MathsCBSE Class 12 · Sample paper, session 2026-27 · 3 marks · Short answer

CBSE Class 12 Mathematics sample paper 2026-27: question 31

From the official CBSE sample question paper for 2026-27, section C, question 31, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q31 · 3 marks · Short answerLinear Programming

Solve the linear programming problem graphically:
Maximize $Z = 40 x + 30 y$
Subject to constraints: $x + y \leq 6$ , $2 x \geq y$ , $y \geq 2$

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2026-27.

Hints

Open one hint at a time and try again before the next.

Hint 1

Draw the three boundary lines and find where each pair meets.

Hint 2

The feasible region is a triangle; evaluate $Z$ at each of its vertices.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Graph: draw the lines $x + y = 6$ (through $(6, 0)$ and $(0, 6)$ ), $y = 2 x$ (through the origin and $(2, 4)$ ) and $y = 2$ . The feasible region is on the origin side of $x + y = 6$ , below or to the right of $y = 2 x$ , and above $y = 2$ . It is the bounded triangle with vertices $A (1, 2)$ [from $y = 2$ , $y = 2 x$ ], $B (4, 2)$ [from $y = 2$ , $x + y = 6$ ] and $C (2, 4)$ [from $y = 2 x$ , $x + y = 6$ ].
1½ marks
Corner point values: $Z_{A} = 40 (1) + 30 (2) = 100$ ; $Z_{B} = 40 (4) + 30 (2) = 220$ ; $Z_{C} = 40 (2) + 30 (4) = 200$
1 mark
Maximum $Z = 220$ at $B (4, 2)$ , i.e. $x = 4, y = 2$ .
½ mark

Answer: Maximum $Z = 220$ at $(4, 2)$

Where the marks are

Draws the graph and shades the correct bounded feasible region with vertices $(1, 2), (4, 2), (2, 4)$ (1½ marks)
Evaluates $Z$ at every corner point (1 mark)
States the maximum value 220 at $(4, 2)$ (½ mark)

Revise the chapter

Chapter: Linear Programming (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Linear Programming, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2026-27: every question, by section.

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CBSE Class 12 Mathematics sample paper 2026-27: question 31

Hints

Step-by-step solution

Revise the chapter

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