If $$\alpha \neq \mathrm{a}, \beta \neq \mathrm{b}, \gamma \neq \mathrm{c}$$ and $$\left|\begin{array}{lll}\alpha & \mathrm{b} & \mathrm{c} \\ \mathrm{a} & \beta & \mathrm{c} \\ \mathrm{a} & \mathrm{b} & \gamma\end{array}\right|=0$$, then $$\frac{\mathrm{a}}{\alpha-\mathrm{a}}+\frac{\mathrm{b}}{\beta-\mathrm{b}}+\frac{\gamma}{\gamma-\mathrm{c}}$$ is equal to :

If the system of equations $$x+4 y-z=\lambda, 7 x+9 y+\mu z=-3,5 x+y+2 z=-1$$ has infinitely many solutions, then $$(2 \mu+3 \lambda)$$ is equal to :

Let $$A=\left[\begin{array}{lll}2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b\end{array}\right]$$. If $$A^3=4 A^2-A-21 I$$, where $$I$$ is the identity matrix of order $$3 \times 3$$, then $$2 a+3 b$$ is equal to

If $$A$$ is a square matrix of order 3 such that $$\operatorname{det}(A)=3$$ and $$\operatorname{det}\left(\operatorname{adj}\left(-4 \operatorname{adj}\left(-3 \operatorname{adj}\left(3 \operatorname{adj}\left((2 \mathrm{~A})^{-1}\right)\right)\right)\right)\right)=2^{\mathrm{m}} 3^{\mathrm{n}}$$, then $$\mathrm{m}+2 \mathrm{n}$$ is equal to :