Do not actually use this recipe...the result will
almost certainly be an inedible disaster)

43 grams sugar
53 grams oil
51 grams flour
32 grams eggs
36 grams water

Find the following

a) mass of sugar to to total mass of ingredients

b) mass of water to total mass of ingredients

In response to customer demand the chef decreases the total mass of this " cullinary masterpiece"
by reducing the mass of each ingredient by 20 grams

Find the answer to (a) and (b)

c) What do you notice?
d) Why?

Increase the mass of each ingredient by the same amount.
Answer (a) (b) (c) (d)

Drecrease the amount of each ingredient by the same amount...but there must always be 5 ingredients.
Answer (a) (b) (c) (d)

Multiply or divide each ingredient by the same
positive integer or mixed numeral.
Answer (a) (b) (c) (d)

23) Factorise

4 x(a to power 4)+ 81×(c to power 4)

24) consider a straight line AB of length m

Using compass, straight edge , sharp pencil
show how to locate a point C on AB such that

(AB) ÷ (AC) = any mixed numeral (2 + 1÷ 3)

25) consider the integers 1, 2, 3....100
What is the sum of the even numbers minus the
sum of the odd numbers?

26) the chocolate problem
Conider a rectangular box containing one layer of circular discs of chocolate of identical diameter and
thickness.The diameter does not necessarily equal to the thickness.Thickness of each disc is constant
irrespective of diameter.

The discs touch each other or the sides of the box.
The discs are packed so that they connot move with respect to each other or sides of box.

Which contains more chocolate

a box containing a large number of small diameter discs


a box containing a small number of large diameter discs



Is there a maximum amount of chocolate ?

Give reasons for your answer.

27) which is larger

99 to power 84

84 to power 99

Do not use logs or calculator
Give reasons

Consider a triangle whose sides are of length

4c + 5



What are the allowable values of c

29) consider a mass m rotating at radius r about
a mass M

a) what is the speed of m relative to M

b) what is the speed of M relative to m

c) according to a clock on m the time taken to
boil an egg is t minutes

What is the time taken for this event as measured
by a clock on M ?

d) according to a clock on M the time taken to
eat an egg on M is j minutes

According to a clock on m what is time taken to eat
this egg?

If the true shape of m and M is spherical what is the shape of M as seen by an observer on m?

30) explain the following dillema

Cathode rays are beams of electrons
Cathode rays are blue red green etc in colour
Therefore electrons are blue red green etc in colour

Do cathode rays travell only in straight lines?
Under what conditions can they travel in curved lines?
Why are electric fields so useful in understanding
the observations of cathode ray tubes exposed to
high voltages?

Give reasons for your answers.

Why does it appear that light in a cathode ray tube is deflected by an electric / magnetic field?
What is happening?

Why does it appear that light causes a paddle wheel inside a cathode ray tube to rotate?

If a cathode ray tube has absolutely no gas presemt
and a high voltage is applied what would you expect
to see?Why?

What is the definition and characteristics of cathode rays?

If a high voltage difference is applied to a gas why does ionisation of gas occur.?Why is emr emitted.

If a high voltage is applied across a gas in a cathode
ray tube and visible light is not observed what conclusions can be drawn?

What are the characteristics and significance of
fluorescence emitted by cathode ray tubes?
Is fluorescence an example of cathode rays?

31) Prove that in general that it is incorrect to say

Square root of ( a xa - bxb) = a - b

Square root of ( axa + bxb) = a+b

Under what conditions are above statement true?

32) Prove that Lenzs law must be true by considering a frictionless horizontal surface,permanent magnet, light globe,a metal coil fixed in place but joined at both ends allowing current to flow..

33) Consider two different investments A and B

In A the principle is invested on the 1/2/18... the interest rate is 5% pa compounded annually.

In B the principle is invested on the same day 1/2/18...the interest rate is 10% pa compounded annually.

The Principle in A Is 3 times the principle in B.

a) A fter how many years is the worth of each investment the same in value?

b) After how many years is the interest earned on each investment the same value?

34)consider thefollowing

7 ÷ (2× square root of 5 - 4×square root of 3 + 3 × square root of 6)

Rationalise the denominator.
Check your answer using calculator.

35) is it possible to construct a circle through all
vertices of

a) any triangle
b) any quadrilateral

Give reasons

35) consider two hemispheres of different diameters
The sum of the diameters is square root of 17
The difference in diameters is square root of 11

Find exact difference of surface area of each hemisphere in terms of pi

36) consider a rectangle ABCD...the length of sides is a and b
the length of each side is increased by one metre to form rectangle EFGH.

ABCD and EFGH have corresponding parallel

Find the differerence in perimeter and area of the rectangles

37) consider two straight lines AB and DE intersecting
at point C.
What is the ratio of areas of triangles and ratio of perimeters of triangles
in terms of AC , CD , EC ,CB
AC = f CD= e CB=g CE= h
and angle DCA = y

38)The barrel of a gun is horizontally aligned and points directly at an apple which is m metres away ( in horizontal direction). At the instant the bullet is fired the apple falls downward from the tree.
Under what circumstances will bullet hit the apple? Hint: draw diagram.

