Class 12 · Chapter 12 · Linear Programming unit (5 of 80 marks)

Linear Programming Class 12: notes and important questions

Q: What is in CBSE Class 12 Maths Chapter 12, Linear Programming?

The topics are: Graphical solution: bounded region, Special cases: multiple optima and infeasibility, Feasible region and corner points, Graphical solution: unbounded region, Terminology and formulation. This page has revision notes for each, and 49 board-style practice questions with step-marked solutions.

Q: How many marks is Linear Programming worth?

It is in the Linear Programming unit, which carries 5 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Q: Where do students lose marks in Linear Programming?

Missing or extra corner points. Fix: list every pair of boundary lines, solve, and keep only intersections that satisfy all constraints. Stating a maximum or minimum on an unbounded region without justification. Fix: write the open half-plane test (e.g. "6x + 4y < 24 has no common point with the region") in words.

Q: Are the Linear Programming important questions free?

Yes. The 10 important questions on this page are free to try, from 1-mark MCQs to a case study. 5 worked answers are open on the page; the others open with a free account (no card).

Q: Are these NCERT questions?

No. Every practice question here is original, written in the board's question styles with step marks. For the textbook itself, use the NCERT link on this page.

Revision notes, 10 free important questions with step-marked answers, and a four-step route from the basics to 95+ through 49 board-style questions.

Practise this chapter Join freeFree account, no card: +5 practice questions on every chapter, with instant feedback.

49 questions
14 multiple choice, 3 assertion–reason, 10 very short answer, 14 short answer, 5 long answer, 3 case study
About 4 hours to master

Unit V Linear Programming: 5 marks of the 80-mark paper — CBSE Curriculum 2025-26 Mathematics (041) (official PDF).

NCERT textbook: Chapter 12 on ncert.nic.in (PDF)

Revision notes

Linear Programming — revision notes

1. Terminology and formulation

Decision variables $x, y \geq 0$ (non-negativity); objective function $Z = a x + b y$ to be maximised or minimised; constraints are linear inequalities.
Formulating: (1) define $x, y$ with units; (2) write $Z$ ; (3) one inequality per resource/requirement (“at most” $\leq$ , “at least” $\geq$ ); (4) add $x, y \geq 0$ .

2. Feasible region and corner points

The feasible region is the common region of all constraints; each point is a feasible solution. Boundary points count.
Test a half-plane with a point not on the line (usually the origin).
Corner points: intersections of boundary lines that satisfy every constraint — solve the pairs of equations algebraically.
Empty region: the problem is infeasible (no solution).

3. Corner-point method

Bounded region: $Z$ has a maximum and a minimum, each at a corner point. Evaluate $Z$ at all corners.
Unbounded region: let $M$ = largest and $m$ = smallest corner value. $M$ is the maximum iff the open half-plane $a x + b y > M$ has no point in common with the region; $m$ is the minimum iff $a x + b y < m$ has no common point. Otherwise that optimum does not exist.
If two adjacent corners give the same optimal value, every point of the edge joining them is optimal (infinitely many solutions).

Worked example 1

Maximise $Z = 2 x + 5 y$ with $x + y \leq 5, y \leq 3, x, y \geq 0$ : corners $(0, 0), (5, 0), (2, 3), (0, 3)$ , $Z = 0, 10, 19, 15$ ; maximum $19$ at $(2, 3)$ .

Worked example 2

Minimise $Z = 3 x + y$ with $x + y \geq 4, x, y \geq 0$ (unbounded): corners $(4, 0), (0, 4)$ , $Z = 12, 4$ . The half-plane $3 x + y < 4$ does not meet the region, so the minimum is $4$ at $(0, 4)$ ; there is no maximum.

Worked example 3

If $Z = p x + q y$ has the same maximum at $(1, 4)$ and $(3, 2)$ : $p + 4 q = 3 p + 2 q \Rightarrow p = q$ .

Common errors

Missing a corner (e.g. where a constraint meets an axis) or including an intersection point that violates another constraint.
Declaring a maximum/minimum on an unbounded region without the half-plane check.
Shading the wrong side of a line — always test a point.
Forgetting the non-negativity constraints, or units and the final interpretation in word problems.

Board-exam tips

The LPP is usually a 5-mark question (or a 3-mark one): a neat, labelled graph with the feasible region shaded, a table of corner points with $Z$ values, and a concluding sentence each earn marks.
Only two variables and up to three non-trivial constraints are examined; types of LPP (diet/transportation classification) are not in the syllabus.

Topics in this chapter: Graphical solution: bounded region · Special cases: multiple optima and infeasibility · Feasible region and corner points · Graphical solution: unbounded region · Terminology and formulation.

