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HSC Physics and Mathematics

 Sydney - Baulkham Hills

Shining light of calculus solves Natures darkest mysteries

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

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

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

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

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

Set homework based only on topics and examples discussed during tuition

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

Working with children check approval from NSW government.



Parents and students

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

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

Advance payments not required.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Mathematics Years 7-10 all levels

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

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

Engineering Studies Years 11-12

Physics 2 U Years 11-12

International Baccalaureate Years 11-12 Mathematics (All levels)

International Baccalaureate Years 11- 12 Physics

Gamsat Physics

UMAT Physics

Ten years teaching High School Mathematics (all levels)

Science Years 7-10

Physics (years 11-12) in High School.

Marking of HSC Physics examinations.


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


1) Find the three smallest consecutive integers such that

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

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

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

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

4)Given that

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

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

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

Three pumps A,B and C extract water.

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

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

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

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

6) Without using calculator find exact value of

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

7) Find the smallest and largest 3 digit number

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

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

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

9) Express as powers of prime numbers

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

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

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

a is less then b is less then c

a + b + c = 5

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

abc = 1

12) Solve

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

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

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

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

16) Factorise

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

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

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

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

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

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

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

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

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

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

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

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

23) Given that

Log( (m^2 ) × (p^2n)) = 1

Log (m^(2n) × (p^2)) = 1

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

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

24) Given that

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

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

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

Note that lcl means absolute value of c.

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

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

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

b) find the length of each side of the triangle

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

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

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

b) find the lenths of each side of the quadrilateral

c) what is the minimum length of each diagnol?

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

30) The addition of three numbers gives Y E S

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

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

a) 2^ 17 + 4 ^ 17

b) 3^20 + 7^ 60

31) Given that

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)

35) Consider the pattern

a) Find the value of the 256 'th term

b) Find the sum of the first 163 terms

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

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

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

39) How many 3 digit numbers are divisible by 14
and 38 ?

40) How many positive integers greater then 100
and less then 1000 can 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) 10 09th digit



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

Must first understand the concept of e

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

Does there exist a value of B such that

df/dx = B (power x)

(Unchanged by differentiation) and if so find its value

Using the fundamental definition of differentiation

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

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

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

h + 1 = B (power h)

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

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

Let h equal a very small number eg 0.000001

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

B = 10 power 0.4329

B= 2.718 (approximately value of e

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

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

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

5×5 + 12×12 = 13× 13

5) Given that m and n are both positive integers

m^3 + n^3 = 854

find the values of m and n using algebra.

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

7) a star has a radius of 42673 4896875 metre

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

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

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

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

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

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

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

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

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

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

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

One fifth of area of ABC

One eleventh area of ABC

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

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

19) consider a triangle
AB= 4m-10

Find the value of
(sinA) ÷ ( sinB)

Find value of all internal angles

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

a) the product of a positive and negative number is

b) the product of two negative numbers is positive

21) Without using calculator find which is larger

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


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

22) a triangle has side lengths

m×m + 1

m×m +7

3m + 1

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

23) The Chefs Problem

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

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

Find the following

a) mass of sugar to to total mass of ingredients

b) mass of water to total mass of ingredients

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

Find the answer to (a) and (b)

c) What do you notice?
d) Why?

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

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

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

23) Factorise

4 x(a to power 4)+ 81×(c to power 4)

24) consider a straight line AB of length m

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

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

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

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

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

Which contains more chocolate

a box containing a large number of small diameter discs


a box containing a small number of large diameter discs



Is there a maximum amount of chocolate ?

Give reasons for your answer.

27) which is larger

99 to power 84

84 to power 99

Do not use logs or calculator
Give reasons

Consider a triangle whose sides are of length

4c + 5



What are the allowable values of c

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

a) what is the speed of m relative to M

b) what is the speed of M relative to m

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

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

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

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

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

30) explain the following dillema

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

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

Give reasons for your answers.

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

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

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

What is the definition and characteristics of cathode rays?

