Friday, October 14, 2016

GRE Math Test - Important Key points and formulas Part - 1

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In this blog, we will see all the important formulas and useful statements which are to be used in Math test of GRE.

1) Integers:

1) Integers are positive numbers,negative numbers and zero like:{ 0, ±1, ±2, ±3, ±4, … }.
2) Product of two positive integers is positive integer.
3) Product of two negative integers is positive integer.
4) Product of one positive integer and one negative integer is negative integer.
5) 1 is a factor of every integer,but not a multiple of any integer except 1 and - 1.
6) 0 is a multiple of every integer,but not a factor of any integer except 0.
7) The sum of two even integers is an even integer.
8) The sum of two odd integers is an even integer.
9) The sum of an even integer and an odd integer is an odd integer.
10) The product of two even integers is an even integer.
11) The product of two odd integers is an odd integer.
12) The product of an even integer and an odd integer is an even integer.
13) An integer grater than 1 is known as Prime Number if it has only two positive divisor,1 and itself.
14) An integer grater than 1 and not a Prime number then it is called a Composite Number.

Rational Numbers

15) A faction is a number of the form a/b,where a and b are integers and  b # 0.  Such numbers are also called as Rational Numbers.
16) In above form a/b, a is known as Numerator and b is known as Denominator.
17) The Numbers which are not Rational Numbers,are called Irrational Numbers.

Real Numbers

18) Union of Rational and Irrational Numbers is known as Real numbers.
19) Set of all real numbers can be represented as as a Number line known as Real Number Line.
20) Every left handed number is smaller than right handed number.
22) There are four types of intervals depending on their end points.
a) a < x < b, it is both side open interval denoted by (a, b).
b) a ≤ x < b, it is left hand closed and right hand opened interval and is denoted by [a, b).
c) a < x ≤ b, it is left hand opened and right hand closed interval and is denoted by (a, b].
d) a ≤ x ≤ b, it is both side closed interval and is denoted by [a, b].
23) Absolute value of a number is the distance of that number from 0 on the number line. It is written as |x| and read as absolute of x. It is always positive. 
       |x| = 0 if x = 0,
       |x| = -x if x < 0,
       |x| = x if x > 0,      
24) Properties of Real Numbers:
      a) Commutative Law:
          1) For Addition:  a + b = b + a  Example:  2 + 3 = 5 = 3 + 2
          2) For Multiplication: a * b = b * a   Example:  2 * 3 = 6 = 3 * 2
      b) Associative Law:    
          1) For Addition:  a + (b + c) = (a + b) + c  Example:  2 + (3 + 4)  = 9 = (2 + 3) + 4
          2) For Multiplication: a * (b * c) = (a * b) * c   Example:  2 * (3 * 4) = 24 = (2 * 3) * 4
      c) Distributive Law:    
          1) a * (b + c) = (a * b) + (a * c)  Example:  2 * (3 + 4)  = 14 = (2 * 3) + (2 * 4)
25) " 0 " is additive identity. This means 0 + a = a, i. e. 0 + 5 = 5.
26) " 1 " is multiplicative identity. This means 1 * a = a, i. e. 1 * 7 = 7.
27) Additive inverse of any number x is - x. That is addition of a number and its additive inverse is additive identity. x + (-x) = 0.
28) Multiplicative inverse of any number x is 1/x. That is multiplication of a number and its multiplicative inverse is multiplicative identity. x * (1/x) = 1.
29) Most important note: 0 * a = 0. Any number multiplied by 0 is always 0. One example will be definitely asked on this basic concept. Please remember this example:

     " There are 26 alphabets in English. Highest power of x in the expression ( x - a) (x - b) is 2. where as in (x - a) (x - b) (x - c) is 3. Like this we used all the 26 alphabets in order say up to z what is the highest power of x ?" 

30) The sum of two positive real numbers is positive.
31) The sum of two negative real numbers is negative.
32) The sum of one positive and one negative numbers is the difference between them with sign of greater number.
33) The product of two positive real numbers is positive.
34) The product of two negative real numbers is positive.
35) The product of one positive and one negative real numbers is negative.

Next part of this important topic will be published in the next blog.
       

Wednesday, August 31, 2016

Different view on Magic Square

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We will get 8 different magic squares simply entering any date. Software will take care of preparing these 8 magic squares in which first row of all these magic squares is the date which you entered.

