The cost of $4\, \mathrm{Kg}$ onion, $3\, \mathrm{kg}$ wheat and $2\, \mathrm{kg}$ rice is $\mathrm{Rs}\, 60 .$ The cost of $2\, \mathrm{kg}$ onion, $4\, \mathrm{kg}$ wheat and $6 \,\mathrm{kg}$ rice is $\mathrm{Rs} \,90 .$ The cost of $6\, \mathrm{kg}$ onion $2 \,\mathrm{kg}$ wheat and $3\, \mathrm{kg}$ rice is $\mathrm{Rs}\, 70 .$

Find cost of each item per $\mathrm{kg}$ by matrix method

If the system of linear equations

$7 x+11 y+\alpha z=13$

$5 x+4 y+7 z=\beta$

$175 x+194 y+57 z=361$

has infinitely many solutions, then $\alpha+\beta+2$ is equal to

If the system of linear equations  $x_1 + 2x_2 + 3x_3 = 6$ ; $x_1 + 3x_2 + 5x_3 = 9$ ; $2x_1 + 5x_2 + ax_3 = b$ is consistent and has infinite number of solutions, then

Solve the system of the following equations  $\frac{2}{x}+\frac{3}{y}+\frac{10}{z}=4$ ; $\frac{4}{x}-\frac{6}{y}+\frac{5}{z}=1$  ; $\frac{6}{x}+\frac{9}{y}-\frac{20}{z}=2$

Let $a, \lambda, \mu \in \mathbb{R}$. Consider the system of linear equations

$a x+2 y=\lambda$

$3 x-2 y=\mu$Which of the following statement($s$) is(are) correct?

($A$) If $a=-3$, then the system has infinitely many solutions for all values of $\lambda$ and $\mu$

($B$) If $a \neq-3$, then the system has a unique solution for all values of $\lambda$ and $\mu$

($C$) If $\lambda+\mu=0$, then the system has infinitely many solutions for $a=-3$

($D$) If $\lambda+\mu \neq 0$, then the system has no solution for $a=-3$