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

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

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

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

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

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

Under what conditions are above statement true?

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

33) Consider two different investments A and B

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

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

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

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

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

34)consider thefollowing

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

Rationalise the denominator.
Check your answer using calculator.

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

a) any triangle
b) any quadrilateral

Give reasons

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

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

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

ABCD and EFGH have corresponding parallel

Find the differerence in perimeter and area of the rectangles

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

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

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

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

The vertical axis is y

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

40) Consider

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


What is domain and range

41) what is the domain of

y = 3 ÷ ( ln ( w x w - 5w + 6))

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

42) consider the following...

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

43) develop a formula for finding the value of

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

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

b) both m and n are mixed numerals

for the equation below

m × m - n× n = 27

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

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

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

Both cars start journey at same time toward each other.

When and where do the cars meet?

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

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

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

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

Find the original dimensions of the rectangular prism.

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

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

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

51) what are the characteristics of a black body.

Is it necessarily black? Give reasons

52) Consider the equation

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

Find g in terms of f

Find f in terms of g

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

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

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

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

b) Find an alternative way to prove( a )

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

How many members are in each Group?

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

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

show using calculus that

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

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

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

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

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

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

Perimeter of A is twice that of B

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

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

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

60) Consider equations

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

Find value of
-7b + 12c

14a - c

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

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

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

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

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

64) Consider coins of following denominations

5 cent 10 cent 20 cent 50 cent

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

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

66) Consider two isoceles triangles

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

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

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

Find total cost of 11 apples and 10 mandarins.

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

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

70) Consider a triangle ABC of area W

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

Sketch a large good quality diagram

Find the following

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

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

Find the dimensions of the triangle and find its

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

y = ex + d

Find the perpendicular distance between them

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

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

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

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

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

83) a) Find the prime factors of 9331

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

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


m^a means m to power of a

84) Prove that

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

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

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

d) if a number is divisible by 8 its last two digits
are divisible by 8

e) if a number is divisible by 24 then the
sum of its digits is divisible by 3 and the last
two digits are divisible by 8

85) Find the value of

5 - 7 + 9 - 11 + 13 -15 ......89 - 91
Using two different methods

86) given that a^n means a to the power n

Develop a formula to evaluate

5^2 - 7^2 + 9^2 - 11^2 ...89^2 - 91^2

87) Solve

m^2 = 4 - mn

n^2 = 5 - mn

88) Five persons A B C D E have a total sum of money equal to $130

A and B have a sum of $50
C and D have a sum of $35
E and A have a sum of $69

How much money does B have?

89) Given that m and n are prime numbers
solve 5m + 9n = 712

90) Factorise 91

Given that and b are positive integers solve

a^ 3 + b^3 = 91

without guesswork.

91) A 2.0 kg mass is acted upon by the following
external forces

11 N east 35 north
15 N east
16 N east 60 south
10 N south
24 N west 65 south
3 N west
8N west 40 north
18N north

Find vector sum of forces in east west direction
by using resolution

Find vector sum of forces in north south direction
by using resolution

Using above answers find net force and acceleration

Using graphical means solve above

92) In a bag there are 4 green balls. The remainder are either yellow or red.
The probability of taking out two green balls without
replacement is 2/7.
Find original number of balls in bag.
What is the probability of taking out (without replacing)
a) three green balls
b) three red balls
c) three yellow balls

93) in a GP the sum of the first three terms is 9
The sum of next three terms is 72
Find the vaue of first term , the common ratio
and the sum of 12th 13th and 14th term.

94 ) if the common ratio in a GP lies between -1 and 1
then the GP has a sum to infinity.
What does this mean?
How should the statement be worded.

95) The product of four prime numbers is 149071
Without using algebra find the numbers.

96) The product of a prime number and15 and added to the product of a different prime number and 8...the total sum is 134

Find value of each prime number.

97) Car A and car B leave point C at the same time
and travell along same road in same direction.
Car A has a speed of 70 kph and car B has a speed
of 60 kph.
At what time and distance will car A overtake
car B?
At what time will car A be 5 km in front of car B.
What is the total distance travelled by each car
in this time?

