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Model Paper 4 — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSModel Paper 44 Marks
Question
Read the following text carefully and answer the questions that follow:
Linear programming is a method for finding the optimal values (maximum or minimum) of quantities subject to the constraints when a relationship is expressed as linear equations or inequations.
i. At which points is the optimal value of the objective function attained? (1)
ii. What does the graph of the inequality $3 x+4 y<12$ look like? (1)
iii. Where does the maximum of the objective function Z = 2x + 5y occur in relation to the feasible region shown in the figure for the given LPP? (2)
Image
OR
What are the conditions on the positive values of p and q that ensure the maximum of the objective function Z = px + qy occurs at both the corner points (15, 15) and (0, 20) of the feasible region determined by the given system of linear constraints? (2? (2)
✓

Answer

When we solve an L.P.P. graphically, the optimal (or optimum) value of the objective function is attained at corner points of the feasible region.
ii. From the graph of 3x + 4y < 12 it is clear that it contains the origin but not the points on the line 3x + 4y = 12Image
iii. Maximum of objective function occurs at corner points
Corner PointsValue of z = 2x + 5y
(0,0)0
(7,0)14
(6,3)27
(4,5)$33 \leftarrow$ Maximum
(0,6)30
OR
Value of $Z=p x+q y$ at $(15,15)=15 p+15 q$ and that at $(0,20)=20 q$. According to given condition, we have $15 p +15 q =20 q \Rightarrow 15 p =5 q \Rightarrow q =3 p$

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