Case study (4 Marks)

Question 14 Marks

On her birthday, Seema decided to donate some money to the children of an orphanage home. If there were $8$ children less, everyone would have got $₹10$ more. However, if there were $16$ children more, everyone would have got $₹10$ less. Let the number of children be $\mathrm{x}$ and the amount distributed by Seema for one child be $\mathrm{y} \ ($in $₹)$.

$(i)$ Represent given information in matrix algebra.
$(ii)$ Find the adjoint of Matrix containing information about of number of children and amount she paid?
$(iii)$ Find the number of children who were given some money by Seema?
OR
How much amount does Seema spend in distributing the money to all the students of the Orphanage?

Answer

$(i) \begin{aligned}& 5 x-4 y=40 \\& 5 x-8 y=-80 \\& {\left[\begin{array}{rr}5 & -4 \\ 5 & -8\end{array}\right]\left[\begin{array}{l}x \\y\end{array}\right]=\left[\begin{array}{c}40 \\-80\end{array}\right]}\end{aligned}$
$(ii) A=\left[\begin{array}{ll}5 & -4 \\ 5 & -8\end{array}\right], \mathrm{X}=\left[\begin{array}{l}x \\ y\end{array}\right] \text { and } \mathrm{B}=\left[\begin{array}{c}40 \\ -80\end{array}\right] $
$ |A|=-40+20=-20 \neq 0 \text { Cofactor matrix } \mathrm{A}$
$=\left[\begin{array}{ll}-8 & -5 \\ 4 & 5\end{array}\right] \text { adj } \mathrm{A}=\left[\begin{array}{cc}-8 & 4 \\ -5 & 5\end{array}\right]$
$(iii) \mathrm{A}=\left[\begin{array}{ll}5 & -4 \\ 5 & -8\end{array}\right],$
$ \mathrm{X}=\left[\begin{array}{l}x \\ y\end{array}\right] \text { and } \mathrm{B}=\left[\begin{array}{c}40 \\ -80\end{array}\right]$
$ \mathrm{A} \mid=-40+20=-20 \neq 0 \text { Cofactor matrix } \mathrm{A}$
$=\left[\begin{array}{ll}-8 & -5 \\ 4 & 5\end{array}\right] \text {, adj } \mathrm{A}=\left[\begin{array}{ll}-8 & 4 \\ -5 & 5\end{array}\right] $
$ \mathrm{X}=\mathrm{A}^{-1} \mathrm{~B} \ldots(\mathrm{i}) $
$ \mathrm{A}^{-1}=\frac{1}{|A|} \cdot \operatorname{adjA} $
$ \mathrm{A}^{-1}=\frac{1}{-20} \cdot\left[\begin{array}{ll}-8 & 4 \\ -5 & 5\end{array}\right] $
From $(i)\ {\left[\begin{array}{l}x \\ y\end{array}\right]=\frac{1}{-20} \cdot\left[\begin{array}{ll}-8 & 4 \\ -5 & 5\end{array}\right]\left[\begin{array}{c}40 \\ -80\end{array}\right]} $
$ \Rightarrow \left[\begin{array}{l}x \\ y\end{array}\right]=\frac{1}{-20}\left[\begin{array}{l}-320-320 \\ -200-400\end{array}\right]=\left[\begin{array}{l}32 \\ 30\end{array}\right] $
$ \mathrm{X}=32 $ and $ \mathrm{y}=30$
Or
There are $32$ Children, and each child is given $₹30$.
Total money spent by Seema $=32 \times 30=₹\ 960$
Hence Seema spends $₹\ 96$0 in distributing the money to all the students of the Orphanage.

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Question 24 Marks

A trust fund has $₹ 35000$ that must be invested in two different types of bonds, say $\mathrm{X}$ and $\mathrm{Y}$. The first bond pays $10 \%$ interest p.a. which will be given to an old age home and second one pays $8 \%$ interest p.a. which will be given to WWA (Women Welfare Association). Let A be a $1 \times 2$ matrix and B be a $2 \times 1$ matrix, representing the investment and interest rate on each bond respectively.

(i) Represent the given information in matrix algebra.

(ii) If ₹ 15000 is invested in bond $\mathrm{X}$, then find total amount of interest received on both bonds?

(iii) If the trust fund obtains an annual total interest of ₹ 3200 , then find the investment in two bonds.

If the amount of interest given to old age home is ₹500, then find the amount of investment in bond Y.

