MATHEMATICAL FORMULAE
Algebra
1.
(a + b)2 = a2 + 2ab + b2 ; a2 +
b2 = (a +
b)2 − 2ab
2. (a − b)2 =
a2 − 2ab + b2 ; a2 +
b2 = (a − b)2 + 2ab
3. (a + b + c)2 = a2 + b2 +
c2 + 2(ab + bc + ca)
4.
(a + b)3 = a3 + b3 +
3ab(a + b); a3 +
b3 = (a +
b)3 − 3ab(a + b)
5. (a − b)3 =
a3 − b3 − 3ab(a − b); a3 − b3 =
(a − b)3 + 3ab(a − b)
6. a2 − b2 =
(a + b)(a − b)
7. a3 − b3 =
(a − b)(a2 + ab +
b2 )
8. a3 +
b3 = (a +
b)(a2 − ab +
b2 )
9. an − bn = (a − b)(an−1 +
an−2 b +
an−3 b2 + ··· + bn−1 )
10. an =
a.a.a . . . n times
11. am .an = am+n
m
12. a
an
= am−n if m > n
=
1
if m = n
1
=
an−m
if m < n; a ∈ R, a = 0
13. (am )n =
amn =
(an )m
14. (ab)n = an .bn
|
15. =
b bn
16. a0 = 1 where a ∈ R, a = 0
17. a−n = 1 , an = 1
an
18. ap/q = √q ap
a−n
19.
If am = an and a = ±1,a = 0 then m = n
20. If an = bn where n = 0, then a = ±b
21. If √x, √y are quadratic surds and if a + √x = √y, then a = 0 and x = y22. If √x, √y are quadratic surds and if a + √x = b + √y then a = b and x = y
23. If a, m, n are positive real numbers and a = 1, then loga mn = loga m+loga n
m
24. If a, m, n are positive real numbers, a = 1, then loga n
= loga m − loga n
25. If a and m are positive real numbers, a = 1 then loga mn = n loga m
logk a
26. If a, b and k are positive real numbers, b = 1,k = 1, then logb a =
1
logk b
|
where a, b are positive real numbers, a = 1,b = 1
b
28. if a, m, n are
positive
real numbers, a = 1
and if loga m = loga n, then
m =
n
Typeset by AMS-TEX
2
|
1, then a = b = 0
|
|
31. The roots of the quadratic equation ax2 +bx+c = 0; a = 0
are −b ± b
2a
− 4ac
The solution set of the equation
is
( −b +
√∆ −b −
√∆ )
|
where
∆ = discriminant = b2 − 4ac
32. The roots are real and distinct if ∆ > 0.
33.
The roots are real and coincident if ∆ = 0.
34.
The roots are non-real
if ∆ < 0.
35. If α and
β
are the roots of the equation ax2 +
bx +
c =
0,a = 0 then coeff. of x
|
a
coeff. of x2
|
a
constant term
coeff. of x2
36. The quadratic equation whose roots are α and β is
(x − α)(x − β) = 0
i.e. x2 − (α +
β)x + αβ = 0
i.e. x2
− Sx + P = 0
where S =Sum of the roots and
P =Product of the roots.
37. For an arithmetic progression (A.P.) whose first term is (a) and the common difference is (d).
i) nth term=
tn = a + (n − 1)d
|
ii) The sum of the first (n) terms = Sn =
where l =last term= a + (n − 1)d.
(a + l) =
{2a + (n − 1)d}
38. For a geometric progression (G.P.) whose first term is (a) and common ratio is (γ ),
i) nth term= tn = aγ n−1.
ii) The sum of the first (n) terms:
a(1 − γ n)
Sn =
1n−
ifγ < 1
γ
= a(γ
− 1)
if γ > 1 .
γ − 1
=
na
if
γ = 1
39. For any
sequence
{tn }, Sn − Sn−1 =
tn where Sn =Sum
of the first (n)
terms.
n
40.
P γ = 1 +2+3+ ··· + n = n (n + 1).
γ=1 2
n
41.
P γ 2 = 12 + 22 + 32 + ··· + n2 = n (n + 1)(2n + 1).
γ=1
6
3
|
P γ 3 = 13 + 23 + 33 + 43 + ··· + n3 = n
(n + 1)2 .
|
43. n! = (1).(2).(3).... .(n − 1).n.
44. n! = n(n − 1)! = n(n − 1)(n − 2)! = .... .
45. 0! = 1.
46. (a + b)n = an + nan−1 b + n(n − 1) an−2 b2 + n(n − 1)(n − 2) an−3 b3 + +
bn ,n > 1.
2! 3!
···
