Important Mathematical Formulas
Maths Formulas
 (a + b)(a – b) = a^{2} – b^{2}
 (a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2(ab + bc + ca)
 (a ± b)^{2} = a^{2} + b^{2}± 2ab
 (a + b + c + d)^{2} = a^{2} + b^{2} + c^{2} + d^{2} + 2(ab + ac + ad + bc + bd + cd)
 (a ± b)^{3} = a^{3} ± b^{3} ± 3ab(a ± b)
 (a ± b)(a^{2} + b^{2} m ab) = a^{3} ± b^{3}
 (a + b + c)(a^{2} + b^{2} + c^{2} ab – bc – ca) = a^{3} + b^{3} + c^{3} – 3abc =
1/2 (a + b + c)[(a – b)^{2} + (b – c)^{2} + (c – a)^{2}]
 when a + b + c = 0, a^{3} + b^{3} + c^{3} = 3abc
 (x + a)(x + b) (x + c) = x^{3} + (a + b + c) x^{2} + (ab + bc + ac)x + abc
 (x – a)(x – b) (x – c) = x^{3} – (a + b + c) x^{2} + (ab + bc + ac)x – abc
 a^{4} + a^{2}b^{2} + b^{4} = (a^{2} + ab + b^{2})( a^{2} – ab + b^{2})
 a^{4} + b^{4} = (a^{2} – √2ab + b^{2})( a^{2} + √2ab + b^{2})
 a^{n} + b^{n} = (a + b) (a ^{n1} – a ^{n2} b + a ^{n3} b^{2} – a ^{n4} b^{3} +…….. + b ^{n1})
(valid only if n is odd)
 a^{n} – b^{n} = (a – b) (a ^{n1} + a ^{n2} b + a ^{n3} b^{2} + a ^{n4} b^{3} +……… + b ^{n1})
{where n ϵ N)
 (a ± b)^{2n} is always positive while (a ± b)^{2n} is always negative, for any real values of a and b
 (a – b)^{2n} = (b – a)^{2}” and (a – b)^{2n+1} = – (b – a)^{2n+1}
 if α and β are the roots of equation ax^{2} + bx + c = 0, roots of cx” + bx + a = 0 are 1/α and 1/β.
if α and β are the roots of equation ax^{2} + bx + c = 0, roots of ax^{2} – bx + c = 0 are α and β.

 n(n + l)(2n + 1) is always divisible by 6.
 3^{2n} leaves remainder = 1 when divided by 8
 n^{3} + (n + 1 )^{3} + (n + 2 )^{3} is always divisible by 9
 10^{2n} ^{+} ^{1} + 1 is always divisible by 11
 n(n^{2}– 1) is always divisible by 6
 n^{2}+ n is always even
 2^{3n}1 is always divisible by 7
 15^{2n1 }+l is always divisible by 16
 n^{3} + 2n is always divisible by 3
 3^{4n} – 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
 Product of n consecutive numbers is always divisible by n!.
 If n is a positive integer and p is a prime, then n^{p} – n is divisible by p.
 x = x if x ≥ 0 and x = – x if x ≤ 0.
 Minimum value of a^{2}.sec^{2}Ɵ + b^{2}.cosec^{2}Ɵ is (a + b)^{2}; (0° < Ɵ < 90°)
for eg. minimum value of 49 sec^{2}Ɵ + 64.cosec^{2}Ɵ is (7 + 8)^{2} = 225.
 among all shapes with the same perimeter a circle has the largest area.
 if one diagonal of a quadrilateral bisects the other, then it also bisects the quadrilateral.
 sum of all the angles of a convex quadrilateral = (n – 2)180°
 number of diagonals in a convex quadrilateral = 0.5n(n – 3)
 let P, Q are the midpoints of the nonparallel sides BC and AD of a trapezium ABCD.Then,
ΔAPD = ΔCQB.