CBSE 2025, Set 2

Question

The feasible region along with corner points for a linear programming problem are shown in the graph. Write all the constraints for the given linear programing problem.

![Diagram LPP Feasible Region](https://www.edvaya.com/images/images/cbse/2025/set2/q28-lpp-feasible-region.webp)

Edvaya · Accepted Answer

The correct answer is Subjective. Let's determine the equations of the boundary lines of the shaded feasible region $OABCD$, using the given corner points $O(0,0)$, $A(0,30)$, $B(15,25)$, $C(30,10)$, and $D(35,0)$. **1. Equation of Line AB (eq2 in graph):** Passes through $A(0, 30)$ and $B(15, 25)$. Slope $m = \frac{25 - 30}{15 - 0} = \frac{-5}{15} = -\frac{1}{3}$. Using point-slope form: $y - 30 = -\frac{1}{3}(x - 0) \implies 3y - 90 = -x \implies x + 3y = 90$. The origin $(0,0)$ lies in the feasible region below this line ($0 \le 90$). Constraint 1: **$x + 3y \le 90$** **2. Equation of Line BC (eq3 in graph):** Passes through $B(15, 25)$ and $C(30, 10)$. Slope $m = \frac{10 - 25}{30 - 15} = \frac{-15}{15} = -1$. Using point-slope form: $y - 10 = -1(x - 30) \implies y - 10 = -x + 30 \implies x + y = 40$. The origin $(0,0)$ lies in the feasible region below this line ($0 \le 40$). Constraint 2: **$x + y \le 40$** **3. Equation of Line CD (eq1 in graph):** Passes through $C(30, 10)$ and $D(35, 0)$. Slope $m = \frac{0 - 10}{35 - 30} = \frac{-10}{5} = -2$. Using point-slope form: $y - 0 = -2(x - 35) \implies y = -2x + 70 \implies 2x + y = 70$. The origin $(0,0)$ lies in the feasible region below this line ($0 \le 70$). Constraint 3: **$2x + y \le 70$** **4. Non-negativity Constraints:** The feasible region is bounded by the non-negative x-axis and y-axis. Constraints 4 & 5: **$x \ge 0, y \ge 0$** **Final set of all constraints:** $x + 3y \le 90$ $x + y \le 40$ $2x + y \le 70$ $x \ge 0, y \ge 0$

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