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Mr David Pollak

 Sydney - Baulkham Hills

Shining light of calculus solves Natures darkest mysteries

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

Service areas
Anywhere as using Whats App

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.



Former HSC marker in Physics.

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 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

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

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

Engineering Studies Years 11-12

Physics 2 U Years 11-12

International Baccalaureate Years 11-12 Mathematics (All levels)

International Baccalaureate Years 11- 12 Physics

Gamsat Physics

UMAT Physics

Ten years teaching High School Mathematics (all levels)

Science Years 7-10

Physics (years 11-12) in High School.

Marking of HSC Physics examinations.



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

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.

Find the value of the area bound by tangent
and curve and the y and w axis in the first

Find the area bound by tangent, curve, y axis, w axis
in the second quadrant.
Find the volume generate by rotating this area
(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


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
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
Find the value of the sector angle and its

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.

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.

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

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
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

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

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
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

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


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

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.

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
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
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
sin 72
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.


6) Without using calculator find exact value of

1÷( 1^0.5 + 2^0.5) + 1, ÷(2^0.5 + 3^0.5)
+ ......1÷ (20^0.5 + 21^ 0.5)

7) Find the smallest and largest 3 digit number

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

8) Solve
1÷ a + 1÷ (ab) + 1÷(abc) = 5÷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

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

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

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

a is less then b is less then c

a + b + c = 5

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

abc = 1

12) Solve

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

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

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

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

16) Factorise

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

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

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

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

( 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
When it is half full of water what is the height
of water surface above ground?

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

20) Consider the sequence
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
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
What is the minimum average of all four

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

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

b) find the length of each side of the triangle

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

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

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

b) find the lenths of each side of the quadrilateral

c) what is the minimum length of each diagnol?

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

30) The addition of three numbers gives Y E S

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

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

a) 2^ 17 + 4 ^ 17

b) 3^20 + 7^ 60

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

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

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

33)Consider the following sequence


Which letter represents the

a) 45 th term
b) 93 rd term

34) Prove that

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

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

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

Find the value of
a) 15 th digit

b) 30th digit

c) 50th digit

d) 109th digit

Find the sum of first 50 digits.

Find the sum of the first 20 even digits.

44) Given M = 12!
How many

a) square numbers are factors of M

b) cubes are factors of M

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

What is value of J?

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

a + (c ÷(cb + 1)) = 45, 7

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

48) Given the following arrangement of numbers


5 7

9 11 13

15 17 19 21

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

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

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

b) the ones digit is twice the tens digit.

c) the ones digit is less then the tens digit

If J is even what are answers to above?

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

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

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

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

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

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

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

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

AB = 109^(0.5)

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

b) perpendicular height from AB to C

Also find lengths of AC and CB

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


For the above multiplication find
the value of each letter.

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

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

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

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

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

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

is greater then

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

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

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

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

a) 100 terms

b) 203 terms

c) 307 terms

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

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

67) consider a 4 digit number J

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

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

68) Express as an exact fraction

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

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

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

70) Given that

a! ÷ b! = 2730

Find the values of a and b.

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

cos 36 = 0.25 (1 + 5^0.5)

Hence find exact value of
Sin72 tan36 cos18 sin81

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

b) regular pentagon

72) Given that

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

Find one solution to the inequality.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

5! + 7! + 9!

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

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

What is the probability that the number chosen

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

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

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

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

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

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

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

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

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

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

Sketch large good quality diagram

What is the value of r?

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

What is radius of circular base and perpendicular
height cone?

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

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

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

96)Consider the following long multiplication.

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

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

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

M A M × M A M = M A D A M

where the capital letters represent integers
from 0 to 9

Find the value of each capital letter.

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

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

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

What is wetted area of inside of cone?

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


Bachelor of Mechanical Engineering (University of New South Wales)

Master of Engineering Science (University of New South Wales)

Diploma of Education (Australian Catholic University)

Qualified experienced teacher.

Working with children check approval from NSW Government.


One to one tuition is now available (using WA)
in 30 minute sessions for $30

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Joined Tutor Finder on 13-Mar-2017 (updated profile on 24-Oct-2020)
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