Matching Students and Tutors around Australia ...
Matching Students and Tutors

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.

Hi

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 question....in 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. etc..how 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

Mathematics

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.

________________________________________

22, 9, 17

The following questions are both unusual

and more difficult then those found in a

normal textbook.

****************************************

SET TWO SET TWO SET TWO

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

which

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

1!2!3!.....10!

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

speed.

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

ground.

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

metres.

What is the change in perimeter of the rectangle?

20) Consider the sequence

3, 4, 5, ......

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

metres.

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

integers?

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

hours.

At what time will Clock A be 800 seconds

ahead of clock B?

30) The addition of three numbers gives Y E S

Y Y

E E

S S

***********

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

DCC + DCC + DCC = BULB

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

JJKLLLMJJKLLLMJJKLLM

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

3, 2, -1, 5, 7, 3, 2, -1, 5, 7....3, 2, -1, 5, 7

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

7891011121314....

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

solve

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

beds?

How many different solutions are there?

48) Given the following arrangement of numbers

3

5 7

9 11 13

15 17 19 21

Find

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

if

a) the ones digit is at least 4 more then the tens

digit.

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

base.

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.

E A J

3

-----------------

C A C J

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

point?

b) the first 851 digits to the right of the decimal

point?

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

digit.

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

0.615384615384615384....

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

Find

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

number?

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

number.

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

a)5

b)6

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

numbers.

The retailer has available for sale only the digits

0 , 7 and 8.

Assume that a house number cannot start with

zero.

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

occurred?

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

respectively

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

1

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

Physics.

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

by

a)100, 000, 000, 00, 002

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

number.

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

numbers

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

3, 6, 14, 21.

What are the next three integers 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 9.2 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 inclusive

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

by

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.

143) Consider a rectangle BCDE such that

BC = 30 km and CD = 20 km

Car J starts at point B at 2.00 pm and travels at

constant speed of 32 km, hour in a clockwise

direction.

Car K starts at point C at 2.00 pm and travels

at a constant speed of 40 km, hr in an

anticlockwise direction.

Find the first three meeting times and

locations .

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

Without finding the individual price of an apple

or mandarin find :

the difference in price between

one apple and one mandarin.

cost of 5 apples plus 5 mandarins

which costs more ...one apple or one mandarin

show that the price of an apple multipled

by the price of a mandarin gives 6300

150) Given that F , E, N and C can be chosen from

2 , 5 , 7 and 8

Find the maximum and minimum value of the

sum of 3 digit numbers

CEF + NEF + FEN

CEF + NEF

151) Bill leaves home at 2.30 pm travelling at an average speed of 65 km, hour on the road to

Bathurst.

At 3.00 pm on the same day Jack leaves the same home travelling at an average speed 75 km, hour on the same road to Bathurst.

After what distance and at what time

does Jack overtake Bill.

152) Consider rectangle BCDE where

BC = 5 km and CD = 3km

Jack leaves point B at 1.00 pm travelling an

average speed of 8 km, hour in a clockwise

direction along the rectangle and Sam leaves

point D at 2.0 pm travelling at 6 km, hour in

an anticlockwise direction along the rectangle.

What is the clock time when they meet on the

first, second and 5th time?

153) given the expansion of

y = k (a + z)^ 6

find the coefficients and sum of coefficients in

the expansion of

w = m( a + z)^5

u = n(a + z)^ 7

h = mn (a + z)^9

154) Rectangle of side lengths a and b has

perimeter of 22m and area 28 square metres.

Without solving for a and b find the lengths

of the diagnol of the rectangle.

164) Given that the first digit cannot be zero

how many three digit numbers are there lk

that

a) have all digits differemt

b) have exactly two identical digits

165) Rectangle has perimeter and area of 38.6m

and 93.1 square metres.

The sides are made of straight sticks of lengths

0.4m and 0.5m and 0.7m

How many of each type of straight stick form

the sides of the rectangle.

166) Consider a three digit number abc.

a) If a, b and c are all different and for given values of a, b and c and first digit may be zero how many such three digit numbers are there?

b) How many such three digit numbers are there if

digits may be identical?

c)Show that the sum of all three digit numbers

(containing the digits a and b and c) is divisible

by 13 and 17.

