How we can connect amazon servers in putty using key pair ec2

In order to connect Amazon  Linux servers we have to use putty from our windows computer. But as other servers amazon will not allow connect with username and password through putty. We have to use the private and public key pair.

During the installation of the server it will ask for generate the key pairs, if there is no key available in that particular region. In a region only one pair of key is enough for many servers. the public key pair we can down load to our local computer during generating the key.

We can generate new key at any time. For generating key we have to go for ec2 dash board–network and security–key pairs, then click on create key par and give name for the key and click generate and down load.



how can disable the ssl mode of webmin and virtualmin

Some time we may not have a valid SSL certificate with us. Then our webmin and virtualmin control panel will always display the security warning in the web browser. But webmin and virtualmin have its default ssl certificate. but it is not valid in the browsers.

Because of the invalid certificate the browser will make hevely delay for webmin and virtualmin. In order to avoid this we can disable the ssl mode of the webmin and virtualmin as described below.

  1. Open the webmin control panel as root user, by typing serverip:10000
  2. Go to the option others —File manager — etc — webmin — mini server.conf
  3. change the lin ssl=1 to ssl=0
  4. Save the file and exit.
  5. reboot the system
  6. Before trying without ssl mode in the browser please delete and clear all browsing data and cookies.

Now your webmin and virtualmin is ready to work without ssl mode.

PHP codes for date and time

creating php Timestamp


Date, month and year format to Timestamp

$time_stamp = strtotime("$year/$month/$day");


$time_stamp = strtotime("$year/$month/$day $hour:$minute:$second");

Time stamp to date,month and year format

$p= date("d/m/y",$timestamp);


$p= date("d/m/y h:i:s",$timestamp);//i for minute

Date function

$d = date(d);
$m = date(m);
$y = date(y);
$h = date(H);
$i = date(i);//Minute
$s = date(s);
echo "$d / $m /$y /$h /$i /$s";

Number system grade-9

Number line

Number line are the line made up with numbers. A line is actually made up with points. Eventually distributed points are called line. So the each point in a  number line have a particular number. that is why we are saying the number line is made up with numbers.

After drawing a line just mark any one point as 0. Then if we mark a particular unit of distance as number 1 in the number line, then the point for the number 2 will come exact same distance from number one. Like this each and every points in a number line have a particular number.

Now we will just start to walk through number line from zero. And we will pick all the numbers like 1,2,3 etc.. Then the collection of all this number can called natural numbers. Denoted by (N).

Now we have to walk back all the way and pick the zro also. So now we have the collection of natural number and zero. And it can be called as whole numbers (w). Whole numbers are 0,1,2,3  etc….. . Now again we walk back we will see lot of negative integer numbers. And we will collect al that numbers also. now we have the collection of positive integer number , zero and negative integer numbers.  The collection of all this number together can call  integer numbers (Z).

Is any numbers left in the number line?. Yes the numbers like 1/2, 3/4,  are left in the number line. So we will collect that numbers also.Including the numbers like 1/2, 3/4 with naturel number is called rational number (R) . Here we can say one definition for rational number.

Any number which can be write in the form of  P/Q is called rational numbers.

Here P  and Q are integers and Q not equal to zero.

Here we cn notice that all natural numbers are rational as per this definition. Because all the natural numbers can be write with denominator 1.

Zero is also a rational number because it can be write with denominator as 1.

And the negative integers also rational number because it can also write with denominator 1. So The numbers so far we studied are all rational numbers.

How ever we now that a number in the form of P/Q is not unique. Example 1/2=2/4= 25/50=47/94. Because these all are equal numbers (fractions.). So when ever we mark a rational number in the form of P/Q then we have to make the number in the lowest form ppf fraction. Or the P and Q have only one as common factor (co-prime numbers).

Are the following are true or false give your reason

  1. Every whole number is natural numbers.
  2. Every integer is rational numbers.
  3. Every rational number is an integer.
Finding rational numbers between two rational numbers

If we have given two rational numbers, then we can find the rational number between them by find the mean value. We have to add the given two numbers and then divided by 2. Then we will get another rational number between them. by repeating this method we can find infinity of rational numbers between two rational numbers.

Example : find the two rational number between 1 and two.

(1+2)/2=3/2 is a rational number between 1 and 2.

(2+3/2)/2 = 4/2+3/2= (7/2 )/2 =7/4 is an another rational number between 1 and 2.

Another method

Suppose  we want to find 5 numbers between 1 and two at a single step, like follows.

