Monday, August 13, 2007

Indian Maths Tutors

Indian Mathematical standards are very high compared to other countries in the world.Indians are well known for intelligence and grasping power.

Learning Mathematics as per indian standards will increase the arthimatic ability and problem solving capability of a child.

Many great theories were discovered in mathematics by great indian mathematicians like Srinivasa Ramanujam.The decimal notation and Number '0' is discovered in india.

The blog is created to publish Indian math topics for kids of all ages from 4 to 15 years of age.

Few Sample Lessons will be posted here and Parents who ever are interested in getting their child trained in indian mathematics online can post a reply. We are planning to open a online tutor program to educate children outside India. There will be a minimal fee for each age group.

For kids of age group 4 to 5 assuming they know a numbers upto 100
the topics are designed to one "How to Count" and what Number Comes before and after etc.

In this topic we skim through the basic number counting and writing the number in words and digits.Counting:Here in the figure below we have some circles on the left hand side and we need to write the number of circles present on the right hand side. We need to write the value in both words and digits.for example

In this way count and fill up the value in adjacent boxes
Exercise A0.1
stars In words in digits
1. ********** ******
2. ********** ********** ********


For Example:For age group 15, the general topics that will be discussed are :

* Sets
*Progressions
*Statements
*Matrices
*Polynomials
Each topic will have a detailed explanation with examples and some practice problems.

For example the Polynomials topic if we take:


1. If a0 ,a1 ,a2 , a3…an are numbers where an not equal to zero then
f(x) = a0 + a1x +a2 x2+…anxn is called a polynomial of order n in x.

a) Non Zero constants are polynomials of zero order.(Ex: 1,2,3….)
b) ax+b is a polynomial in x of order one. Here a ,b are any two constants (Ex: 2x+1,5x+6….).
c) ax2+bx+c where a not equal to zero is a quadratic expression in x.

2. The General form of Quadratic Equation (above one is expression) is ax2+bx+c = 0 where a not equal to zero.
So when a quadratic expression equated to zero it becomes quadratic equation.
3. Roots of a quadratic equation means the possible values for the variable ‘x’ that will satisfy the Quadratic equation. That means when ‘x’ is replaced by these values in
ax2+bx+c=0 then the equation becomes true.
For a Quadratic equation there exists two roots.
4. Formulae to find the 2 roots of the quadratic equation
ax2+bx+c=0 are
Let P, Q represent the two roots then

P= -b + squareroot(b2 – 4ac)/ 2a



Q= -b - squareroot(b2 – 4ac)/ 2a

Here a,b,c are constants taken from ax2+bx+c=0 and

b2 – 4ac is called Discriminant denoted by D.

5. Nature of Roots
The nature of roots of a quadratic equation depend on the value of the
Discriminant D.

1. If the value of D > 0 then the roots are Real and different.
That means the values of the roots belong to set of real numbers and both roots are not equal.

2. If the Value of D = 0 then the roots are Real and Equal.
In this case the roots belongs to Real numbers and also equal.
We can find the roots with simple formula
P=Q= -b/2a

3. If the Value of D is < p =" 2a" q =" 2a">
P + Q = -2b/2a = -b/a = - coefficient of x / coefficient of x2
7. Product of the roots of a quadratic equation


-b + squareroot(b2 – 4ac)/
P = 2a

X

-b - squareroot(b2 – 4ac)/
Q = 2a

ð P X Q = c/a = constant term / coefficient of x2

8. We can find a quadratic equation if the two roots P,Q are given by using the formula
x2 – x(P+Q)+(PXQ)=0
=> x2 – x(Sum of the roots) + (Product of the roots)=0.


Example problems:
1. What is the Nature of roots of the quadratic equation 2x2+4x+1=0
Here a = 2 , b = 4 , c= 1.
Lets find Discriminant d = b2-4ac = 4*4- 4*2*1
16-8=8 >0 so D > 0
this implies the roots are Real and not equal.

2. Find the roots of the quadratic equation 3x2+5x+2=0
Here a= 3, b= 5, c= 2
Then D = b2-4ac=5*5-4*3*2
=25-24
=1 >0 roots are real and not equal
Lets find roots using formula

-b + squareroot(b2 – 4ac)
P = 2a

= -5+squareroot(1)
2 * 3

= -5 +1
= -4/6 = -2/3=> first root = -2/3
6


-b - squareroot(b2 – 4ac)
Q = 2a

= -5-squareroot(1)
2 * 3

= -5 -1
= -6/6 = -1=> Second root = -1
6
So the roots of the above equation are (-2/3,-1).

3. Find the quadratic equation whose roots are (2 ,2)
So we can find a quadratic equation if the roots P, Q are given using the formula
x2 – x(P+Q)+(PXQ)=0
=> x2 – x(Sum of the roots) + (Product of the roots)=0.

Here P= 2 and Q=2 ( Note: Here roots are equal this implies D = 0 )
So the equation is
x2 – x(2+2)+(2*2)=0
=> x2 – x(4)+4=0
=> x2 – 4x+4=0 is the required equation.









Practice Exercises:
1. Find the nature of the roots of the following quadratic equations.
1. 4x2+7x+2=0
A. Real & Different B. Real &amp;amp;amp; Equal C. Complex
2. x2+6x+9=0
A. Real & Different B. Real & Equal C. Complex
3. 5x2+3x+1=0
A. Real & Different B. Real & Equal C. Complex
4. 3x2+7x+2=0
A. Real & Different B. Real & Equal C. Complex
5. 7x2+4x+1=0
A. Real & Different B. Real & Equal C. Complex
6. 4x2+5x+1=0
A. Real & Different B. Real & Equal C. Complex

2. Find the roots of following QE
1. 2x2+7x+3=0
A. 1,1/2 B. 2/3 ,2/3 C. -3,-1/2

2. 3x2+7x+2=0
A. -4,3/2 B. -3 ,3 C. -2,-1/3

3. 3x2-5x-2=0
A. 2,-1/2 B. 2 ,-1/3 C. -2,-1/3

4. x2-x-20=0
A. 3,4 B. 2 ,-4 C. 5,-4

3. Find Sum of the roots Using the formula of following QE (Hint : -b/a)
1. 3x2+7x-5=0
A. 3/5 B. -7/3 C. 5/3

2. x2-5x+1=0
A. 5 B. -5 C. 1

3. 2x2-5x-1=0
A. 1/2 B.-5/2 C. 5/2

4. x2-x-20=0
A. 1 B. 20 C. 1/20

4. Find product of the roots Using the formula of following QE (Hint : c/a)
1. 3x2+7x-5=0
A. 3/5 B. -7/3 C. -5/3

2. x2-5x+1=0
A. 5 B. -5 C. 1

3. 2x2-5x-1=0
A. 1/2 B.-5/2 C. 5/2

4. x2-x-20=0
A. 1 B. -20 C. 1/20
5. Find the quadratic equations whose roots are
1. 2, 3
2. 1/3, 3
3. 5, -2
4. 7, 1/7


The topics will be provided in a pdf format and the student will be refering to this material while learning the specific topic. The tutoring will be done online using a whiteboard utility and chatting software with Webcam.Parents or the student can send emails incase of any difficulty in learning or for clarifications.




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We will be launching the program quite soon and we are interested to know how many are interested in the same.