Animesh's Math: Mathematical formulas
In higher algebra what we know about binomial theorem, in lower classes, we teach this as the result of special product. It forced the student to memorize it without knowing what is actually what. Due to this two problems arises first in present state students has no proper idea that why they memorize these formulas secondly, the science aspirant students when get binomial theorem they think that it’s a complete new thing so they again memorize it without knowing what is actually what. We don’t teach binomial theorem in this state that in these state students has no idea about permutation and combination so they can’t grasp it. But there is a palatable form to teach them binomial theorem in early state also. This I try to describe bellow:
First of all we should discuss that why we have so importance about a + b & a – b? Because these represent the sum and difference of two unknown quantities. And algebra is a mathematical science where a rule of Arithmetic is become generalized so, we deal here various unknown quantities. Hence a + b & a – b is so important to measure various quantities.
The rules of expansion of any binomials i.e.(a±b)
Is as follows:
i) The no. of terms in expansion will be n+1
ii) The first and last term will be a & b in same power of (a±b)
iii)The coefficient of2ndterm is the power of (a ± b).
iv) The coefficient of any term of the expansion will be the product of the coefficient of previous term with the power of a in that term and divided by the no. of previous term in expansion. i.e.
Coefficient of previous term X power of a in that term
= -------------------------------------------------------------------
The no. of previous term in expansion
v) The rule of sign: All term of expansion a+b will be positive. And sign
Of expansion a – b will be alternatively positive and negative.
Another point, generally we teach our students about (a +b + c +d +..)² that sum up the squares of all the terms + twice of all the term taken twice at a time. But it is a general rule act only to the power 2. It is better to teach them:
(a + b + c - d +e)² ={ (a+b+c) – (d – e )}² = (a+b+c)² + 2. (a+b+c). (d – e ) + (d – e )²
= a² + b² +c² +2ab+2bc+2ca + 2ad +2bd + +2cd – 2ae – 2be – 2ce
Algebra
(a+b) ² = a² + 2ab + b²
(a- b) ² = a² - 2ab + b²
a² + b² = (a+b) ² - 2ab
= (a-b) ² +2ab
a² - b² = (a+b) (a-b)
(a + b)² = (a – b)² + 4ab
(a – b )² = (a + b)² - 4ab
(a + b)² + (a – b)² = 2(a² + b²)
(a + b)² – (a – b)² = 4ab
4ab = (a + b)² – (a – b)²
(a+b) ³ = a³ + 3a²b + 3ab² +b³
Or, = a³ +b³ +3ab (a+b)
(a-b) ³ = a³ - 3a²b + 3ab² - b³
Or, = a³ - b³ -3ab (a-b)
a³ +b³ = (a+b) ³ - 3ab (a+b)
= (a+b)(a² -ab +b²)
a³ - b³ = (a-b) ³ + 3ab(a-b)
= (a-b)(a² +ab +b²)
(a + b + c) ² = a² + b² + c² +2ab +2bc +2ca
a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² +
c² -ab –bc – ca)
or, = ½( a +b +c){ (a-b)² + (b-c)²
+ (c-a)² }
Factorization
1. Take common out.
2. Use formulae i) a² - b² , ii) a² + b² if ab is a perfect square, iii) a³ + b³ iv) a³ - b³
v) a³ + b³ + c³ - 3abc
3. In case of quadratic equation:
i) Middle term factor: factorize the
Product (with its sign) of the co-efficient of x² & x° so that their algebraic sum be the co-efficient of x
ii) Using the Sridhar Archery’s formulae:
- b ± √b² - 4ac
X = --------------------
2a
3. In case of 3rd degree equation: Reminder Theorem: If f(x) is divisible by ax + b, then reminder will be f( - b/a)
MENSURATION
1. Perimeter of rectangle: 2 X (length + breath)
2. Area of rectangle: Length X breath.
3. Diagonal of rectangle: √ length² + breath².
4. Perimeter of Square: 4 X side.
5. Area of Square: Side ².
6. Diagonal of Square: √2 X Side.
7. Perimeter (s) of Triangle: Sum of 3 Sides.
8. Area of triangle: ½ X base X altitude.
Or √s(s-a)(s-b)(s-c)
9. Area of parallelogram: (sum of two parallel sides) X altitude.
10. Area of Rhombus: Product of its diagonals.
11. Area of Trapezium: ½ (sum of two parallel sides) X altitude.
12. Altitude of a equilateral triangle:
√3/2 X side
13. Sum of internal angles of a Polygon:
(2n – 4) Right angels//Sum of external angles: 4 Rt.s
14. Numbers of diagonals of a Polygon:
15. The ratio between circumference and the diameter of a circle is constant.
Circumference ` diameter = π
Circumference = π X 2 X radius (r)
Circumference = 2. π.r
16. Area of a circle = π.r²
17. Total surface area of cuboids:
2 X (lb + lh + bh)
18. Volume of cuboids: l X b X h
19. Total surface area of cube: 6.side²
20. Volume of cube: Side³
21. Curved surface area of cylinder:
= 2 π.r.h
22. Total surface area of cylinder:
= 2π(r + h)
23. Volume of cylinder = π.r².h
24. Slant height (l) of Cone = √h² + r²
25. Curved surface area of Cone = π.r.l
26. Total surface area of Cone: π.r(r+l)
27. Volume of Cone: 1/3 .π.r².h
28. Surface area of sphere: 4.π.r²
29. Volume of sphere: 4/3 π.r³
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