A) Prove with the help of Property:
1) 1 x x²
1 y y² = (x-y)(y-z)(z-x)
1 z z²
2) b+c a b
c+a c a =(a+b+c)(a- c)²
a+b b c
3) 1 a b+ c
1 b c+a = 0
1 c a+ b
4) 1+a²-b² 2ab -2b
2ab 1- a²+b² 2a
2b -2a 1- a²- b² = (1+a²+b²)³
5) a²+1 ab ac
ab b²+1 bc = 1+a²+ b²+ c²
ac bc c²+1
6) y+z x-y x
z+x y-x y = 3xyz-x³-y³-z³
x+y z-x z
7) x y z
x² y² z²
yz zy xy =(x -y)(y-z)(z-x)(xy+ yz+ zy)
8) -a² ab ac
ba -b² bc = 4a²b²c²
ac bc -c²
9) 1+ a b c
a 1+ b c = 1+a+b+c
a b 1+ c
10) 1 x x²-yz
1 y y²- zx = 0
1 z z²- xy
11) a+b+2c a b
c b+c+2a b
c a c+a+2b = 2(a+b+c)³
12) a b c
a² b² c²
b+c c+a a+b = (a-b)(b-c)(c-a)(a+b+c)
13) 1+a 1 1
1 1+b 1
1 1 1+ c = abc(1+ 1/a + 1/b + 1/c)
14) If x,y,z are different and
A= x x² 1+ x³
y y² 1+ y³ = 0
z z² 1+ z³ then show that 1+ xyz = 0
15) Find the integral value of x if
x² x 1
0 2 1 = 28
3 1 4
16) a-b-c 2b 2c
2a b-c-a 2c
2a 2b c -a- b = (a+b+ c)³
17) a b c
a- b b- c c- a
b+ c c+a a+ b = a³+ b³+ c³ - 3abc
18) (a²+ b²)/c c c
a (b²+c²)/a a
b b (c²+a²)/b = 4abc
19) y+ z z y
z z+ x x = 4xyz
y x x+ y
20) a a+ b a+ 2b
a+ 2b a a+ b =9b³(a+b)
a+ b a+ 2b a
B) Solve by Cramer's rule:
1) 3x - 4y= 1; 2x - 7y= 3. -5/13,-7/13
2) x + 2y - 3z= -4; 2x + 3y+ 2z= 2, 3x - 3y- 4z= 11. 3,-2,1
3) 3x - y= 7; 2x+ 3y = -1. 2, -1
C)
1) Find the minor and cofactor of -2 in the deteminant 2 -1 1
3 -2 4
1 1 2 3, 3
2)
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