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Question 34 Marks

Three car dealers, say A, B and C, deals in three types of cars, namely Hatchback cars, Sedan cars, SUV cars. The sales figure of 2019 and 2020 showed that dealer A sold 120 Hatchback, 50 Sedan, 10 SUV cars in 2019 and 300 Hatchback, 150 Sedan, 20 SUV cars in 2020; dealer B sold 100 Hatchback, 30 Sedan, 5 SUV cars in 2019 and 200 Hatchback, 50 Sedan, 6 SUV cars in 2020; dealer C sold 90 Hatchback, 40 Sedan, 2 SUV cars in 2019 and 100 Hatchback, 60 Sedan, 5 SUV cars in 2020.

(i) Write the matrix summarizing sales data of 2019 and 2020.

(ii) Find the matrix summarizing sales data of 2020.

(iii) Find the total number of cars sold in two given years, by each dealer?

If each dealer receives a profit of ₹ 50000 on sale of a Hatchback, ₹100000 on sale of a Sedan and ₹200000 on sale of an SUV, then find the amount of profit received in the year 2020 by each dealer.

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Question 44 Marks

Three friends Ravi, Raju and Rohit were doing buying and selling of stationery items in a market. The price of per dozen of pen, notebooks and toys are Rupees $\mathrm{x}, \mathrm{y}$ and $\mathrm{z}$ respectively.Ravi purchases $4$ dozen of notebooks and sells $2$ dozen of pens and $5$ dozen of toys. Raju purchases $2$ dozen of toy and sells $3$ dozen of pens and $1$ dozen of notebooks. Rohit purchases one dozen of pens and sells $3$ dozen of notebooks and one dozen of toys.
In the process, Ravi, Raju and Rohit earn $₹\ 1500, ₹\ 10$0 and $₹ \ 400$ respectively.

$(i)$ Write the above information in terms of matrix Algebra.
$(ii)$ What is the total price of one dozen of pens and one dozen of notebooks?
$(iii)$ What is the sale amount of Ravi?
$OR$
What is the amount of purchases and sales made by all three friends?

Answer

$(i) \ 2 x-4 y+5 z=1500$
$ 3 x+y-2 z=100$
$ -x+3 y+z=400 \ {\left[\begin{array}{ccc}2 & -4 & 5 \\3 & 1 & -2 \\-1 & 3 & 1\end{array}\right]\left[\begin{array}{l}x \\y \\z\end{array}\right]=\left[\begin{array}{c}1500 \\100 \\400 \end{array}\right]}$
$(ii) \mathrm{A}=\left[\begin{array}{ccc}2 & -4 & 5 \\ 3 & 1 & -2 \\ -1 & 3 & 1\end{array}\right], \mathrm{X}=\left[\begin{array}{l}x \\ y \\ z\end{array}\right], \mathrm{B}=\left[\begin{array}{c}1500 \\ 100 \\ 400\end{array}\right]$
$ \mathrm{X}=\mathrm{A}^{-1} \mathrm{~B} \ldots(\mathrm{i})$
$ |\mathrm{A}|=2(1+6)+4(3-2)+5(9+1)=68 \neq 0 $
$ \text { co-factor matrix } \mathrm{A}=\left[\begin{array}{ccc}7 & -1 & 10 \\ 19 & 7 & -2 \\ 3 & 19 & 14\end{array}\right] \text {, adj } \mathrm{A}=\left[\begin{array}{ccc}7 & 19 & 3 \\ -1 & 7 & 19 \\ 10 & -2 & 14\end{array}\right] $
$ \mathrm{A}-1=\frac{1}{|A|} \cdot \operatorname{adj} A $
$ \mathrm{~A}-1=\frac{1}{68}\left[\begin{array}{ccc}7 & 19 & 3 \\ -1 & 7 & 19 \\ 10 & -2 & 14\end{array}\right] $
From $(i)$
$\Rightarrow\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\frac{1}{68}\left[\begin{array}{ccc}7 & 19 & 3 \\ -1 & 7 & 19 \\ 10 & -2 & 14\end{array}\right]\left[\begin{array}{c}1500 \\ 100 \\ 400\end{array}\right] $
$ \Rightarrow\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\frac{1}{68}\left[\begin{array}{c}10500+1900+1200 \\ -1500+700+7600 \\ 15000-200+5600\end{array}\right] $
$ \Rightarrow\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\frac{1}{68}\left[\begin{array}{c}13600 \\ 6800 \\ 20400\end{array}\right]$
$x=200, y=100 $ and $ z=300$
Total price of one dozen of pens and one dozen of notebooks $=200+100=₹ 300$
$(iii)$ The sale amount of Ravi $=2 x+5 z=2 \times 200+5 \times 300=400+1500=₹ 1900$
Or
The amount of purchases made by all three friends
$4 y+2 z+x=4 \times 100+2 \times 300+200=₹ 1200$
The amount of sales made by all three friends
$2 x+5 z+3 x+y+3 y+z=5 x+4 y+6 z=5 \times 200+4 \times 100+6 \times 300=₹ 3200$