Click on the following button to go to the software.


Different view on Magic Square

Step-1

Click on above link. The following page will open.


Wednesday, August 24, 2016

Math Riddles 01

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Dear Students,
Today we will see some riddles. Actually the riddles are very simple. Only we need to think on the riddle with the basic concept of the topic.

We all know the decimal system very well. There are 10 digits say 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. In this riddle, we changed all the values of digits 0 to 9. On the basis of these changed values of all these digits, following problems are solved. Study all these problems very carefully.

Let us discuss some steps of the solution of such problems.

Solved Example:

Key-1

      Sum of all the digits from 0 to 9 is 58.

Key-2

     853
  + 388
-----------
     818

Key-3

     5381
  + 1573
-------------
    26501

Key-4

     9
  x 6
-----------
   75

Step-1

Using Key 1:

As we know that the sum of all the digits from 0 to 9 is 45 in actual case. Here in the problem, for changed values of the digits, the sum is 58 so original digit for 5 is 4 and that of 8 is 5. This means, for new value 5, the original value of digit is 4 and for new value 8, the original value of digit is 5.

Thursday, June 16, 2016

Software to calculate Degree and Radians

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a) Click here for "01-Degree-2-Degree-Minute-Second-n-Radian-converter"
b) Click here for "02-Degree-Minute-Second-2-Degree-n-Radian-converter"
c) Click here for "03-Radian-2-Degree-n-Degree-Minute-Second-converter"

1) Directed Angles:


The initial arm rotates through certain amount of rotation in clockwise or anti-clockwise direction to the terminal arm then this amount of rotation is called as the measure of directed angle and such an angle is called as directed angle.



The directed angle AOB has ray OA as an initial arm and ray OB as the terminal arm. O is called as vertex of an angle AOB. 
Note: 
1) Here, Angle AOB ≠ Angle BOA even if they have same amount of rotation.
2) If the rotation of initial arm is anticlockwise, the directed angle is positive and if it is clockwise then the angle is negative.


See the figure carefully to understand the concept.

Monday, March 14, 2016

Albert Einstein’s 137th birthday (Pi Day)

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Dear Einstein, Happy Birthday

(Happy Pi Day)



Today is March, 14. It is celebrated as Pi day. The closest calendar date on Pi.  Pi represents the ratio of the circumference of a circle to its diameter. This ratio is approximately 3.14159... 

March, 14 is also Albert Einstein's birthday. Today, it is his 137th birthday. So let's celebrate his 137th birthday with two different magic squares. 5 numbers of his birth date with 137th birthday is arranged in the first row of both the magic squares. 03/14/18/79/137 are the numbers in the first row of these two magic squares as shown bellow.

Magic square 1

Magic square 2

Thursday, July 30, 2015

Magic Square Software (5 x 5) part-3

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Third Part of Magic Square Software (5 x 5). 


1) Click here to read "Magic Square Software (5 x 5) part-1"

2) Click here to read "Magic Square Software (5 x 5) part-2"

This is the third part of 5x5 magic square software.

You can enter any 5 numbers between -9999 to 9999 (actually you can enter any number, but for the betterment of the look of the sheet, this restriction is implemented in the software) of your choice in the first row. Software will give you two different magic squares with your chosen numbers are in 1 st row. These two magic squares have 120 types of same addition.

Now the software is uploaded and ready to use.

Rows
5
Columns
5
Diagonals
2
Broken Diagonals
8
Different Patterns
100
Total
120

Click the following button for the software of magic squares of order 5 x 5.

5x5 magic square software

Now we will see all the properties of 5 x 5 magic squares one by one. Let us see the magic squares in which we choose the first row with the numbers as 9, 21, 34, 45 and 56. Following two magic squares will be obtained by the software. 


Now in continuation of the previous blogs (Blog-99) and (Blog-100), we will see remaining types of addition:

Sunday, July 26, 2015

Magic Square Software (5 x 5) part-2

BLOG-100


Second Part of Magic Square Software (5 x 5). Click here to read "Magic Square Software (5 x 5) part-1"


Prepare your own two magic squares of 5 rows and 5 columns in which first row is of your choice. You can enter any 5 numbers between -9999 to 9999 (actually you can enter any number, but for the betterment of the look of the sheet, this restriction is implemented in the software) of your choice in the first row. Software will give you two different magic squares with your chosen numbers are in 1 st row. These two magic squares have 120 types of same addition.