98) Find the average of a and b if
31-4a= 4b + 18
Find the value of
a^2 + 2ab + b^2

99) The four edges of a square are licated on a circle
of radius r.
If the difference between the area of square and
area of circle is 8 square cm find the radius of
circle and length of side of square.

100) using induction show that

-1 + 2 (power 2n) is divisible by 3
for all positive integers n

Is there a faster way to prove above without using

101) consider two trains travelling in opposite direction
Train A ( of length 2.2 km) leaves station C at 9.00 am travelling east at a
constant speed of 65 kph.
Train B ( of length 1.8 km) leaves station D at 9.00 am travelling west at a
constant speed of 80 kph

Station C and D are separated by a a straight line distance of 450 km.

a) when is the front of train A in same position
as front of train B

b) when is front of train A in same position
as rear of train B

c) when is rear of train A in same position as
front of train B

d) when is rear of train A in same position as rear
of train

e) for what time period is any part of train A next
to any part of train B

102) Consider a car travelling at night.
If the reaction time of driver to apply brakes
is 0.7 sec and the deceleration of the car is
0.8 m per scond squared .
The headlight beam can only see 80 m into
the distance.
What is maximum allowable speed of car in order to
avoid crashing into unexpected obstacle on rhe road?

What are the implications for the setting of safe
speed limits for cars?

Cosider a train 1.5 km long travelling at 70 kph.
For how long will any part of the train be in a tunnel
of length 2.0 km?

104) The hypotenuse of a right angled triangle
is of length

(2ab) to power 0.5

The other side lengths are a and b

What are the lengths of and b
How many solutions exist ?

105) consider an analogue clock.
How many times ( during a period of 8 hours )
commencing at 3.0 pm) do the hour and second hand

106) consider the sequence
3,4,7, -2 ,-5,-8, 3,4,7, -2,-5,-8....
Find the sum of the first
a) 98 terms
b) 99 terms
c) 100 terms

107) What is the last digit of

( 32 to power 53) plus ( 8 to power 51) ?

108) Consider the following
4132431442 × 10,000,000,002 = y
What is the sum of the digits for y?

109) The pages of a book are numbered from 1 to 40
a) what is the sum of these numbers
b) what is the sum of the digits of all of these pages.

110) An apple, bannana and pear cost 150 cents.
A bannana pear and lemon costs 170 cents.
A lemon and apple costs 90 cents.

What is the total cost of
a) 3 lemons 3apples 3pears 3 bananas

b) 5 apples,2 bananas, 2 pears and 5 lemons

111) Without using calculator find which is larger

( 5 to the power 0.5 ) + (7 to the power 0.5)


( 6 to the power 0.5) + (3 to the power 0.5 )

112) A rectangle has a diagnol of length 17 and
and area of 120.
The dimension of the rectangle are a and b

Without solving for a and b find the value of the

a ( power 2) + b (power 2)


a (power 6) - b (power 6)

113) consider the first n consecutive multiples of 12

eg 12 24 36 ........12n

Find the average of these n multiples in terms of n

114) Consider a triangle ABC of area 43.65
square cm.

CE/CB= 1/2 and AD/AB = 5/11 and CF/CA =4/13
Points D , E , F lie on AB , BC and CB respectively.
Draw diagram showing above information.

Without finding numerical value of any angles or
lengths find

Areas of triangles ADF , DBE ,CEF, FDE

Area of quadrilateral FDBC

Given that

4(y-2) to power 3 - 5(y-2) to power 2 - 6(y-2) + 8

= a ( y to power 3) + b (y to power 2 ) + cy +d

find value of a , b , c and d

without expanding .without using simultaneous

116) Consider the following five scores

17 15 20 16 J

The mean mode and median are equal to each
Find the value of J

117) A circle of radius r has a chord AB of
length c.
The point C is on the circle and is opposite AB.
The angle ACB is eqal to m

In terms of a and r find the value of
sin m and cos m

118) Consider the arithmetic sequence
1/4 , e , f , g , 13/12

Find the values e, f and g.

