(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)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.

(i) Let x be the number of people belive in morning walk.
Let y be the number of people belive in yoga.
Let z be the number of people joining gym
x + y + z = 70
20x + 30y + 40z = 2,100
\(\Rightarrow\)2x + 3y + 4z = 210 and
500y + 400z = 23,000
\(\Rightarrow\) 5y + 4z = 230
This system of equation can be written in the matrix form as
\(\left[ \begin{matrix} 1 & 1 & 1 \\ 2 & 3 & 4 \\ 0 & 5 & 4 \end{matrix} \right] \left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] =\left[ \begin{matrix} 70 \\ 210 \\ 230 \end{matrix} \right] \)
Let AX = B
where A = \(\left[ \begin{matrix} 1 & 1 & 1 \\ 2 & 3 & 4 \\ 0 & 5 & 4 \end{matrix} \right] \)
X = \(\left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] \)
and B = \(\left[ \begin{matrix} 70 \\ 210 \\ 230 \end{matrix} \right] \)
X = A^-1B
|A| = \(\left[ \begin{matrix} 1 & 1 & 1 \\ 2 & 3 & 4 \\ 0 & 5 & 4 \end{matrix} \right] \)
= 1(12-20)-1(8-0)+1(10-0)
= -8-8+10 = -6 \(\neq \) 0
\(\therefore\) A^-1 exists
adj A= \(\left[ \begin{matrix} -8 & -8 & 10 \\ 1 & 4 & -5 \\ 1 & -2 & 1 \end{matrix} \right] \)
=\(\left[ \begin{matrix} -8 & 1 & 1 \\ -8 & 4 & -2 \\ 10 & -5 & 1 \end{matrix} \right] \)
\(A^{ -1 }=\frac { 1 }{ |A| } adjA\)
\(\Rightarrow\) A^-1 = \(\frac { 1 }{ -6 } \left[ \begin{matrix} -8 & 1 & 1 \\ -8 & 4 & -2 \\ 10 & -5 & 1 \end{matrix} \right] \)
X = \(-\frac { 1 }{ 6 } =\left[ \begin{matrix} -8 & 1 & 1 \\ -8 & 4 & -2 \\ 10 & -5 & 1 \end{matrix} \right] \left[ \begin{matrix} 70 \\ 210 \\ 230 \end{matrix} \right] \)
= \(-\frac { 1 }{ 6 } \left[ \begin{matrix} -560+210+230 \\ -560+840-460 \\ 700-1050+230 \end{matrix} \right] \)
= \(-\frac { 1 }{ 6 } \left[ \begin{matrix} -120 \\ -180 \\ -120 \end{matrix} \right] \)
\(\Rightarrow\) \(\left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] =\left[ \begin{matrix} 20 \\ 30 \\ 20 \end{matrix} \right] \)
The number of people believe in morning walk = 20
The number of people believe in yoga = 30
The number of people joining gym = 20
(iii) Exercise keeps a person fit and healthy.

Let Rs. x crores, Rs. y crores and Rs. z crores be spent on three Schemes.
By the question.
x + y + z = 600 .....(1)
x + 2z = 500 ......(2)
3x + y + z = 1200 .....(3)
Solving (1), (277444) and (3) as usual we get:
x = Rs. 300crores, y = Rs. 200 crores and z = Rs.100 crores.
(i) In our country, male population is more than female population
(ii) It is essential for a human being to save the life of all.