Q.Refer to Exercise 11. How many of circuits of Type A and of Type B, should be produced by the manufacturer so as to maximise his profit? Determine the maximum profit.

Exercise 11 (electronic circuits). A stock of $200$ resistors, $120$ transistors, $150$ capacitors makes Type A (uses $20,10,10$; profit ₹50) and Type B (uses $10,20,30$; profit ₹60). With $x$ = Type A, $y$ = Type B:

(The resource rows $20x+10y\le200$, $10x+20y\le120$, $10x+30y\le150$ reduce to these.)

Corners and $Z$:

• $(0,0):0$
• $(10,0):500$
• $(0,5):300$
• $(6,3):480$ — from $x+2y=12$, $x+3y=15$ …

The referenced problem (Exercise 11)

A manufacturer has $200$ resistors, $120$ transistors and $150$ capacitors. Type A needs $20$ resistors, $10$ transistors, $10$ capacitors (profit ₹50); Type B needs $10$ resistors, $20$ transistors, $30$ capacitors (profit ₹60). Let $x$ = number of Type A, $y$ = number of Type B.

which simplify to

Step 1 — Corner points

• $(0,0)$.
• $2x+y=20$ with $y=0\Rightarrow(10,0)$ (check $x+2y=10\le12$, $x+3y=10\le15$).
• $x+3y=15$ with $x=0\Rightarrow(0,5)$ (check $x+2y=10\le12$, $2x+y=5\le20$).
• $x+2y=12$ and $x+3y=15$: subtract $\Rightarrow y=3,\,x=6\Rightarrow(6,3)$ (check $2x+y=15\le20$).
• $2x+y=20$ and $x+2y=12$: $y=20-2x$, so $x+2(20-2x)=12\Rightarrow -3x=-28\Rightarrow x=\dfrac{28}{3},\,y=\dfrac{4}{3}$ (check $x+3y=\dfrac{40}{3}\le15$).

Step 2 — Evaluate $Z=50x+60y$

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: Treating every line intersection as a corner without checking feasibility.

Why it's wrong: two constraint lines can cross at a point that violates the third constraint, so it lies outside the feasible region and must not be used. Correct approach: substitute each intersection into all three simplified constraints ($2x+y\le20$, $x+2y\le12$, $x+3y\le15$) and keep only those that pass.

Mistake 2: Rounding the fractional optimum and calling that the LPP answer. …