As we need 5 numbers we have to make  denominator as 6 for both number. Here

1= 6/6 and 2 = 12/6

Now we can write 7/6, 8/6, 9/6, 10/6, 11/6. these all are in between 6/6 and 12/6 and hence between 1 and 2.

Notes: Also remember that there will be infinite number of rational numbers between any two rational numbers.

Do the exercise 1.1

Irrational numbers

As we agai check on the number line still there is lot of numbers left in between rational numbers. So now we will see how that numbers is different from rational numbers.

If a number is irrational then it cannot be written in the form of p/q, where p and q are integers and q not equal to zero.

Some examples are √2, √3, √15, π, 0.123400678…… .

There will be infinite number of irrational numbers in the number line.

After collecting all the irrational numbers from the number line, then the number line will empty as there is any more number. So the collection of all these numbers we studies so for together can called the real number. (R).

The proof for √2 as an irrational number will study in the nest class. But now we will study how we can mark the √2 in a number line.

Locate √2 on the number line
√2 on a number line
√2 on a number line

Consider the unit square OABC. As shown in the fig.1.7. Now consider the diagonal OB. And we can consider the triangle OAB, it is aright angle triangle. Therefore we can say that

OB= OA2 +AB2

OB=√OA2 +AB2

OB= √1+1 =√2

Now consider the vertex O of the square are coincide with 0 of the number line. And the OB which is equal to √2 can now marked on the number line by draw an arc with radius OB. The ar will intersect on the number line at the point P And the distance from 0 to p is √2.

Locate √3 on the number line

This can also mark on the number as same as √2. For that first we have to draw one more right angle triangle in the previous fig. One side of this right angle triangle is OB (√2) and other side DB is unit length and OD is the hypotenuse. see the below fig.

real number and their decimal expansion

A we study the real number having rational number and irrational numbers. In this section we will analyze the decimal expansion of both type of number and understand the difference of the rational number and irrational number.

First we will check the decimal expansion of three different rational number.  They are 10/3, 7/8, 1/7. See the below fig for there division.

Division of rational number
Division of rational number

In the first case  10/3 we can see that the quotient is not ending. But it is repeating as a block . This type of decimal expansion can call non terminating recurring. It can write as 3.3(bar). That means the decimal places are repeating with 3,3,3,3..

In the second case 7/8 we can see that the quotient is terminating after a finite number. This type of decimal expansion can call terminating decimal expansion.

In the third case 1/7 the quotient is not ending but it is also repeating as block of  142857, 142857,…… so it can write as 0. 142857(bar). So this can also call non terminating recurring decimal expansion.

Now we have to proof the above decimals are rational numbers by wring in the form of P/Q . As P and Q are integers. it should be a perfect number ( cannot be repeated or require in their decimal expansion).

First we will consider a terminating decimal number 3.142678.  Here the decimal places are 6 digits. Therefore we can multiply and divide the 3.142678 with 1000000, so that it will become 3142678/1000000 so now it is in the form of p/q  and hence it is a rational number.

In this second case we will show that the non terminating recurring number is a rational number, by writing the number as p/q.

consider 0.3(bar), here 0.3333… we don’t know the exact number. so we will take

x= 0.333333….  ( we can consider multiplying x with 10 because the nuber have 1 digit requiring number)

10x= 10 *( 0.33333…..) = 3.333333….

3.3333….=3+x (since x= 0.3333…) (here the obtained number after multiplication will write as sum of a perfect number and the given number x)


9x=3 and x=3/9= 1/3 this is in the form p/q. and Hence it is a rational number.

Let us try with other number 1.272727 her 27 is repeating.

Here also we don’t know the exact number so we will take

x= 1.2727…..

100x = 100 * 1.2727… =127.2727…

100x =126+x (here the obtained number after multiplication will write as sum of a perfect number and the given number x)

99x=126 so x=126/99. which is 14/11 so it is also a rational number.

From the above studies we can understand the decimal number with terminating decimal places and the number with non terminating recurring decimal places are rational number.

Irrational number

From the above studies we can say that any number is irrational if it cannot be written in the form of p/q. Where p and q are integers and q not equal to 0.

In the decimal expansion we see the terminating and non terminating recurring decimal number are rational numbers. Because they can write in the form of p/q.

But some decimal number like 0.1011001111000001 are not terminating and not requiring. this type of decimals cannot be written in the form of p/q. And hence it is not rational number. ex. √2, Π.

Finding irrational numbers.