Formulas for this chapter: Chapter 12 on the Class 12 formula sheet (free PDF and A3 poster).

Linear Programming Class 12: important questions

10 original questions in the board paper's question types, from 1-mark MCQs to a case study, each with a step-marked worked answer. Try each one first. 5 worked answers open here; the other 5 open with a free account (no card).

Download these questions (PDF) 10 questions, 5 worked answers · 7 pages · 756 KB

Section A style · Multiple choice and assertion–reason (1 mark each)

Q1·1 mark·Multiple choiceGraphical solution

A bounded feasible region has corner points $(0, 0)$ , $(6, 0)$ , $(4, 3)$ and $(0, 5)$ . What is the greatest value of $Z = 2 x + 3 y$ on it?

Shaded bounded feasible region with corner points (0, 0), (6, 0), (4, 3) and (0, 5). — Feasible region with corners (0, 0), (6, 0), (4, 3), (0, 5)

Show worked answer

$Z$ at the corners: $0$ , $12$ , $17$ , $15$ ; the greatest is $17$ at $(4, 3)$ A1
Correct option: (b) $17$

Q2·1 mark·Multiple choiceFeasible region and corner points

Which point lies in the region given by $x + y \leq 4$ , $2 x + y \leq 6$ , $x \geq 0$ , $y \geq 0$ ?

Q3·1 mark·Assertion–reasonFeasible region and corner points

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): The point $(2, 3)$ lies in the feasible region of the constraints $x + y \leq 4$ , $x \geq 0$ , $y \geq 0$ .

Reason (R): A point is in the feasible region only if it satisfies every constraint.

Section B style · Short answer (2 marks each)

Q4·2 marks·Short answerGraphical solution

The corner points of a bounded feasible region are $O (0, 0)$ , $A (0, 4)$ , $B (2, 3)$ and $C (4, 0)$ . Find the maximum and minimum values of $Z = 4 x + 6 y$ .

Shaded bounded feasible region OABC with corner points O(0, 0), A(0, 4), B(2, 3) and C(4, 0). — Feasible region OABC

Show worked answer

Answer: Max

26

, min

0

$Z (O) = 0$ , $Z (A) = 24$ , $Z (B) = 8 + 18 = 26$ , $Z (C) = 16$ M1
Maximum $26$ at $B (2, 3)$ ; minimum $0$ at $O (0, 0)$ A1

Q5·2 marks·Short answerGraphical solution

The feasible region for $x + y \geq 2$ , $x + 3 y \geq 3$ , $x \geq 0$ , $y \geq 0$ is unbounded, with corner points $(0, 2)$ , $(\frac{3}{2}, \frac{1}{2})$ and $(3, 0)$ . Find the minimum of $Z = 3 x + 5 y$ .

Unbounded shaded feasible region above the lines x + y = 2 and x + 3y = 3 in the first quadrant, with corner points (0, 2), (3/2, 1/2) and (3, 0) marked. — Unbounded region for x + y ≥ 2, x + 3y ≥ 3

Section C style · Short answer (3 marks each)

Q6·3 marks·Short answerGraphical solution

Use the corner-point method to find the maximum of $Z = 5 x + 3 y$ , given $x + 2 y \leq 10$ , $3 x + y \leq 15$ and $x, y \geq 0$ .

Graph of the lines x + 2y = 10 and 3x + y = 15 in the first quadrant, with the feasible region below both lines shaded. — Feasible region for x + 2y ≤ 10, 3x + y ≤ 15

Show worked answer

Answer:

29

(4, 3)

Corner points: $(0, 0)$ , $(5, 0)$ , $(0, 5)$ and the intersection of $x + 2 y = 10$ and $3 x + y = 15$ , which is $(4, 3)$ M1
$Z$ : $0$ , $25$ , $15$ , $20 + 9 = 29$ A1
Maximum $Z = 29$ at $x = 4$ , $y = 3$ A1

Q7·3 marks·Short answerGraphical solution

Minimise $Z = 2 x + 3 y$ subject to $x + y \geq 5$ , $x + 3 y \geq 9$ , $x \geq 0$ , $y \geq 0$ .

Graph of the lines x + y = 5 and x + 3y = 9 in the first quadrant, with the unbounded feasible region above both lines shaded. — Unbounded region for x + y ≥ 5, x + 3y ≥ 9

Section D style · Long answer (5 marks each)

Q8·5 marks·Long answerFormulating LPPs

A workshop makes two kinds of lamp, A and B. Each lamp A needs $2$ hours of machine time and $1$ hour of finishing; each lamp B needs $1$ hour of machine time and $2$ hours of finishing. There are at most $24$ machine hours and $24$ finishing hours a week. The profit is ₹ $40$ on each lamp A and ₹ $50$ on each lamp B. How many of each should be made for the greatest profit?

