Prove that the value of each the following determinants is zero: `|a=bb-cc-a x-y y-z z-x p-q q-r r-p|`

" 1."|[a-b,b-c,c-a],[x-y,y-z,z-x],[p-q,q-r,r-p]|" is equal to "

Using the property of determinants and without expanding, prove that: |b+c q+r y+z c+a r+p z+x a+b p+q x+y|=2|a p x b q y c r z|

Let a ,x, y ,z b be in AP where x+y+z=15 Let a ,p,q ,r ,b be in HP where p^(-1) +q^(-1)+r^(-1)=5/3 What is the value of ab What is the value of xyz What is the value of pqr

In a family there are 6 members P,Q,R,X,Y and Z. R is the sister of Z. X is the father of P and grandfather of Z. Q is the brother of Y's husband . There are three brother , one mother and two fathers. Which of the following is the group of brothers?

Without expansion, prove the following: |(a+b, b+c, c+a),(p+q, q+r, r+ p),(x+y, y+z, z+x)|=2|(a,b,c),(p,q,r),(x,y,z)|

Suppose that vec p, vec q and vec r are three non-coplanar vectors in R^3. Let the components of a vector vec s along vec p, vec q and vec r be 4, 3 and 5, respectively. If the components of this vector vec s along (-vec p+vec q +vec r), (vec p-vec q+vec r) and (-vec p-vec q+vec r) are x, y and z, respectively, then the value of 2vec x+vec y+vec z is

Show that: |b+cc+a a+b q+r r+p p+q y+z z+xx+y|=2|a b c p q r x y z|

The sides of triangle are denoted by x , y nd z . Area of the triangle and semiperimeter of the triangle are denoted by P and q respectively .If P= sqrtq (q-x)(q-y)(q-z) and x+y-z=y+z-x=z+x-y=4 Find P (in square units)