167) Given that

(a + b + c + d)÷d = 2.3

and (a +b + c)÷b = 3.25

Find the value of

(( a + b + c + d)÷ ( a + b + c)) × (b ÷d)

Show that (d÷ b) = -35÷13

168) A set contains M counting numbers all different in value.None of the numbers is a multiple

of any other number

Find the minimum sum of these counting

numbers if M equals

a) 4

b) 5

c) 6

169) How many digits are in the following

5 ^17 x 4^8 × 49

What is the sum of the digits

170) A laboratory has masses each of which are a whole number of grammes. What are the minimum

number of masses needed to make a total mass

between 1 and 10 grams and how many of each such mass are there?

171) An exam conists of 25 mutiple choice questions

The marking scheme is:

Correct answer 5 marks

Incorrect answer - 3 marks

Answer not given 0 marks

What is the highest and lowest possible mark?

How many questions were correctly answered

and not answered if the exam mark was

a) 68

b) 41

c) -7

Are there outcomes in which the marking

scheme does not indicate the students

knowledge.

172) In 30 consecutive days what are the

greatest and least number of Sundays

What are the greatest and least number

of consecutive Sundays and Mondays?

173) A solid wooden cube , edge length 12 cm

is painted green on the outside. It is then cut

entirely into an integral number of smaller

cubes each of length 2cm. Apart from saw

dust there is no loss of volume.

What is the total unpainted surface area of

all the smaller cubes?

How many of the cubes have

a) 3 sides painted

b) 2 sides painted

c) 1 side painted

d) no sides painted

174)Consider the counting numbers

1, 2, 3 ..38, 39, 40

What is the sum of all the digits?

If none if the numbers contain the digit 2

what is sum of all the digits?

If none of the digits contain a zero or 2 what is

sum of all digits?

175) Consider a three digit number M in

which none of the digits is zero and all the

digits are different.

The order of the digits is reversed to create

the number N.

What is the largest possible difference

in M- N ?

176) Bill bought apples at 25 cents each and

bananas at 16 cents each for a total

of $3.78.

What are the least and greatest number

of bannanas purchased?

177)Consider a rectangle ABCD of perimeter

40 cm. Length AB = a is greater then

length BC=b

Point E lies on AB.

Point F lies on BC.

EBF is a right angled triangle and all its

side lengths are integers.

What are the possible perimeters of AEFCD ?

178) If the integer m is divided by 2 or 3 or 4

or 5 the remainder is 1.

m has a factor of 7

What is minimum value of m?

179) Given that

1÷15 = (1 ÷ (15 +a)) +( 1 ÷ (15 + b))

find the value of ab.

180) Find the next three numbers of the sequences

below

a) 9 61 52

b) 001 121 441 961

c) 7 26 63

d) -2 5 24 61

181) In a school there are 96 more female

students then male students

20% of the students are male.

How many male and female students are there?

182) A queue of persons is formed.Anthony is in centre of queue. Samuel is 4 places in front of

Anthony.Elizebeth is 7 places behind Samuel and

Klara is 6 places in front of Elizebeth.Klara is the fourth person from front of queue.

How many persons are in the queue?

What positions in queue are each of the persons named?

183) Given that a and b are integers and the fractions are in simplest possible forms and

(a÷6) + ( b÷8) = (53÷24)

Find the values of a and b

184) The number 2abcd is divisible by 81, 5

and 4 where a, b, c, d

are digits which are not necessarily different

find all the

possible values of each digit.

185) The number 378efg is divisible by 937.

Find the value of digits e, f and g which

are not necessarily different.

186) Goldbachs Conjecture (unproven)

a) Every even number can be written as the

sum of two primes.

b) Every odd number greater then 5 can be

written as the sum of three primes.

Find 2 primes which add to 84, 210

Find three primes which add to 95, 131

Show that if ( a) is true (b) is true.

187) Find the remainder when

a) 3^26 is divided by 7

b) 9^24 is divided by 7

188) At what time of the day is the angle between

the hour hand and minute hand

equal to 60 degrees?

189) The integer J has first and last digits equal

to 1. All other digits equal zero.

Show that J^2 is palindromic.

190) Jack is paid at the following rate

The first 35 hours at $22, hour

The next 4 hours at $33, hour

The next 6 hours at $40, hour

How many hours did Jack work if

his total payment was

a) $902

b) $992

191) Marbles are numbered 2, 3, 4, 5, 7, 8 and placed in a box.

Three marbles are withdrawn without

replacement.

