Prove that the value of determinants is zero: `|a-b b-c c-a x-y y-z z-x p-q q-r r-p|`

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]|

If p+q+r=0=a+b+c =0, then the value of the determnalnt |p a q b r c q c r a p b r b p c q a|i s a 0 b. p a+q b+r c c. 1 d. none of these

The determinant |x p+y x y y p+z y z0x p+y y p+z|=0 if x ,y ,z

Prove that |a x-b y-c z a y+b x c x+a z a y+b x b y-c z-a x b z+c y c x+a z b z+c y c z-a x-b y|=(x^2+y^2+z^2)(a^2+b^2+c^2)(a x+b y+c z)dot

Prove that =|a c c-a +b c b b-c b+c a-b b-c 0 a-c x y z 1+x+y|=0 implies that a ,b ,c are in A.P. or a ,c ,b are in G.P.

In a three-dimensional coordinate system, P ,Q , and R are images of a point A(a ,b ,c) in the x-y ,y-z and z-x planes, respectively. If G is the centroid of triangle P Q R , then area of triangle AOG is ( O is the origin) (A) 0 (B) a^2+b^2+c^2 (C) 2/3(a^2+b^2+c^2) (D) none of these

Let ABC be an equilateral triangle inscribed in the circle x^2+y^2=a^2 . Suppose pendiculars from A, B, C to the ellipse x^2/a^2+y^2/b^2=1,(a > b) meets the ellipse respectivelily at P, Q, R so that P, Q , R lies on same side of major axis as A, B, C respectively. Prove that the normals to the ellipse drawn at the points P Q nad R are concurrent.

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)]|

If alpha,beta,gamma are the roots of p x^2+q x^2+r=0, then the value of the determinant |alphabetabetagammagammaalphabetagammagammaalphaalphabetagammaalphaalphabetabetagamma| is p b. q c. 0 d. r

Given a=x//(y-z), b=y//(z-x),a n dc=z//(x-y),w h e r ex , y , za n dz are not all zero, then the value of a b+b c+c a is 0 b. 1 c. -1"" d. none of these