Case study chapter 4(Determinants)

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.

(i) If monthly income of Reshma and Ritam are ₹3x and ₹4x and their monthly expenditure are ₹5x and ₹7y respectively, then the information provided in problemmay be written in system of linear equation.

(a) x – y = 15,000, x + y = 15,000

(b) 3x + 5y = 15,000, 4x + 7y = 15,000

(d) 5x – 3y = 15,000, 7x – 4y = 15,000

(ii) System of linear equations

3x – 5y = 15,000

4x – 7y = 15,000

may be written in matrix equation as

(a) $AX = B, \text{ where } A = \begin{bmatrix} 3 & -5\\ 4 & -7 \end{bmatrix}, X = \begin{bmatrix} x \\ y\end{bmatrix},B = \begin{bmatrix} 15,000 \\ 15,000\end{bmatrix}$

(b) $BX = A, \text{ where } A = \begin{bmatrix} 3 & -5\\ 4 & -7 \end{bmatrix}, X = \begin{bmatrix} x \\ y\end{bmatrix},B = \begin{bmatrix} 15000 \\ 15000\end{bmatrix}$

(c) $AB = X, \text{ where } A = \begin{bmatrix} 3 & -5\\ 4 & -7 \end{bmatrix}, X = \begin{bmatrix} x \\ y\end{bmatrix},B = \begin{bmatrix} 15000 \\ 15000\end{bmatrix}$

(d) $AB = I, \text{ where } A = \begin{bmatrix} 3 & -5\\ 4 & -7 \end{bmatrix}, I = \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix},B = \begin{bmatrix} 15000 \\ 15000\end{bmatrix}$

(iii) If AX = B, where A,X,B are matrices then X should be

(a) X = AB (b) $X = BA^{-1}$

(iv) If $A = \begin{bmatrix} 3 & -5\\ 4 & -7 \end{bmatrix}$ then $A^{-1}$ is

(a) $\begin{bmatrix} 7 & 5\\ 4 & 3 \end{bmatrix}$

(b) $\begin{bmatrix} -7 & 5\\ -4 & 3 \end{bmatrix}$

(d) $\begin{bmatrix} 7 & -5\\ 4 & -3 \end{bmatrix}$

(v) Monthly income of Reshm,a and Ritam is

(a) ₹90000 each (b) ₹ 90000 and ₹12,000 respectively

Solution:(i) Answer (c)

Monthly savings of Reshma

= Monthly income of Reshma – Monthly expenditure of Reshma

15000 = 3x – 5y

Simmilarly for Ritam

15000 = 4x – 7y

Hence, required system of linear equation

3x – 5y = 15,000

4x – 7y = 15,000

(ii) Answer (a)

We have system of linear equations

3x – 5y = 15,000

4x – 7y = 15,000

This may be written in matrix system as

$AX = B, \text{ where } A = \begin{bmatrix} 3 & -5\\ 4 & -7 \end{bmatrix}, X = \begin{bmatrix} x \\ y\end{bmatrix},B = \begin{bmatrix} 15,000 \\ 15,000\end{bmatrix}$

(iii) Answer (d)

We have

AX = B

Pre multiplying $A^{-1}$ both sides, we have

$A^{-1}(AX) = A^{-1}B$

$\Rightarrow (A^{-1}A)X = A^{-1}B$

$\Rightarrow IX = A^{-1}B$

$\Rightarrow X = A^{-1}B$

(iv) Answer (d)

We have

$A = \begin{bmatrix} 3 & -5\\ 4 & -7 \end{bmatrix}$

⇒ |A| = -21 + 20 = -1 ≠ 0

NOW,

$A_{11} = -7, A_{12} = -4$

$A_{21}= 5, A_{22} = 3$

$adj(A) = \begin{bmatrix} -7 & -4\\ 5 & 3 \end{bmatrix}^T$

$=\begin{bmatrix} -7 & 5\\ -4 & 3 \end{bmatrix}$

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

$= \frac{1}{-1}\begin{bmatrix} -7 & 5\\ -4 & 3 \end{bmatrix}$

$=\begin{bmatrix} 7 & -5\\ 4 & -3 \end{bmatrix}$

(v) Answer (b)

$X = A^{-1}B$

$\Rightarrow \begin{bmatrix} x \\ y\end{bmatrix}=\begin{bmatrix} 7 & -5\\ 4 & -3 \end{bmatrix}\begin{bmatrix} 15000 \\ 15000\end{bmatrix}$

$\Rightarrow \begin{bmatrix} x \\ y\end{bmatrix} = \begin{bmatrix} 105000-75000 \\ 60000-45000\end{bmatrix}$

$\Rightarrow \begin{bmatrix} x \\ y\end{bmatrix}= \begin{bmatrix} 30000 \\ 15000\end{bmatrix}$

⇒ x = 30000, y = 15000

Hence

Reshma’s monthly income = 3x = 3×30000 = ₹90000

Ritam’s monthly income = 4x = 4×30000 = ₹ 120000

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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 4:– Read the following and answer the question(Case study problem determinant 4)

On the occasion of children’s day. Class teacher of class XII shri singh, decided to donate some money to students of class XII.

If there were 8 studens less, every one would have got ₹10 more, however if there were 16 students more, everyone would have got ₹10 less.

Solution: For solution click here

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.

Solution: For solution click here