Determinants Te...

Find $$ \begin{vmatrix}\log e & \log e^{2} & \log e^{3} \\ \log e^{2} & \log e^{3} & \log e^{4} \\ \log e^{3} & \log e^{4} & \log e^{5}\end{vmatrix}$$.

If the lines $$p_{ 1 }x+q_{ 1 }y=1, p_{ 2 }x+q_{ 2 }y=1$$ and $$p_{3}x+q_{3}y=1$$ be concurrent, show that the points $$(p_{1},q_{1}), (p_{2}, q_{2})$$ and $$ (p_{3}, q_{3})$$ are collinear.

The value of $$\dfrac { 1 }{ x-y } \begin{vmatrix} 1 & 0 & 0 \\ 3 & { x }^{ 3 } & 1 \\ 5 & { y }^{ 3 } & 1 \end{vmatrix}$$ is-

Find the determinant of given matrix $$\left[ \begin{matrix} a-b-c & 2a & 2a \\ 2b & b-c-a & 2b \\ 2c & 2c & c-a-b \end{matrix} \right] $$

If $$\triangle =\begin{bmatrix} { a }_{ 1 } & { b }_{ 1 } & { c }_{ 1 } \\ { a }_{ 2 } & { b }_{ 2 } & { c }_{ 2 } \\ { a }_{ 3 } & { b }_{ 3 } & { c }_{ 3 } \end{bmatrix}$$ and $${A}_{2},{B}_{2},{C}_{2}$$ are respectively cofactors of $${a}_{2},{b}_{2},{c}_{2}$$ then $${a}_{1}{A}_{2}+{b}_{1}{B}_{2}+{c}_{1}{C}_{2}$$ is equal to ?

If $${x}^{a}{y}^{b}={e}^{m},{x}^{c}{y}^{d}={e}^{n}, { \triangle  }_{ 1 }=\begin{vmatrix} m & b \\ n & d \end{vmatrix},{ \triangle  }_{ 2 }=\begin{vmatrix} a & m \\ c & n \end{vmatrix}$$ and $${ \triangle  }_{ 3 }=\begin{vmatrix} a & b \\ c & d \end{vmatrix}$$ the value of $$x$$ and $$y$$ are respectively

For distinct numbers $$a,b,c,x,y,z\ \epsilon R$$ if $${ \Delta  }_{ 1 }\left| \begin{matrix} \left( a-x \right) ^{ 2 } & \left( b-x \right) ^{ 2 } & \left( c-x \right) ^{ 2 } \\ \left( a-y \right) ^{ 2 } & \left( b-y \right) ^{ 2 } & \left( c-y \right) ^{ 2 } \\ \left( a-z \right) ^{ 2 } & \left( b-z \right) ^{ 2 } & \left( c-z \right) ^{ 2 } \end{matrix} \right| { \Delta  }_{ 2 }\left| \begin{matrix} (ax+1)^{ 2 } & (bx+1)^{ 2 } & (cx+1)^{ 2 } \\ (ay+1)^{ 2 } & (by+1)^{ 2 } & (cy+1)^{ 2 } \\ (az+1)^{ 2 } & (bz+1)^{ 2 } & (cz+1)^{ 2 } \end{matrix} \right| $$ then $$\frac { { \Delta  }_{ 1 }^{ 2 } }{ { \Delta  }_{ 2 }^{ 2 } } +\frac { { \Delta  }_{ 2 }^{ 2 } }{ { \Delta  }_{ 1 }^{ 2 } } =$$

Three straight lines $$2x+11y-5=0, 24x+7y-20=0$$ and $$4x-3y-2=0$$

$$\Delta =\left| \begin{matrix} 0 & i-100 & i-500 \\ 100-i & 0 & 1000-i \\ 500-i & i-1000 & 0 \end{matrix} \right|$$ is equal to

If $$p{\lambda ^4} + p{\lambda ^3} + p{\lambda ^2} + s\lambda  + t = $$ $$\left| {\begin{array}{*{20}{c}}{{\lambda ^2} + 3\lambda } & {\lambda  + 1} & {\lambda  + 3}\\{\lambda  + 1} & {2 - \lambda } & {\lambda  - 4}\\{\lambda  - 3} & {\lambda  + 4} & {3\lambda }\end{array}} \right|$$, then value of t is