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One to one tuition in the comfort and convenience of students home at $30 / minute in 30 minute sessions using Whats App.

Due to cv19 all one to one tuition is online using WA.

During tuition there is a constant exchange of

photographs of questions, student solutions and my solutions.

My contact number is 0412 995 933.

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 teachers or other sources (e.g., 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.

I set homework based only on topics and examples discussed during tuition.

I focus tuition on future examinable topics as outlined in exam notification sheet. I find out the date of next exam and topics and prepare appropriately.

Working with children check approval from NSW government.

High school Mathematics, Physics and Science teacher of ten years teaching years 7-12

I have been tutoring for 21 years.

For year 7 - 12 students fee based on $60/hour of High School or IB (Mathematics , Calculus, Physics)

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 maximum 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.

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.

As there are no other students present, the student need not feel embarrassed 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 embarrassed about asking questions....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 - whenever reasonably possible using 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 than receive it without background explanation and derivations.

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

Experiments in Mathematics help students bridge the gap between theory and practice and better able to understand the more abstract theories (e.g., Simpsons rule/integration, 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, Galileo's experiment, how to calculate, radius and mass of earth using three simple measurements, Newtons Law of Universal Gravitation, Lenz's 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 revised (2018) Physics syllabus 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 or Science at a university level.

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 tuition. 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 demoralising 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 include formulas examples and an index of topics.

A separate smaller note book containing index and summary of topics as written by student immediately after successfully answering homework questions. These contain formula and examples. 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.

________________________________________

SAMPLE CHALLENGE QUESTIONS DEVISED TO MAKE STUDENTS THINK AT A DEEPER LEVEL

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

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

SET FOUR

1) The present age of Bill is m^2. The present age

of his brother is 2m. When Bill is 7 years of age the brother is 3m years of age. What is

present age of Bill and his brother?

2)Find the sum of the digits of the first

a)33 odd counting numbers.

b) 34 even counting numbers

c) 16 numbers which are divisible by 3

90) Rectangle BCDE has following data

BC = DE = 1

Point F lies on BC

Point G lies on CD

Angle GED = m

Angle FEG = n

Angle FGE = 90

Sketch.

Find expressions for lengths of all intervals.

Hence show that

tan (m + n) = ( tan m + tan n) /( 1 -- tan m tan n)

91) Solve the following inequalities

| 4m³ -- 2| > -- 30

| 9m¹² -- 5m⁶ | > --8

| 6m⁵ -- 2m⁴ + 5m³ -- 11m +1| < -3

| 6m² + 11m| < 7

92) Find the equation of the tangent and normal to the curve

y = w³ +2 at w= 1

Find the point of intersection of tangent to curve.

Sketch

Find the value of the area bound by tangent

and curve and the y and w axis in the first

quadrant.

Find the area bound by tangent, curve, y axis, w axis

in the second quadrant.

Find the volume generate by rotating this area

about

(i) w axis

(ii) y axis

93) Volume V of an object is generated by

rotating a curve y = f (w) about w axis isp

V=( 0.8w⁵ + 3w⁴ + (13/3)w³ + 3w² + w) × pil

What is the formula for y = f(w) ?

94) Sketch the curve

y = w³ -- 5w² + 6w

Show intercepts on w axis.

Find the maximum and minimum values

of y accurring between w= 0 and w = 6

Find value of area bound by w axis, the cuve.

and w =1 and w = 4.

Find the value of volume generated by rotating the curve

(i) about w axis and between w = 0.5 and w = 4

(ii) about y axis and 0.4 < y < 0.6

95) Find points of intersection of

circle of radius 3, centre (--2, 1) and

and straight line y = --2w

Sketch

Find the area bound by the straight line

y = --2 and the arc below same straight line

using integration.

Find the volume generated if

the area above the straight line

and below the arc is rotated about

the straight line.

96) Consider a six digit number L of the form

abc, def

The first three digits are exchanged with the

last three digits to form the six digit number M

of the form def, abc

It is known that L + M = 1, 270, 269

What are two sets of the values of digits

a, b , c, d, e, f ?

97) There are 32 students in a class and all

students study maths.

