By using properties of determinants , show that : ` {:|( 1,x,x^(2) ),( x^(2) ,1,x) ,( x,x^(2), 1) |:} =( 1-x^(3)) ^(2) `

Prove that |(1,x,x^2),(x^2, 1,x),(x,x^2 ,1)| =(1-x^3)^2

Prove that |(1,x,x^2),(x^2,1,x),(x,x^2,1)|=(1-x^2) .

using properties of determinant prove that {:[( x,x^(2) , 1+ px^(3) ),( y,y^(2) , 1+ py^(2)),( z,z^(2) , 1+pz^(2)) ]:} =( 1+pxyz ) ( x-y) ( y-z ) (z-x) , where p is any scalar .

Using Properties of determinants, prove that: {:|(x^2+1,xy,yz),(xy,y^2+1,yz),(xz,yz,z^2+1)|=1+x^2+y^2+z^2

Using Properties of determinants, prove that {:|(x+lamda,2x,2x),(2x,x+lamda,2x),(2x,2x,x+lamda)|=(5x+lamda)(lamda-x)^2

Solve for x: 2(x^(2) -1) = x( 1-x)

The sum of first 10 terms of the series : (x+(1)/( x))^(2) + ( x^(2) + ( 1)/( x^(2)))^(2) + ( x^(3)+ ( 1)/( x^(3)))^(2) + "......" is :

If y=(tan^(-1)x)^(2) , show that (x^(2)+1)^(2)y_(2)+2x(x^(2)+1)y_(1)=2 .

The constant term of the polynomial |(x+3,x,x+2),(x,x+1,x-1),(x+2,2x,3x+1)| is

If x,y,z in R , then the value of determinant |((5^(x)+5^(-x))^(2) , (5^(x)-5^(-x))^(2),1),((6^(x)+6^(-x))^(2),(6^(x)-6^(-x))^(2), 1),((7^(x)+7^(-x))^(2),(7^(x)-7^(-x))^(2),1)| is