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Question 54 Marks

The nut and bolt manufacturing business has gained popularity due to the rapid Industrialization and introduction of the Capital-Intensive Techniques in the Industries that are used as the Industrial fasteners to connect various machines and structures. Mr. Suresh is in Manufacturing business of Nuts and bolts. He produces three types of bolts, $\mathrm{x}, \mathrm{y}$, and $\mathrm{z}$ which he sells in two markets. Annual sales (in ₹) indicated below:

(i) If unit sales prices of $x, y$ and $z$ are $₹ 2.50$, ₹ 1.50 and $₹ 1.00$ respectively, then find the total revenue collected from Market-I \&II.

(ii) If the unit costs of the above three commodities are ₹2.00, ₹ 1.00 and 50 paise respectively, then find the cost price in Market I and Market II.

(iii) If the unit costs of the above three commodities are ₹2.00, ₹1.00 and 50 paise respectively, then find gross profit from both the markets.

If matrix $\mathrm{A}=\left[a_{i j}\right]_{2 \times 2}$ where $\mathrm{a}_{\mathrm{ij}}=1$, if $\mathrm{i} \neq \mathrm{j}$ and $\mathrm{a}_{\mathrm{ij}}=0$, if $\mathrm{i}=\mathrm{j}$ then find $\mathrm{A}^2$.

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Question 64 Marks

Three schools A, B and C organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold handmade fans, mats, and plates from recycled material at a cost of ₹ 25 , ₹ 100 and ₹ 50 each. The number of articles sold by school A, B, C are given below.

(i) Represent the sale of handmade fans, mats and plates by three schools A, B and C and the sale prices (in ₹) of given products per unit, in matrix form.

(ii) Find the funds collected by school A, B and C by selling the given articles.

(iii) If they increase the cost price of each unit by $20 \%$, then write the matrix representing new price.

Find the total funds collected for the required purpose after $20 \%$ hike in price.

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Question 74 Marks

Consider 2 families A and B. Suppose there are 4 men, 4 women and 4 children in family A and 2 men, 2 women and 2 children in family B. The recommended daily amount of calories is 2400 for a man, 1900 for a woman, 1800 for children and 45 grams of proteins for a man, 55 grams for a woman and 33 grams for children.

(i) Represent the requirement of calories and proteins for each person in matrix form.

(ii) Find the requirement of calories of family A and requirement of proteins of family B.

(iii) Represent the requirement of calories and proteins If each person increases the protein intake by $5 \%$ and decrease the calories by $5 \%$ in matrix form.

If $\mathrm{A}$ and $\mathrm{B}$ are two matrices such that $\mathrm{AB}=\mathrm{B}$ and $\mathrm{BA}=\mathrm{A}$, then find $\mathrm{A}^2+\mathrm{B}^2$ in terms of $\mathrm{A}$ and $\mathrm{B}$.

Answer

(i) Let $\mathrm{F}$ be the matrix representing the number of family members and R be the matrix representing the requirement of calories and proteins for each person. Then

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Question 84 Marks

Read the following passage and answer the questions given below.

In an elliptical sport field the authority wants to design a rectangular soccer field with the maximum possible area. The sport field is given by the graph of $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

(i) If the length and the breadth of the rectangular field be $2 x$ and $2 y$ respectively, then find the area function in terms of $x$.

(ii) Find the critical point of the function.

(iii) Use First derivative Test to find the length $2 x$ and width $2 y$ of the soccer field (in terms of $a$ and b) that maximize its area.

(iii) Use Second Derivative Test to find the length $2 x$ and width $2 y$ of the soccer field (in terms of $a$ and $b$ ) that maximize its area.

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Maths Case study (4 Marks)

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