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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

I travel to these locations
Anywhere as using Whats App

One to one tuition in the comfort and convenience of students home at $60 / hour using Whats App or Skype.
Due to risk of transmission of Coronavirus I can no longer offer tuition where I am physically in same premises as student.

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

3) A two digit number is added to a three digit
number to give a value of 393
The twp digit number is subtracted from the three
digit number to give 307.
What is the sum of the digits of the three digit number?
4) The number A is a four digit number whose digits consecutively decrease from left to right.
If the order of digits is reversed in A then the
number B is formed.
Show that A--B is divisible by 9.
What is lowest value of B?
Show that
A + B + 3333
is divisible by 2222.

5) Solve the following equations
7/m --5/n = 3
m/2 + n/5 = 10

6) Given that
(a/b) ^m = 1
what can be deduced?
7) Find one solution for
m + mn + mnp = 2019
where m, n and p are positive integers and
m , n and p are in ascending order.
8) Solve the following equations
7/m -- 5/n = 3
2/m + 25/(2n) = 12

9) Prove that the product of 4 consecutive integers
when added to 1 give a perfect square.

10) Consider an AP whose members are not
necessarily integers but having a common
difference of one.
Show that the product of any 4 cosecutive
members of this AP when added to 1
has a rational square root.
11) Consider a right angled triangle EFG where
EF is hypotenuse.
H lies on EG.
J lies on FG.
I lies on EF.
GHIJ is a square of side length d.
EH = a
IF = b
Find the equation for d in terms of a and b.

12) Cosider a row containing 12 single
digits. In going from left to right the first digit
is 7 and third digit is 5. The sum of any
three consecutive digits is 20.
What is the value of the twelfth digit?

13) Rectangle has an area of 60 square cm..
What is its
a) maximum perimeter
b) maximum length of diagnol.

14) Using induction show that

1^3 + 2^3 +3^3...+ n^3 = (n^2 (n+1)^2) × 0.25
where n is a positive integer.
15) Consider three positive integers p, q ans r.

p is less then q is less then r

p + pq + pqr = 2019

Find one solution for above.

16) Find the sum of all counting numbers between
12 and 55 which are not divisible by 3.
17) Find the smallest positive number which
is divisible by 3, 5, 7 but not divisible by
4 , 6 , 8

18) Hardware store has numbers 4, 6 and 7
for sale.
How many different
a) one digit numbers
b) two digit numbers
c) three digit numbers
can be formed?

19) Given that the area of a rectangle is ab
where a and b are the the side lengths of the
Prove that the area of a triangle is:
base length × perpendicular height × 0.5

20) Bill and David have $126 and $192 in their
bank accounts respectively.

Bill deposits $6 each week every Friday.
David withdraws $4 each week every Friday.
When will Bills account have $24 more then
Davids account?

At the end of how many weeks will they
have same amount of money in their
bank accounts?

21)Coloured beans are placed on a necklace
in the following order from left to right:

3 blue 4 green 6 red 5 blue 6 green 7 red
7 blue 8 green 8 red....

How many of the first 500 beads ( starting
from the left are
a) blue
b) red
c) green

What is the order of the last 5 beads
on the right? What is the order of last
12 beads on the right?

22) Ships A, B and C sound their horn every
30 minutes, 80 minutes and 120 mimutes
respectively. At 3.00 pm on Tuesday they
all sound their horns. By Saturday 10.00 am
how many times will any
a) two ships have
sounded their horns simultaneously?
b) three ships have sounded their horns

23) Given that
y = (3n^2 + 7n -- 2) ( -- 8 + 5cos (2n--1))
you are asked to find dy/dn

The book answer differs from your answer.
What technique would you use to establish
with a very high degree of certainty as to which
answer must be wrong?