What is the probability that the

a) sum of the numbers is 15?

b) product of numbers is a square ?

192) In a family John has 5 more brothers then

sisters.

How many more brothers then sisters

does his sister Elizabeth have?

How many more boys then girls are in family?

193) In an unusually large family one of the

children Jack has 17 brothers and sisters.

There are 8 more boys then girls.

How many children are boys?

194) Rectangle BCDE such that

BC = 20 km and CD = 15 km

Sam starts at B and rides his bike clockwise

at a constant speed of 25km, per hour.

Mick starts at C and rides his bike at a

constant speed of 20km per hour in

anticlockwise direction.When and where

do they meet for second time?

195) Three racing cars B, C and D leave the starting

point at the same time 2.00 pm and travel the

same distance to a finishing point.

Cars B , C and D take 1.5 hours, 1.333hours

and 1.2 hours respectively to arrive at

finishing point F.The speed of B differs from

speed of D by 20km per hour.Each car is driven

at constant speed..

Find the speed of each car and the distance

from start to finish.

196) Sector A has a perimeter of 18cm

and sector B has a perimeter of 25 cm.

Area of Sector B is 3 times the area

of Sector A.

Find the radius and subtended angle

of each sector

197)

The speed of light is constant. This means that the relative velocity between the source of light

and the measuring device will not change the measured speed of light ( by measuring device) which remains constant at 3× 10^8 m per sec.

Why?

To the best of my knowledge the development of

time dilation equation in general is based on behaviour of light meaning visible light only.

Visible light is part of the electromagnetic spectrum. What is true for full electromagnetic

spectrum must be true for visible light.

Assume that the laws of physics are true for all inertial reference frames and as a consequence

for identical inputs the outcome (outputs) of an experiment (conducted on any inertial frame of

reference) will be identical.

An output is any observation or measurement or reading taken at any time after the experiment has commenced.

Any difference in outputs of experiments A and B even for a short length of time will mean that

experiments A and B have different outcomes.

Consider two hypothetical experiments A and B in two different inertial frames of reference having identical inputs eg the variation of temperature or

power wirh time...distance between thermometer and source of electromagnetic radiation (EMR) etc

Consider an experiment in which

Platform of length H has two identical thermometers. One is located at the extreme left hand end of the platform... the other is located at the extreme right hand end. The platform has at its centre 0.5H a source of EMR which a constant

intensity of I watts per square metre.

The distribution of watts per square metre versus wavelength etc may vary with time but the intensity distribution does not vary with angle as measured from source of EMR. T seconds after the power is turned on the source emits constant radiation.

In hypothetical experiment A the platform is stationery relative to some point D located on earth (which itself is approximately an inertial reference frame) Hence both thermometers and EMR source are stationery relative to point D. After the source of radiation is switched on the speed of EMR moving to the left or right are both equal to C where C is speed of EMR that is 3 × 10^8 metres per sec.

Hence would expect that both the left and right hand

thermometers would show same reading at any given time after the source of radiation is switched on.The simple reason is that

the same magnitude and type of power emitted

reaches both thermometers at the same time.

The graphs of both thermometer readings will

be identical and will show an increase in temperature with time until steady state temperature has been reached.

In hypothetical experiment B the source of radiation is switched on after the speed of

platform reaches a constant speed of V

to the right. Immediately after the source of radiation is switched on the experimental

results are recorded.

Let us assume that in experiment B the speed of

EMR is affected by the speed of source of EMR

.. .more specifically the speed of EMR is C +V

to the right which is greater then the speed

, of EMR of C- V to the left.

The magnitude and type of power emitted is the same in the left and right hand directions but takes a longer time to reach the left hand thermometer then the right hand thermometer.

But there is a problem!

For the same inputs and differing inertial frames of reference the outputs must be identical in all respects including magnitude and magnitude versus time. The outputs refer to any measurements taken after the experiment has commenced. This includes temperature versus time...watts per square metre versus time etc.

For experiments A and B having identical inputs

but located on different inertial frames of reference

the outputs of thermometer readings were not the same.

One way of resolving this dilemma is to state

that the speed of EMR emanating from a source

is not influenced by speed of source.

Therefore speed of EMR and hence speed of

visible light is constant.

If there is any part of above analysis with which

you disagree please let me know.