39)Sketch the following ( do not use calculus)...
do not find co ords of local minima and
local maxima.Show the z interceps.

y =( ( z-2) to power 3 ) x ((z + 3) to power 2)

The vertical axis is y

How many local minimums are there.
How many local maximums are there.

40) Consider

y =( (W - 3) to power 2) × (( 2W + 5) to power 6) + 5


What is domain and range

41) what is the domain of

y = 3 ÷ ( ln ( w x w - 5w + 6))

Y =( ( 2f + 3) ( 2f + 3) (3f - 5) (f + 6) (f + 6)(f+ 6) (f + 6)) to power 0.5

42) consider the following...

(11×11) - (13 × 13) + (15 × 15) ..((97×97) -(99 × 99))
Isi an AP or GP or neither?
Find its value by developing an appropriate formula.

43) develop a formula for finding the value of

20 x 20 + 21 x 21 +22× 22.......90× 90

44) Given that m and n are both rational and greater then zero find values for m and n such that
a) both m and n are integers

b) both m and n are mixed numerals

for the equation below

m × m - n× n = 27

45) consider two cars separated by a straight line
distance of 75000m and both are initially stationery.

Car A has an initial speed
20 m/ sec and acceleration of 3m/ sec/sec
and travells to the right

Car B has an initial speed
25 m/sec and acceleration of 4 m/sec/sec
and travells to the left.

Both cars start journey at same time toward each other.

When and where do the cars meet?

If Car A has a constant speed of 18 m/sec to the right and
car B has a constant speed of 14 m/ sec the left when and where do they meet?

46) consider a sector having a uniform thickness
and an angle at centre between zero and 180 degrees.
The area of the sector is ( 32 pi) ÷3
The total area of the two rectangular sides is 32
The area of the curved surface is ( 16 pi) ÷ 3

Find the volume of the sector WITHOUT finding the
values of angle at cenre.radius,thickness of sector.

47) Consider a rectangular prism .The side lengths
consist of 3 consecutive positive integers.
If one metre is added to each side length
its volume increases by 2 cubic metres.

Find the original dimensions of the rectangular prism.

48) consider a circular cylinder of height h and thickness 1.0 m.
The base of the cylinder is annulus.
What is difference between the outside curved
surface area and the inside curved surface area
Do not consider the area of the two annuli.

49) Using the product rule for differentiation derive the quotient rule for differentiation.

50) Given that y = sin m and dy/dm = cos m find the first derivative of cosm and tan m.

51) what are the characteristics of a black body.

Is it necessarily black? Give reasons

52) Consider the equation

(Square root of g) (1 + 3(square root of f))
-f - (6÷(square root of g))

Find g in terms of f

Find f in terms of g

53) Given that n is greater then or equal to 3
show by induction that

29 to the power (n×n -3n + 2) - 1 is divisible by 14

54) a)Using induction show that for all integers n
greater then or equal to 7 that

(2 n -5) (2n-4) (2n -3) is divisible by 6

b) Find an alternative way to prove( a )

55) Group A has an average of 9
Group B has an average of 11 and 2 more
members then Group A
When Group A and Group B are combined
the average is 121/ 6

How many members are in each Group?

56) consider a mass m attatched to end of taut
string radius r and rotating at a tangential
speed of v m/ sec.
Ignore gravitational field.

Given that the initial vertical and horizontal
displacement are respecively
r sin u and r cos u
where u is initial angle in radians

show using calculus that

centripetal acceleration = (v x v) ÷ r
and this acceleration points toward
the circle centre.

57) consider a triangle whose perimeter is 18 cm
Its side lengths are 2w-7....w+2 ..w +3

Consider another triangle whose perimeter is 36 cm
its side lengths are 2y- 10...y+6 ....3y - 8

Are the triangles similar....Give reasons
Find the lengths of all sides of both triangles.
Find area of each triangle.

58)Triangle A has side lengths
3w-2 and 14-2w and 14 -w

Triangle B has side lengths
2y - 7 and y - 1 and 3y - 9

Perimeter of A is twice that of B

Find side lengths of B...find area of triangle B

What are maximum and minimum
side lengths of A
What are maximum and minimum areas of triangle A?

59) Given that the area of a rectangle is LB prove
that area of a triangle is 0.5 LB where B is base
length and L is perpendicular height of triangle.

60) Consider equations

2a - 3b + 5c = 6
4a +b - 2c = 7

Find value of
-7b + 12c

14a - c

Find value of
3 to the power ( 24a - b)

61) The original volume and suface area of a cone
is V and S
If its radius is halved and its height is tripled what
is its new volume and surface area.

If the radius is doubled and the height is halved
what is new volume and surface area?

62) An equilateral triangle has sides of length w.
It contains three identical discs of radius r. Ech disc
touches each of the other discs and the sides of the triangle. The discs cannot move with respect to the friangle or the other discs.
What is the equation of r in terms of w.
What is area outside the discs but inside the
equilateral triangle? What is its perimeter?

