UPSC 2014 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

Solve graphically: Maximize $Z=6x_1+5x_2$ subject to

2x_1+x_2\le 16,\;x_1+x_2\le 11,\;x_1+2x_2\ge 6,\;5x_1+6x_2\le 90,\;x_1,x_2\ge 0.

Technique

Standard 2-variable graphical LPP; enumerate vertices, check feasibility, evaluate objective.

Solution

Strategy. Identify feasible-region vertices via intersection of constraint boundaries; evaluate $Z$ at each.

Step 1 — Enumerate candidate vertices

Constraint lines:

$L_1: 2x_1+x_2=16$
$L_2: x_1+x_2=11$
$L_3: x_1+2x_2=6$
$L_4: 5x_1+6x_2=90$
$x_1$ -axis, $x_2$ -axis.

Check pairwise intersections for feasibility (all 6 constraints must hold):

Pair	Intersection	Feasible?
$L_1\cap L_2$	$(5,6)$	✓ (all constraints check)
$L_1\cap L_4$	$(6/7,100/7)$	✗ ( $L_2$ : $15.1>11$ )
$L_2\cap L_4$	$(-24,35)$	✗ ( $x_1<0$ )
$L_1\cap\{x_2=0\}$	$(8,0)$	✓
$L_2\cap\{x_1=0\}$	$(0,11)$	✓
$L_3\cap\{x_2=0\}$	$(6,0)$	✓
$L_3\cap\{x_1=0\}$	$(0,3)$	✓

Feasible vertices: $(6,0),\,(0,3),\,(0,11),\,(5,6),\,(8,0)$ .

Step 2 — Boundary trace

Walking the feasible region counter-clockwise:

(6,0)\to(0,3)\to(0,11)\to(5,6)\to(8,0)\to(6,0).

The 5-gon has edges along $L_3$ (from $(6,0)$ to $(0,3)$ ), $x_1=0$ (from $(0,3)$ to $(0,11)$ ), $L_2$ (from $(0,11)$ to $(5,6)$ ), $L_1$ (from $(5,6)$ to $(8,0)$ ), $x_2=0$ (from $(8,0)$ to $(6,0)$ ).

Step 3 — Evaluate $Z$ at each vertex

Vertex	$Z=6x_1+5x_2$
$(6,0)$	$36$
$(0,3)$	$15$
$(0,11)$	$55$
$(5,6)$	$\mathbf{60}$
$(8,0)$	$48$

Maximum: $Z=60$ at $(x_1,x_2)=(5,6)$ .

Answer

\boxed{\;Z_{\max}=60\;\text{at}\;(x_1,x_2)=(5,6).\;}

UPSC 2014 Maths Optional Paper 2 Q1e — Step-by-Step Solution

Question

Technique

Solution

Step 1 — Enumerate candidate vertices

Step 2 — Boundary trace

Step 3 — Evaluate ZZZ at each vertex

Answer

This solution is part of the Maths Coverage Map — 13 years, mapped. Get the take-away PDF free.

Step 3 — Evaluate $Z$ at each vertex