Without expanding the determinant at any stage, prove that \(\left| \begin{matrix} x+1 & x+2 & x+a \\ x+2 & x+3 & x+b \\ x+3 & x+4 & x+c \end{matrix} \right| =0\), where a, b, c are in A.P.

Prove that:\(\left| \begin{matrix} -a({ b }^{ 2 }+{ c }^{ 2 }-{ a }^{ 2 }) & { 2b }^{ 3 } & { 2c }^{ 3 } \\ { 2 }a^{ 3 } & -b({ c }^{ 2 }+{ a }^{ 2 }-{ b }^{ 2 }) & { 2c }^{ 3 } \\ { 2a }^{ 3 } & { 2b }^{ 3 } & -c({ a }^{ 2 }+{ b }^{ 2 }-{ c }^{ 2 }) \end{matrix} \right| =abc({ a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }{ ) }^{ \\ 3 }\)

Find \(fof^{ -1 }\) and \(f^{ -1 }\) of for the function:
\(f(x)=\frac { 1 }{ x } ,x\neq 0\). Also prove that \(fof^{ -1 }\)= \(f^{ -1 }\) of .

If \({ x }^{ 2 }+{ y }^{ 2 }=t-\frac { 1 }{ t } \)and \({ x }^{ 4 }+{ y }^{ 4 }={ t }^{ 2 }+\frac { 1 }{ { t }^{ 2 } } \) show that \(\frac { dy }{ dx } =\frac { 1 }{ { x }^{ 3 }y } \)

Use Lagrange's Theorem to determine a point P on the curve \(f\left( x \right) =\sqrt { x-2 } \)defined in the interval [2,3], where the tangent is parallel to the chord joining the end points on the curve.

Let the * be the binary operation on N be defined a*b = H.C.F of a and b.Is * commutative? Is * associative? does there exist identify for this operation of N?  
(i) 5 * 7, 20 * 16
(ii) Is * commutative?
(iii) Is * associative?
(iv) Find the identity of * in N
(v) Which elements of N are invertible for the operation *?

For what velues of x and y are the following matrices equal \(A=\left[ \begin{matrix} 2x+1 & 3y \\ 0 & { y }^{ 2 }-5y \end{matrix} \right] \)  \(B=\left[ \begin{matrix} x+3 & { y }^{ 2 }+2 \\ 0 & -6 \end{matrix} \right] \)