Case study (4 Marks)

Question 14 Marks

Two farmers Ramakishan and Gurucharan Singh cultivate only three varieties of rice namely Basmati, Permal and Naura. The sale (in ₹) of these varieties of rice by both the farmers in the month of September and October are given by the following matrices $A$ and $B$

September sales (in Rupees).
$
A=\left[\begin{array}{lll}
10,000 & 20,000 & 30,000 \\
50,000 & 30,000 & 10,000
\end{array}\right] \begin{array}{l}
\text { Ramakrishan } \\
\text { Gurcharan }
\end{array}
$
October sales (in Rupees)
$
B=\left[\begin{array}{lll}
5,000 & 10,000 & 6,000 \\
20,000 & 10,000 & 10,000
\end{array}\right] \begin{array}{l}
\text { Ramakrishan } \\
\text { Gurcharan }
\end{array}
$
Q.1. The total sales in September and October for each farmer in each variety can be represented as __________.
(A) $A+B$
(B) $A - B$
(C) $A > B$
(D) $A < B$
Q.2. What is the value of $A_{23}$ ?
(A) $10,000$
(B) $20,000$
(C) $30,000$
(D) $40,000$
Q.3. The decrease in sales from September to October is given by __________.
(A) $A+B$
(B) $A - B$
(C) $A > B$
(D) $A < B$
Q.4. If Ramkishan receives $2 \%$ profit on gross sales, compute his profit for each variety sold in October.
(A) ₹ $100,$ ₹ $200$ and ₹ $120$
(B) ₹ $100,$ ₹ $200$ and ₹ $130$
(C) ₹ $100,$ ₹ $220$ and ₹ $120$
(D) ₹ $110,$ ₹ $200$ and ₹ $120$

Answer

(1) (A) $A + B$
Explanation: Combined sales in September and October for each farmer in each variety is given by $ A+B=$
Basmati Permal Naura
$\left[\begin{array}{lll}15,000 & 30,000 & 36,000 \\70,000 & 40,000 & 20,000\end{array}\right] \begin{array}{l}\text { Ramkrishan } \\\text { Gurcharan singh }\end{array}$
(2) (A) $10,000$
Explanation: $A_{23}=10,000$
(3) (B) $A - B$
Explanation: Change in sales from September to October is given by
$A-B=$
Basmati Permal Naura
$\left[\begin{array}{ccc}5000 & 10,000 & 24,000 \\30,000 & 20,000 & 0\end{array}\right] \begin{array}{l}\text { Ramkishan } \\\text { Gurcharan Singh }\end{array}$
(4) (A) ₹ $100,$ ₹ $200$ and ₹ $120$
Explanation:
$\begin{aligned}2 \% \text { of } B & =\frac{2}{100} \times B \\& =0.02 \times B \\& =0.02\end{aligned}$
Basmati Permal Naura $\left[\begin{array}{ccc}5000 & 10,000 & 6000 \\ 20,000 & 10,000 & 10,000\end{array}\right] \begin{array}{l}\text { Ramkishan } \\ \text { Gurcharn Singh }\end{array}$
Thus, in October Ramkishan receives ₹ $100,$ ₹ $200$ and ₹ $120$ as profit in the sale of each variety of rice, respectively.

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Question 24 Marks

A manufacture produces three stationery products Pencil, Eraser and Sharpener which he sells in two markets. Annual sales are indicated below.

Market	Products (in numbers)
Market	Pencil	Eraser	Sharpener
$A$	$10,000$	$2,000$	$18,000$
$B$	$6,000$	$20,000$	$8,000$

If the unit Sale price of Pencil, Eraser and Sharpener are ₹ $ 2.50,$ ₹ $1.50$ and ₹ $1.00$ respectively, and unit cost of the above three commodities are ₹ $ 2.00,$ ₹ $1.00$ and ₹ $0.50$ respectively, then,
Q.1. Total revenue of market $A$ :
(A) ₹ $64,000$
(B) ₹ $60,400$
(C) ₹ $46,000$
(D) ₹ $40600$
Q.2. Total revenue of market $B$ is:
(A) ₹ $35,000$
(B) ₹ $53,000$
(C) ₹ $50,300$
(D) ₹ $30,500$
Q.3. Cost incurred in market $A$ :
(A) ₹ $13,000$
(B) ₹ $30,100$
(C) ₹ $10,300$
(D) ₹ $31,000$
Q.4. Profits in market $A$ and $B$ respectively are:
(A) (₹ $15,000,$ ₹ $17,000$)
(B) (₹ $17,000,$ ₹ $15,000$)
(C) (₹ $51,000,$ ₹ $71,000$)
(D) (₹ $10,000,$ ₹ $20,000$)