119) Without using simultaneous equations or expanding (1) solve for a, b, c and d

(4m -1)^ 3 + ( 3m -2)^2 +2m -5
= a(m^3 ) + b(m^2 ) + cm + d .....(1)

Note : a(m^b) means a ×[ ( m) to power b]

120) consider a staircase.
Jack climbs 2/5 of the total number of stairs.
He then pauses for a short break and than
climbs 1/7 of the remainder and then climbs
2/3 of the the remaining number if stairs.

What is minimum number of stairs in staircase?

121) The average mark of 25 students in an English class is 64%.
The average mark of 18 students in a different
English class is 70%.
What is the average mark of all students?

122) Given that
a/b = c/ d = 2/ 3
Find the value of

(2a + 9c)/ (3b + 5d)

( a - 6c) / ( 4b - 9d)

(a + mc) / ( b- md)

123) The cost of a single blue ,green ,red pencil
are 45 cents , 74 cents and 63 cents respectively
At least one pencil of different colour was purchased.
A $10 note was used to pay and change of
$2.20 was given.
What was the number of each different coloured pencil sold?

124) The product of 19008 and 8036 is divided by the
sum of 940 and 1060.
Without using calculator find the remainder.

125) The pages of a book are numbered consecutively
from 1 to 119
What is the probability that the sum of the digits
a page number is

a) 8


126) The cost of 3 pizzas is shared equally between a group of friends.

If each friend pays $7 the change will be $8.
If each friend pays $4.50 there will be $12 too little.

What is the cost of each pizza?
How many friend in the group?

127) in a group of 42 students
23 play cricket
15 play football
8 do not play cricket or football

Draw the Venn diagram.

How many students play
a) only cricket
b) only football
c) cricket and football

128) Consider the sequence

1 1 2 3 5 8 13 21 34 55

After the second term each term of the sequence
is equal to the sum of the two previous terms.

What is the value of the the 15 th term and 20th term?

129) Develop the formula for finding
1 + 4 + 9 + 16 + 25 +....n^2
where n is even

130) consider the series

1× 2 , 2× 3 , 3×4 , 4×5 , 5×6 ....
= 2 , 6 , 12 , 20 , 30.....

What is the value of
a) 30 th term
b) n' th term
c) the sum of first 30 terms
Hint : use results of (129)
d) the sum of the first n terms
131) A circle of radius 5 units has centre at ( 1,3)
a) lFind co ordinates of circle where it cuts x
and y axis

b) What is area of sector
i) below x axis
i) to the left of y axis

132) The points
A (7.3) B (5,1) C(2,4) D(6,8) lie on the circumference of a circle. Draw a large good quality

a) co ordinates of the circle centre E and the radius r.

b)area of triangle ECD

133) Originally a sphere has a surface area of A
If the radius of this sphere is reduced to 1/3
of its original radius to form a new sphere ;
This new sphere is divided into 4 identical
quarter spheres .

Find the ratio of :

a) new sphere surface area to original surface
area of sphere

b) new sphere volume to original sphere

c) surface area of new sphere quarter to
to surface area of original sphere

134) the parabola
y = 3 ( n - 2 )^2 + 4

is rotated about y = 1.5
between n = 1 and n = 3
The horizontal axis is n.

By relocating the position of the curve find
the volume generated.

135) consider a red dice and a black dice each with
eight faces numbered
2, 3 ,4 ,5 ,6 ,7 , 8 ,9 rolled simultaneously
What is the probability that the sum of the numbers on top face is

a)multiple of 3
b) is prime
c) greater then 17

What is rhe probability that each dice shows same

What is probability that the product of numbers
shown is divisible by 7

136) Square ABCD lies in first quadrant.
Vertice A lies on y axis
Vertice D lies on x axis
Vertice C has co ordinates (8,9)
Sketch a large good quality diagram.
Find area of square and co ordinates
of A,B,D

137) if 3^20 is divided by 5 what is the remainder

If 7^20 is divided by 5 what is the remainder

138) An interger when divided by
2 or 3 or 4 or 5 or 6 leaves a remainder
of 1 in each case.
What is the least value of this integer?