24) Consider a mass moving up and down in vertical direction y = (t^2 --4t +3)(t--4) where
t is in seconds and y is in metres.
What is distance moved by mass in first
a) 3 seconds
b) 4 seconds
c) 5 seconds

25) Without using a calculator deduce which is greater in magnitude

2 × (7)^0.5 + 3 (11)^0.5 --(5)^0.5


4(6)^0.5 --2(3)^0.5 + 4(8)^0.5

26) Given that y = (p)^p
and that y = (p)^q
and p = p(x) and q = q (x)

In each case find dy/dx

If w = p^q -- q^p find dw/dx

Given that z = (sin w) ^sin w find dz/dw
How would you check your answer?
Given that r = (3cost) ^ (4 sin 2t) find dr/dt
27) Given that
Z= 4 ÷ (2× (6)^0.5 +3 × (7)^0.5 --8 × (2)^0.5)
rationalise the denominator.
How would you check your answer?
28) Given that u = u(x) and v = v(x)
y = (u/v) ^n
y = (u --v)^n
y = u^n -- v^n
y = ln( (uv)^n)
y = ( ln(uv))^n
y = ln(u/v)
In each case find dy/dx

29) Given that 2p + 4q --5r is a multiple of 17
show that 3p+ 6q -- r is also a multiple of 17

30) Consider two points A and B on Cartesian
The midpoint is (8, 1)
The slope is 3
Straight line distance AB is (40)^0.5
Find co ordinates of A and B by solving
four equations simultaneously.

Is there an easier way to find co ordinates
of A and B? If so what is it and check your

31) A box in the form of a rectangular prism
has a base 18cm by 7cm and height 5cm
but no top.
It is fully packed with cubes of side length 1 cm
in horizontal layers. There is no gap between adjacent cubes.
How many cubes actually touch the sides or base
of the box? What is the volume of these cubes.

32) Pyramid has a square base side of length
n cm and is formed of successive square layers each of length 1.0 cm less then the lower layer.
The top layer is a square 2cm by 2 cm.
What is volume of all the layers.

33) Consider the consecutive integers 1, 2, 3...100.
How many of these numbers are divisible
by 3 or 7?
How many numbers are divisible by 3 or 7 or 4?

34) Pump A fills an empty tank with liquid
in 4 hours.

Pump B fills an empty tank with liquid
in 3 hours

Pump C empties a full tank of liquid
in 6 hours.

If the tank is initially empty , how long will it take
to fill the tank to 0.75 of its maximum volume

a)Pumps A and B are switched on

b) Pumps A, B, C are all switched on

c) Pumps A and C are switched on

If the tank is initially filled to 2/3 of its maximum volume and Pumps B and C are both switched on
what fraction of tank will be occupied in 45

35) Given that the curve y =f(w)) obeys
dy/ dw = 5w^2 -- 6w +10
The equation of the tangent to y = f(w)
is y = 18w -- 9 The point of tangency lies in the
first quadrant.
Find the equation of the curve.

36) The line y = 3w^2 -- 12 is rotated about the w
axis between w =1 and w= 3.
Find the volume generated by considering
two separate integrations from limits 1 to 2
and then limits 2 to 3

Is this equal to integration from limits 1 to 3?

37) Consider the following

( 3 + 2/w) ^ 11 × ( 6w - 2)^8
Find the coefficient of the 3rd term.
Is there a term independent of w? If so
find its value.

38) Show by induction that
1^3 + 2^3 + 3^3 +...n^3 = (1 + 2 +...n)^2
where n is a positive integer greater then
or equal to 1

39) Lucas developed the formula for the value
of the n' th Fibbonacci number
F(n) = ((a +b)^n -- (a --b)^n ) × ( 1/ (2b))

The Fibonnacci sequence is
1, 1, 2, 3, 5, 8....

Find the value of a and b

Hence find the value of the 20 th Fibbonacci

A modified Fibbonacci sequence is
8, 13, 21, 34...
What is the value of the 12 th , 13 th and 14 th term ?
Check your result.

