# important formulae

1.

**(a + b)(a – b) = a**^{2}– b^{2}
2.

**(a + b + c)**^{2}= a^{2}+ b^{2}+ c^{2}+ 2(ab + bc + ca)
3.

**(a ± b)**^{2}= a^{2}+ b^{2}± 2ab
4.

**(a + b + c + d)**^{2}= a^{2}+ b^{2}+ c^{2}+ d^{2}+ 2(ab + ac + ad + bc + bd + cd)
5.

**(a ± b)**^{3}= a^{3}± b^{3}± 3ab(a ± b)
6.

**(a ± b)(a**^{2}+ b^{2}m ab) = a^{3}± b^{3}
7.

**(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}]
8.

**when a + b + c = 0, a**^{3}+ b^{3}+ c^{3}= 3abc
9.

**(x + a)(x + b) (x + c) = x**^{3}+ (a + b + c) x^{2}+ (ab + bc + ac)x + abc
10.

**(x – a)(x – b) (x – c) = x**^{3}– (a + b + c) x^{2}+ (ab + bc + ac)x – abc
11.

**a**^{4}+ a^{2}b^{2}+ b^{4}= (a^{2}+ ab + b^{2})( a^{2}– ab + b^{2})
12.

**a**^{4}+ b^{4}= (a^{2}– √2ab + b^{2})( a^{2}+ √2ab + b^{2})
13.

**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)**

14.

**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})**{where n Ïµ 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.

**if Î± and Î² are the roots of equation ax**

if Î± and Î² are the roots of equation ax^{2}+ bx + c = 0, roots of cx” + bx + a = 0 are 1/Î± and 1/Î².if Î± and Î² are the roots of equation ax

^{2}+ bx + c = 0, roots of ax^{2}– bx + c = 0 are -Î± and -Î².
18.

o

**n(n + l)(2n + 1) is always divisible by 6.**
o

**3**^{2n}leaves remainder = 1 when divided by 8
o

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

**10**^{2n}^{+}^{1}+ 1 is always divisible by 11
o

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

**n**^{2}+ n is always even
o

**2**^{3n}-1 is always divisible by 7
o

**15**^{2n-1 }+l is always divisible by 16
o

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

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

**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 n**^{p}– n is divisible by p.
21.

**|x| = x if x ≥ 0 and |x| = – x if x ≤ 0.**
22.

**Minimum value of a**^{2}.sec^{2}ÆŸ + b^{2}.cosec^{2}ÆŸ is (a + b)^{2}; (0° < ÆŸ < 90°)**for eg. minimum value of 49 sec**

^{2}ÆŸ + 64.cosec^{2}ÆŸ 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,**

Î”APD = Î”CQB.Î”APD = Î”CQB.

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