Maths project for class 10 th
-
Upload
chandragopal-yadav -
Category
Education
-
view
2.246 -
download
5
description
Transcript of Maths project for class 10 th
![Page 1: Maths project for class 10 th](https://reader033.fdocuments.in/reader033/viewer/2022061202/547c6bafb47959b6508b465b/html5/thumbnails/1.jpg)
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
![Page 2: Maths project for class 10 th](https://reader033.fdocuments.in/reader033/viewer/2022061202/547c6bafb47959b6508b465b/html5/thumbnails/2.jpg)
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 .
![Page 3: Maths project for class 10 th](https://reader033.fdocuments.in/reader033/viewer/2022061202/547c6bafb47959b6508b465b/html5/thumbnails/3.jpg)
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 .
![Page 4: Maths project for class 10 th](https://reader033.fdocuments.in/reader033/viewer/2022061202/547c6bafb47959b6508b465b/html5/thumbnails/4.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,
![Page 5: Maths project for class 10 th](https://reader033.fdocuments.in/reader033/viewer/2022061202/547c6bafb47959b6508b465b/html5/thumbnails/5.jpg)
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
![Page 6: Maths project for class 10 th](https://reader033.fdocuments.in/reader033/viewer/2022061202/547c6bafb47959b6508b465b/html5/thumbnails/6.jpg)
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
![Page 7: Maths project for class 10 th](https://reader033.fdocuments.in/reader033/viewer/2022061202/547c6bafb47959b6508b465b/html5/thumbnails/7.jpg)
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
![Page 8: Maths project for class 10 th](https://reader033.fdocuments.in/reader033/viewer/2022061202/547c6bafb47959b6508b465b/html5/thumbnails/8.jpg)
TRIANGLES
![Page 9: Maths project for class 10 th](https://reader033.fdocuments.in/reader033/viewer/2022061202/547c6bafb47959b6508b465b/html5/thumbnails/9.jpg)
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
![Page 10: Maths project for class 10 th](https://reader033.fdocuments.in/reader033/viewer/2022061202/547c6bafb47959b6508b465b/html5/thumbnails/10.jpg)
Pro
vin
g th
e T
hale
s’
Th
eore
m
![Page 11: Maths project for class 10 th](https://reader033.fdocuments.in/reader033/viewer/2022061202/547c6bafb47959b6508b465b/html5/thumbnails/11.jpg)
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
![Page 12: Maths project for class 10 th](https://reader033.fdocuments.in/reader033/viewer/2022061202/547c6bafb47959b6508b465b/html5/thumbnails/12.jpg)
Pro
vin
g th
e c
on
vers
e o
f T
hale
s’ T
heore
m
![Page 13: Maths project for class 10 th](https://reader033.fdocuments.in/reader033/viewer/2022061202/547c6bafb47959b6508b465b/html5/thumbnails/13.jpg)
Trigonometry means
“Triangle” and “Measurement”
Introduction Trigonometric Ratios
![Page 14: Maths project for class 10 th](https://reader033.fdocuments.in/reader033/viewer/2022061202/547c6bafb47959b6508b465b/html5/thumbnails/14.jpg)
There are 3 kinds of trigonometric ratios we will learn.
sine ratio
cosine ratio
tangent ratio
Three Types Trigonometric Ratios
![Page 15: Maths project for class 10 th](https://reader033.fdocuments.in/reader033/viewer/2022061202/547c6bafb47959b6508b465b/html5/thumbnails/15.jpg)
SINE RATIOS Definition of Sine Ratio. Application of Sine Ratio.
![Page 16: Maths project for class 10 th](https://reader033.fdocuments.in/reader033/viewer/2022061202/547c6bafb47959b6508b465b/html5/thumbnails/16.jpg)
Definition of Sine Ratio.
1
If the hypotenuse equals to 1
Sin = Opposite sides
![Page 17: Maths project for class 10 th](https://reader033.fdocuments.in/reader033/viewer/2022061202/547c6bafb47959b6508b465b/html5/thumbnails/17.jpg)
Definition of Sine Ratio.
For any right-angled triangle
Sin = Opposite side
hypotenuses
![Page 18: Maths project for class 10 th](https://reader033.fdocuments.in/reader033/viewer/2022061202/547c6bafb47959b6508b465b/html5/thumbnails/18.jpg)
Cosine Ratios Definition of Cosine. Relation of Cosine to the sides of right
angle triangle.
![Page 19: Maths project for class 10 th](https://reader033.fdocuments.in/reader033/viewer/2022061202/547c6bafb47959b6508b465b/html5/thumbnails/19.jpg)
Definition of Cosine Ratio.
1
If the hypotenuse equals to 1
Cos = Adjacent Side
![Page 20: Maths project for class 10 th](https://reader033.fdocuments.in/reader033/viewer/2022061202/547c6bafb47959b6508b465b/html5/thumbnails/20.jpg)
Definition of Cosine Ratio.
For any right-angled triangle
Cos =
hypotenuses
Adjacent Side
![Page 21: Maths project for class 10 th](https://reader033.fdocuments.in/reader033/viewer/2022061202/547c6bafb47959b6508b465b/html5/thumbnails/21.jpg)
Tangent Ratios
Definition of Tangent. Relation of Tangent to the sides of
right angle triangle.
![Page 22: Maths project for class 10 th](https://reader033.fdocuments.in/reader033/viewer/2022061202/547c6bafb47959b6508b465b/html5/thumbnails/22.jpg)
MATHS PROJECT
MADE BY- CHANDRAGOPAL YADAV
CLASS – 10th
![Page 23: Maths project for class 10 th](https://reader033.fdocuments.in/reader033/viewer/2022061202/547c6bafb47959b6508b465b/html5/thumbnails/23.jpg)
Definition of Tangent Ratio.
For any right-angled triangle
tan = Adjacent Side
Opposite Side
![Page 24: Maths project for class 10 th](https://reader033.fdocuments.in/reader033/viewer/2022061202/547c6bafb47959b6508b465b/html5/thumbnails/24.jpg)
CONCLUSION
hypotenuse
side oppositesin
hypotenuse
sidedjacent acos
sidedjacent a
side oppositetan
Make Sure that the
triangle is right-angled