26 students pass the

first maths test and 29 pass the second maths

test.

25 students pass both tests.

How many students fail both tests?

98) The volume V of a full tank is 800 litres.

It releases liquid at the rate

dv/dt = 30t -- 150

when tap is fully open

a) What is rate at which liquid is released

when t = 0, 1, 2, 3, 4, 5 seconds when tap

is fully open?

b) How much liquid is discharged at the end

of 3 , 4 , and 5 seconds?

c) Can the fully open tap fully drain the tank?

Give reasons for your amswer.

b) Find an expression for the volume V in

terms of t, given that at t= 0, V = 600

c) Sketch the volume versus time equation

d) What happens when t > 5 seconds?

99) The combined surface area of a sphere and a

cube is 24 cm². Let the radius of sphere be r and

the length of one edge of the cube be w.

a) What is the domain of r and w ?

b) Is there a maximum total volume and if

so find its value? Find the corresponding

values of r and w.

c) Is there a minimum total volume and if

so find its value? Find the corresponding

values of r and w.

100) A sphere of radius r and a cube of side

length w have a total fixed volume

of 48 cm³.

a)What is the domain of r and w?

b) Is there a minimum total surface area

and if so find its value? What are the

corresponding values of r and w?

c) Is there a maximum total surface area

and if so find its value? What are the

corresponding values of r and w?

101) The pay per hour for a given employee

during a given week is as follows

Zero to 35 hours at $28/hour

For 35 to 40 hours at $42/hour. This rate is

only applied after 35 hours of work during

the week.

For 40 to 46 hours at $54/ hour. This rate is

only applied after forty hours of work during

the week.

For example an employee who

works a total of 44 hours during a

given week would receive following

gross pay

35 × 28 + 5 × 42 + 4 × 54

An employee can work a maximum of 46 hours

in a week.

How many hours did an employee work during a given week if the gross pay was

a) $800

b) $ 1000

c) $1100

d) $ 1200

e) $ 1300

f) $ 1400

102) Referrng to previous question in the second week an employee works for 3 more hours then in the the first week.

His gross pay for two weeks work is $2350.

How many hours did he work in first and second week?

103) A sector has an angle of 30 degrees.

The area of the minor segment is 0.5 cm²

The extermal perimeter of the sector is 17.5 cm

Sketch.

Find the value of the sector angle and its

radius.

104) A circle of radius 4 has a centre O at (3, --2)

and intersects the straight line y = -- 0.5 w + 1

at points B and D. The horizontal axis is w.

Sketch large diagram...w axis is horizontal.

Find

a) co ordinates of B and D

b) value of angle BOD

c) area of triangle BOD and minor segment

and major segment.

d) perimeter and area of sector

105) Given that 0 < w < 2 pi

find the range of

a) y = sinw cos w

b) y = 4 sin 2w × cos w

Sketch the curves.

106) Given that the domain

--5 < w < 7

Find the range of

y= ( 2w--5)( w + 1) ( w + 3)

Sketch the curve.

107) Consider the cubic

y = aw³ + bw² + cw + d

Why is it incorrect in general to say that if

a> 0

then the curve must be concave up?

108) Given that

y = (4w--2) ( 2w + 3) ( w + 1)³ (w -- 5)² = f(w)

Sketch. How many maximums are there?

How many minimums are there.Do not find

co ords of maximums / minimums.

a) y versus w

b) dy/dw

c) d²y/ dw²

d) |y|

e) 1/y

f) f( --w)

g) square root of y

h) f ( |w|)

i) y²

k) |f(-w)|

109) Triangle BCD has side lengths of

8 cm, 12 cm and 16 cm.

Without using trigonometry find area of

triangle BCD.

A circle

a) passes through all three vertices B, C, D.

b) touches all three sides of triangle BCD

Find area of circle in ( a) and (b)

What is equation of circle in (a) and (b)

110) in the previous problem let

BD = 8 BC = 12 CD = 16

and co ordinates of B and D

be (0, 0) and (8, 0) respectively .

Find the co ordinates of C and area of

triangle BCD without using trigonometry.

Triangle BCD is rotated about BD.

Find the value of the volume generated.