Finding irrational number between 1/7 and 2/7. First find the value of 1/7 which is 0.142857(bar). And the value of 2/7 = 0.285714(bar).

Now easily we can write the irrational number in between them ex.

0.156578911112. we can write such infinity of irrational number between two. By this same method we can write irrational numbers between any two real numbers.

representing real number on the number line

We can represent any real number in a number line. Suppose we ant to mark 5.3775 (5 decimal places) as the number is between 5 and 6 we will make the portion of the number line with 5 and 6. Then we will divide that portion in 10 equal parts. The we can mark the first decimal digit 5.3. A our number is greater than 5.3 and less than 5.4 we will draw anew portion of number line with 5.3 and 5.4. this process can repeat until we mark the complete given number. Please see the below fig for the details.

real number on number line
real number on number line

Do some exercises for practice.

Arithmetical operation on real numbers

I earlier class we studied the rational numbers are satisfy the commutative, associative and distributive laws for addition and multiplication. If we add, subtract,  multiply and divide (Except zero) two rational numbers we will get a a rational number . So rational number are closer.

The irrational number also tisfy the commutative, associative and distributive laws for addition and multiplication.

How ever the sume , difference, product and quotient are not always irrational.

ex. (√6)+(-√6), (√2)-(√2), (√3)*(√3),(√17)/(√17)

By doing some exercise we can summarize following  points.

  1. The sum or difference of a rational number and an irrational number is irrational.
  2. The product or quotient of a non zero rational number with an irrational number is irrational.
  3. If we add, subtract, multiply or divide two irrationals, then the result may be rational or irrational.
Geometrical method to find the square root of a positive real number

Finding the √3.5 geometrically.

finding root by geometry
finding root by geometry

Procedure: Draw a line AB with 3.5 unit length. From B extend one more unit length and mark C. Now find the midpoint of AC and mark that point as O. Now draw the semicircle with the radius AO. A s shown in the figure. Draw a perpendicular passing through B and intersec the semi circle. And then the length of BD= √3.5

Proof : Not required for grade 9

We can also use the above method to mark root of any number on the number line. See the below figure, in that the BC is the unit length And we can extend the line with more unit lent and mark B AS 0, C as 1 and so on… Now we can draw an arc with center B and radius BD, then it will intersect on the number lane.

root x in a number line
root x in a number line
Identities for roots

identity for roots

Identity for roots
Identity for roots

The above are the 6 identity for roots. By this identity we can do various mathematical operation on roots. Do some exercises for more exposure.

Laws of exponent
Laws of exponents
Laws of exponents

I addition we can also say that  nth root of a= a1/n.

And  am/n = nthroot of am

That means if a number have the power with fractional number, then the denominator of the fraction will become the power of the root and the numerator will the power of all thing.

Practice some exercises

Lines and angles Grade-7

Eventually distributed points in any direction can called  line.

Eventually distributed points in particular direction  can called   straight line.

See below fig to understand different type of lines.


When two lines are intersect  then there will be one corner formed this corner is called vertex. In the above fig1-e two line AB and CD are intersecting at the point O. Due to the intersection  of two lines there are four angles. Try to name that four angles.

complementary angles

If the sum of two angle are 90º then the are called complementary angles.

Now try to answer following questions.

  1. Can two acute angles be complement to each other?
  2. can two obtuse angles be complement to each other?
  3. Can two right angles be complement to each other?

From the below figure identify which pair of angles are complementary.

complementary angles
complementary angles

Supplementary angles : If the sum of 2 angles are equal to 180°, then that angles are called Supplementary angles.

Answer the following questions

  1. Can two obtuse angles be supplementary ?
  2. Can two acute angles be supplementary ?
  3. Can two right angles be supplementary ?

Check for some exercises in the test book.

Adjacent angles: If two angles are adjacent angles then they have a common vertex .a common arm and there non -common arm are the different side of the common arm. In the below figure-2 we can see a adjacent angles.

Here the angle ∠ABD and ∠DBC are adjacent angle. But also note that ∠ ABC is not adjacent angle. More over when there is two adjacent angle, then the sum is always equal to the angle formed by two non-common arms

that is  ∠ABD+∠CBD=∠ABC


If the non-common arm of an adjacent angle are opposite ray the angles are called linear pair of angle. As shown in the fig3, AB and BC are non common arm and the angle ABD and angle CBD are linear pair of angle. Linear pair are supplementary.

TRy to answer the following questions

  1. Can two obtuse angles be linear pair ?
  2. Can two acute angles be linear pair ?
  3. Can two right angles be linear pair ?

