## Posts

Showing posts from October, 2013

### JSC English Dialogue Suggestion

Junior School Certificate (JSC) Examination- 2013 JSC English Dialogue SuggestionOnly for Jessore Board 01. Preparation for the J.S.C Examination
02. Future Plan/ Choice of career/ Your aim in life
03. Environment Pollution/ Air Pollution
04. Opening a bank account
05. Good results

07. Illiteracy Problem
08. Between a doctor and a patient
09. Between a seller and a customer
10. Between a librarian and a student

Quick navigation:JSC/ JDC/ Equivalent Exam - 2015 Result

### JSC English Paragraph Suggestion

Junior School Certificate (JSC) Examination- 2013 JSC English Paragraph Suggestion Only for Jessore Board
01. Tree Plantation
03. A Rainy Day
04. A Tea Stall
05. A Book Fair

07. Nakshi Kantha
08. The river gypsies of Bangladesh
09. The ethnic people in Bangladesh
10. Bangladeshi food culture/ Good Food

11. The impotence of reading newspaper/ The impotence of Paper
12. Early rising
13. How to maintain good health/ Health and Hygiene
14. A Moonlit Night
15. A Railway Station/ A Bus Stand

17. A Street Accident
18. Your Favourite Teacher/ Parents/ Country/ Native Village

⇛ Quick navigation:JSC/ JDC/ Equivalent Exam - 2015 Result

### Prove that the square root of 2 is an irrational number

Proposition: √2 is an irrational number.Proof:
We Know, 1 < 2 < 4
∴ √1 < √2 < √4
or, 1 < √2 < 2

Here, 12 = 1, (√2)2 = 2, 22 = 4, (√2)2 = 2
∴ Therefore, the value of √2 is greater than 1 and less than 2.
∴ √2 is not an integer.

∴ √2 is either a rational number or a irrational number. If √2 is a rational number
Let, √2 =p/q; where p and q are natural numbers and co-prime to each other and q > 1
or, 2 = p2/q2; squaring
or, 2 = 2p2/q; multiplying both sides by q.

Clearly 2q is an integer but p2/q is not an integer because p and q are co-prime natural numbers and q > 1
Clearly 2q is an integer but p2/q2 is not an integer because p and q are co-prime natural numbers and q > 1

∴ 2q and p2/q2 cannot be equal, i.e., 2q≠ p2/q
∴ Value of √2 cannot be equal to any number with the form p/q , i.e., √2≠ p/q

∴ √2 is an irrational number. (Proved)

### Prove that the diameter is the greatest chord of a circle

Proposition:Let O be the centre of the circle ABCD. Let AB be the diameter and CD be a chord other than diameter of the circle. It is required to prove that AB > CD.

Construction:Join O,C and O,D.

Proof:OA = OB = OC = OD (radius of the same circle)
Now, in ΔOCD,
OC + OD > CD
or, OA + OB > CD
Therefore, AB > CD. (Proved)

### Chords equidistant from the centre of a circle are equal

Chords equidistant from the centre of a circle are equalProposition:Let AB and CD be two chords of a circle with centre O. OE and OF are the perpendiculars from O to the chords AB and CD respectively. Then OE and OF represent the distances from centre to the chords AB and CD respectively.
If OE = OF, it is to be proved that AB = CD.

Construction:Join O,A and O,C.

Step-1: Since OE⊥AB and OF⊥CD (right angles)
Therefore, ∠OEA = ∠OFC = 1 right angle.

Step-2: Now, between the right-angled
ΔOAE and ΔOCF
hypotenuse OA = hypotenuse OC (radius of same circle)
and OE = OF. (supposition)
∴ ΔOAE ≅ ΔOCF
∴ AE = CF.

Step-3: AE = 1/2 AB and CF = 1/2 CD. (Perpendicular from the centre)

Step-4: Therefore, 1/2 AB = 1/2 CD
i.e., AB = CD. (Proved)

### Equal chords of a circle are equidistant from the centre.

Proposition:Let AB and CD be two equal chords of a circle with centre O. It is to be proved that the chords AB and CD are equidistant from the centre.

Construction:Draw from O, the perpendiculars OE and OF to the chords AB and CD respectively. Join O,A and O,C.

Step-1: OE⊥AB and OF⊥CD (Perpendicular from the centre bisects the chord)
Therefore, AE = BE and CF = DF.
∴ AE = 1/2 AB
and CF = 1/2 CD

Step-2: But AB = DC (supposition)
∴ AE = CF.

Step-3: Now b etween the right-angled ΔOAE and ΔOCF (radius of same circle)
hypotenuse OA = hypotenuse OC and AE = CF.
∴ ΔOAE ≅ ΔOCF
∴ OE = OF.

Step-4: But OE and OF are the distances from O to the chords AB and CD respectively.
Therefore, the chords AB and CD are equidistant from the centre of the circle. (Proved)

### Two diagonals of a rhombus bisect each other at right angles

Two diagonals of a rhombus bisect each other at right anglesProposition:Let the diagonals AC and BD of the rhombus ABCD intersect at O. It is required to prove that,
(i) ∠AOB = ∠BOC = ∠COD = ∠DOA = 1 right angle
(ii) AO = CO, BO = DO.

