**(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^{n-1}– a^{n-2}b + a^{n-3}b^{2}– a^{n-4}b^{3}+…….. + b^{n-1})(valid only if n is odd)**a**^{n}– b^{n}= (a – b) (a^{n-1}+ a^{n-2}b + a^{n-3}b^{2}+ a^{n-4}b^{3}+……… + b^{n-1}){were 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.**

**20.**

**3**

^{2n}leaves remainder = 1 when divided by 8**21.**

**n**

^{3}+ (n + 1 )^{3}+ (n + 2 )^{3}is always divisible by 9**22.**

**10**

^{2n}^{+}^{1}+ 1 is always divisible by 11 23.

**n(n**^{2}- 1) is always divisible by 6 24.

**n**^{2}+ n is always even**25.**

**2**

^{3n}-1 is always divisible by 726.

**15**

^{2n-1 }+l is always divisible by 1627.

**n**

^{3}+ 2n is always divisible by 3**28.**

**3**

^{4n}– 4^{3n}is always divisible by 17**29.**

**n! + 1 is not divisible by any number between 2 and n(where n! = n (n – l)(n – 2)(n – 3)…….3.2.1)**

**30.**

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

31.

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

**32.**

**If n is a positive integer and p is a prime, then n**

^{p}– n is divisible by p.**33.**

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

**34.**

**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 +^{2}= 225.**35.**

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

**36.**

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

37.

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

38.

**number of diagonals in a convex quadrilateral = 0.5n(n – 3)**

**39.**

**let P, Q are the midpoints of the nonparallel sides BC and AD of a trapezium ABCD.Then,**

ΔAPD = ΔCQB.

ΔAPD = ΔCQB.