David P..l.k

25 July 2019

Further qualifications to above

a) the Michelson Morley experiment gave the

null result...the speed of light was the same

in any direction...whether the light was

travelling parallel or perpendicular to the direction

of the eaths velocity etc

b) Maxwell derived the speed of EMR and showed

it was constant 1 ÷( kn)^0.5 where k and n

are constants

c) the time in inertial reference frame A is

measured by a clock fixed in position to

reference frame A

Similarly the time in inertial referece B is

measured by a clock fixed in position to

reference frame B

Consider the following example

Platform A of length 6 metres acting as an inertial

reference frame. Point D on earth ( which is also considered an approximate inertial reference

frame is such that the relative velocity berween point D and platform A is zero.

At the centre of platform A is a source of EMR

which can be switched on or off at will.

At the extreme left hand end of platform A is

thermometer J and at extreme right hand is

thermometer K. The thermometers are identical.

At time t equals zero the source of EMR is

switched on.

The time taken for EMR to reach both thermometers

will be 3÷(3× 10^8) = 1 x 10^(-8) seconds.

Hence this will be the time taken for both thermometers to start to register an identical

increase in temperature over any time as

measured by a clock fixed to said platform A.

The temperature time graph is information

that can be considered as one of the many

experimental outputs.

Now consider an identical platform B of length

6.0 metres travelling at a constant velocity

of 2 × 10^(8) metres per second to the right

(relative to point D)

Platform B has two identical thermometers one

located at the extreme left of the platform (J)

and the other (K) at the extreme right of the

platform B.

Assume that the speed of light is not a constant

and show this will lead to a contradiction.

The speed of EMR travelling to the left is

(3 -2) × 10^8 metres per second which equals

1 × 10^8 metres per second.

Immediately after the source of EMR is switched

on it takes EMR a time of

3÷ (1 × 10^8) = 3 × 10 ^(-8) seconds according to a clock fixed to platform B to reach left hand thermometer. In other words the left hand thermometer will only register an increase in temperature at 3 × 10^(-8) seconds ( according to a clock fixed to platform B ) after the source

of EMR is switched on.

But in different inertial frames of reference

A and B for identical inputs the outputs

(temperature versis time) for left hand thermometers are different.

This contradicts the concept that the laws of physics are the same in all inertial reference frames and as a consequence identical experimental inputs on different inertial reference frames A and B must have identical outputs.

The obvious solution to this conflict is to

assume that the speed of EMR emanating from

its source is not affected by the speed of the EMR

source.

In summary

Postulate 1

The laws of physics are the same in all inertial

frames of reference.

As a consequence of this postulate an experiment

with identical inputs carried out on different inertial

frames of reference A and B will have absolutely

identical outcomes.

As a consequence it can be shown that the speed

of EMR is constant...it is not changed by the speed

of the source of EMR.

In other words Postulate 1 logically implies Postulate 2.

DP

3, 8 2019

198) a) Consider 4 boys, 5 girls and 6 teachers.

In how many different ways can they form a straight line queue.

b) Consider 4 black billiard balls, 5 green

billiard balls and 6 tan billiard balls.In how many different ways can they form a straight line queue?

c) are the answers in (a) and (b) the same?

What assumptions have you made?

199) Jack , Bill and Tom have the same amount

of money of 70 cents but none of them have the

same number of coins. The coins are either ten

or twenty cent denominations. Each person

must possess two types of denomination.

What is the total number of coins of each type

held by each person.

200) In a restaurant 38 customers eat vegetables,

39 customers eat soup, 44 customers eat bread,

25 eat vegetables and soup, 15 eat vegetables and

bread, 13 eat soup and bread, 8 eat vegetables , soup and bread. 21 customers do not eat bread,

soup or vegetales. How many customers eat only soup?How many customers are in

the restaurant?

201)Find the sum of the first 100 terms and first

87 terms.

a) 5, 7, -3, 8, 10, 0, 11, 13, 3...

b) 3, -1, 8, 6, -2, 16, 12, -4, 32...

202) Green , purple , red , yellow lights have a frequency of 50, 51, 52 and 53 hz respectively.

At time t = 0 all of the lights are switched on.

At what is the first and second time that all lights

are on?

203) The analog clock shows a time of

8.27pm and later during same day a

time of 9.48 pm

What was angle travelled by minute hand

and the hour hand ?

After 12.00 midday what is first time at which

angles between hour and minute is 30 degree?

****************************************

SET ONE SET ONE

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

Rearranging

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

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