Rows
5
Columns
5
Diagonals
2
Broken Diagonals
8
Different Patterns
100
Total
120

The software of magic squares of order 5 x 5 will be uploaded shortly.


Now we will see all the properties of 5 x 5 magic squares one by one. Let us see the magic squares in which we choose the first row with the numbers as 9, 21, 34, 45 and 56. Following two magic squares will be obtained by the software. 


Now in continuation of the previous blog (Blog-99) we will see remaining types of addition:

Wednesday, July 22, 2015

Magic Square Software (5 x 5) part-1

BLOG-99


Prepare your own two magic squares of 5 rows and 5 columns in which first row is of your choice. You can enter any 5 numbers between -9999 to 9999 (actually you can enter any number, but for the betterment of the look of the sheet, this restriction is implemented in the software) of your choice in the first row. Software will give you two different magic squares with your chosen numbers are in 1 st row. These two magic squares have 120 types of same addition.

Rows
5
Columns
5
Diagonals
2
Broken Diagonals
8
Different Patterns
100
Total
120

The software of magic squares of order 5 x 5 will be uploaded shortly.

Now we will see all the properties of 5 x 5 magic squares one by one. Let us see the magic squares in which we choose the first row with the numbers as 9, 21, 34, 45 and 56. Following two magic squares will be obtained by the software. 


Now we will see all types of addition:

1) Addition of all the numbers in the Rows: (1 to 5)

     a) First Row: (1/5)



                             09 + 21 + 34 + 45 + 56 = 165          09 + 21 + 34 + 45 + 56 = 165

     b) Second Row: (2/5)



                             44 + 55 + 13 + 20 + 33 = 165          36 + 47 + 53 + 11 + 18 = 165

     c) Third Row: (3/5)



                              24+ 32 + 43 + 54 + 54 = 165          55 + 47 + 20 + 38 + 44 = 165

     d) Forth Row: (4/5)



                              55+ 11 + 23 + 36 + 42 = 165          22 + 35 + 46 + 52 + 10 = 165

    e) Fifth Row: (5/5)



                              35+ 46 + 52 + 10 + 22 = 165          43 + 54 + 12 + 19 + 37 = 165

2) Addition of all the numbers in the Columns: (6 to 10)

     a) First Column: (6/10)



                              09+ 44 + 24 + 53 + 35 = 165          09 + 36 + 55 + 22 + 43 = 165

     b) Second Column: (7/10)


                           
                              21+ 55 + 32 + 11 + 46 = 165          21 + 47 + 08 + 35 + 54 = 165


     c) Third Column: (8/10)



                             34+ 13 + 43 + 23 + 52 = 165          34 + 53 + 20 + 46 + 12 = 165

     d) Forth Column: (9/10)



                             45 + 20 + 54 + 36 + 10 = 165          45 + 11 + 38 + 52 + 19 = 165

     e) Fifth Column: (10/10)



                              56 + 33 + 12 + 42 + 22 = 165          56 + 18 + 44 + 10 + 37 = 165

3) Addition of all the numbers in the Diagonals: (11 to 12)

     a) First Diagonal: (11/12)



                             09 + 55 + 43 + 36 + 22 = 165          09 + 47 + 20 + 52 + 37 = 165

     b) Second Diagonal: (12/12)



                             56 + 20 + 43 + 11 + 35 = 165          56 + 11 + 20 + 35 + 43 = 165

8 types of broken diagonal addition and 100 types of different patterns of addition will be published in the next blog.

Thursday, July 2, 2015

Code of Click-Button for Blog or Website

BLOG-98


While writing the blogs, I want to place "button" to open the corresponding site, I found one interesting method to write the code for this button.

Click on the following button to go to the software to get the code.




For the software, follow the following instructions step by step.

Step-1


Tuesday, June 23, 2015

New Concept of Learning Mathematics

Blog-97

1)       About Students:

 Every child is Brilliant. They need to love studying every subject. First of all they need to find the technical meaning of each word from every statement while reading. If you confirm about knowing the meaning of that statement the child need to proceed further. Now the child will understand that particular line/Statement. Like this, statement by statement if child is reading & understanding each statement then definitely he/she will understand the concept of the paragraph.   Think this way that why the Author/Scientist/Mathematician had written the statement which one you are reading. Then the Concept of entire topic will be cleared and definitely you will understand the topic very easily.

