CUET UG 2024 — Mathematics Algebra

The constraints for the linear programming problem are:

$$4x + y \geq 80$$4x+y≥80

$$x + 5y \geq 115$$x+5y≥115

$$3x + 2y \leq 150$$3x+2y≤150

$$x, y \geq 0$$x,y≥0

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To identify the region, find the intercepts for each boundary line:

For $4x + y = 80$4x+y=80:
When $x = 0, y = 80$x=0,y=80
When $y = 0, x = 20$y=0,x=20

For $x + 5y = 115$x+5y=115:
When $x = 0, y = 23$x=0,y=23
When $y = 0, x = 115$y=0,x=115

For $3x + 2y = 150$3x+2y=150:
When $x = 0, y = 75$x=0,y=75
When $y = 0, x = 50$y=0,x=50

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Test the origin $(0,0)$(0,0) against the inequalities to determine the direction of the feasible area:

$4(0) + (0) \geq 80 \implies 0 \geq 80$4(0)+(0)≥80⟹0≥80 (False; region is away from the origin)

$0 + 5(0) \geq 115 \implies 0 \geq 115$0+5(0)≥115⟹0≥115 (False; region is away from the origin)

$3(0) + 2(0) \leq 150 \implies 0 \leq 150$3(0)+2(0)≤150⟹0≤150 (True; region is toward the origin)

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The intersection of these half-planes, constrained by $x, y \geq 0$x,y≥0 in the first quadrant, forms the bounded polygonal area labeled as Region C.