139) A square is divided into 16 identical squares.
How many squares are there?

140) A right angled triangle has a perimeter of 340

3 times the length of one perpendicular
side minus 13 gives the length of the
other perpendicular side.

What are the dimensions of the right angled

141) Solve the following equation

Log ( 64m^3 + 22265) to the base 8


Log (4m + 5) to the base 2

142) Given two straight lines.

y = cx + d

y = ex + f

Show that the lines are perpendicular if

ce = - 1

143) Which is larger

5^96 or 7^ 60

Do not use calculator.

144) Consider a parallelogram
A pair of opposite sides are of length
4m + 3 and 7m -5

The other length is 2m + 11

Find the perimeter of the parallelogram and its
maximum area.

145) Consider a sequence of 12 integers

The first integer equals 6
The last integer equals 11

The sum of any three consecutive integers is 24

Find the value of the remaining 10 integers

146) A rectangular block of land (BCDE) is 9m
by 12 m and is enclosed by a very high lizard proof fence using 4 vertical posts at B, C, D, and E.

A rope of length 16m is attattched at ground level
to post D and the other end of rope is attattched
to a pet lizard which only moves along the ground.

Given length of BE and CD are both equal to 12m
find the area over which the lizard can roam.
Assume the rope is always at ground level.

147) Referring to Q 146 the pet lizard is attatched to a shorter rope UK and to Post D...however
the lizard is now confined to partially roam inside
rectangle BCDE whose dimensions are unchanged.

Find the area over which the lizard can roam if
the lengths of attatched rope are

a) 8m
b) 10 m
c) 14m
d) 16 m

148) If a+b = 10 and ab = 7

Without finding actual values of a and b find the value of :

a ^2 + b^2

(a - b) ^2

1/a + 1/ b

a^3 - b^3

a^3 + b^3

a^4 - b^4

149) A triangle has the following side lengths

4 x 15^12 and 105 x 15^11 and 8 x 15 ^ 12

Find its area and the size of all its internal angles

150) In travelling from A to B a train has a constant
speed of u m/sec.When travelling from B to A it
has a constant speed of 1.5u m/sec .
What is the average speed of the train in going from A to B and then B to A?

151) simplify the following

6 ^(5m+3) × 8^(2m-5) ÷ 24^ (1-2m)

How would you check your answer?

152) Triangle ABC has vertices
A( 1,0) B(5,0) C(3,7)
Using A as circle centre an arc of radius 4 units
is drawn to intersect AB at D and CB at E.

a) co ordinates of E
b) area of sector ADE
c) area of EBD

153) Triangle ABC has dimensions

AB = 7
BC = 8
CA = 10

If the triangle is placed on x y plane
and A (2,0) and B (9,0) find coordinates of C.

154) Show that a five digit number ( having the same or different digits ) minus the
sum of its digits is divisible by 9.

155) In a quadrilateral ABCD the diagnols BD and AC
intersect at right angles at E.

Draw the appropriate diagram.

Show that
(AB)^2 - (BC)^2 = (AD)^2 - (CD)^2

156) In a quadrilateral ABCD under what
condition does
AB × BC = CD × DB
Give reasons for your answer.

157) Why is statistics so important in the
the study of the causes of diseases
and the method of transmission.

eg cause of cholera
reducing deaths due to unsanitary
hospital conditions unclean hands
touching and infecting wounds

158) Consider 3 Points

A( 4,2) B(5,6) C(2,1)

Find co ordinates of point which is equally distant
from A ,B and C.

159) A biased coin can show only a head or tail.
P(H) = 1/3

For n consecutive throws of the coin for a single given
combination of heads and tails the probability
is ( 4÷ 2187)

Find the value of n the number of heads thrown
and the number of tails thrown.
Use algebra ...not guesswork.