Graph of the lines 2x + y = 24 (machine hours) and x + 2y = 24 (finishing hours) for x lamps A and y lamps B, with the feasible region below both lines shaded. — Feasible region for 2x + y ≤ 24, x + 2y ≤ 24

Show worked answer

Answer:

8

of each; ₹

720

Let $x$ lamps A and $y$ lamps B. Maximise $Z = 40 x + 50 y$ M1
subject to $2 x + y \leq 24$ , $x + 2 y \leq 24$ , $x \geq 0$ , $y \geq 0$ A1
Corners: $(0, 0)$ , $(12, 0)$ , $(0, 12)$ and $(8, 8)$ (where $2 x + y = 24$ meets $x + 2 y = 24$ ) A1
$Z$ : $0$ , $480$ , $600$ , $320 + 400 = 720$ M1
Make $8$ of each; greatest profit ₹ $720$ A1

Q9·5 marks·Long answerFormulating LPPs

A diet must contain at least $18$ units of vitamin A and at least $10$ units of iron. Food P gives $3$ units of vitamin A and $1$ unit of iron per kg and costs ₹ $6$ per kg; food Q gives $2$ units of vitamin A and $2$ units of iron per kg and costs ₹ $10$ per kg. Find the cheapest mix and its cost.

Section E style · Case study (4 marks)

Q10·4 marks·Case studyFormulating LPPs

Bakery plan. A bakery makes $x$ cakes and $y$ trays of pastries a day. A cake needs $200$ g of flour and a tray $100$ g; $5$ kg of flour is available. The oven holds at most $30$ items (cakes and trays together) a day. The profit is ₹ $30$ a cake and ₹ $20$ a tray.

Write the constraints. [1 mark]
Find the corner points of the feasible region. [1 mark]
Find the plan that gives the greatest profit. [2 marks]

Show worked answer

$2 x + y \leq 50$ (flour, in $100$ g units), $x + y \leq 30$ , $x \geq 0$ , $y \geq 0$ A1
$(0, 0)$ , $(25, 0)$ , $(0, 30)$ and $(20, 10)$ A1
$Z = 30 x + 20 y$ : $0$ , $750$ , $600$ , $600 + 200 = 800$ M1
$20$ cakes and $10$ trays: ₹ $800$ A1

Next: practise by Route to 95 step · the revision notes · free MCQs and case studies · all Class 12 assertion–reason questions · the Class 12 syllabus checklist.

Route to 95: four steps

Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.

Step 1

Secure the basics

Use LP vocabulary, test points against constraints, formulate a simple LPP and read optimal values from given corner points.

Read first: 1. Terminology and formulation; 2. Feasible region and corner points 6 practice questions · checkpoint: 3 questions, 5 marks, pass 80%

Practise step 1

Step 2

Board standard

Draw feasible regions, find corner points algebraically and solve bounded and unbounded problems, including the half-plane check.

Read first: 3. Corner-point method; Worked examples 1-2 20 practice questions · checkpoint: 4 questions, 8 marks, pass 75%

Practise step 2

Step 3

Full marks on long answers

Write full-mark 5-mark LPP answers from a word problem: formulation, labelled graph, corner-point table and conclusion.

Read first: 3. Corner-point method; Board-exam tips 5 practice questions · checkpoint: 3 questions, 15 marks, pass 70%

Practise step 3

Step 4

95+ stretch (HOTS)

Handle parameters in the objective function, multiple optimal solutions and existence questions for unbounded regions.

Read first: 3. Corner-point method (unbounded regions, multiple optima); Worked example 3 6 practice questions · checkpoint: 3 questions, 13 marks, pass 60%

Practise step 4

Practice questions

Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 11 of the 49 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceGraphical solution: bounded region

The corner points of the bounded feasible region of an LPP are $(0, 0)$ , $(5, 0)$ , $(3, 4)$ and $(0, 6)$ . The maximum value of $Z = 3 x + 2 y$ is

(a) $17$
(b) $15$
(c) $12$
(d) $18$

Q2·1 mark·Multiple choiceFeasible region and corner points

Which of the following points lies in the feasible region of $x + 2 y \leq 8, 3 x + y \leq 9, x \geq 0, y \geq 0$ ?

