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Consider the following… (Class 12 Maths SQP 2024-25, Q30)

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MathsCBSE Class 12 · Sample paper, session 2024-25 · 3 marks · Short answer

CBSE Class 12 Mathematics sample paper 2024-25: question 30

From the official CBSE sample question paper for 2024-25, 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

Consider the following Linear Programming Problem:
Minimise
Subject to , , .
Show graphically that the minimum of occurs at more than two points

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2024-25.

Hints

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

Hint 1

Plot both lines; note they meet the y-axis at the same point.

Hint 2

Find at the corner points — what do you notice? Which line is the objective line ?

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.

  1. Graph: draw through and , and through and . In the first quadrant the region satisfying both constraints lies on and above (since for ); it is unbounded with corner points and .

    1 mark

  2. Values of at corners:

    Corner point
    6
    6
    1 mark

  3. Since the region is unbounded, draw the open half-plane ; it has no point in common with the feasible region, so the minimum value of is 6.

    ½ mark

  4. at every point of segment on , e.g. at : . Hence the minimum 6 occurs at more than two points (all points of segment ).

    ½ mark

Answer: Minimum , attained at every point of segment joining and (e.g. (2,2)) — more than two points.

Where the marks are

  • Correct graph of feasible region with corners , (1 mark)
  • Corner-point table with at both (1 mark)
  • Checks has no common point, so min (½ mark)
  • Shows another point such as gives 6, so more than two points (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 2024-25: every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.