Solve graphically: Maximise Z=2x+y subject to x+y≤1200 x+y≥600 y/2 x≥0, y≥0.

MathsCBSE Class 12 · Sample paper, session 2025-26 · 3 marks · Short answer

CBSE Class 12 Mathematics sample paper 2025-26: question 30

From the official CBSE sample question paper for 2025-26, section C, question 30, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q30 · 3 marks · Short answerLinear Programming

Solve graphically:
Maximise $Z = 2 x + y$ subject to
$x + y \leq 1200$
$x + y \geq 600$
$y \leq \frac{x}{2}$
$x \geq 0, y \geq 0$ .

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2025-26.

Hints

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

Hint 1

Plot each constraint as a line and decide which side satisfies it.

Hint 2

Find the corner points of the feasible region (solve pairs of lines) and evaluate $Z$ at each.

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 $x + y = 1200$ (through $(1200, 0), (0, 1200)$ ), $x + y = 600$ (through $(600, 0), (0, 600)$ ) and $y = \frac{x}{2}$ (through $(0, 0), (800, 400)$ ). The feasible region lies between the two parallel lines, below $y = x /2$ , in the first quadrant: the quadrilateral ABCD (bounded) shown shaded. Intersections: $x + y = 1200, y = x /2 \Rightarrow C (800, 400)$ ; $x + y = 600, y = x /2 \Rightarrow D (400, 200)$ .
1½ marks
Corner points: $A (600, 0)$ , $B (1200, 0)$ , $C (800, 400)$ , $D (400, 200)$ .
½ mark
$Z = 2 x + y$ : $Z_{A} = 1200$ , $Z_{B} = 2400$ , $Z_{C} = 2000$ , $Z_{D} = 1000$ .
½ mark
Maximum $Z = 2400$ at $x = 1200, y = 0$ .
½ mark

Answer: Max $Z = 2400$ at $(1200, 0)$

Where the marks are

Correct graph with feasible region shaded (1½ marks)
Correct corner points (½ mark)
Values of Z at corners (½ mark)
Maximum 2400 at (1200, 0) (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 2025-26: every question, by section.

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CBSE Class 12 Mathematics sample paper 2025-26: question 30

Hints

Step-by-step solution

Revise the chapter

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