Determinants Te...

When the determinant $$\begin{vmatrix} \cos { 2x }  & \sin ^{ 2 }{ x }  & \cos { 4x }  \\ \sin ^{ 2 }{ x }  & \cos { 2x }  & \cos ^{ 2 }{ x }  \\ \cos { 4x }  & \cos ^{ 2 }{ x }  & \cos { 2x }  \end{vmatrix}$$ is expanded in powers of $$\sin { x }$$, then the constant term in that expression is

In triangle $$ABC$$, if $$\begin{vmatrix} 1 & 1 & 1 \\ \cot { \cfrac { A }{ 2 }  }  & \cot { \cfrac { B }{ 2 }  }  & \cot { \cfrac { C }{ 2 }  }  \\ \tan { \cfrac { B }{ 2 }  } +\tan { \cfrac { C }{ 2 }  }  & \tan { \cfrac { C }{ 2 }  } +\tan { \cfrac { A }{ 2 }  }  & \tan { \cfrac { A }{ 2 }  } +\tan { \cfrac { B }{ 2 }  }  \end{vmatrix}=0$$, then the triangle must be

The value of the determinant $$\begin{vmatrix} 1 & 1 & 1 \\ { _{  }^{ m }{ C } }_{ 1 } & { _{  }^{ m+1 }{ C } }_{ 1 } & { _{  }^{ m+2 }{ C } }_{ 1 } \\ { _{  }^{ m }{ C } }_{ 2 } & { _{  }^{ m+1 }{ C } }_{ 2 } & { _{  }^{ m+2 }{ C } }_{ 2 } \end{vmatrix}$$ is equal to

If $$p+q+r=0=a+b+c$$, then the value of the determinant $$\begin{vmatrix} pa & qb & rc \\ qc & ra & pb \\ rb & pc & qa \end{vmatrix}$$ is

The value of $$\sum _{ r=2 }^{ n }{ { \left( -2 \right)  }^{ r } } \begin{vmatrix} { _{  }^{ n-2 }{ C } }_{ r-2 } & { _{  }^{ n-2 }{ C } }_{ r-1 } & { _{  }^{ n-2 }{ C } }_{ r } \\ -3 & 1 & 1 \\ 2 & -1 & 0 \end{vmatrix}(n>2)$$

If $$a,b,c$$ are different, then the value of $$x$$ satisfying $$\begin{vmatrix} 0 & { x }^{ 2 }-a & { x }^{ 3 }-b \\ { x }^{ 2 }+a & 0 & { x }^{ 2 }+c \\ { x }^{ 4 }+b & x-c & 0 \end{vmatrix}=0$$ is

If $${ A }_{ 1 },{ B }_{ 1 },{ C }_{ 1 },..$$ are, respectively, the cofactors of the elements $${ a }_{ 1 },{ b }_{ 1 },{ c }_{ 1 },..$$ of the determinant $$\quad \Delta =\begin{vmatrix} { a }_{ 1 } & { b }_{ 1 } & { c }_{ 1 } \\ { a }_{ 2 } & { b }_{ 2 } & { c }_{ 2 } \\ { a }_{ 3 } & { b }_{ 3 } & { c }_{ 3 } \end{vmatrix}$$, $$\Delta\neq 0$$, then the value of $$\begin{vmatrix} { B }_{ 2 } & { C }_{ 2 } \\ { B }_{ 3 } & { C }_{ 3 } \end{vmatrix}$$ is equal to

If $$a>0$$ and discriminant of $$a{ x }^{ 2 }+2bx+c$$ is negative, then $$\Delta =\begin{vmatrix} a & b & ax+b \\ b & c & bx+c \\ ax+b & bx+c & 0 \end{vmatrix}$$ is

If $$\begin{vmatrix} { a }^{ 2 }+{ \lambda  }^{ 2 } & ab+c\lambda  & ca-b\lambda  \\ ab-c\lambda  & { b }^{ 2 }+{ \lambda  }^{ 2 } & bc+a\lambda  \\ ca+b\lambda  & bc-a\lambda  & { c }^{ 2 }+{ \lambda  }^{ 2 } \end{vmatrix}\begin{vmatrix} \lambda  & c & -b \\ -c & \lambda  & a \\ b & -a & \lambda  \end{vmatrix}={ \left( 1+{ a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 } \right)  }^{ 3 }$$, then the value of $$\lambda$$ is

If $$\begin{vmatrix} 6i & -3i & 1 \\ 4 & 3i & -1 \\ 20 & 3 & i \end{vmatrix}=x+iy$$, then