Algebra

29. Algebra Identities:
(i) (a + b)2 + (a – b)2 = 2 (a2 + b2) (ii) (a + b)2 – (a – b)2 = 4ab
(iii) a3 + b3 = (a + b) (a2 – ab + b2) (iv) a3 – b3 = (a – b) (a2 + ab + b2)
(v) a4 + a2 + 1 = (a2 + a + 1) (a2 – a + 1) (vi) If a + b + c = 0, then a3 + b3 + c3=3abc
(vii)
(a b)2 (a b)2
4
ab
+ - -
= (viii)
( )2 ( )2
2 2
a b a b
2
a b
+ + -
=
+
(ix) ( ) a b d e g h j k a d g j b e h k
c f i l c f i l
+ + - = + + - + æ + + - ö ç ÷
è ø
(x) If a + b + c = abc, then
2 2 2
2a 2b 2c
1 a 1 b 1 c
æ ö æ ö æ ö
ç ÷ + ç ÷ + ç ÷
è - ø è - ø è - ø
= 2 2 2
2a . 2b . 2c
1 a 1 b 1 c
æ ö æ ö æ ö
çè - ÷ø çè - ÷ø çè - ÷ø and
3 3 3
2 2 2
3a a 3b b 3c c
1 3a 1 3b 1 3c
æ - ö æ - ö æ - ö
çç ÷÷ + çç ÷÷ + çç ÷÷ è - ø è - ø è - ø
=
3 3 3
2 2 2
3a a . 3b b . 3c c
1 3a 1 3b 1 3c
æ - ö æ - ö æ - ö
ç ÷ ç - ÷ ç ÷
è - ø è - ø è - ø
30. If a1x + b1y = c1 and a2x + b2y = c2, then
(i) If 1 1
2 2
a b
a b
¹ , one solution. (ii) If 1 1 1
2 2 2
a b c
a b c
= = , Infinite many solutions.
(iii) If 1 1 1
2 2 2
a b c
a b c
= ¹ , No solution
31. If a and b are roots of ax2 + bx + c = 0, then
1
a and
1
b are roots of cx2 + bx + a = 0
32. If a and b are roots of ax2 + bx + c = 0, then
(i) One root is zero if c = 0.
(ii) Both roots zero if b = 0 and c = 0.
(iii) Roots are reciprocal to each other, if c = a.
(iv) If both roots a and b are positive, then sign of a and b are opposite and sign of c and a are same.
(v) If both roots a and b are negative, then sign of a, b and c are same.
( ) b , c
a a
a + b = - ab = , then
a - b = (a + b)2 - 4ab
a4 +b4 = (a2 +b2 )2 - 2a2b2 = ( ) ( ) 2 2 2 é a + b - 2abù - 2 ab
ë û
33. Arithmetic Progression:
(i) If a, a + d, a + 2d, ..... are in A.P., then, nth term of A.P. an = a + (n – 1)d
Sum of n terms of this A.P. = n ( )
S n 2a n 1 d
2 =
éë + - ùû = n [a l]
2
+ wherel = last term
a = first term
d = common difference
(ii) A.M. =
a b
2
+
[Q A.M. = Arithmetic mean]
34. Geometric Progression:
(i) G.P. ® a, ar, ar2,.........
Then, nth term of G.P. an = arn–1
Sn =
( )
( )
nar 1
,r 1
r 1
-
>
-
a(1 rn ),r 1
(1 r)
-
= <
-
S¥ = 1- r [where r = common ratio, a = first term]
(ii) G.M. = ab
35. If a, b, c are in H.P.,
1 , 1 , 1
a b c are in A.P.
nth term of H.M. = th
1
n term of A.P.
H.M. 2ab
a b
=
+
Note : Relation between A.M., G.M. and H.M.
(i) A.M. × H.M. = G.M.2
(ii) A.M. > G.M. > H.M.
A.M. ® Arithmetic Mean
G.M. ® Geometric Mean
H.M. ® Harmonic Mean