(i) Let Rs. x and Rs. y by charges for typing 1  English and 1 Hindi page respectively.
By the question 10x + 3y = 145
and  3x + 10y = 180
These can be written AX = B
where \(A=\begin{bmatrix}10&3\\3&10 \end{bmatrix},X=\begin{bmatrix} x\\y\end{bmatrix}, B=\begin{bmatrix} 145\\180\end{bmatrix}\)
Now \(|A|=\begin{bmatrix} 10&3\\3&10\end{bmatrix}=100-9=91\)
= A^-1exists
Now adj A = \(\begin{bmatrix}10&-3\\-3&10 \end{bmatrix}=\begin{bmatrix} 10&-3\\-3&10\end{bmatrix}\)
\(\therefore\ A^{-1}={1\over 91}\begin{bmatrix} 10&-3\\-3&10\end{bmatrix}\)
From(1) \(X=A^{-1}B\Rightarrow X={1\over91}\begin{bmatrix}10&-3\\-3&10 \end{bmatrix}\begin{bmatrix}145\\180 \end{bmatrix}\)
\(\Rightarrow \begin{bmatrix}x\\y \end{bmatrix}={1\over91}\begin{bmatrix}1450-540\\-435+1800 \end{bmatrix}\)
\(={1\over91}\begin{bmatrix}910\over1365 \end{bmatrix}=\begin{bmatrix} 10\\15\end{bmatrix}\)
⇒ x = 10 and y = 15
Hence charges for one English and one Hindi pages are Rs. 10 and Rs. 15 respectively.
(ii) Shyam is to pay = 5x2 = Rs. 10
Less charged = Rs. (5x15-10) = Rs. 65
Value: Poor should be charged subsidised rates.

(i) Here x, y, z refer to honesty, punctuality and obedience respectively.
By the question,
\(\left( \begin{matrix} 5 & 4 & 3 \\ 4 & 3 & 5 \\ 1 & 1 & 1 \end{matrix} \right) \left( \begin{matrix} x \\ y \\ z \end{matrix} \right) =\left( \begin{matrix} 11000 \\ 10700 \\ 2700 \end{matrix} \right) \)
\(\left( \begin{matrix} 5x+4y+3z \\ 4x+3y+5z \\ x+y+z \end{matrix} \right) =\left( \begin{matrix} 11000 \\ 10700 \\ 2700 \end{matrix} \right) \)
The 5x+6y+3z = 11000
      4x+3y+5z = 10700
           x+y+z = 2700
(ii) Let \(A=\left( \begin{matrix} 5 & 4 & 3 \\ 4 & 3 & 5 \\ 1 & 1 & 1 \end{matrix} \right) \)
\(\therefore\ |A|=\left| \begin{matrix} 5 & 4 & 3 \\ 4 & 3 & 5 \\ 1 & 1 & 1 \end{matrix} \right| \)
= 5(3-5)-4(4-4)+3(4-3)
= 5(-2)-4(-1)+3(1)|
= -10+4+3 = -3 ≠ 0
A^-1 exists ⇒ equations have a unique solution
Yes, it is possible to solve the system of equations, using matrix multiplication
(iii) We prefer to be awarded most the value of honesty (x) because it has highest value

The given system of equations is:
x + 2y + z = 7
x - y + z = 4
x + 3y + 2z = 10
Solving usual we get : x = 3, y = 1, z = 2
Food, taken at home is always the best way.

(i)Let 'x', 'y' and 'z' be the amount invested in three investments.
Then
x + y + z = 65000  ......(1)
\({6x\over100}+{8y\over100}+{9z\over100}=4800\)
\(\Rightarrow\) 6x + 8y + 9z = 480000  ...(2)
\({9z\over100}=600+{8y\over100}\)
\(\Rightarrow\) 0x - 8y + 9z = 60000
These can be written as AX = B where:
\(A=\begin{bmatrix} 1&1&1\\6&8&9\\0&-8&9\end{bmatrix},X=\begin{bmatrix} x\\y\\z\end{bmatrix}\)
and \(B=\begin{bmatrix} 650000\\480000\\60000\end{bmatrix}\)
(ii) Now \(|A|=\begin{bmatrix}1&1&1\\6&8&9\\0&-8&9 \end{bmatrix}\)
= 1.(72 + 72) - 6(9+8)
= 144 - 102 = 42 ≠ 0
Hence, the equations have a unique solution.
(iii) No.We are not fools because most of such companies are frauds.

Let Rs. x, Rs. y and Rs. z be the cost of reach carry bag.
By the question,
20x + 30y + 40z = 250
30x + 40y + 20z = 270
40x + 20y + 30z = 200
i.e. 2x + 3y + 4z = 25
3x + 4y + 2z = 27
4x + 2y + 3z = 20
Solving a susual, we will get : x = 1, y = 5, z = 2.
Shopkeeper (A) is better for environmental conditions because he is using least number of polythene.
Shopkeeper (B) is better for social conditions because he using more made bags, prepared by prisoners.