Case study Chapter 4 (Determinant)

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.

(i) If length of recangular plot of land be x m and breadth y m then the situation described in problem may be written in system of linear equation as

(a) x + y = 50, x + 2y = 550

(b) x – y = 100, 2x + y = 500

(d) x + y = 50, 2x + y = 550

(ii) System of linear equations

x – y = 50,

2x + y = 550

may be written in matrix equation as

(a) $AX = B, \text{ where } A = \begin{bmatrix} 1 & 50\\ 2 & 350 \end{bmatrix}, X = \begin{bmatrix} x \\ y\end{bmatrix},B = \begin{bmatrix} -1 \\ 1\end{bmatrix}$

(b) $X = AB, \text{ where } A = \begin{bmatrix} 1 & -1\\ 2 & 1 \end{bmatrix}, X = \begin{bmatrix} x \\ y\end{bmatrix},B = \begin{bmatrix} 50 \\ 550\end{bmatrix}$

(c) $AX = B, \text{ where } A = \begin{bmatrix} 1 & -1\\ 2 & 1 \end{bmatrix}, X = \begin{bmatrix} x \\ y\end{bmatrix},B = \begin{bmatrix} 50 \\ 550\end{bmatrix}$

(d) $BX = A, \text{ where } A = \begin{bmatrix} 1 & -1\\ 2 & 1 \end{bmatrix}, X = \begin{bmatrix} x \\ y\end{bmatrix},B = \begin{bmatrix} 50 \\ 550\end{bmatrix}$

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

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

(iv) If $A = \begin{bmatrix} 1 & -1\\ 2 & 1 \end{bmatrix}$ then $A^{-1}$ is

(a) $\frac{1}{3}\begin{bmatrix} 1 & 1\\ 2 & 1 \end{bmatrix}$

(b) $\frac{1}{3}\begin{bmatrix} 1 & 1\\ -2 & 1 \end{bmatrix}$

(d) $\frac{1}{5}\begin{bmatrix} 1 & 1\\ 2 & 1 \end{bmatrix}$

(v) The length and breadth of recangular plot donated by Reshma for school is

(a) 200 m× 100 m (b) 200 m× 150 m

Solution (i) Answer (c)

From question

(x – 50)(y + 50) = xy

⇒ 50 x – 50 y = 2500

⇒ x – y = 50 ——–(i)

Also,

(x – 10)(y – 20) = xy -5300

⇒ xy – 20 x – 10y + 200 = xy – 5300

⇒ -20x – 10y = -5300 – 200

⇒ 2x + y = 550 ——-(ii)

Hence, the represented system of linear equations is

x – y = 50,

2x + y = 550

(ii) Answer (c)

We have system of linea equations,

x – y = 50,

2x + y = 550

We can write matrix system as

$AX = B, \text{ where } A = \begin{bmatrix} 1 & -1\\ 2 & 1 \end{bmatrix}, X = \begin{bmatrix} x \\ y\end{bmatrix},B = \begin{bmatrix} 50 \\ 550\end{bmatrix}$

(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 (b)

We have

$A = \begin{bmatrix} 1 & -1\\ 2 & 1 \end{bmatrix}$

⇒ |A| = 1 + 2 = 3 ≠ 0

NOW,

$A_{11} = 1, A_{12} = -2$

$A_{21}= 1, A_{22} = 1$

$adj(A) = \begin{bmatrix} 1 & -2\\ 1 & 1 \end{bmatrix}^T$

$=\begin{bmatrix} 1 & 1\\ -2 & 1 \end{bmatrix}$

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

$= \frac{1}{3}\begin{bmatrix} 1 & 1\\ -2 & 1 \end{bmatrix}$

(v) Answer (b)

$X = A^{-1}B$

$\Rightarrow \begin{bmatrix} x \\ y\end{bmatrix}=\frac{1}{3}\begin{bmatrix} 1 & 1\\ -2 & 1 \end{bmatrix}\begin{bmatrix} 50 \\ 550\end{bmatrix}$

$\Rightarrow \begin{bmatrix} x \\ y\end{bmatrix} = \frac{1}{3}\begin{bmatrix} 600 \\ 450\end{bmatrix}$

$\Rightarrow \begin{bmatrix} x \\ y\end{bmatrix}= \begin{bmatrix} 200 \\ 150\end{bmatrix}$

⇒ x = 200, y = 150

Hence dimension is

200 m×150 m

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 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.

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

Case study application of derivative 4 — These days chinese and Indian troops are engaged in aggressive melee, face-offs skirmishes

Case study application of integral 3 — A student designs an open air Honeybee nest on the branch of a tree

Case study linear programming 3 chapter 12 class 12 — A bakery in an establishment produces and sells flour-based food baked in an oven such as bread

For more question

Case study problem determinant 2 chapter 4 class 12

Case study Chapter 4 (Determinant)

Some other Case study problem

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