UPSC 2023 Maths Optional Paper 2 Q1e — Step-by-Step Solution

10 marks · Section A

Graphical method · Linear Programming · asked 5× in 13 yrs · Read the full method →

Question

A person requires 24, 24 and 20 units of chemicals $A,B$ and $C$ respectively for his garden. Product $P$ contains 2, 4 and 1 units of chemicals $A,B$ and $C$ respectively per jar and product $Q$ contains 2, 1 and 5 units of chemicals $A,B$ and $C$ respectively per jar. If a jar of $P$ costs ₹30 and a jar of $Q$ costs ₹50, then how many jars of each should be purchased in order to minimize the cost and meet the requirements?

Technique

Two-variable LPP, graphical method — enumerate corner-point feasible solutions.

Solution

Formulation. Let $x_1=$ jars of $P$ , $x_2=$ jars of $Q$ . Minimize

Z=30x_1+50x_2

subject to

\begin{aligned}\text{(A)}&\quad 2x_1+2x_2\ge 24\;\;\Leftrightarrow\;\; x_1+x_2\ge 12,\\ \text{(B)}&\quad 4x_1+x_2\ge 24,\\ \text{(C)}&\quad x_1+5x_2\ge 20,\\ &\quad x_1,x_2\ge 0.\end{aligned}

This is a two-variable LPP — solve graphically by identifying vertices of the feasible region (an unbounded polygon in the first quadrant) and evaluating $Z$ at each.

Step 1 — Find candidate vertices (pairwise intersections of boundary lines).

Lines intersected	Solution	All other constraints?
A ∩ B: $x_1+x_2=12,\;4x_1+x_2=24$	$(4,8)$	C: $4+40=44\ge 20$ ✓
A ∩ C: $x_1+x_2=12,\;x_1+5x_2=20$	$(10,2)$	B: $40+2=42\ge 24$ ✓
B ∩ C: $4x_1+x_2=24,\;x_1+5x_2=20$	$(100/19,56/19)$	A: $156/19\approx 8.21<12$ ✗
A ∩ axes: $(12,0),(0,12)$	—	C and B fail respectively ✗
B ∩ axes: $(6,0),(0,24)$	—	$(6,0)$ : A fails ( $6<12$ ); $(0,24)$ : A ✓, C: $120\ge 20$ ✓
C ∩ axes: $(20,0),(0,4)$	—	$(20,0)$ : A ✓, B ✓; $(0,4)$ : A fails

Feasible vertices: $(4,8),\;(10,2),\;(0,24),\;(20,0)$ .

Step 2 — Evaluate $Z$ at each feasible vertex.

Vertex	$Z=30x_1+50x_2$
$(4,8)$	$120+400=520$
$(10,2)$	$300+100=\mathbf{400}$
$(0,24)$	$0+1200=1200$
$(20,0)$	$600+0=600$

Step 3 — Optimal solution.

Minimum is at $(x_1,x_2)=(10,2)$ with $Z=400$ .

Answer

\boxed{\;\text{Buy 10 jars of }P\text{ and 2 jars of }Q;\text{ minimum cost }=₹400.\;}