Proof:Step-1: A rhombus is a parallelogram. Therefore, AO = CO, BO = DO. (Diagonals of a parallelogram bisect each other)

Step-2: Now in ΔAOB and ΔBOC,
AB = BC (sides of a rhombus are equal)
AO = CO
and OB = OB. (common side)
So ΔAOB = ΔBOC.
Therefore, ∠AOB = ∠BOC.
∠AOB + ∠BOC = 1 straight angle = 2 right angles.
∠AOB = ∠BOC = 1 right angle.
Similarly, it can be proved that, ∠COD = ∠DOA = 1 right angle. (Proved)

### The diagonals of a parallelogram bisect each other.

The diagonals of a parallelogram bisect each other.Proposition:Let the diagonals AC and BD of the parallelogram ABCD intersect at O. It is required to prove that AO = CO, BO = DO.

Proof: Step-1: The lines AB and DC are parallel and AC is their transversal.
Therefore, ∠BAC = alternate ∠ACD. (Alternate angles are equal)

Step-2: The lines BC and AD are parallel and BD is their transversal
Therefore, ∠BDC = alternate ∠ABD. (Alternate angles are equal)

Step-3: Now, between ΔAOB and ΔCOD
∠OAB = ∠OCD, ∠OBA = ∠ODC and AB = DC .
So ΔAOB ≅ ΔCOD.
Therefore, AO = CO and BO = DO. (Proved)

### Converse of Pythagoras Theorem

If the square of a side of any triangle is equal to the sum of the squares of other two sides, the angle between the latter two sides is a right angle.

Proposition:
Let in ΔABC, AB2 = AC2 + BC2
It is required to prove that ∠C is a right angle.

Construction:Draw a triangle ΔDEF so that ∠F = 1 right angle.
EF = BC and DF = AC.

Proof:DE2 = EF2 + DF2 [Since in ΔDEF, ∠F is aright angle]
= BC2 + AC2 = AB2
∴ DE = AB

Now, in ΔABC and ΔDEF , BC = EF, AC = DF and AB = DE. [supposition]
∴ ΔABC ≅ ΔDEF; ∴ ∠C = ∠F
∴ ∠F =1 right angle.
∴ ∠C= 1 right angle. (Proved)

### Pythagoras Theorem

In a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the two other sides.
Proposition:Let in the triangle ABC, B = 90°, hypotenuse AC = b, AB = c and BC = a. It is required to prove that, AC2 = AB2 + BC 2, i.e. b2 = c2 + a2.

Construction:Produce BC up to D such that CD = AB = c. Also draw perpendicular DE at D on BC produced, so that DE = BC = a. Join C, E and A, E.

Proof:Steps-1: In ΔABC and ΔCDE, AB = CD = c, BC = DE = a and included ∠ABC = included ∠CDE. [each right angle]
Hence, ΔABC ≅ ΔCDE.
∴ AC = CE = b and ∠BAC = ∠ECD.

Steps-2: Again, since AB⊥BD and ED⊥BD, AB ll ED.
Therefore, ABDE is a trapezium.

Steps-3: Moreover, ∠ACB + ∠BAC = ∠ACB + ∠ECD = 1 right angle.
∴ ∠ACE = 1 right angle.
Now, area of the trapezium ABDE = area of (Δ region ABC + Δ region CDE + Δ region ACE)

N.B:

1. Area of trapezium =1/2 Χ (sum of parallel sides Χ distance between parallel sides).

2. Pythagoras Theorem: In 6th century B.C. Greek philosopher Pythagoras discovered …

### Measurement

The units of measuring length:
N.B: Unit of measurement of length: Metre
Relation between British and Metric System:
Metric Units of Measurement of Weights:
N.B: There are two more units used for measurement in metric system. The units quintal and metric ton are used to measure large quantity of goods.
100 kilograms = 1 quintal 1000 kilograms = 1 metric ton Metric Units for measurement of Volume of Liquids:
N.B:The unit of measuring volume of liquid : litre and Weight of 1 litre of pure water is 1 kilogram

Metric Units in Measuring Area:
N.B:Unit of measure of area : square metre

Relation between Metric and British System in Measuring Area:
Relation between Metric, British and National Units in Measuring Area:
Metric Units of Measuring Volume:
Relation between Metric and British Systems of Volume:

### Profits : Class Eight Mathematics: Chapter Two : Example 6

Example 6. Profit-principal of some principal becomes Tk. 5500 in 3 years and profit is 8/3 of the principal. What are the principal and rate of profit?
Solution:
We know, principal + profit = profit-principal

∴ Profit = profit-principal principal
= Tk. (5500 4000)
= Tk. 1500

Again, we know, I = Prn

### Find the next four consecutive numbers of the following list

Class: Eight (VIII) Exercise: 1 (Patterns)
1. a) Find the next four consecutive numbers of the following list: 1, 3, 5, 7, 9, ... .. .Solution:

Numbers in the list: 1, 3, 5, 7, 9, ... .. .
Difference:                2  2  2  2

Note that every time the difference is 2. Hence the next four numbers are 9 + 2 = 11, 11 + 2 = 13, 13 + 2 = 15 and 15 + 2 = 17.

