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Mathematics · 2024 · 3 marks
Linear ProgrammingLinear Programming Problem and its Mathematical Formulation3 marksApplyshort answer
CBSE 2024 · Region 4 · Set 3 · Q30
The corner points of the feasible region determined by the system of linear constraints are as shown in the following figure: (i) If Z =3 x-4 y be…
The corner points of the feasible region determined by the system of linear constraints are as shown in the following figure:
(i)
If $\displaystyle \mathrm{Z}=3 x-4 y$ be the objective function, then find the maximum value of Z .
(ii)
If $\displaystyle \mathrm{Z}=\mathrm{p} x+\mathrm{qy}$ where $\displaystyle \mathrm{p}, \mathrm{q}>0$ be the objective function. Find the condition on $\displaystyle \mathrm{p}$ and $\displaystyle \mathrm{q}$ so that maximum value of $\displaystyle \mathrm{Z}$ occurs at $\displaystyle B(4,10)$ and $\displaystyle C(6,8)$.
Marking-scheme solution
| corner points | $\displaystyle \mathrm{Z}=3 x-4 y$ | |
| \cline { $\displaystyle 1$ - $\displaystyle 1$ }$\displaystyle A(0,8)$ | -$\displaystyle 32$ | |
| $\displaystyle B(4,10)$ | -$\displaystyle 28$ | |
| $\displaystyle C(6,8)$ | -$\displaystyle 14$ | |
| $\displaystyle D(6,5)$ | -$\displaystyle 2$ | |
| $\displaystyle E(4,0)$ | $\displaystyle 12$ (Maximum) | |
| $\displaystyle O(0,0)$ | $\displaystyle 0$ |
Maximum value of Z is $\displaystyle 12$ at E when $\displaystyle x=4, y=0$
(ii)
$\displaystyle \mathrm{Z}_{B}=\mathrm{Z}_{C} \Rightarrow 4 \mathrm{p}+10 \mathrm{q}=6 \mathrm{p}+8 \mathrm{q}$ Thus, $\displaystyle \mathrm{p}=\mathrm{q}$
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