Q.Refer to Exercise 12. What will be the minimum cost?

Exercise 12 (vans). $1200$ packages go by large vans (capacity $200$, cost ₹400) and small vans (capacity $80$, cost ₹200); at most ₹3000 may be spent and large vans $\le$ small vans. With $x$ = large, $y$ = small:

which reduce to $5x+2y\ge30$, $2x+y\le15$, $x\le y$.

Corners and cost:

• $(0,15):3000$
• $(5,5):3000$ …

The referenced problem (Exercise 12)

$1200$ packages must be transported. A large van carries $200$ (cost ₹400), a small van carries $80$ (cost ₹200). At most ₹3000 may be spent, and the number of large vans cannot exceed the number of small vans. Let $x$ = large vans, $y$ = small vans.

which simplify to

Step 1 — Corner points

• $5x+2y=30$ and $2x+y=15$: from the second $y=15-2x$, so $5x+2(15-2x)=30\Rightarrow x=0,\,y=15\Rightarrow(0,15)$.
• $2x+y=15$ and $x=y$: $3x=15\Rightarrow(5,5)$.
• $5x+2y=30$ and $x=y$: $7x=30\Rightarrow x=y=\dfrac{30}{7}\Rightarrow\left(\dfrac{30}{7},\dfrac{30}{7}\right)$. …

Method: Graphical (Corner-Point) Method for a Two-Variable LPP

Once a problem is written as "optimise $Z = ax + by$ subject to linear inequalities in $x$ and $y$," this is the standard CBSE technique for finding the optimum.

Steps

Step 1: Draw every constraint as a line.

Replace each inequality by an equation and plot the line from its axis intercepts. Include $x = 0$ and $y = 0$.

Step 2: Shade the correct half-plane.

Test the origin $(0,0)$ in each inequality: if it is satisfied, keep the origin's side; if not, take the other side. Non-negativity confines you to the first quadrant.

Step 3: Identify the feasible region.

It is the single region where all the shaded half-planes overlap.

Step 4: Find the corner points.

Each vertex is the intersection of two boundary lines, found by solving those two equations together. Keep a candidate only if it satisfies every other constraint -- an intersection that breaks a third inequality is not a corner of the region.

Step 5: Evaluate $Z$ at each corner. …

Common Mistakes

Mistake 1: Omitting the "large vans cannot exceed small vans" constraint.

Why it's wrong: dropping $x \le y$ enlarges the feasible region and can give a spuriously lower cost at an inadmissible point. Correct approach: include $x \le y$ alongside the capacity and budget constraints.

Mistake 2: Minimising the budget line instead of the cost. …