Friday, June 12, 2015

Derivative-Integration Calculator

Blog 96

Dear Students,
We know that, everyone of us have list of formulae of so many mathematical theory from so many sources. This blog will give you the derivative integration calculator, which display the derivative and integration of the function which you choose from the drop down menu of the calculator.

Click on the following button:



The list of all the functions and their derivative and integration is give below.

Friday, May 22, 2015

Area Volume Calculation Tool

BLOG-95

Special Note: Following link will help you to get all the calculations which are based on two values given by you.

Most amazing part of this blog is this that you will get all calculated values such as perimeters, area volume, curved surface area and total surface area, using the software of which the link is provided here.

Click on the following Button for Area Volume Calculator:


Area Volume Calculator

Friday, May 8, 2015

Magic square for some particular date (Part-3)

Blog-94

Special Note: Two links are given bellow to get your own Magic Squares.

Most amazing part of this blog is this that you can prepare your own two Magic Squares using the software of which the link is provided here.

You can prepare magic squares simply by clicking the following images.

A) Magic Square Maker for any Date:

Date Magic Square Maker

Friday, May 1, 2015

Magic square for some particular date (Part-2)

Blog-93

Click here to see the previous Blog on this subject.

In previous blog, we observed, addition of all the numbers in each row, column, diagonal and broken diagonal. Today we will discuss some new patterns of the same magic square.


4) Addition of all the numbers in other pattern-01: 


Thursday, April 30, 2015

Magic square for some particular date

Blog-92

Dear Friends,
Today we will discuss something new about the Magic square prepared for some specific date.
See the following examples for the date 30/04/1968.



Here 1st row of both the squares is 30/04/1968.

The addition of all the numbers in each rows , columns, diagonals is same. Like this all together we 26 kind of groups in which addition of all these numbers is same. We will discuss all these groups one by one. Let us observe all these groups in the following diagrams.

1) Addition of all the numbers in each row: 


Monday, February 23, 2015

5 Colors and 5 Cubes

Blog-91


Dear Students,

Today, we will see something new, other than our regular study.

We will take 5 colors and 5 cubes. We will paint them in such a way that in some particular position, no color is repeated on any cube to that side. If we look to all the 5 cubes, from front, we will see that no color is repeated. In the same way the same situation is there from all the remaining 5 sides.  Now we will see the following diagram.





Diagram-01

Thursday, August 28, 2014

Basics of Trigonometry - 09

Blog-90

Dear Students,

Now we will study the next part of trigonometry. 
Every triangle has three sides so there will be six ratios of the lengths of the sides of a triangle which are already known to us. These are the trigonometric ratios. In the adjacent diagram, we have the following ratios. (Let us take Angle A for all the trigonometric ratios).:

1) BC/AC = sin A
2) AB/AC = cos A
3) BC/AB = tan A
4) AC/BC = csc A
5) AC/AB = sec A
6) AB/BC = cot A

In a right angled triangle, by theorem of Pythagoras, we have, BC AB 2   = AC 2   [ Please see the adjacent diagram]
If we divide above equation by AC 2 , AB 2  and BC 2 , 
we get three different identities.
1) Divide equation (1) by AC 2 , we get 
BC AB 2   = AC 2      ----------  (1)

   (BC 2/AC 2) + (AB 2/AC 2)  = (AC 2/AC 2)
    (BC/AC)2 + (AB/AC)2   = (AC/AC)2
    (sin A)2 + (cos A)2   = (1)2
   sin 2 A + cos 2 A   = 1

   sin 2 A + cos 2 A   = 1 

Tuesday, January 21, 2014

Basics of Trigonometry - 08

Blog-89

Dear Students,
Now we will study the next part of trigonometry. 

Today we will study trigonometric ratios of three groups as shown bellow.
Group-03: 150° (90° + 60°)240° (180° + 60°)330° (270° + 60°).

3) Trigonometric ratios of Group-03: 150° (90° + 60°)240° (180° + 60°)330° (270° + 60°).

a) An angle q = 150°
Here angle XOP is of 150° (Anti-clock-wise-direction).

