MathsCBSE Class 12 · Sample paper, session 2023-24 · 3 marks · Short answer

CBSE Class 12 Mathematics sample paper 2023-24: question 30(b)

From the official CBSE sample question paper for 2023-24, section C, question 30(b), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q30(b) · 3 marks · Short answerLinear Programming

Solve the following Linear Programming Problem graphically:
Maximize: $z = - x + 2 y$ ,
subject to the constraints: $x \geq 3, x + y \geq 5, x + 2 y \geq 6, y \geq 0$ .

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2023-24.

Hints

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

Hint 1

Find the corner points, but notice that the region is not closed.

Hint 2

For an unbounded region, test whether $- x + 2 y > 1$ overlaps the feasible 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 $x = 3$ , $x + y = 5$ and $x + 2 y = 6$ with $y \geq 0$ . The feasible region lies to the right of $x = 3$ and above both lines. It is unbounded, with corner points $A (3, 2)$ (from $x = 3$ , $x + y = 5$ ), $B (4, 1)$ (from $x + y = 5$ , $x + 2 y = 6$ ) and $C (6, 0)$ .
1 mark
Values of $Z = - x + 2 y$ :
Corner point $Z = - x + 2 y$
$A (3, 2)$ 1
$B (4, 1)$ $- 2$
$C (6, 0)$ $- 6$
½ mark
Because the region is unbounded, $Z = 1$ may not be the maximum. Draw the open half-plane $- x + 2 y > 1$ . It has points in common with the feasible region; for example $(3, 5)$ is feasible and gives $Z = 7 > 1$ . Hence $Z$ has no maximum value.
1½ marks

Corner point	$Z = - x + 2 y$
$A (3, 2)$	1
$B (4, 1)$	$- 2$
$C (6, 0)$	$- 6$

Answer: $Z$ has no maximum value (the feasible region is unbounded).

Where the marks are

Correct graph of the unbounded feasible region (1 mark)
Corner points and values of $Z$ (½ mark)
Checks the half-plane $- x + 2 y > 1$ and concludes there is no maximum (1½ marks)

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 2023-24: every question, by section.

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CBSE Class 12 Mathematics sample paper 2023-24: question 30(b)

Hints

Step-by-step solution

Revise the chapter

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