Case study (4 Marks)

Question 14 Marks

A fruit grower can use two types of fertilizer in his garden, brand P and brand Q. The amounts (in kg) of nitrogen, Phosphoric acid, potash, and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs at least 240 kg of phosphoric acid, at least 270 kg of potash and at most 310 kg of chlorine.

kg per bag
	Brand P	Brand Q
Nitrogen	3	3.5
Phosphoric acid	1	2
Potash	1	1.5
Chlorine	1.5	2

Q. 1. The objective function to minimise the amount of nitrogen added to garden ?
(A) Maximise Z = 3x + 4y
(B) Maximise Z = 3x + 3.5y
(C) Maximise Z = 4x + 3.5y
(D) Maximise Z = 3x + 4y
Q. 2. If the grower wants to minimise the amount of nitrogen added to the garden, how many bags of brand P should be used?
(A) 40 (B) 50 (C) 100 (D) 60
Q. 3. If the grower wants to minimise the amount of nitrogen added to the garden, how many bag of brand Q should be used ?
(A) 40 (B) 50(C) 100 (D) 60
Q. 4. What is the minimum amount of nitrogen added
(A) 595 kg (B) 550 kg (C) 400 kg (D) 470 kg

Answer

Ans. 1. (B) Maximise Z = 3x + 3.5y
Ans. 2. (A) 40
Ans. 3. (C) 100
Ans. 4. (D) 470 kg

View full question & answer→

Question 24 Marks

A dealer in rural area wishes to purchase a number of sewing machines. He has only ₹ 5,760 to invest and has space for at most 20 items for storage. An electronic sewing machine cost him ₹360 and a manually operated sewing machine ₹ 240. He can sell an electronic sewing machine at a profit of ₹ 22 and a manually operated machine at a profit of ₹ 18. Assume that the electronic sewing machines he can sell is x and that of manually operated machines is y.

Q. 1. The objective function is
(A) Maximise Z = 360x + 240y
(B) Maximise Z = 22x + 18y
(C) Minimise Z = 360x + 240y
(D) Minimise Z = 22x + 18y
Q. 2. The maximum value of x + y is
(A) 5760 (B) 18 (C) 22 (D) 20
Q. 3. Which of the following is not a constraint?
(A) $x+y \geq 20$
(B) $360 x+240 y \leq 5,760$
(C) $x \geq 0$
(D) $y \geq 0$
Q. 4. The profit is maximum when (x, y) = _____.
(A) (5,15) (B) (8, 12) (C) (12,8) (D) (15,5)

Answer

Ans. 1. (B) Maximise Z = 22x + 18y
Ans. 2. (D) 20
Ans. 3. (A) $x+y \geq 20$
Ans. 4. (B) (8, 12)

View full question & answer→

Question 34 Marks

The feasible region for an LPP is shown in the following figure. The CB is parallel to OA.

Q. 1. The equation of line OA is
(A) x - 2y = 0 (B) y - 2x = 0 (C) x + 2y = 0 (D) 2x + y = 0
Q. 2. The equation of line BC is
(A) y - 2x = 2 (B) x - y = 2 (C) 2y - x = 2 (D) x + 2y = 2
Q. 3. The co-ordinates of the point B are
(A) (5,7) (B) (7,5) (C) (5, 12) (D) (12, 5)
Q. 4. The constraints for the L.P.P. are
(A) $y \geq 2 x, y-2 x \leq 2, x \leq 5, x \geq 0, y \geq 0$
(B) $y \leq 2 x, y-2 x \leq 2, x \leq 5, x \geq 0, y \geq 0$
(C) $y \geq 2 x, y-2 x \geq 2, x \leq 5, x \geq 0, y \geq 0$
(D) $y \leq 2 x, y-2 x \geq 2, x \leq 5, x \geq 0, y \geq 0$

Answer

Ans. 1. (B) y - 2x = 0
Explanation: The point A(5, 10) lies on the equation y - 2x = 0 therefore the equation of line OA is y - 2x = 0.
Ans. 2. (A) y - 2x = 2
Explanation: Point on line BC i.e., C(0,2) lies on the equation y - 2x = 2 therefore equation of line BC is y - 2x = 2.
Ans. 3. (C) (5, 12)
Explanation: Point B is the intersection point of line BC and BD.
So, substituting x = 5 in y - 2x = 2
we get y = 12
Thus, required coordinates are (5, 12).
Ans. 4. (A) $y \geq 2 x, y-2 x \leq 2, x \leq 5, x \geq 0, y \geq 0$
Explanation: The required constrains for L.P.P. are
$\begin{array}{l}y \geq 2 x \\ y-2 x \leq 2 \\ x \leq 5 \\ x \geq 0, y \geq 0\end{array}$

View full question & answer→

Question 44 Marks

A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding/cutting machine and a sprayer. It takes 2 hours on grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp. It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer to manufacture a shade. On any day, the sprayer is available for at the most 20 hours and the grinding/cutting machine for at the most 12 hours. The profit from the sale of a lamp is ₹ 25 and that from a shade is ₹15.
If x is the number of lamps and y is the number of shades manufactured. Assuming that the manufacturer can sell all the lamps and shades that he produces.
Q. 1. In order to maximize the profit, which of the flowing objective function is correct:
(A) Max. z = 25x + 15y
(B) Min. Z = 25x + 15y
(C) Max. z = 25x - 15y
(D) Max. z = 15x + 25y
Q. 2. Which of the following constraint are related to the given LPP?
(A) $2 x+y \geq 12 ; 3 x+2 y \leq 20$
(B) $2 x+y \leq 12 ; 3 x+2 y \leq 20$
(C) $2 x-y \geq 12 ; 3 x+2 y \leq 20$
(D) $2 x+y \leq 12 ; 3 x-2 y \leq 20$
Q. 3. The non-negative constraints associative to the given LPP are:
(A) $x \geq 0, y \geq 1$
(B) $x \geq 1, y \geq 0$
(C) $x \geq 0, y \geq 0$
(D) $x \geq 1, y \geq 1$
Q. 4. Which of the following is not the vertices of feasible region:
(A) (6,4) (B) (0,0) (C) (6,0) (D) (4, 4)

Answer

Ans. 1. (A) Max. z = 25x + 15y
Explanation: Since profit from the sale of a lamp = ₹25
And profit from the sale of a shade = ₹15
The associative objective function is
Max. Z = 25x + 15y
Ans. 2. (B) $2 x+y \leq 12 ; 3 x+2 y \leq 20$
Explanation: