IMPORTANT MATHS FORMULA 1. (a + b)(a – b) = a2 – b 2. (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca) 3. (a ± b)2 = a2 + b2± 2ab 4. (a + b + c + d)2 = a2 + b2 + c2 + d2 + 2(ab + ac + ad + bc + bd + cd) 5. (a ± b)3 = a3 ± b3 ± 3ab(a ± b) 6. (a ± b)(a2 + b2 m ab) = a3 ± b3 7. (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] 8. when a + b + c = 0, a3 + b3 + c3 = 3abc 9. (x + a)(x + b) (x + c) = x3 + (a + b + c) x2 + (ab + bc + ac)x + abc 10. x – a)(x – b) (x – c) = x3 – (a + b + c) x2 + (ab + bc + ac)x – abc 11. a4 + a2b2 + b4 = (a2 + ab + b2)( a2 – ab + b2) 12. a4 + b4 = (a2 – v2ab + b2)( a2 + v2ab + b2) 13. 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) 14. an – bn = (a – b) (a n-1 + a n-2 b + a n-3 b2 + a n-4 b3 +……… + b n-1) {where n ? N) 15. (a ± b)2n is always positive while -(a ± b)2n is always negative, for any real values of a and b 16. (a – b)2n = (b – a)2” and (a – b)2n+1 = – (b – a)2n+1 17. f ? and ? are the roots of equation ax2 + bx + c = 0, roots of cx” + bx + a = 0 are 1/? and 1/?. if ? and ? are the roots of equation ax2 + bx + c = 0, roots of ax2 – bx + c = 0 are -? and -?. 18. * 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 19. Product of n consecutive numbers is always divisible by n!. 20. If n is a positive integer and p is a prime, then np – n is divisible by p. 21. |x| = x if x ? 0 and |x| = – x if x ? 0. 22. Minimum value of a2. sec2O + b2. osec2O is (a + b)2; (0° < O < 90°) for eg. minimum value of 49 sec2O + 64. cosec2O is (7 + 8)2 = 225. 23. among all shapes with the same perimeter a circle has the largest area. 24. if one diagonal of a quadrilateral bisects the other, then it also bisects the quadrilateral. 25. sum of all the angles of a convex quadrilateral = (n – 2)180° 26. number of diagonals in a convex quadrilateral = 0. 5n(n – 3) 27. let P, Q are the midpoints of the nonparallel sides BC and AD of a trapezium ABCD. Then, ? APD = ? CQB.