63) If 1019 is multiplied 981 what are last 4 digits?

64) Consider coins of following denominations

5 cent 10 cent 20 cent 50 cent

In how many different ways is it possible to have a total of 155 cents?
All denominations must be used in any given selection.

Given that y! equals an integer whose last 4 digits are all zero what is minimum value of y?
Note that y! means y factorial
5! = 5 x 4 x 3 x 2 x 1

66) Consider two isoceles triangles

Triangle A has lengths c,c and d
Triangle B has lengths e,e and f
Assume that c,d , e , f are all different positive
Can the triangles A and B have same area and if
so under what conditions?

Two apples , three bannanas and four mandarins
have a total cost of 570 cents

Five apples , two bannanas and six mandarins have a total cost of 795 cents.

Find total cost of 11 apples and 10 mandarins.

68) The sum of lengths of three sides of a rectangle is 65 cm.
The sum of lengths of all four sides is 90 cm.
What are the dimensions of the rectangle?

69) Consider a trapezium whose area is 8 square cm
It is modified so that one of the parallel sides is increased by 3 cm and the other parallel
side is decreased by 3 cm. The perpendicular distance between the parallel sides is doubled.
What is the area of the modified trapezium?

70) Consider a triangle ABC of area W

D lies on AB and AD/AB = 1/3
E lies on BC and BE/BC = 1/4
F lies on AC and AF/AC = 2/7

Sketch a large good quality diagram

Find the following

( area FAD ÷ area BAC)
( area DBE ÷ area BAC)
( area FED ÷ area BAC)
( area DBE ÷ area FAD)

71) consider a triangle whose lengths are
3m - 2
2m+ 8
8m - 11
The sum of a pair of different lengths is 42.
All side lengths are integrals.

Find the dimensions of the triangle and find its

72) Given that
(a + b) × ( a + b) = axa + 2×a×b + b× b

Find expansion of
(a-b) (a-b)
(m^3 -n^2) (m^3 + n^2)

73) Consider a cylinder.
After it is filled with water the total mass is 820 grammes.
When it is 2/7 full of water the total mass is 490 grammes.
What is mass of empty cylinder?

74) consider an analogue clock.
What is the angle between the minute and
hour hand at 4.38 pm?

At what times will the angle between hour and
minute hand be 65 degrees

For the next two occasions only at what time
will the hour and minute hand be in alignment?

75) consider the analogue clock in (74)
If it loses 6 minutes per day what is the true
time when it displays 5.00 pm?

76) Consider three consecutive sets of traffic lights A , B and C all in a straight line.

Traffic light A shows green for 50 seconds and not green for 100 seconds.

Traffic light B shows green for 75 seconds and not green for 225 seconds.

Traffic light C shows green for 25 seconds and not green for 125 seconds.

At t = 0 seconds all three sets of traffic lights have tuned green.
What is the next time taken for all three sets of lights to turn green simultaneously?

If the straight line distance between TL A and TL C is 1350 metres
what is the minimum speed if car driver needed to
pass through all 3 sets of lights showing green without stopping given that

the car driver leaves TL A at t = 0.
the velocity of car driver is constant and the reaction time of the driver is zero.

77) The product of five consecutive numbers when divided by the middle number equals 17920.
Find the value of these number

What is the sum of these five numbers in terms
of the middle number?

78) The diagnol length of a square is 8 cm
What is the area and perimeter of the square?

79) Consider a square of diagnol length w.
a) If the diagnol length is tripled by what factor
is the new area and perimeter increased?
b) if the diagnol length is halved by what factor
is the new area and perimeter decreased

80) Consider two straight parallel lines
y = mx + b

y = ex + d

Find the perpendicular distance between them

81) consider a circle of radius 4 whose centre is at
the origin.
The straight line
y = 2x + 3 intersect circle at A and B
y = - 0.5x - 1 intersect the circle at D and E

Find the co ordinates of A ,B,D and E

Show that the perpendicular bisectors of AB
and DE intersect at circle centre.

82) The straight line y= mx + b intersects the circle whose radius is r and whose centre is (0,0) at A and B.
Find the equation of the perpendicular bisector of AB and show it passes through (0,0)

Show that the perpendicular bisector of any two chords on same circle must pass through circle centre

83) a) Find the prime factors of 9331

b) If m is a positive integer solve the following equation using results of (a) and without guess work

m^5 + m^ 4 + m^3 + m^2 + m + 1 = 9331


m^a means m to power of a

84) Prove that

a) if number is divisible by 3
the sum of its digits is divisible by 3

b) if a number is divisible by 9 the sum of
its digits is divisible by 9

c) if a number is divisible by 4 its last
two digits digits are divisible by 4

d) if a number is divisible b


Bachelor of Mechanical Engineering (University of New South Wales)

Master of Engineering Science (University of New South Wales)

Diploma of Education (Australian Catholic University)

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Joined Tutor Finder on 13-Mar-2017 (updated profile on 26-May-2019)
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