Case study Chapter 4(Determinant)

Case study 5:– In coaching institutes, the students not only get academic guidance but also they get to know about career options and right goals as per their interest and academic record.(Case study problem determinant 5)

Case study problem determinant 5 — In coaching institutes, the students not only get academic guidance but also they get to know about career options

A coaching institute conduct classes in two sections A and Band fees for rich and poor children are different. In section A, there are 20 poor and 5 rich children and total monthly collection is ₹9,000, where as in section B, there are 5 poor and 25 rich children and total monthly collection is ₹ 26, 000.

Based on the above information answer the following question.

(i) (a) If ₹x and ₹ y be the fees for rich and poor children respectively then the information provided in problem in system of linear equation express.

(b) Express the system of linear equations obtained in (a) as matrix equation.

(ii) Find th fees for rich and poor children.

Solution:(i) (a) From question

5x + 20y = 9000 ——-(i)

and 25x + 5y = 26000 —–(ii)

(b) We have system of linear equations,

5x + 20y = 9000 ——-(i)

25x + 5y = 26000 —–(ii)

We can write matrix system as

$AX = B, \text{ where } A = \begin{bmatrix} 5 & 20\\ 25 & 5 \end{bmatrix}, X = \begin{bmatrix} x \\ y\end{bmatrix},B = \begin{bmatrix} 9000 \\ 26000\end{bmatrix}$

$\Rightarrow \begin{bmatrix} x \\ y\end{bmatrix} = \begin{bmatrix} 5 & 20\\ 25 & 5 \end{bmatrix}\begin{bmatrix} 9000 \\ 26000\end{bmatrix}$

(ii) $A = \begin{bmatrix} 5 & 20\\ 25 & 5 \end{bmatrix}$

⇒ |A| = 25 – 500 = -475 ≠ 0

NOW,

$A_{11} = 5, A_{12} = -25$

$A_{21}= -20, A_{22} = 5$

$adj(A) = \begin{bmatrix} 5 & -25\\ -20 & 5 \end{bmatrix}^T$

$=\begin{bmatrix} 5 & -20\\ -25 & 5 \end{bmatrix}$

$\therefore A^{-1} = \frac{adj(A)}{|A|}$

$= \frac{1}{-475}\begin{bmatrix} 5 & -20\\ -25 & 5 \end{bmatrix}$

We have

$X = A^{-1}B$

$\Rightarrow \begin{bmatrix} x \\ y\end{bmatrix}=\frac{1}{-475}\begin{bmatrix} 5 & -20\\ -25 & 5 \end{bmatrix}\begin{bmatrix} 9000 \\ 26000\end{bmatrix}$

$\Rightarrow \begin{bmatrix} x \\ y\end{bmatrix} = -\frac{1}{475}\begin{bmatrix} 45000-520000 \\ -225000+130000\end{bmatrix}$

$\Rightarrow \begin{bmatrix} x \\ y\end{bmatrix}= -\frac{1}{475}\begin{bmatrix} -475000\\ -95000\end{bmatrix}$

$\Rightarrow \begin{bmatrix} x \\ y\end{bmatrix}= \begin{bmatrix} 1000 \\ 200\end{bmatrix}$

⇒ x = 1000, and y = 200

Fees of rich and poor students are respectively ₹ 1000 and ₹ 200.

Some other Case study problem

Case study 1:– Read the following and answer the questions(Case study problem determinant 1)

Three friends Rahul, Ravi and Rakesh went to a vegetable market to purchase vegetable. From a vegetable shop Rahul purchased 1 kg of each potato, onion and Brinjal for a total of ₹ 21. Ravi purchased 4 kg of potato, 3 kg of onion and 2 kg of Brijal for ₹ 60 while Rakesh purchased 6 kg potato, 2 kg onion and 3 kg Brinjal for ₹70

Solution : For solution click here

Case study 2:- Read the following and answer the question(Case study problem determinant 2)

Reshma wants to donate a rectangular plot of land for a school of her village. When she was asked by construction agency to give dimensions of the plot, she said that if its length is decreased by 50 m and breadth is increased by 50m, then its area will remain same, but if length is decreased by 10 m and breadth is decreased by 20m, then its area will decreased by 5300 m².

Solution: For solution click here

Case study 3:– Read the following and answer the question(Case study problem determinant 3 )

The monthly incomes of two sister Reshma and Ritam are in the ratio 3:4 and their monthly expenditures are in the ratio 5;7. Each sister saves ₹ 15,000 per month.