111) Given the sequence of numbers

230, 554, 878, 290, 534, 858 etc

What are the values of 7th , 8th and 9th term

Find the sum of the first

a) 8 numbers

b) 11 numbers

c) 50 numbers

Find the sum of the digits for (a) , (b) , (c)

112) Show that

sec(w--1) sec w = [tan w -- tan (w--1)] ÷ sin 1

All angles are in rads.

Hence find expression for

a) sec (w--2) sec (w --1)

b) sec (w--3) sec(w --2)

c) hence find

sec (w--2)sec (w--1) + sec(w--3)sec(w--2) +

.....sec(w--9)sec(w-- 8)

113) Consider the inequality

[ ( 5/6) to the power n] > 0.18

What is the least integral value of n

for which this is true?

This is normally solved by

n ln (5/6) > ln 0.18

However , some caution is now required.

Why?

114) Consider the following sequence

3, 5, 6, 15, 12, 45...

Find the value of the 12 th and n' th term

If n is an even number find the sum of the

first n terms.

115) From first principles find the acute angle between the two straight lines

3w -- 2y + 8 = 0

--6w --3y --5 = 0

The vertical axis is y and horizontal axis is w.

Do not use the conventional formula for finding the

angle between the two lines.

116) Given that T = A + [B/( e to index kt)]

where A, B and k are all non zero

positive constants and T is temperature

and t is time find

z= dT/dt and w= dT²/dt²

If w is the vertical axis and z is the

horizontal axis sketch z versus w

If z is the vertical axis and t is the

horizontal axis sketch dT/dt versus t

117) Given that

dy/dw = (5w -- 17)/[(w -- 5) (w -- 1)]

a) sketch dy/dw versus y showing

asymptotes and co ordinates of maximum,

minimum value of y

b) Find equation for y in terms of w and

c) Sketch y versus w.

118) Sketch

y = [ (2w --1)(w + 3)] ÷[(w -- 3)² (3w + 5)]

Use y as vertical axis and w as

horizontal axis.

Sketch y versus w.

showing asymptotes and zeroes

Sketch 1/y versus w

Sketch y to index 0.5 versus w

Sketch dy/dw versus w

Do not find co- ordinates of max/ min

for above sketches.

119) Show that

sin ( a +b) sin (a -- b) = sin² a -- sin² b

If sin (a +b) = 0.7 and sin (a -- b) = 0.3

find the vaue of a and b given that

a and b lie in first quadrant.

Find the expression for

sin (5b) sin (3b)

If b is constant and w varies find the

expression for indefinite integral of

sin (w + b) sin (w-- b) dw

120) How many terms of the sequence

50 , 47 , 44 . 41......

a)have squares less then 300 ?

b) have squares between 500 and 950 ?

c) have cubes between 7 and 999 ?

121) Find the value of the constant term in

( 1 + w)⁵

( 2w + 3/w)⁵

(1 + w + 1/w) ⁵

(2 + 3w)² ( 4 -- 1/w)⁵

Knowing the expansion for (1 + w)⁵ find

(without using binomial theorem)

the expansion for

(1 + w)⁶

(1 + w)⁴

( a + w)⁶

How would you check your answer for the above

expansions?

122) solve the following

n! + m! = 126

( n!)² -- (m!)² = 14, 948

123) Express

2! 3! 4!......10! = m

Express m as product of integral powers of

the prime numbers 3, 5, 7 ...etc

If m/n = r find the maximum and minimum

values of integer n

such that r is integral square number.

124) A fair dice has 6 faces numbered 1 to 6.

What is the probability of a total of 9

twice in 8 tosses of a pair of such dice

What is the probability of

a total of 9 at least twice in 8 tosses of a pair

of fair dice.

125) a) Consider the sequence of four terms

shown

5/ 7........15/9 ..........45/ 11......

What are the values of the 4'th, 5'th

and 6'th term?

What is the formula for n' th term

b) Consider the sequence of four terms

shown

5/ 7.... ( 5 + 15)/(7 + 9).....

(5 + 15 + 45)/(7 + 9 + 11)....

What is the value of 5'rh and 6'th term

What is the formula for n' th term?