1. b) Find the next four consecutive numbers of the following list: 4, 8, 12, 16, 16, 20 ... .. .Solution:
Numbers in the list: 4, 8, 12, 16, 20, ... .. .
Difference:                 4  4  4  4  4

Note that every time the difference is 4. Hence the next four numbers are 20 + 2 = 24, 24 + 4 = 28, 28 + 4 =32 and 32 + 4 = 36.

1. c) Find the next four consecutive numbers of the following list: 5, 10, 15, 20, 25, ... .. .Solution:
Numbers in the list: 5, 10, 15, 20, 25, ... .. .
Difference:                 5    5    5    5

Note that every time the difference is 5. Hence the next four numbers are 25 + 5 = 30, 30 + 5 = 35, 35 + 5 =40…

### A Village Market

A Village MarketIntroduction; A village market is a public place in a village where people come from different parts of the village to buy and sell commodities which are necessary for everyday life. It is an important buying and selling center for the villagers.

Kinds of village markets: There are two kinds of village markets. They are the daily market and the weekly or bi-weekly market. The daily market is called bazar and it sits daily in the morning. The weekly market is called 'hat'. It sits once or twice in a week in the forenoon and continues up to late in the evening.

Sit/ place/ where held/ location: Usually a village market sits beside a river, canal or dighi and under some big trees.

Nature and arrangement of shops and things available: Most of the village markets sit one or twice in a week. They sit in the afternoon and break up at night. These markets have permanent shops which sell rice, salt, oil, pepper, cloth and many other things of daily use. Fish, vegetables …

### Tree Plantation

Tree PlantationIntroduction: Tree is of great importance in human life. Tree can be used in different ways in life. It remains with us as a daily companion from the cradle to the grave. There is no other alternative than plant more trees in order to improve life and environment.

Its Importance:
Tree plays a vital role in our day-to-day life. Tree is the only source of food of all animals including human beings. It keeps balance in the environment and prevents environment from pollution. It saves animal world from danger. It gives us not only food but also wood, furniture, fuel etc. Tree takes carbon-di-oxide as its food and leaves oxygen. Again, man takes oxygen and leaves carbon-di-oxide. Oxygen is the most important for mn's survival on earth. The limitless gift of tree has made the world people happy. Tree and its fruits are used as various elements of industry.

Condition of Forest in Bangladesh: If we consider the great importance of tree, the condition of the tree Bangladesh …

### The Value of Time

The Value of TimeIntroduction: Time is the most valuable in a man's life. Life is nothing but a periods of time. So to use time properly is to make the most of life. Time is more valuable than health and wealth; for we can regain lost health and lost wealth but we can never get back lost time. Time once gone is gone for ever. It always goes its way and waits for none, nor can it ever be called back for all the wealth of the earth.

Life short but work long: Man is born to do something in life. He has some duties to do in every stage of his life. But the span of life is short and the work he has to do is long. So he has to make right use of every moment of his life. So he should the right thing at the right moment. If he does not do a thing at the right time he may not be able to do it at all.

How to make right use of time: In order to make the proper use of time it should be regulated. There should be time for study, time for other work., time for…

### Basic Rules of Trigonometry

Some common angles: TrigonometryThe following table shows the conversions for some common angles:

Reciprocal Identities:
Quotient Identities:
Pythagorean Identities:
Co-Function Identities:
Even-Odd Identities:
Sum-Difference Formulas:
Double Angle Formulas:
Sum-to-Product Formulas:
Product-to-Sum Formulas:
Half Angle Formulas:

### Basic Rules of Logarithm

Basic Rules of Logarithm» loga (1) = 0
Because, a 0 = 1 and given that a>0

» logaa = 1
Because, a 1 = a and given that a>0

» logaa (x) = n => an = x

» loga (xn) = n loga (x)

» loga (x.y) = loga (x) + loga (y)

» loga (x/y) = loga (x) - loga (y)

» loga (x) = logb (x) / logb (a)

» logaa = logab . logba

Note:
» loga (x+y) ≠ loga (x) + loga (y)

» loga (x-y) ≠ loga (x) - loga (y)

### BD e-Newspapers

Newspapers plays a vital role in this modern world. It has removed global distance. The world has become much smaller than before. One part of the world has been linked up with another through newspaper. Not only that, newspaper gives us both pleasure and knowledge. Through newspaper we can have the current political situation, economics, culture, events, games and sports, speeches of the whole world. It is said that newspaper is a store-house of knowledge and information.

At present thousands of newspapers are being printed daily all over the day. Besides there are many online newspapers, are called e-newspapers. In the modern age e-newspaper is more popular than printed newspaper. A means of reading e-newspaper we can know what is happening around the whole world in a very short time. We can read e-newspaper from online, there have no cost to buy it.

There are following BD e-Newspapers: