1.Algebra Booster

7.54 Algebra Booster
= ([x] + [y] + [z] + 1)
= (–1 + 0 + 1 + 1)
= 1
= [z]
12. We have,
a b - c b + c a b - c b + c
a + c b c - a = c c -( a + b)
a - b a + b c - b a + c – b
(R 2
Æ R 2
– R 1
, R 3
Æ R 3
– R 1
)
a b- c – a b+
c – a
= c 0 -( a+ b+
c)
- b a + c+
b 0
(C 2
Æ C 2
– C 1
, C 3
Æ C 3
– C 1
)
= b(a + b + c)(b – c – a)
+ (a + b + c){a(a + b + c) + c(b + c – a)}
= (a + b + c) (b 2 – bc – ab + a 2 + ab + ac + bc + c 2 – ac)
= (a + b + c) (a 2 + b 2 + c 2 )
13. We have,
2
( b+
c)
2
a bc
2
( c+
a)
2
b ca
2
( a + b)
2
c ab
2 2 2
b + c a bc
2 2 2
= c + a b ca ( C1ÆC1-2 C3)
2 2 2
a + b c ab
2 2 2 2
a + b + c a bc
2 2 2 2
= b + c + a b ca ( C1Æ C1+
C2)
2 2 2 2
a + b + c c ab
1 a bc
2 2 2
= ( a + b + c )1
2
b ca
1
2
c ab
2
2
1 a bc
2 2 2
= ( a + b + c ) 0
0
2 2
b -a 2 2
c - a
c( a – b)
b( a – c)
(R 2
Æ R 2
– R 1
, R 3
Æ R 3
– R 1
)
2 2
2 2 2
b – a c( a – b)
=( a + b + c )
2 2
c – a b( a – c)
2 2 2
( b+
a) – c
=( a + b + c )( b – a)( c – a) ( c+
a ) – b
2 2 2 2 2
= ( a + b + c )( b – a)( c – a)( c + ac – b – ab)
2 2 2
= ( a + b + c )( b – a)( c – a)( c – b)( c+ a + b)
2 2 2
=( a – b)( b – c)( c – a)( a+ b+ c)( a + b + c )
14. We have,
3 a + b + c a + b + c
a + b + c a + b + c a + b + c
a + b + c a + b + c a + b + c
3 3 3
2 2 2 4 4 4
2 2 2 3 3 3 5 5 5
3 3 3
3 a + b + c a + b + c
= a + b + c a + b + c a + b + c
a + b + c a + b + c a + b + c
2 2 2 4 4 4
2 2 2 3 3 3 5 5 5
1 1 1 1 1 1
= a b c ¥ a b c
2
a
2
b
2
c
3
a
3
b
3
c
= (a – b)(b – c)(c – a)
¥ (a – b)(b – c)(c – a) (a + b + c)
= (a – b) 2 (b – c) 2 (c – a) 2 (a + b + c)
15. We have,
- 1 + cos B cos B + cos C cos B
cos A+ cos C - 1 + cos A cos A
- 1 + cos B - 1 + cos A -1
-1 cos C cos B
= cos C -1 cos A
cos B cos A -1
2
(C 2
Æ C 2
– C 1
, C 3
Æ C 3
– C 1
)
=-(1 -cos A) -cos C( -cos C -cos Acos B)
+ cos B(cos Acos C + cos B)
2
=-sin A- cos C[cos ( A+ B) -cos Acos B]
+ cos B[cos Acos C - cos ( A+
C)]
2
=-sin A-cos C (– sin Asin B)
+ cos B(sin Asin C)
2
=- sin A+ sin A(sin Bcos C + cos Bsin C)
2
=- sin A+ sin A[sin ( B + C)]
2
=-sin A+
sin Asin( p – A)
= –sin 2 A + sin A ◊ sin A
= –sin 2 A + sin 2 A
= 0
a b c
16. Let D= b c a
c a b
The determinant of the co-factors of D is
2
bc - a
2
ca -b 2
ab - c
2
ca -b 2
ab - c
2
bc - a
2
ab - c
2
bc - a
2
ac -b
3-1 2
=D =D