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.

If a, b, c are all positive and are pth, qth, rth terms respectively of a G > P, then prove that : \(\left| \begin{matrix} loga & p & 1 \\ logb & q & 1 \\ logc & r & 1 \end{matrix} \right| =0\)

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

\(\left| \begin{matrix} { yz }-x^{ 2 } & { zx }-y^{ 2 } & { xy-z }^{ 2 } \\ { zx-y }^{ 2 } & { xy }-z^{ 2 } & { yz-x }^{ 2 } \\ { xy-z }^{ 2 } & { yz }-x^{ 2 } & { zx-x }^{ 2 } \end{matrix} \right| \)is divisible by (x+y+z) and here, find the quotient.

An equilateral triangle has each side equal to a. If the co-ordinates of its vertices are (x₁,y₁), (x₂,y₂) and (x₃, y₃), show that \(\left| \begin{matrix} x_1 &y_1 &1 \\x_2 &y_2 &1 \\x_3 &y_3 &1 \end{matrix} \right| ^2={3\over4}a^4\)