Maths project for class 10 th
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The document discusses the basic proportionality theorem in geometry. It states that if a line is drawn parallel to one side of a triangle, intersecting the other two sides, the lengths of the segments of those two sides will be divided in the same ratio. This is also known as Thales' theorem, after the Greek mathematician who discovered it. The document also provides proofs of both the theorem and its converse.
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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 .](https://image.slidesharecdn.com/mathsprojectforclass10th-131014092924-phpapp01/85/Maths-project-for-class-10-th-2-320.jpg)
![(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,](https://image.slidesharecdn.com/mathsprojectforclass10th-131014092924-phpapp01/85/Maths-project-for-class-10-th-4-320.jpg)
Maths project for class 10 th
- 1. Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.
- 2. 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 .
- 3. 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 .
- 4. (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,
- 5. 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
- 6. 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
- 7. 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
- 9. Basic Proportionality Theorem 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.
- 11. If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side
- 12. Proving the converse of Thales’ Theorem
- 13. Introduction Trigonometric Ratios Trigonometry means “Triangle” and “Measurement”
- 14. There are 3 kinds of trigonometric ratios we will learn. sine ratio cosine ratio tangent ratio
- 15. Definition of Sine Ratio. Application of Sine Ratio.
- 16. Definition of Sine Ratio. 1 If the hypotenuse equals to 1 Sin = Opposite sides
- 17. Definition of Sine Ratio. For any right-angled triangle Opposite side Sin = hypotenuses
- 18. Definition of Cosine. Relation of Cosine to the sides of right angle triangle.
- 19. Definition of Cosine Ratio. 1 If the hypotenuse equals to 1 Cos = Adjacent Side
- 20. Definition of Cosine Ratio. For any right-angled triangle Adjacent Side Cos = hypotenuses
- 21. Definition of Tangent. Relation of Tangent to the sides of right angle triangle.
- 23. Definition of Tangent Ratio. For any right-angled triangle Opposite Side tan = Adjacent Side
- 24. opposite side sin hypotenuse adjacent side cos hypotenuse opposite side tan adjacent side Make Sure that the triangle is right-angled