Answer

(1) (C) ₹ $46,000$
Explanation: Total revenue of
$=\left[\begin{array}{lll}10,000 & 2,000 & 18,000\end{array}\right]\left[\begin{array}{l}2.50 \\ 1.50 \\ 1.00\end{array}\right]$
$=2.50 \times 10,000+1.50 \times 2,000+1.00 \times 18,000$
= ₹ $46,000$
(2) (B) ₹ $53,000$
Explanation: Total revenue of market B
$=\left[\begin{array}{lll}6,000 & 20,000 & 8,000\end{array}\right]\left[\begin{array}{l}2.50 \\ 1.50 \\ 1.00\end{array}\right]$
= ₹ $53,000$
(3) (D) ₹ $31,000$
Explanation: Cost incurred in market A
$=\left[\begin{array}{lll}10,000 & 2,000 & 18,000\end{array}\right]\left[\begin{array}{l}2.00 \\ 1.00 \\ 0.50\end{array}\right]$
$=2.00 \times 10,000+1.00 \times 2,000+0.50 \times 18,000$
= ₹ $31,000$
(4) (A) (₹ $15,000,$ ₹ $17,000$)
Explanation: Cost incurred in market B
$=\left[\begin{array}{lll}6,000 & 20,000 & 8,000\end{array}\right]\left[\begin{array}{l}2.00 \\ 1.00 \\ 0.50\end{array}\right]$
$\begin{array}{l}=[2.00 \times 6,000+1.00 \times 20,000+0.50 \times 8,000] \\ =[36,000]\end{array}$
Profit of market $A$ & $B=$ total revenue of $A$ and $B-$ Cost increased in market $A$ and $B$
$\begin{aligned} {\left[\begin{array}{l}A \\ B\end{array}\right] } & =\left[\begin{array}{l}46,000 \\ 50,000\end{array}\right]-\left[\begin{array}{l}31,000 \\ 36,000\end{array}\right] \\ & =\left[\begin{array}{l}15,000 \\ 17,000\end{array}\right]\end{aligned}$
i.e., (₹ $15,000,$ ₹ $17,000$)

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Question 34 Marks

There are three families $A, B$ and $C$. The number of members in these families are given in the table below.

	Men	Women	Children
Family A	3	2	1
Family B	2	4	2
Family C	4	3	2

The daily expenses of each man, woman and child are respectively ₹ $200,$ ₹ $100$ and ₹ $50.$

Q.1. The total daily expense of family $A$ is __________.
(A) $850$ $\quad\quad$ (B) $900$
(C) $1,200$ $\quad$ (D) $2,950$
Q.2. The total daily expense of family C is __________.
(A) $850$ $\quad\quad$ (B) $900$
(C) $1,200$ $\quad$ (D) $2,950$
Q.3. The combined daily expense of all the women is __________.
(A) $850$ $\quad\quad$ (B) $900$
(C) $1,200$ $\quad$ (D) $2,950$
Q.4. The family with highest expense is __________.
(A) $A$ $\quad$ (B) $B$
(C) $C$ $\quad$ (D) All have same expense

Answer

(1) (A) $850$
Explanation: Expense of family $A$
$=\left[\begin{array}{lll}3 & 2 & 1\end{array}\right]\left[\begin{array}{c}200 \\100 \\50\end{array}\right]=[850]$
(2) (C) $1,200$
Explanation: Expense of family $C$
$=\left[\begin{array}{lll}4 & 3 & 2\end{array}\right]\left[\begin{array}{c}200 \\100 \\50\end{array}\right]=[1200]$
(3) (B) $900$
Explanation: Combined expense of women
$=\left[\begin{array}{lll}2 & 4 & 3\end{array}\right]\left[\begin{array}{l}100 \\100 \\100\end{array}\right]=[900]$
(4) (C) $C$
Explanation: Most expensive family is $C$ with an expense of ₹ $1200.$

