Euclid’s Division Lemma Given positive

integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.

REAL NUMBERS

POLYNOMIALS

RELATIONSHIP BETWEEN ZEROS AND COEFFICIENT OF A POLYNOMIAL

relationship between zeros and coefficient of a polynomial in case of quadratic and cubic polynomial is stated as follows

(1) QUADRATIC POLNOMIAL

Let ax² +bx +c be the quadratic polynomial and α and β are its zeros ,then

Sum of zeros = α + β = -b/a = - (coefficient of x)/ (coefficient of x²)

Product of zeros = αβ = c/a = constant term / (coefficient of x²)

If we need to form an equation of degree two ,when sum and products of the roots is given ,then K[x²-( α + β )x + αβ ]=0 is the required equation ,where k is constant .

Procedure for finding zeros of a quadratic polynomial

·         Find the factors of the quadratic polynomial .

·         Equate each of the above factors (step 1) with zero.

·         Solve the above equation (step 2)

·         The value of the variables obtained (step 3) are the required zeros .

(2) CUBIC POLYNOMIAL

Let axᶟ +bx² +cx +d be the cubic polynomial and α , β and γ are its zeros ,then

Sum of zeros = α + β + γ = -b/a = - (coefficient of x²)/ (coefficient of xᶟ)

Sum of Product of zeros taken two at a time = αβ +βγ +γα = c/a = (coefficient of x)/ (coefficient of xᶟ)

Product of zeros = αβγ = -d/a = - constant term / (coefficient of xᶟ)

When sum of zeros , Sum of Product of zeros taken two at a time , Product of zeros is given , then K[xᶟ-( α + β + γ )x² + (αβ +βγ +γα)x – αβγ ]=0 is the required equation ,where k is constant,

Procedure for finding zeros of a cubic polynomial

·         By hit and trial method find one zeros of the polynomial using remainder theorem

·         Now if we know one zero , then we know one factor of the polynomial . divide the cubic polynomial by this factor to obtain quadratic polynomial

·         Now , solve this quadratic polynomial to obtain the other two zeros of the cubic polynomial .

·         These three zeros are the required

PAIR OF LINEAR EQUATINS IN TWO VARABLES

ALGEBRAC METHOD OF SOLVING A PAIR OF LINEAREQUATIOS

One algebraic method is the substitution method. In this case, the value of one variable is expressed in terms of another variable and then substituted in the equation. In the other algebraic method – the elimination method – the equation is solved in terms of one unknown variable after the other variable has been eliminated by adding or subtracting the equations. For example, to solve:

8x + 6y = 16

-8x – 4y = -8

Using the elimination method, one would add the two equations as follows:8x + 6y = 16

-8x – 4y = -8

2y = 8 y = 4 The variable "x" has been eliminated. Once the value for y is known, it is possible to solve for x by substituting the value for y in either equation:

8x + 6y = 16

8x + 6(4) = 16

8x + 24 = 16

8x + 24 – 24 = 16 – 24

8x = -8

X = - 1

TRIANGLES

Basic Proportionality Theorem states that if a line is drawn parallel to one side of a triangle to intersect the other 2 points , the other 2 sides are divided in the same ratio.

It was discovered by Thales , so also known as Thales theorem.

Basic Proportionality Theorem

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Converse of the Thales’ Theorem

If a line divides any two sides of a triangle in the same ratio, then

the line is parallel to the third side

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Trigonometry means

“Triangle” and “Measurement”

Introduction Trigonometric Ratios

There are 3 kinds of trigonometric ratios we will learn.

sine ratio

cosine ratio

tangent ratio

Three Types Trigonometric Ratios

SINE RATIOS Definition of Sine Ratio. Application of Sine Ratio.

Definition of Sine Ratio.

1

If the hypotenuse equals to 1

Sin = Opposite sides

Definition of Sine Ratio.

For any right-angled triangle

Sin = Opposite side

hypotenuses

Cosine Ratios Definition of Cosine. Relation of Cosine to the sides of right

angle triangle.

Definition of Cosine Ratio.

1

If the hypotenuse equals to 1

Cos = Adjacent Side

Definition of Cosine Ratio.

For any right-angled triangle

Cos =

hypotenuses

Adjacent Side

Tangent Ratios

Definition of Tangent. Relation of Tangent to the sides of

right angle triangle.

MATHS PROJECT

MADE BY- CHANDRAGOPAL YADAV

CLASS – 10th

Definition of Tangent Ratio.

For any right-angled triangle

tan = Adjacent Side

Opposite Side

CONCLUSION

hypotenuse

side oppositesin

hypotenuse

sidedjacent acos

sidedjacent a

side oppositetan

Make Sure that the

triangle is right-angled