40) Pyramid has a rectangular base
70 cm by 90 cm in the horizontal
130 cm above the mdpoint of the
rectangular base is the tip of the

Find the volume and surface area of
the pyramid.

41) The sum of three consecutive terms of
an arithmetic progression is 60.
and product is 7820.
What is the value of each term?

42) The sum of 5 consecutive members of an
arithmetic progression is 40.
The product is 12320.
Find the value of rach of the five terms.

43) For all positive integers n greater then or equal
to 1 show that
10 × (8)^(n-- 1) + (3)^(3n -- 1)
is divisible by 19

44) If n is a positive integer and also a multiple
of 3 show that
5 × (4)^(n + 1) + (5) ^(2n)
is a multiple of 105

45) Consider a GP. The product of the 3rd , 7th, 11th
15th amd 19th term is 4× (2)^0.5
The sum of the 3rd and 19th term is
(2)^0.5 x 82/9
Find the value of the 7th , 8th and 9th term.

46) Given that y = log (4w + 1) to base (2w--3)
find dy/dw

47) Rationalise the denominator of
1 ÷ ( (6)^0.5 + (7)^0.5 + (8)^0.5)
Check your answer using calculator.

48) Solve the following
4m^2 + 2my + y^2 = 189
2m -- (2my)^0.5 + y = 9

49)Find the domain of w given that y is a
function of w and
w^2 + wy + y^2 = 9

50) In a right angled triangle the hypotenuse
is of length 6 units and the other two sides
are of length:

log ((w^2 + 3)^2) to base 7

log ( w^2 + 3) to base 3

Find the value of w.
51)Given the sequence of numbers
8 , 3 , 12 , 5, 18, 7, 27 Find the sum
of the first 30 terms and the first 19 terms

Find the equation for sum of first n terms
a) when n is even
b) when n is odd


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 silution 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 integer.
What is the minimum average of all four

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

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

b) find the length of each side of the triangle

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

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

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

b) find the lenths of each side of the quadrilateral

c) what is the minimum length of each diagnol?

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

30) The addition of three numbers gives Y E S

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

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

a) 2^ 17 + 4 ^ 17

b) 3^20 + 7^ 60

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

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

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

33)Consider the following sequence


Which letter represents the

a) 45 th term
b) 93 rd term

34) Prove that

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

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

Are the following odd or even ? Give reasons.

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

f(t) ÷ f(-t)

f( y) × f(-y)

( g( -t))^2

35) Consider the pattern

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.

100) Consider a rectangular prism BCDEFGHI.

BCDE and FGHI both lie in the vertical plane.

Base DEHI lies in horizontal plane.

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

Prism is 1, 4 full of liquid.

Prism is rotated 30 degrees anticlockwise
about axis EH.

What is new height of water surface above EH

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

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

Find the perimeter of triangles EAB and EDC.

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

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

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

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

104) M and N are positive integers

M^ 0.5 and N^ 0.5 are both surds

Given that
M^0.5 + N ^ 0.5 = P

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

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

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

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

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

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

How can this be done?

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

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

Explain why the procedure used is valid.

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

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

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

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

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

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

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

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

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

117) Consider the following

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

a) 11111 × 11111

b) 11111 × 11111

c) 11112 × 11111

d) 111114 × 111115

e) 111109 × 111113

f) 2222 × 3333

g) 8888 × 2222 + 44444

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

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

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

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

120) Consider three classes of English, Maths and

A total of

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

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

121) The number 51368 is multiplied
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

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

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

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

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

133) The number 109, 032 is divisible by factors
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
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
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

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

b) cost of 5 apples plus 5 mandarins

c) which costs more apple or one mandarin

d) 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


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

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
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

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


Bachelor of Mechanical Engineering (University of New South Wales)

Master of Engineering Science (University of New South Wales)

Diploma of Education (Australian Catholic University)

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