Class 12 Maths · Chapter 12
Unit: Linear Programming
Tentative Study Share: 5.0 Marks
Chapter 12: Linear Programming — NCERT Solutions, Notes, Formula Sheet & PYQs
Complete NCERT exercise solutions, revision notes, formula cards, printable mind map, and solved board questions for Class 12 Linear Programming.
NCERT Exercises in Chapter 12: Linear Programming
📘 NCERT Concept Notes — Linear Programming
01 · Translate the story before drawing
Choose decision variables with clear units. The objective is the quantity to maximise or minimise; constraints express limits or requirements. Nonnegativity is itself a constraint. The chapter uses continuous graphical optimisation.
02 · Vertices reduce a bounded search
For a nonempty bounded feasible polygon, evaluate the linear objective at each vertex. The chapter states the corner-point theorems without proving them, so these cards teach the method rather than inventing a textbook derivation.
03 · Unbounded does not mean no optimum
An unbounded feasible region can still have a finite minimum or maximum. After finding the best vertex candidate, check whether the feasible region contains points beyond that objective level. Empty feasible regions have no solution.
⚡ Key Formulas & Results — Linear Programming
Objective and constraints
What makes a two-variable linear programming model?
\(Z=ax+by,\qquad\text{linear inequalities in }x,y\)
Feasible region
Which part of the graph is feasible?
\(R=\bigcap_i\{(x,y):\text{constraint }i\text{ holds}\}\)
Corner-point method
Where should you test a linear objective on a bounded feasible polygon?
\(Z_i=ax_i+by_i\quad\text{at every feasible vertex}\)
Work through the textbook corner-point example
How is the maximum of 4x+y found under the textbook constraints?
\(Z_{max}=120\text{ at }(30,0)\)
Check beyond a vertex candidate
What extra test is needed in an unbounded region?
\(R\cap\{ax+by>M\}=\varnothing\quad\text{for a maximum candidate }M\)
No common feasible point
How do you recognise contradictory constraints?
\(R=\varnothing\ \Rightarrow\ \text{no optimum}\)
Why an entire segment can be optimal
What happens when two feasible optimal vertices have the same objective?
\(Z((1-t)P+tQ)=(1-t)Z(P)+tZ(Q)\)
🎯 CBSE 2026 Board PYQs from Linear Programming
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q17
The feasible region of a linear programming problem with objective function Z = 5x + 7y is shown below…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q18
The degree of an objective function of a linear programming problem is
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q30
Solve the following linear programming problem graphically: Minimize Z = 13x − 15y subject to…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q10
The feasible region of a linear programming problem with objective function Z = 5x + 7y is shown below…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q11
The degree of an objective function of a linear programming problem is
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q28
Solve the following linear programming problem graphically: Minimize Z = 13x − 15y subject to…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q3
The feasible region of a linear programming problem with objective function Z = 5x + 7y is shown below…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q4
The degree of an objective function of a linear programming problem is
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q31
Solve the following linear programming problem graphically: Minimize Z = 13x − 15y subject to…
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q16
What is the linear function that a linear programming problem maximizes or minimizes?
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q17
Which pair gives the nontrivial constraints for the shaded feasible region?
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q30
Graphically maximize Z=10500x+9000y for x+y≤50, 2x+y≤80 and x,y≥0.
CBSE Class 12 Mathematics 2026 Main Paper 65/2/2 · Q9
What is the linear function that a linear programming problem maximizes or minimizes?
CBSE Class 12 Mathematics 2026 Main Paper 65/2/2 · Q10
Which pair gives the nontrivial constraints for the shaded feasible region?
CBSE Class 12 Mathematics 2026 Main Paper 65/2/2 · Q28
Graphically maximize Z=8000x+12000y subject to 3x+4y≤60, x+3y≤30 and x,y≥0.
CBSE Class 12 Mathematics 2026 Main Paper 65/2/3 · Q2
What is the linear function that a linear programming problem maximizes or minimizes?
CBSE Class 12 Mathematics 2026 Main Paper 65/2/3 · Q3
Which pair gives the nontrivial constraints for the shaded feasible region?