160) A hat contains red ,yellow and green marbles.
After a ball is drawn from hat it is replaced.

The probability of drawing two consecutive
red balls is 0.09

The probability of drawing three consecutive
green balls is 0.008

Find the probability of drawing
a) 3 yellow balls
b) a yellow green red ball in this order
c) a red ball and a green ball in this order

What are the minimum total number of balls in the
hat.?Based on this number and assuming replacement
does not occur answer (a) (b) (c).

161) Consider a quadrilateral ABCD

AB= 7
BC = 9
CD= 5
For differing shapes of ABCD what is minimum length of diagnols BD and AC?

162) Given a , b ,c,d are all positive and that
a/b > c/d show that

b/a < d/c

a/ (a+b) > c/ (c+d)

a/ (a-b) > c/(c-d)

ABC is a right angle triangle
Angle ABC = 90 degrees

AB = 0.461538461538... (Recurring decimal)
BC = 0.285714285714....(Recurring decimal)
Find the exact
a) area of triangle

b)length of hypotenuse

c) value of inverse tan ( CAB)

164) The integers a , b and c are positive
and a < b < c
Find a single set of values of a ,b and c such that
1/ a + 1/b + 1/c = 1

165) prove that if the last 3 digits of a number
is divisible by 8 then the entire number is divisible
by 8.

For example the number 47643523168
must be divisible by 8 as 168 is divisible
by 8.

166) Consider two triangles

ABC has a base length of 5× 7^0.5 + 2× 3^0.5
height of 4× 6^0.5 - 3× 5^0.5

DEF has a base length of 6×7^0.5 - 3 × 2^0.5
height of 4×3^0.5 - 3 × 5^0.5

Which area is larger and why.
Do not use calculator.

167) Consider the 4 digit number.
A = 6mn5
What is the smallest value of A such
thay A is divisible by
a) 19


c) 873

168) How many numbers between 51328 and 52763
are divisible by 421 ?
What is their sum?

169) Given that

DE × F = DDD
where DE is a two digit number
F is a single digit number
DDD is a three digit number

Find the value of each digit.

170) Consider a right angled triangle ABC
BC = 2 × 12^0.5 + 2^0.5

CA = 2 × 8^0.5 - 3^0.5

angle BCA = 90 degrees

a) Find the length of hypotenuse, area and perimeter
of triangle.

b) Find inverse sin ( BAC)

171) A quarter circle , centre (0,0) and radius 5
lies in first quadrant.
The straight lines x = 2 and y = 3 intersect
the arc at A and C respectively.

The straight lines x = 2 and y = 3 intersect
at B.

a) Draw an appropriate large good quality sketch.

b) Find the value of area ABC where AC
is an arc.

172) An arc centre (0,0) and radius 6 units cuts x axis
at C (6,0)
An arc centre (9,0) and radius 4 are drawn
in first quadrant and cuts x axis at A (5,0).
The above arcs intersect at B.

Show above information on a large good quality

Find the co ordinates of B

Find the area of of ABC where
AB and BC are both arcs and AC
is a straight line.

173) The sum of two five digit palindromic
numbers is 103539
Find the value of each palindromic number.
Note that there is more then one solution...
only one solution is needed.

174) Show that
1 ÷ ((n)(n+ 1) = (1÷n ) - (1÷ (n+1))
Hence find the value of

1/(1×2) + 1/(2×3) + 1/(3×4) +...1/(99×100)

( Check your answer using sum of GP)

175) Prove using induction that
1 × 1 + 2× 2 + 3× 3+....n × n
= n (n + 1) (2n +1) ÷ 6
a) sum of first 30 terms
b) sum of first 18 terms
c) sum of terms between n = m and n = 2m + 3
including m'th and (m+1)' th term

176) Prove using induction that
1 ×1 + 3×3 + 5×5......+ (2n - 1) ^2
= (4n^3 - n) /3

Find sum of first 25 terms and sum of first 11 terms
Hence find sum of terms from n= 12 to n= 25

177) Prove using induction that
1^3 + 2^3 + 3^3 +... n^3 = ( 1+ 2+ 3 ...n)^2
= n^2 × 0.25 (1+ n)^2

Find the value of
10^3 + 11^3 + .....24^3

Given that
1^3 + 2^3 + 3^3 +.....n^3 = 6864
find the value of n

178) Each of the numbers from 1 to 40 is written
separately on a ball which is then placed
in a box.

