Mathematics Basic Formulae Part - 1

                                     

         Mathematics Basic Formulae Part - 1

Algebraic Expansions:

1. (a + b)² = a² + 2ab + b²

2. (a - b)² = a² - 2ab + b²  

3. (a + b) (a - b) = a² - b²

4. (x + a)(x + b) = x² + (a + b)x + ab

5. (x + a)(x - b) = x² + (a - b)x - ab   

6. (x - a)(x + b) = x² + (b - a)x - ab 

7. (x - a)(x - b) = x² - (a + b)x + ab 

8. (a + b)³ = a³ + b³ + 3ab(a + b)

9. (a - b)³ = a³ - b³ - 3ab(a - b)  

10. (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2xz 

11. (x + y - z)² = x² + y² + z² + 2xy - 2yz - 2xz

12. (x - y + z)² = x² + y² + z² - 2xy - 2yz + 2xz 

13. (x - y - z)² = x² + y² + z² - 2xy + 2yz - 2xz 

14. x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz -xz)

15. x² + y² = 1212 [(x + y)² + (x - y)²]

16. (x + a) (x + b) (x + c) = x³ + (a + b + c)x² 

+ (ab + bc + ca) x + abc

17.  x³ + y³ = (x + y) (x² - xy + y²)

18.  x³ - y³ = (x - y) (x² + xy + y²)

  

2D and 3D figures

Figure	Formula for surface area	Formula for volume
Rectangular Prism	SA = 2ab + 2bc + 2ca  sq. Units  where a, b,c are the sides of the cube.	V = abc  cubic units
Cylinder	SA = 2πrh  sq. units  TSA =  2πr(h+r)  sq. units  r = radius of the cylinder  h – height of the cylinder	V = πr2h  cubic units
Cube	SA = 6a2  sq. units  a = sides of the cube	V = a3   cubic units
Sphere	SA = 4πr2  sq. units  r = radius of the sphere	V = 4343 πr3   cubic units
Ellipsoid	SA = 4π[(apbp+apcp+bpcp3)]1p4π[(apbp+apcp+bpcp3)]1p  p = 1.6075  a,b,c are semi axis of ellipsoid	V = 4343  π r1,r2,r3   cubic units
Cone	CSA = πr1 sq. units	V = 1313  πr2h   cubic units
Pyramid	SA = a + 1212*p*1   p = perimeter of pyramid   l = slant height   a= area of the base of the pyramid	V = 1313*a*h    cubic units
Hemisphere	CSA = 2πr2  TSA = 3πr2  r = radius	V = 2323 πr3    cubic units
Triangle	SA = √s(s-a) (s-b) (s-c)  where s is the perimeter of the triangle  a, b, c are the sides of the triangle

Simple interest =  PTR / 100                      

Pythagorean Theorem:   a² + b² = c²

^{Rational Number}

§  n(n + l)(2n + 1) is always divisible by 6.

§  3²ⁿ leaves remainder = 1 when divided by 8 

§  n³ + (n + 1 )³ + (n + 2 )³ is always divisible by 9

§  10^2n + 1 + 1 is always divisible by 11

§  n(n²– 1) is always divisible by 6

§  n²+ n is always even

§  2³ⁿ-1 is always divisible by 7

§  15^2n-1 +l is always divisible by 16

§  n³ + 2n is always divisible by 3

§  3⁴ⁿ – 4 ³ⁿ is always divisible by 17

 

Angles

1.   Product of n consecutive numbers is always divisible by n!. 

2.   If n is a positive integer and p is a prime, then n^p – n is divisible by p.

3.   |x| = x if x ≥ 0 and |x| = – x if x ≤ 0.

4.   Minimum value of a².sec²Ɵ + b².cosec²Ɵ is (a + b)²; (0° < Ɵ < 90°) for eg. minimum value of 49 sec²Ɵ + 64.cosec²Ɵ is     (7 + 8)² = 225 

5.   Among all shapes with the same perimeter a circle has the largest area.

6.   If one diagonal of a quadrilateral bisects the other, then it also bisects the quadrilateral.

7.   Sum of all the angles of a convex quadrilateral = (n – 2)180°

8.   Number of diagonals in a convex quadrilateral = 0.5n(n – 3)

 

● Laws of Indices:

(i) aᵐ ∙ aⁿ = aᵐ + ⁿ 

(ii) aᵐ/aⁿ = aᵐ - ⁿ 

(iii) (aᵐ)ⁿ = aᵐⁿ 

(iv) a = 1 (a ≠ 0). 

(v) a-ⁿ = 1/aⁿ 

(vi) ⁿ√aᵐ = aᵐ/ⁿ 

(vii) (ab)ᵐ = aᵐ ∙ bⁿ. 

(viii) (a/b)ᵐ = aᵐ/bⁿ 

(ix) If aᵐ = bᵐ (m ≠ 0), then a = b. 

(x) If aᵐ = aⁿ then m = n.