(a) $(3, 1)$
(b) $(2, 3)$
(c) $(0, 5)$
(d) $(4, 0)$

Q3·1 mark·Multiple choiceSpecial cases: multiple optima and infeasibility

The feasible region for the constraints $x + y \geq 6, x + y \leq 4, x \geq 0, y \geq 0$ is

(a)empty
(b)bounded
(c)unbounded
(d)a single point

Worked solution·2 marks·Very short answerSpecial cases: multiple optima and infeasibility

The feasible region of an LPP is the quadrilateral with corners $(0, 0)$ , $(6, 0)$ , $(4, 3)$ and $(0, 6)$ . Show that $Z = 3 x + 4 y$ has infinitely many points of maximum, and describe them.

Full worked solution Free, no sign-in

$Z = 0, 18, 24, 24$ at the four corners: the maximum $24$ occurs at both $(4, 3)$ and $(0, 6)$ M1
Hence every point of the segment joining $(4, 3)$ and $(0, 6)$ (the line $3 x + 4 y = 24$ ) gives $Z = 24$ : infinitely many optimal points A1

Every question here has a full worked solution. See them all →

Stretch yourself: original algebra problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S93 · Problem S28 · Problem S92 · Problem S27. All 22 →

Knowledge organiser: Linear Programming on one page (key facts, formulas and mistakes to avoid; printable PDF)

Where marks are lost in Linear Programming

Missing or extra corner points. Fix: list every pair of boundary lines, solve, and keep only intersections that satisfy all constraints.
Stating a maximum or minimum on an unbounded region without justification. Fix: write the open half-plane test (e.g. "6x + 4y < 24 has no common point with the region") in words.
Shading the wrong side of a constraint. Fix: test the origin (or another point off the line) for every inequality and write the result.
Wrong inequality sign in formulation ("at least" vs "at most"), or forgetting x, y ≥ 0. Fix: underline the key phrase in the question and translate it before writing the constraint.
No final answer sentence in a word problem. Fix: end with "Make 10 kurtas and 20 shirts; maximum profit ₹8000".
Graph without labelled lines or scale. Fix: label each line with its equation and mark all corner points with coordinates.

Exam strategy: how answers are step-marked · competency-based questions

Examiner Insights: common mistakes in CBSE Class 12 Maths, with fixes (our analysis of public sources) →

Linear Programming in our sample papers

Once step 3 is passed, test the chapter inside a full timed paper on the 2026-27 pattern (original papers by us, not official CBSE papers).

Sample paper 1 (Board standard): questions 16, 17, 31
Sample paper 2 (Board standard): question 32
Sample paper 3 (Above board): questions 16, 17, 31
Sample paper 4 (Above board): question 32
Sample paper 5 (95+ challenge): questions 16, 17, 31

Want it against the clock? Take a timed 30-mark chapter test on Linear Programming, new questions each time, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Pro or the free trial).

Preparing for CUET? Try the CUET-style questions on linear inequalities and linear programming.

Linear Programming: common questions

What is in CBSE Class 12 Maths Chapter 12, Linear Programming?

The topics are: Graphical solution: bounded region, Special cases: multiple optima and infeasibility, Feasible region and corner points, Graphical solution: unbounded region, Terminology and formulation. This page has revision notes for each, and 49 board-style practice questions with step-marked solutions.

How many marks is Linear Programming worth?

It is in the Linear Programming unit, which carries 5 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Where do students lose marks in Linear Programming?

Missing or extra corner points. Fix: list every pair of boundary lines, solve, and keep only intersections that satisfy all constraints. Stating a maximum or minimum on an unbounded region without justification. Fix: write the open half-plane test (e.g. "6x + 4y < 24 has no common point with the region") in words.

Are the Linear Programming important questions free?

Yes. The 10 important questions on this page are free to try, from 1-mark MCQs to a case study. 5 worked answers are open on the page; the others open with a free account (no card).

Are these NCERT questions?

No. Every practice question here is original, written in the board's question styles with step marks. For the textbook itself, use the NCERT link on this page.

Work through the Route to 95 with your progress saved

Save your answers and track your progressA free account remembers the questions you have done and brings you back to this page.

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Revision notes

Linear Programming — revision notes

1. Terminology and formulation

2. Feasible region and corner points

3. Corner-point method

Worked example 1

Worked example 2

Worked example 3

Common errors

Board-exam tips

Linear Programming Class 12: important questions

Section A style · Multiple choice and assertion–reason (1 mark each)

Section B style · Short answer (2 marks each)

Section C style · Short answer (3 marks each)

Section D style · Long answer (5 marks each)

Section E style · Case study (4 marks)

Route to 95: four steps

Secure the basics

Board standard

Full marks on long answers

95+ stretch (HOTS)

Practice questions

Full worked solution Free, no sign-in

Where marks are lost in Linear Programming

Linear Programming in our sample papers

Linear Programming: common questions

Work through the Route to 95 with your progress saved