CBSE Class 12 Mathematics 2026 Main Paper 65/2/3 · Q31
Graphically maximize Z=4500x+5000y subject to x+y≤250, 25x+40y≤7000 and x,y≥0.
📝 Chapterwise Practice Questions — Linear Programming (75 Questions)
Class 12 Linear Programming · Mcq Single Correct
How many feasible solutions satisfy 3x+y≤12, x+2y≤10, x,y≥0?
Class 12 Linear Programming · Mcq Single Correct
The shaded first-quadrant region lies above both x+2y=4 and x+y=3. Which constraints describe it?
Class 12 Linear Programming · Mcq Single Correct
Z=ax+by has maximum 42 at (4,6) and minimum 19 at (3,2). Find a,b.
Class 12 Linear Programming · Mcq Single Correct
A bounded feasible polygon has vertices (2,72),(15,20),(40,15). For Z=18x+9y, select the correct…
Class 12 Linear Programming · Mcq Single Correct
How many vertices does x−y≥0, 2y≤x+2, x,y≥0 have?
Class 12 Linear Programming · Mcq Single Correct
A bounded feasible polygon has vertices (0,3),(3,2),(0,5). Find the minimum of Z=11x+7y.
Class 12 Linear Programming · Mcq Single Correct
How many solutions satisfy x+2y≤3,3x+4y≥12,x≥0,y≥1?
Class 12 Linear Programming · Mcq Single Correct
Find max Z=3x+4y subject to x+y≤1,x,y≥0.
Class 12 Linear Programming · Mcq Single Correct
The shaded first-quadrant region contains the origin and is bounded by lines through (0,104),(52,0) and…
Class 12 Linear Programming · Mcq Single Correct
Maximize Z=3x+4y subject to x−y≥−1,x≤3,x,y≥0. The maximum is:
Class 12 Linear Programming · Mcq Single Correct
A bounded feasible polygon has vertices (0,0),(5,0),(6,5),(6,8),(4,10),(0,8). Where is Z=3x−4y minimum?
Class 12 Linear Programming · Mcq Single Correct
A linear objective attains its maximum at two distinct feasible vertices. How many optimal points are…
Class 12 Linear Programming · Mcq Single Correct
Vertices are (0,3),(1,1),(3,0), and Z=px+qy with p,q>0. For minima at (3,0) and (1,1), which condition…
Class 12 Linear Programming · Mcq Single Correct
For a nonempty bounded feasible polygon, a linear objective attains an optimum at:
Class 12 Linear Programming · Mcq Single Correct
For x≥3,x≤9,y≥0,x−y≥0,x+y≤14, which describes the feasible region?
Class 12 Linear Programming · Mcq Single Correct
Vertices are P(0,5),Q(1,5),R(4,2),S(12,0). Where is Z=2x+5y minimum?
Class 12 Linear Programming · Mcq Single Correct
For Z=2x−y+5 on 3x+4y≤60,x+3y≤30,x,y≥0, which statement is true?
Class 12 Linear Programming · Mcq Single Correct
For a,b>0, vertices P(0,5),Q(3,5),R(5,0),S(4,1), suppose Z=ax+2by has equal values at Q and S. Which…
Class 12 Linear Programming · Mcq Single Correct
For a bounded polygon O(0,0),P(0,40),Q(30,20),R(40,0), and Z=4x+3y, which statement is true?
Class 12 Linear Programming · Mcq Single Correct
Find max Z=4x+3y subject to x+y≤10,x,y≥0.
Class 12 Linear Programming · Assertion Reason
Assertion (A): The maximum of 5x+3y on 5x+2y≤10,x,y≥0 is 15. Reason (R): A feasible region may be…
Class 12 Linear Programming · Assertion Reason
Assertion (A): The maximum of x+3y on 2x+y≤20,x+2y≤20,x,y≥0 is 30. Reason (R): The unknown quantities…
Class 12 Linear Programming · Assertion Reason
Assertion (A): The polygon with vertices (0,0),(20,0),(20/3,20/3),(0,16) represents…
Class 12 Linear Programming · Conceptual Answer
The boundary x+y=1 divides the plane into A (the origin side) and B (the other side). Which side solves…