Maths Formulas

1. (a + b)(a – b) =  a2 – b2

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

1. (a ± b)2 =  a2 + b2± 2ab

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

1. (a ± b)3 =  a3 ± b3 ± 3ab(a ± b)

1. (a ± b)(a2 + b2 m ab) =  a3 ± b3

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

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

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

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

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

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

1. (a – b)2n = (b – a)2” and (a – b)2n+1 = – (b – a)2n+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.

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

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

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

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.

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

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

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

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.