Important Mathematical Formulas

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Maths Formulas

1. 1. (a + b)(a - b) = a2 - b2

1. 1. (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)

1. 1. (a ± b)2 = a2 + b2± 2ab

1. 1. (a + b + c + d)2 = a2 + b2 + c2 + d2 + 2(ab + ac + ad + bc + bd + cd)

1. 1. (a ± b)3 = a3 ± b3 ± 3ab(a ± b)

1. 1. (a ± b)(a2 + b2 m ab) = a3 ± b3

1. 1. (a + b + c)(a2 + b2 + c2 -ab - bc - ca) = a3 + b3 + c3 - 3abc =
      

       

      1/2 (a + b + c)[(a - b)2 + (b - c)2 + (c - a)2]

1. 1. when a + b + c = 0, a3 + b3 + c3 = 3abc

1. 1. (x + a)(x + b) (x + c) = x3 + (a + b + c) x2 + (ab + bc + ac)x + abc

1. 1. (x - a)(x - b) (x - c) = x3 - (a + b + c) x2 + (ab + bc + ac)x - abc

1. 1. a4 + a2b2 + b4 = (a2 + ab + b2)( a2 - ab + b2)

1. 1. a4 + b4 = (a2 - √2ab + b2)( a2 + √2ab + b2)

1. 1. an + bn = (a + b) (a n-1 - a n-2 b +  a n-3 b2 - a n-4 b3 +........ + b n-1)

(valid only if n is odd)

1. 1. an - bn = (a - b) (a n-1 + a n-2 b +  a n-3 b2 + a n-4 b3 +......... + b n-1)

{where n ϵ N)

1. 1. (a ± b)2n is always positive while -(a ± b)2n is always negative, for any real values of a and b

1. 1. (a - b)2n = (b - a)2" and (a - b)2n+1 = - (b - a)2n+1

1. 1. if α and β are the roots of equation ax2 + bx + c = 0, roots of cx" + bx + a = 0 are 1/α and 1/β.
      if α and β are the roots of equation ax2 + bx + c = 0, roots of ax2 - bx + c = 0 are -α and -β.

1. 1. • • 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

      
      

      • • 102n+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

      
      

      • • 152n-1 +l is always divisible by 16

      
      

      • • n3 + 2n is always divisible by 3

      
      

      • • 34n - 4 3n is always divisible by 17

      
      

      • n! + 1 is not divisible by any number between 2 and n
        

      (where n! = n (n - l)(n - 2)(n - 3).......3.2.1)

      for eg 5! = 5.4.3.2.1 = 120 and similarly 10! = 10.9.8.......2.1= 3628800
      

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

1. 1. If n is a positive integer and p is a prime, then np - n is divisible by p.

1. 1. |x| = x if x ≥ 0 and |x| = - x if x ≤ 0.

1. 1. 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.

   2. among all shapes with the same perimeter a circle has the largest area.

1. 1. if one diagonal of a quadrilateral bisects the other, then it also bisects the quadrilateral.

1. 1. sum of all the angles of a convex quadrilateral = (n - 2)180°

1. 1. number of diagonals in a convex quadrilateral = 0.5n(n - 3)

1. let P, Q are the midpoints of the nonparallel sides BC and AD of a trapezium ABCD.Then,
   ΔAPD = ΔCQB.