Number of triplets of $(a, b, c)$ for which the system of equations $ax - by = 2a - b$ and $(c + 1)x + cy = 10 - a + 3b$ has infinitely many solutions and $(x = 1, y = 3)$ is one of the solutions,is:

$A$ trust fund has Rs. $30,000$ that must be invested in two different types of bonds. The first bond pays $5 \%$ interest per year,and the second bond pays $7 \%$ interest per year. Using matrix multiplication,determine how to divide Rs. $30,000$ among the two types of bonds if the trust fund must obtain an annual total interest of Rs. $1800$.

Let $a, b, c, d, e$ be five numbers satisfying the system of equations:
$2a + b + c + d + e = 6$
$a + 2b + c + d + e = 12$
$a + b + 2c + d + e = 24$
$a + b + c + 2d + e = 48$
$a + b + c + d + 2e = 96$
Then $|c|$ is equal to:

If the solution of the system of simultaneous equations $\frac{1}{x}+\frac{2}{y}-\frac{3}{z}-1=0$,$\frac{2}{x}-\frac{4}{y}+\frac{3}{z}-1=0$ and $\frac{3}{x}+\frac{6}{y}-\frac{6}{z}-4=0$ is $x=\alpha, y=\beta, z=\gamma$,then $\alpha^2+\gamma^2=$

In a legislative assembly election,a political group hired a public relations firm to promote its candidate in three ways: telephone,house calls,and letters. The cost per contact (in paise) is given in matrix $A$ as $A = \begin{bmatrix} 40 \\ 100 \\ 50 \end{bmatrix} \begin{matrix} \text{Telephone} \\ \text{Housecall} \\ \text{Letter} \end{matrix}$. The number of contacts of each type made in two cities $X$ and $Y$ is given by $B = \begin{bmatrix} 1000 & 500 & 5000 \\ 3000 & 1000 & 10000 \end{bmatrix} \begin{matrix} \text{Telephone} & \text{Housecall} & \text{Letter} \\ \to X \\ \to Y \end{matrix}$. Find the total amount spent by the group in the two cities $X$ and $Y$.

The number of solutions of the equations $x + 4y - z = 0,$ $3x - 4y - z = 0,$ and $x - 3y + z = 0$ is