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# Maths project for class 10 th

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### Maths project for class 10 th

1. 1. Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.
2. 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. 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. 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. 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. 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. 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
8. 8. 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.
9. 9. Proving the Thales’ Theorem
10. 10. If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side
11. 11. Proving the converse of Thales’ Theorem
12. 12. Introduction Trigonometric Ratios Trigonometry means “Triangle” and “Measurement”
13. 13. There are 3 kinds of trigonometric ratios we will learn. sine ratio cosine ratio tangent ratio
14. 14.   Definition of Sine Ratio. Application of Sine Ratio.
15. 15. Definition of Sine Ratio. 1  If the hypotenuse equals to 1 Sin = Opposite sides
16. 16. Definition of Sine Ratio.  For any right-angled triangle Opposite side Sin = hypotenuses
17. 17.   Definition of Cosine. Relation of Cosine to the sides of right angle triangle.
18. 18. Definition of Cosine Ratio. 1  If the hypotenuse equals to 1 Cos = Adjacent Side
19. 19. Definition of Cosine Ratio.  For any right-angled triangle Adjacent Side Cos = hypotenuses
20. 20.   Definition of Tangent. Relation of Tangent to the sides of right angle triangle.
21. 21. Definition of Tangent Ratio.  For any right-angled triangle Opposite Side tan = Adjacent Side
22. 22. opposite side sin   hypotenuse adjacent side cos   hypotenuse opposite side tan   adjacent side Make Sure that the triangle is right-angled