126) Consider the following discrete probability

distribution

w 5 6 7 8 9

P(W = w) 0.1 2m 3m 4m -- 1 0.2

Find value of

a) m

b) E(W) and E(W²)

c) variance and standard deviation for above

original discrete probability distribution

d) E (3W --2)

e) is it possible to find exact value for

P(W = 6.5)

Give reasons.

127) what is meant by a uniform probability

distribution?

What is its mean and standard deviation?

Give an example.

128) given that

y = (t ln t) -- t

Show that dy/dt = ln t

Find the integral of

a) -5 ln t³

b) (2t -- 1) ln (2t--1) -- 3t + 8

c) (4t -- 3) ln (4t -- 3) + 5/(4t -- 3)

129) Given that

f(w) = 3w² -- aw

Find the value of a if the tangent at the vertex

has y intercept of - 8 and then sketch

f(w) versus w.

Find the equation of tangent and normal

at w = --2

130) Consider the point J(a, b)

The vertical axis is y and horizontal axis is w

Show that the reflection of J about y = w

is the point (b, a)

131) Consider y = f( w) passing through J (a, b)

Show that the slope of the tangent at point J

has the same sign as the slope of the tangent on the inverse but passing through N( b, a)

The verical axis is y and the horizontal axis is w.

132) Consider the equation

y = 2w ( w + 3) where y is vertical axis and w

is horizontal axis.Sketch

Is original equation a function or relation ?

Find the equation for the inverse and sketch.

Is it a function or relation?

Given that w is greater then or

equal to zero find domain and range

of the original equation and its inverse.

The straight line

y= --w + 4 intersects the original

equation at points P and R respectively.

Find the area bound by the straight line PR,

the origin, the original equation and its

inverse .

143) Given that z = p(w) q(w) r(w)

find the general formula for dz/dw

I f r(w) = a where a is constant

find general formula for dz/dw

If y = b/ { p(w) q(w) r(w)}

where b is constant

find general formula for dy/dw

144) Given that

y = (sin w + cos w)⁴ ( sinw -- cos w)⁶

Find dy/dw

145) Consider two cars A and B separated

by a straight road of length CD =50 km.

Car A and car B are both initially stationery

at points C and D repectively.

At 7.00 am both cars C and D leave A and B

respectively, travelling at constant speed

toward each other . The speed of car A is

2/3 that of car B.

a) At what distance from

their respective starting points do the

cars meet?

b) A fly sits on car B. When both cars

start moving the fly maintains a constant

horizontal speed of 4 times that of car B.

When the fly reaches car A it reverses direction

travels toward B with unchanged speed.

When fly reaches car B it reverses speed and

moves at constant speed toward car A

The fly constantly reverses direction of speed

after reaching a car but always moves at

constant speed.

What is distance travelled by fly when

cars A and B meet?

When cars A and B are 10 km from each other

what is position of fly and how far has

it travelled?

146)Find the equation of the circle of

radius 2 and circle centre (3, 1)

Hence find the integral of

[ 1 + { (-- w² + 6w -- 8) }¹/² ]dw

between limits of

w = 1 to w= 5

and w is the horizontal axis.

147) An isosceles triangle has a perimeter of

24 cm . Find its maximum and minimum

area.

148) Force of 4i + 2j acts on a mass located

on a ramp 3i -- 5j where i and j are the

perpendicular unit vectors and and there is

zero gravity Find the magitude and

direction of force

acting parallel and perpendicular to the ramp.

149)Given that

u = (a ⁰•⁵ )+ (b⁰•⁵ )

z = ( a⁰•⁵) -- ( b⁰•⁵)

show that u² + z² = 2(a + b)

and that uz = a -- b

SET THREE SET THREE SET THREE

5)Sketch the following over one period showing the co ordinate of the first maximum and first minimum to the right of the y axis.

Horizontal axis is w.

g equals pi.

y = 4 cos ( 3w - 0.25g)

y = -2 sin (2w + 0.2g)

y = 5 + 3 cos( -4w + 0.3g)

6) Consider the following inequalities

a + b is less then c

d + e is less then f

Therefore

a - d + b - e is less then c - f

Is last inequality true? Give reasons.

