Solve the following Linear Programming Problem graphically : Maximize Z = 2x/5 + 3y/10 subject to constraints 2x + y ≤ 1000 x + y ≤ 800 x, y ≥ 0.

MathsCBSE Class 12 · Board exam 2026, set 65/5/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2026: question 29

Set in the CBSE Class 12 board exam of 2026, paper 65/5/1, section C, question 29, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q29 · 3 marks · Short answerLinear Programming

Solve the following Linear Programming Problem graphically :

Maximize $Z = \frac{2 x}{5} + \frac{3 y}{10}$

subject to constraints

$2 x + y \leq 1000$

$x + y \leq 800$

$x, y \geq 0$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 65/5/1.

Hints

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

Hint 1

Plot each constraint line using its intercepts, then shade the side containing the origin.

Hint 2

Find the intersection of $2 x + y = 1000$ and $x + y = 800$ by subtracting the equations.

Hint 3

Evaluate $Z$ at every corner of the bounded region.

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.

Draw the lines $2 x + y = 1000$ (through $(500, 0)$ and $(0, 1000)$ ) and $x + y = 800$ (through $(800, 0)$ and $(0, 800)$ ). The lines meet where $x = 200$ , $y = 600$ .
The feasible region is bounded and lies in the first quadrant, below both lines (towards the origin). It is shaded with vertices $O (0, 0)$ , $B (500, 0)$ , $C (200, 600)$ and $D (0, 800)$ .
1½ marks
Evaluate $Z = \frac{2 x}{5} + \frac{3 y}{10}$ at the corner points:
Corner point $Z$
$(0, 0)$ 0
$(500, 0)$ 200
$(200, 600)$ 80 + 180 = 260
$(0, 800)$ 240
1 mark
Maximum $Z = 260$ at $x = 200$ , $y = 600$ .
½ mark

Corner point	$Z$
$(0, 0)$	0
$(500, 0)$	200
$(200, 600)$	80 + 180 = 260
$(0, 800)$	240

Answer: $Z_{m a x} = 260$ at $(200, 600)$

Where the marks are

Correct lines and shaded feasible region with its corner points (1½ marks)
Correct values of Z at all corner points (1 mark)
Maximum 260 at (200, 600) (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 paper2026 board paper, set 65/5/1: every question, by section.

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CBSE Class 12 Mathematics 2026: question 29

Hints

Step-by-step solution

Revise the chapter

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