Here, inclination of ray OP is 150°, so angle AOP 30° and angle OPA is 30°

We know that the side opposite to 30° is 1/2 times the hypotenuse.
So if hypotenuse OP = r, then  AP = r/2 (side opposite of 30°).

As point A is to the negative side of X-axis, x-coordinate of point A will be - 3r/2. In the same way point P is in the 2nd quadrant, so y-coordinate of point P will be r/2. So, the coordinates of point P will be (-3r/2r/2).

So, all the trigonometric ratios of q = 150° with
 x = 3r/2,
 y = r/2,
 r = r.

a) sin 150° = y/r
    sin 150° = (r/2)/r   
    sin 150° = 1/2
b) cos 150° = x/r
    cos 150° = (-√3 r/2)/r  
    cos 150° = - √3/2
c) tan 150° = y/x
    tan 150° = (r/2)/(-√3 r/2)         tan 150° = - 1/√3
d) csc 150° = r/y
    csc 150° =  r/(r/2)    
     csc 150° = 2
e) sec 150° = r/x
    sec 150° = r/(-√3 r/2)         sec 150° = - 2/√3
f) cot 150° = x/y
    cot 150° = (-√3 r/2)/(r/2)   
    cot 150° = - √3

b) An angle q = 240°
Here angle XOP is of 240° (Anti-clock-wise-direction).

Here, inclination of ray OP is 240°, so angle AOP 60° and angle OPA is 30°

We know that the side opposite to 30° is 1/2 times the hypotenuse.


So if hypotenuse OP = r, then  AO = r/2 (side opposite to 30°)
& AP = √3 r/2 (side opposite of 60°).

As point A is to the negative side of X-axis, x-coordinate of point A will be - r/2. In the same way point P is in the 2nd quadrant, so y-coordinate of point P will be √3 r/2. So, the coordinates of point P will be (-r/2, √3 r/2).

So, all the trigonometric ratios of q = 150° with
 x = - r/2,
 y = - √3 r/2,
 r = r.

a) sin 240° = y/r
    sin 240° = [(-√3 r)/2]/r   
    sin 240° = -√3/2
b) cos 240° = x/r
    cos 240°  = (-r/2)/r       cos 240° = -1/2
c) tan 240° = y/x
    tan 240° = [(-√3 r)/2]/(-r/2)
     tan 240° =  √3
d) csc 240° = r/y
    csc 240° =  r/[(-√3 r)/2]  
    csc 240° = -2/√3
e) sec 240° = r/x
    sec 240° = r/(-r/2)        sec 240° = -2
f) cot 240° = x/y
    cot 240° = (-r/2)/[(-√3 r)/2]
    cot 240° = 1/√3

c) An angle q = 330°
Here angle XOP is of 330° (Anti-clock-wise-direction).

Here, inclination of ray OP is 330°, so angle AOP 30° and angle OPA is 60°

We know that the side opposite to 30° is half the hypotenuse and side opposite to 60° is √3/2 times the hypotenuse.
So if hypotenuse OP = r, then  AP = r/2 (side opposite of 30°)
and OA = (√3 r)/2 (side opposite of 60°).

As point A is to the positive side of X-axis, x-coordinate of point A will be (√3 r)/2. In the same way point P is in the 4th quadrant, so y-coordinate of point P will be -r/2. So, the coordinates of point P will be (√3 r/2, -r/2)

So, all the trigonometric ratios of q = 330° with
 x = (√3 r)/2,
 y = - r/2,
 r =  r.
a) sin 330° = y/r
    sin 330° = (-r/2)/r   
    sin 330° = -1/2
b) cos 330° = x/r
    cos 330= (√3 r)/2/ r        cos 330° = √3/2
c) tan 330° = y/x
    tan 330° = (-r/2)/(√3 r)/2
     tan 330° =  -1/√3
d) csc 330° = r/y
    csc 330° =  r/(-r/2) 
   csc 330° = -2
e) sec 330° = r/x
    sec 330° = r/(√3 r)/2        sec 330° = 2/√3
f) cot 330° = x/y
    cot 330° = (√3 r)/2/(-r/2)
    cot 330° = -√3










In the next Blog, we will see some more important proofs and formulae about trigonometry.

Anil Satpute