Now try to do some exercises from textbook


Comparing Quantities-grade 08


Ratios can be use to compare quantities. Example. If we have 5 orange and 20 apple the we can write the ratio 5:20. If we want to write the ratio of apple and orange the it will write as 20:5. The ratio is writing respectively.

Ratio as fraction

The ratio can also write as in terms of fractional number. Here the first ratio will come as numerator and and the second ratio will come as denominator. So in our previous example we can write the ratio of orange and apple is 5:20  = 5/20.

If we are saying the ratio of orange and apple is 5:20, that means if there is 5 orange then there is 20 apple. The ratio will always write as simplest form. So 5:20 will write as 1:4. That means there will be 1 orange for 4 apple. Like this we can compare the quantity.

Ratio as percentage

From the ratio we can find the percentage of quantities.For example. 5 orange and 20 apple can write as 5/20. now we will make the denominator as 100. then it will become (5×5)/(20×5)= 25/100.  So the percentage of orange is 25%. (that means there will be 25 orange when apple is 100). We can also find the fraction by unitary method  like ( 5/20)x100  = 500/20= 25%.

Finding the increase or decrease percentage

To find the increase percentage of thing, for example. The price of a scooter was 34000 in last year. if the price increased by 20 % what will be the price of the scooter this year.

Method -1: Here first we will find the 20% of the current price and then add that to the existing price then we will get the new price.

(20/100) x 34000 = 6800

increased price = 34000+6800= 40800

Method2: Unitary method.  (120/100)*34000 =40800 ( here the 120 = 100% + the increased 20% =120%)

To find the decrease  percentage of thing, for example. The price of a scooter was 34000 in last year. If the price decreased by 20 % what will be the price of the scooter this year. Here also we can find the 20% of the current price and can subtract the amount form the current price, then we will get the present price.

In unitary method we can find the 80 % of the current value (because the decreased percentage is 20%). which is (80/100)*34000.

finding discounts

Discount is the difference of the actual price(MRP) and the sales price.

Discount= market price- sales price.

If an item Have the market price 840 and it is selling at 714 then

Discount = 840-714=126.

we can also find the discount % by

Discount %= (discount amount / market price) *100.

(remember (% /100)*total=value)

We can also find the discount amount if the discount % is given by

Discount amount= (discount%*market price)/100.




WordPress Elementor Astra basic tutorial

By the use of elemetor and astra theme we can make very stylish wordpress site. Please follow below instruction to install and work with elementor and astra.


First we have to install wordpress in our domain. I Am using virtualmin control panel to manage my server. In virtual min we can install wordpress in single click from the install script option. After the installation, we can do the preliminary setup for the site by the link provide.

Installing elementor

After completing the wordpress installation we can login into wordpress using the username and password we created during the installation. The login link will be domain name/wp-admin.

After log in from the dashboard, go for the plugin option and click on add new plugin.  Then search for the elements in side the plugin option. you can see that the elementor page builder  plugin. And click on install and the activate. your element is ready.

Installing astra theme

The astra is the stylish theme. But in order to run astra we need elementor. That is the reason we install elementor previously. To install astra theme we have to go appearance then theme.  Click on add new. then search for astra. Then install and activate astra. Now everything is finish to work with elementor and astra.

creating new pages

For creating new pages just go to page option from the dashboard. In the new installation we can see two default page , sample and privacy page. By placing the cursor over the page name edit delete and other option will pop up and we can use that if needed. We can also use bulk action option after selecting multiple pages.

To create a new page just click on add new and give the page title and click to publish an select the public option. So that the page will saved and publish. And now, again we can go to the pages option and click on the page title and use edit with elementor . So that we can start to build the page with elementor. In order to exit the elementor, just click on the menu bar on the left top corner and we can see exit elementor.

Creating the ready made site from template

In astra we have option to create readymade site from template. In order to do this we have to go to appearance and asta option.  In astra option page we can see the install starter site card. in the bottom of the card we can see the install importer plugin. install it and activate.

After installing the plugin, under the appearance, Astra option we can see the starter site option. from the starter ste we can get lot of free template. We can also get ecommerce site also.

After installing ecommerce site we can see the woocommerce menu in the dashboard. Through this we can setup the payment option. There is also the product option available. through this we can add or edit our product.