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Question 44 Marks

Amit, Biraj and Chirag were given the task of creating a square matrix of order 2.
Below are the matrices created by them. A, B, C are the matrices created by Amit, Biraj and Chirag respectively.
$A=\left[\begin{array}{cc}1 & 2 \\ -1 & 3\end{array}\right] B=\left[\begin{array}{ll}4 & 0 \\ 1 & 5\end{array}\right] C=\left[\begin{array}{cc}2 & 0 \\ 1 & -2\end{array}\right]$
If $a=4$ and $b=-2$
Q.1. Sum of the matrices $A, B$ and $C, A+(B+C)$ is
(A) $\left[\begin{array}{ll}1 & 6 \\ 2 & 7\end{array}\right]$$\quad$(B) $\left[\begin{array}{ll}6 & 1 \\ 7 & 2\end{array}\right]$
(C) $\left[\begin{array}{ll}7 & 2 \\ 1 & 6\end{array}\right]$$\quad$(D) $\left[\begin{array}{ll}2 & 1 \\ 7 & 6\end{array}\right]$
Q.2. $\left(A^T\right)^T$ is equal to
(A) $\left[\begin{array}{cc}1 & 2 \\ -1 & 3\end{array}\right]$$\quad$(B) $\left[\begin{array}{cc}2 & 1 \\ 3 & -1\end{array}\right]$
(C) $\left[\begin{array}{cc}1 & -1 \\ 2 & 3\end{array}\right]$$\quad$(D) $\left[\begin{array}{cc}2 & 3 \\ -1 & 1\end{array}\right]$
Q.3. $(b A)^T$ is equal to
(A) $\left[\begin{array}{cc}-2 & -4 \\ 2 & -6\end{array}\right]$$\quad$(B) $\left[\begin{array}{cc}-2 & 2 \\ -4 & -6\end{array}\right]$
(C) $\left[\begin{array}{cc}-2 & 2 \\ -6 & -4\end{array}\right]$$\quad$(D) $\left[\begin{array}{cc}-6 & -2 \\ 2 & 4\end{array}\right]$
Q.4. $A C-B C$ is equal to
(A) $\left[\begin{array}{cc}-4 & -6 \\ -4 & 4\end{array}\right]$$\quad$(B) $\left[\begin{array}{cc}-4 & -4 \\ 4 & -6\end{array}\right]$
(C) $\left[\begin{array}{cc}-4 & -4 \\ -6 & 4\end{array}\right]$$\quad$(D) $\left[\begin{array}{cc}-6 & 4 \\ -4 & -4\end{array}\right]$

Answer

(1) (C) $\left[\begin{array}{ll}7 & 2 \\ 1 & 6\end{array}\right]$
Explanation:
$\begin{aligned} A+(B+C) & =\left[\begin{array}{cc}1 & 2 \\ -1 & 3\end{array}\right]+\left(\left[\begin{array}{ll}4 & 0 \\ 1 & 5\end{array}\right]+\left[\begin{array}{cc}2 & 0 \\ 1 & -2\end{array}\right]\right) \\ & =\left[\begin{array}{cc}1 & 2 \\ -1 & 3\end{array}\right]+\left[\begin{array}{ll}6 & 0 \\ 2 & 3\end{array}\right] \\ & =\left[\begin{array}{cc}7 & 2 \\ 1 & 6\end{array}\right]\end{aligned}$
(2) (A) $\left[\begin{array}{cc}1 & 2 \\ -1 & 3\end{array}\right]$
Explanation :
$\begin{aligned}\left(A^T\right) & =\left[\begin{array}{cc}1 & -1 \\2 & 3\end{array}\right] \\\left(A^T\right)^T & =\left[\begin{array}{cc}1 & 2 \\-1 & 3\end{array}\right]\end{aligned}$
(3) (B) $\left[\begin{array}{cc}-2 & 2 \\ -4 & -6\end{array}\right]$
Explanation:
$\begin{aligned} & b A\end{aligned}=-2 A=\left[\begin{array}{cc}-2 & -4 \\ 2 & -6\end{array}\right]$
$(b A)^T=\left[\begin{array}{cc}-2 & 2 \\ -4 & -6\end{array}\right]$
(4) (C) $\left[\begin{array}{cc}-4 & -4 \\ -6 & 4\end{array}\right]$
Explanation:
$\begin{aligned}A C & =\left[\begin{array}{cc}1 & 2 \\-1 & 3\end{array}\right]\left[\begin{array}{cc}2 & 0 \\1 & -2\end{array}\right] \\& =\left[\begin{array}{cc}4 & -4 \\1 & -6\end{array}\right]\end{aligned}$
$\begin{aligned} B C & =\left[\begin{array}{ll}4 & 0 \\ 1 & 5\end{array}\right]\left[\begin{array}{cc}2 & 0 \\ 1 & -2\end{array}\right] \\ & =\left[\begin{array}{cc}8 & 0 \\ 7 & -10\end{array}\right]\end{aligned}$
$\begin{aligned} A C-B C & =\left[\begin{array}{ll}4 & -4 \\ 1 & -6\end{array}\right]-\left[\begin{array}{cc}8 & 0 \\ 7 & -10\end{array}\right] \\ & =\left[\begin{array}{cc}-4 & -4 \\ -6 & 4\end{array}\right]\end{aligned}$

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Question 54 Marks

A manufacture produces three stationery products Pencil, Eraser and Sharpener which he sells in two markets. Annual sales are indicated below.