On how many balls does the digit

a) 6 appear
b) 8 appear
c) 5 appear

179) show that
21^n + 19^n - 8^n - 6^ n is divisible by 13
if n is an integer greater then or equal to 1

a ) by using induction

b) by using another method

181) Sketch the curve not calculate the
co ordinates of max or min.

y = (2t + 3)^3 × (5 - t)^2 × ( 7 + t)
find the area below the horizontal t axis but
above the curve.

182) The vertices A,B,C,D,E,F,G and H of a regular
8 sided figure are all located on a circle centre 0
of radius r.

Thea area of BCDE is 24 square centimetres.
Without using calculator find

a) radius of the circle
b) perimeter of the 8 sided figure

c)area of 0FG

183) Consider an empty tank.
Pump A working alone can fill the tank
in 5 minutes .

Pump B working alone can fill the tank
in 8 minutes.

Pump C working alone can fill the tank
in 10 minutes.

If all 3 pumps are working how long will it
take to completely fill the tank?

184) Pumps A,B,C, and D add water to a tank.

Pumps A,Band C working together fill
tank completely in 10 minutes.

Pumps B,C and D working together fill
tank completely in 13 minutes.

Pumps D,A and B working together fill
tank completely in 15 minutes.

How long will it take to fill 3/4 of the
tank if all 4 pumps work together?

185) Four pumps E F,G and H are used for filling a tank with oil. Pump J is used for emptying tank.
If pumps E,F and G are switched on the tank
is filled in 5 minutes.

If pumps F,G and H are switched on the tank
is filled in 10 minutes.

If pumps H and J are switched on the tank
is filled in 20 minutes.

If pumps E and J are switched on the tank
is filled in 25 minutes.

If all 5 pumps are switched on what is time
taken to fill 2/3 of tank?

186) The following sequence is given

Is it an AP or GP or neither?

Find the equation for the n'th term if
a) n = 3k where k = 0,1,2...
b) n = 3k + 2 where k=0,1,2.....
c) n = 3k + 1 where k = 0,1 2....

Find the sum of the first
d) 28 terms
e) 29 terms
f) 30 terms

187) A horizontal circular road has a radius of 5 km
and circle centre is located at 0.

Pushbike A leaves point D at a speed of 70 kph in clockwise direction at 6.00 am
Pushbike B leaves point E at a speed of 85 kph in a clockwise direction at 6.00 am

The points D and E subtend an angle of 120 degrees
at circle centre 0.

Draw a large good quality diagram.

At what time do the riders first meet at point F and
what is angle subtended at circle centre by DF?

At what time do the riders next meet at G?
What is angle subtended at circle centre by GF?

188) A circle of radius 6 and centre (5,3) is
intersected at two points H and J by a straight
y = -3w + 2
and w is the horizontal axis.
Find the area bound by HJ and the minor arc.

189) If y = k(w^a ) and a is integer
then dy/dw = ka(w ^ (a- 1))

Show that if y = k ( w^ (a/b)) and a and b
are integers having no common factors then
using function of a function rule
prove that

dy/dw =(ka/b ) w^ (a/b - 1)

190) What is the number of palindromic numbers
which are
a) 3 digit
b) 4 digit
c) 5 digit
d) 6 digit and even
e) 7 digit and the first digit is 3 or 4

191) A three digit palindromic number is added to a four digit palindrom


Bachelor of Mechanical Engineering (University of New South Wales)

Master of Engineering Science (University of New South Wales)

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