Thanks for reading. For more details please contact at


Triangles grade-09

Triangl is a three sided polygon. Triangle is made up with the parts of sides, angles and vertex.

congruence of triangle

If an object is Congruence with other  object, then the both object can be perfectly site one over another. So if two triangle are congruent then all of the parts of the two triangle should be same.

congruent triangle
congruent triangle

Consider the triangle i, ii, we can say both triangle are congruent because all the side and angle are same for bot triangle. But here we have to take care to say the name of both congruent triangle. Because in the figure i and ii,  the two triangles will be congruent  only if we keep the side AB of the one triangle over the side PQ of the another triangle. So the name of both congruent triangle should be in the congruent order.

here ΔABC≅ΔPQR. It can be write in many ways corect as well as wrong. so please take care and write the name of other congruent triangle in the fig. with the triangle ABC.

Important Notes. Corresponding parts of congruent triangle are same (CPCT).

In Order to find two triangles are congruent we have to check by keep both triangle one over another.  This is not possible in all time, so now we can study some criterias to check the congruence  of triangles.

SAS Congruence rule

if two side and one include angle are same for two triangle then the two triangle will be in congruence

This can’t be proven by previously known result. So it is accepted true as an axiom.


ASA congruence rule

If two angles and included side of two triangles are same then they are in congruent.

This can be proved so that it can be called as a theorem.

Theorem 7.1 (ASA) :


In the Above fig ∠B=∠E,  ∠C= ∠F and BC=EF (Given)

consider AB=DE Then we can say that ΔABC≅ΔDEF (SAS RULE)

now jus we have to prove that AB=DE

Suppose BA greater than DE, Then we can construct a line PC  as shown in the fig. 7.12

Then  PB=DE (by construction)

∠B=∠E (given)

BC=EF (given)

Ther for  ΔPCB≅ΔDEF (SAS)

As we know CPCT are same


∠ACB=∠DEF (given) there for ∠PCB=∠ACB and this is only possible when P is coincide with A

There for BA =ED and   ΔABC≅ΔDEF (SAS RULE) so ASA rule is proved for the congruence of triangles.


This congruence rule says that if any two angle and any one side of two triangle is same then the triangle will be in congruence.

This is happening because, we know that the sum of all the angle of a triangle is 180º.  There for if two angle is same for two triangle then the third angle will also same. (same thing is subtracted from equal then the result will also same). If three angle and one side is same the these triangles will in congruence by ASA rule. And hence we can say that AAS rule is valid.

Some property of a triangle

As we know in an isosceles  triangle the two side are equal. By constructing many Isosceles triangle we can check the angle opposite to the same side.  then we can see that the opposite angles of the same sides are equal. This one can be prove by the congruence rule.

Theorem 7.2 : Angles opposite to equal side of an isosceles triangle are equal

isosceles triangle
isosceles triangle

Consider the triangle ABC where AB=AC , Now constructing the bisector of the angle A with the line AD. Now consider the two triangles BAD and CAD

AB=AC (Given)

∠BAD=∠CAD (By construction)

AD=AD (common line)

ther for ΔBAD≅ΔCAD

Ther for ∠B=∠C (CPCT)

So it is proved.

By the observation, trial and error we can say that converse  of this theorem is also true. That is if two angle of any triangle  equal then the opposite side will also equal. It can also proof as a theorem.

Theorem 7.3 : The side opposite to equal angle of a triangle are equal.


isosceles triangle
isosceles triangle

Consider the triangle ABC. And draw the bisector of the ∠A  by the line AD, is also perpendicular to BC.

∠BAD=∠CAD (By construction)

AD=AD (Common Side)

∠ADB=CDA=90 So,

ΔABD≅ΔACD (ASA rule)  So,


SSS congruence rule

Theorem  7.4 : If three side of one triangle are equal to three side of other triangle then two triangle are congruent

As we know if three side of two triangle are equal then there three angle are also equal. so this theorem can proven by SAS SSA rules.

RHS congruence rule

Theorem 7.5 : In two right angle triangle if the hypotenuse and one side is same then two triangle are congruent.

Here RHS stands for Right angle, Hypotenuse, Side. But here this three can be any order.

Inequalities in triangle

By observing  we can understand that any one side of the triangle increased then the opposite angle also increase.

Theorem 7.6 : If two side of the triangle are unequal, the angle opposite to the larger side is larger.


From this fig. we can see that, when the one side of the triangle increased, the angle opposite to that side will also increase. thus the theorem is proved.

Theorem 7.7 We can also say that the side opposite to the biggest angle will be the logestside. 

Theorem 7.8 The sum of any two side of a triangle is greater than the third side.