Prove that \(\left| \begin{matrix} 1! & 2! & 3! \\ 2! & 3! & 4! \\ 3! & 4! & 5! \end{matrix} \right| =4!\)

3

(a)

If \(f(x)=\left| \begin{matrix} a & -1 & 0 \\ ax & a & -1 \\ { a }x^{ 2 } & ax & a \end{matrix} \right| \), using properties of determinants, find the value of f(2x)-f(x).

3

(a)

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

3

(a)

Using the properties of determinants, solve the following for 'x' \(\left| \begin{matrix} x+2 & x+6 & x-1 \\ x+6 & x-1 & x+2 \\ x-1 & x+2 & x+6 \end{matrix} \right| =0\)

3

(a)

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

3

(a)

Prove that \(\left| \begin{matrix} 2y & y-z-x & 2y \\ 2z & 2z & z-x-y \\ x-y-z & 2x & 2x \end{matrix} \right| =(x+y+z)^{ 3 }\)

3

(a)

If the area of the triangle is 35sq.units (2, -6), (5, 4) and (k, 4). Find the value of 'k'.

3

(a)

Prove that the area of the triangle whose vertices are: \(\left( { at }_{ 1 }^{ 2 },{ 2at }_{ 1 } \right) :({ at5 }_{ 2 }^{ 2 },{ 2at }_{ 2 });({ at }_{ 3 }^{ 2 },{ 2at }_{ 3 })\quad is\quad { a }^{ 2 }(t_1-t_2)(t_2-t_3)(t_3-t_2).\)

3

(a)

If\(A=\left| \begin{matrix} 1 & -2 & 3 \\ 0 & -1 & 4 \\ -2 & 2 & 1 \end{matrix} \right| \), find (A')^-1

3

(a)