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MATHEMATICAL FORMULAE
Algebra
1
...
(a − b)2 = a2 − 2ab + b2 ; a2 + b2 = (a − b)2 + 2ab
3
...
(a + b)3 = a3 + b3 + 3ab(a + b); a3 + b3 = (a + b)3 − 3ab(a + b)
5
...
a2 − b2 = (a + b)(a − b)
7
...
a3 + b3 = (a + b)(a2 − ab + b2 )
9
...
an = a
...
a
...
am
...
n = am−n if m > n
a
=1
if m = n
1
= n−m if m < n; a ∈ R, a = 0
a
13
...
(ab)n = an
...
= n
b
b
16
...
a−n = n , an = −n
a
a
√
18
...
If am = an and a = ±1, a = 0 then m = n
20
...
If x, y are quadratic surds and if a + x = y, then a = 0 and x = y
√ √
√
√
22
...
If a, m, n are positive real numbers and a = 1, then loga mn = loga m+loga n
m
24
...
If a and m are positive real numbers, a = 1 then loga mn = n loga m
logk a
26
...
logb a =
where a, b are positive real numbers, a = 1, b = 1
loga b
28
...
if a + ib = 0
√
−1, then a = b = 0
√
where i = −1, then a = x and b = y
where i =
30
...
The roots of the quadratic equation ax2 +bx+c = 0; a = 0 are
The solution set of the equation is
−b ±
√
b2 − 4ac
2a
√
√
−b + ∆ −b − ∆
,
2a
2a
where ∆ = discriminant = b2 − 4ac
32
...
33
...
34
...
35
...
of x
=−
i) α + β =
a
coeff
...
of x2
36
...
e
...
e
...
37
...
P
...
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
...
For a geometric progression (G
...
) whose first term is (a) and common ratio
is (γ),
i) nth term= tn = aγ n−1
...
if γ = 1
39
...
n
n
(n + 1)
...
γ 2 = 12 + 22 + 32 + · · · + n2 = (n + 1)(2n + 1)
...
γ = 1 + 2 + 3 + ··· + n =
3
n
42
...
4
43
...
(2)
...
(n − 1)
...
44
...
45
...
46
...
n(n − 1) n−2 2 n(n − 1)(n − 2) n−3 3
a
b +
a
b +···+
2!
3!