1.Algebra Booster

7.12 Algebra Booster
46. Expand the determinant
a b c
b c a.
c a b
1 a
47. Evaluate: 1 b b .
1 c
2
c
1 a b+
c
48. Evaluate: 1 b c+
a.
1 c a + b
sin a cos b cos ( a + q)
49. Evaluate: sin b cos b cos ( b + q) .
a
2
2
sin g cos g cos ( g + q)
1 bc a( b + c)
50. Prove that the value of 1 ca b( a + c)
is independent
of a, b, c. 1 ab c( a + b)
51. Prove that
a + b+
2c a b
c b+ c+ 2a b = 2( a + b+
c) .
c a c+ a + 2b
b+
c a a
52. Prove that b c + a b = 4 abc.
c c a + b
2 2 2 2
b + c a a
2 2 2 2 2 2 2
53. Prove that b c + a b = 4 abc.
2
c
2
c
2 2
a + b
a a+ b a+ b+
c
54. Prove that 2a 3a+ 2b 4a+ 3b+ 2 c
3
= a .
3a 6a + 3b 10a + 6b+
3a
2 2
1+ a -b 2ab -2b
55. Prove that 2ab 2 2
1- a + b 2a
2b -2a 2 2
1-a -b
= (1 + a 2 + b 2 ) 3 .
2
a + 1 ab ac
56. Prove that ab
2
b + 1 bc = (1 + a 2 + b 2 + c 2 ).
57. Prove that
ac cb c + 1
x x x
C1 C2 C3
y y y
1 2 3
z z z
C1 C2 C3
2
xyz
C C C = ( x-y)( y-z)( z-x).
12
3
58. Prove that
1+
a 1 1
1 1+
b 1
1 1 1+
c
Ê 1 1 1ˆ
= abcÁ1 + + + = abc + bc + ca + ab.
Ë
˜
a b c¯
59. Prove that
2
( b + c)
2
a
2
a
2
b
2
( c + a)
2
b = 2abc(a + b + c) 3 .
2
c
2
c
2
( a + b)
60. Prove that
( a + 1)( a + 2) ( a + 2) 1
( a + 2)( a + 3) ( a + 3) 1 =-2.
( a + 3)( a + 4) ( a + 4) 1
b+ c c+ a a+
b a b c
61. Prove that a + b b+ c c+ a = 2 c a b.
c+ a a + b b+
c b c a
62. Solve the following system of equations:
2x + 3y = 4
3x – 2y = 5.
63. Solve the following system of equations:
x + 3y = 4
2x + 6y = 10.
64. Solve the following system of equations:
2x + 5y = 6
6x + 15y = 18.
65. Find the number of triplets of a, b and c for which the
system of equations
ax – by = 2a – b
(c + 1)x + cy = 10 – a + 3b.
has infinitely many solutions.
66. Solve for x, y, z:
x + y + z = 1
ax + by + cz = d
a 2 x + b 2 y + c 2 z = d
67. Solve for x, y, z:
1 1 1 1
+ - =
x y z 4
2 1 3 9
- + =
x y z 4
1 2 4
- - + = 1.
x y z
68. Find the equation of the parabola y = ax 2 + bx + c, which
passes through the points (2, 4), (–1, 1) and (–2, 5).
69. Find the value of k, for which the system of equations
2x
+ ky = 5
3x
- 4y
= 7
has a unique solution.