Prove that the value of each the following determinants is zero: `|[(a^x+a^(-x))^2,(a^x-a^(-x))^2 ,1],[(b^y+b^(-y))^2,(b^y-b^(-y))^2 ,1],[(c^z+c^(-z))^2,(c^z-c^(-z))^2, 1]|`

Prove that the value of each the following determinants is zero: |[a-b,b-c,c-a],[ x-y, y-z, z-x],[ p-q, q-r ,r-p]|

If a, b,c> 0 and x,y,z in RR then the determinant |((a^x+a^-x)^2,(a^x-a^-x)^2,1),((b^y+b^-y)^2,(b^y-b^-y)^2,1),((c^z+c^-z)^2,(c^z-c^-z)^2,1)| is equal to:

Factorize each of the following expressions: a x y+b c x y-a z-b c z

Without expanding, show that "Delta"=|(a-x)^2(a-y)^2(a-z)^2(b-x)^2(b-y)^2(b-z)^2(c-x)^2(c-y)^2(c-z)^2|=2(a-b)(b-c)(c-a)(x-y)(y-z)(z-x)

Without expanding,FIND "Delta"=|(((a-x)^2),((a-y)^2),((a-z)^2)),(((b-x)^2),((b-y)^2),((b-z)^2)),(((c-x)^2),((c-y)^2),((c-z)^2))|

Factorize each of the following expressions: (a) 2p(a-b)+3q\ (5a-5b)+4r(2b-2a) (b) a b(a^2+b^2-c^2)+b c(a^2+b^2-c^2)-c a(a^2+b^2-c^2) (c) x(x^2+y^2-z^2)+y(-x^2-y^2+z^2)-z\ (x^2+y^2-z^2)

The value of the determinant |[log_a(x/y), log_a(y/z), log_a(z/x)], [log_b (y/z), log_b (z/x), log_b (x/y)], [log_c (z/x), log_c (x/y), log_c (y/z)]|

Let a ,b , c be real numbers. The following system of equations in x ,y and z (x^2)/(a^2)+(y^2)/(b^2)-(z^2)/(a^2)=1,(x^2)/(a^2)-(y^2)/(b^2)+(z^2)/(a^2)=1,-(x^2)/(a^2)+(y^2)/(b^2)+(z^2)/(a^2)=1 has a. no solution b. unique solution c. infinitely many solutions d. finitely many solutions

Identify the terms, their coefficients for each of the following expressions: 3x^2y^2-5x^2y^2z^2+z^2 (ii) 9-a b+b c-c a

Factorize each of the following expressions: x a^2+x b^2-y a^2-y b^2 (2) x^2+x y a+x z+y z (3) 2a x+b x+2a y+b y (4) a b-b y-a y+y^2