1.Algebra Booster

7.38 Algebra Booster
1 a a
47 The given determinant = 1 b
2
b
1 c
2
c
2
1 a a
2 2 ÊR2Æ R2 - R1ˆ
= 0 b – a b - a Á
ËR 2 2
3Æ R3-
R ˜
1¯
0 c – a c - a
2 2
b - a b - a
=
2 2
c – a c - a
1 b + a
= ( b - a)( c - a) 1 c + a
= (b – a)(c – a)(c + a – b – a)
= (b – a)(c – a)(c – b)
= (a – b)(b – c)(c – a)
1 a b+
c
48. The given determinant = 1 b c+
a
1 c a+
b
1 a a + b + c
= 1 b b + c + a ( C3Æ C2+
C3)
1 c c + a + b
1 a 1
= ( a + b + c)1 b 1
1 c 1
= (a + b + c) ¥ 0
= 0
49. The given determinant is
sin a cos b cos ( a + q)
sin b cos b cos ( b + q)
sin g cos g cos ( g + q)
sin a cos b 0
= sin b cos b 0
sin g cos g 0
[ C3ÆC3- ( C1cosq
+ C2sin q)]
= 0
50. The given determinant =
2
1 bc a( b + c)
1 ca b( a + c)
1 ab c( a + b)
1 bc bc + ab + ac
= 1 ca ac + ab + bc ( C3Æ C2+
C3)
1 ab ab + ac + bc
1 bc 1
= ( ab + bc + ca) ¥ 1 ca 1
1 ab 1
= (ab + bc + ca) ¥ 0
= 0
51. The given determinant
a + b + 2c a b
= c b + c + 2a b
c a c + a + 2b
2( a + b + c)
a b
= 2( a + b + c) b + c + 2a b
2( a + b + c) a c + a + 2b
(C 1
Æ C 1
+ C 2
+ C 3
)
1 a b
=2( a + b + c) ¥ 1 b + c + 2a b
1 a c + a + 2b
1 a b
=2( a + b + c) ¥ 0 b + c + a 0
0 0 c + a + b
ÊR2Æ R2 - R1ˆ
Á
ËR3Æ R3-
R ˜
1¯
1 a b
3
= 2( a + b + c) ¥ 0 1 0
0 0 1
= 2(a + b + c) 3 ¥ 1
= 2(a + b + c) 3
52. The given determinant
b + c a a
= b c + a b
c c a + b
2( b + c) 2( c + a) 2( a + b)
= b c + a b
c c a + b
( b + c) ( c + a) (R 1
Æ R 1
+ R 2
+ R 3
)
( a + b)
= 2 b c + a b
c c a + b
b + c ( c + a) ( a + b)
= 2 – c 0 – a
– b – a 0
ÊR2Æ R2 - R1ˆ
Á
ËR Æ R - R ˜
¯
3 3 1
0 c b
= 2– c 0 – a ( R1Æ R1+ R2 + R3)
– b – a 0
= 2[–c(–ab – 0) + b(ca – 0)]