7) A bag contains

3 identical blue balls

4 identical green balls

Two balls are removed one at a time

without replacement and placed into a box.

How many combinations are possible .

What is probability of a green and blue ball

in the box.

How many permutations are possible ?

What is probability of drawing a green and

blue in this order?

8)Solve the following

(m^2 -- 4m --21) × (m^2 -- 6m -- 16) = 264

Do not guess.

9)Solve

a) [sin ( 2m + p)]^2 + [ cos (2m +p)]^2 = 1

b) [ sin ( 3n + g)]^2 + [ cos (3n -- g)]^2 = 1

where g = 2pi rads.

c) [ sin ( 2r -- 0.5g) ^2 + [cos( 2r + 0.5g)]^2 = 1

10) An isosceles right angled triangle ABC

has side lengths AC = CB = n and perimeter p.

What is the radius of the largest area circle

that can fit inside this triangle?Where is the

centre of the circle located?

11) Set A contains 45 odd numbers and Set B

contains 29 even numbers.

Determine if the sum, difference, product of

numbers in Set A and Set B

is even or odd.

12)While at work Jack observed that the time that had elapsed since 2.30 pm Tuesday was equal to the half the time that remained until 10.00 am of

Wednesday , the following day.

At what time did Jack make his observation?

13) Triangle has side lengths

5n and 3n--1 and n+8

What is domain of n?

What is its area in terms of n?

What is maximum area?

14) using a compass , sharp pencil, straight edge

only show how to divide a stroight line of length

L into two portions

One portion of length (La)÷b

Other portion of length (L (b--a)) ÷ b

where a and b are integers having no

common factors.

For example given a straight line AB on a

sheet of paper show how to locate point

C on line AB such that

AC × (19 ÷7) = AB

15) Show that the product of any four consecutive

positive integers plus one gives a perfect

square.

16) Find one solution for the following three

simultaneous equations

2m = p(3m^2 + 3n)

2 = p(3m +3n^2)

5 = m^3 + 3mn + n^3

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

and a^2 + b^2 = c^2

show that at least one of the integers

has a factor of 5

18) coside the points

A(7, 3) and B(5, 9)

The diameter of a circle is AB.

Point D lies on the circle and angle DAB is 30

degrees.

Sketch above.

What is value of area of

a)triangle DAB?

b) DAB where AB is the arc of circle.

c) without using calculus find the equation

of the tangent and normal to the circle

at point C

19) The sum of the 4th, 6th and 8th term

of a GP is 3.3125. The fifth term

equals 0.75.

Find the value of the 12th term.

20) Solve without guessing

m^2 + mn +n^2 =133

m -- (mn)^(0.5) + n = 7

21) Show that

(tan 36)^2 = 5 -- 2(5)^0.5

Hence find exact value of

sin 36

cos 36

tan36

sin 72

cos72

tan 72

22) Given that

(tan36)^2 = 5 -- 2(5)^0.5

Using sharp pencil, straighr edge , compass only

show how to construct an angle of

a) 36 degrees

b) 81 degrees

c) 33 degrees

23) Consider the right angled triangle ABC and

angle ACB = 90 degrees.

D is a point on BC such that DA = BD

Let angle ABC = w

Using above tiangles show that

Sin 2w = 2 sinw cos w

Hence find an equation for sin 8w

in terms of w

24) Using a compass, sharp pencil and straight

edge only construct a triangles whose

side lengths are in ratio of 5:6:8

25) Four identical circles having radius 5 cm

and centres A, B, C and D touuch each other

ABCD is a square of side length 10 cm.

a) Sketch

b) Find the area between the four circles

26) Two circles are of radius 5 cm and 8 cm

and a distance of 14 cm between centres

A and B.

Tangents CD and EF are drawn to touch

both circles at C, D, E and F. The tangents

do not intersect.

Find the area and perimeter of ACDBEF.

SET TWO SET TWO

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÷A26

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

( 2cos m - 3sin m) ^2 -- (cos m + 5-sin m)^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.

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

c) [ f(t) + f(--t)]⁵ is even

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.

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