MathsCBSE Class 12 · Board exam 2023, set 65/1/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2023: question 29

Set in the CBSE Class 12 board exam of 2023, paper 65/1/1, 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 graphically the following linear programming problem :
Maximise $z = 6 x + 3 y$ ,
subject to the constraints
$4 x + y \geq 80$ ,
$3 x + 2 y \leq 150$ ,
$x + 5 y \geq 115$ ,
$x \geq 0, y \geq 0$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 65/1/1.

Hints

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

Hint 1

Plot each constraint line using its intercepts and shade the correct side.

Hint 2

Find the corner points by solving pairs of lines, then 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 the lines
- $4 x + y = 80$ through $(20, 0)$ and $(0, 80)$ ;
- $3 x + 2 y = 150$ through $(50, 0)$ and $(0, 75)$ ;
- $x + 5 y = 115$ through $(115, 0)$ and $(0, 23)$ .
The feasible region lies above $4 x + y = 80$ , below $3 x + 2 y = 150$ and above $x + 5 y = 115$ , in the first quadrant. It is the bounded triangle with corner points:
- $(2, 72)$ , from $4 x + y = 80$ and $3 x + 2 y = 150$ ;
- $(15, 20)$ , from $4 x + y = 80$ and $x + 5 y = 115$ ;
- $(40, 15)$ , from $3 x + 2 y = 150$ and $x + 5 y = 115$ .
2 marks
Values of $z = 6 x + 3 y$ at the corner points:
Corner point $z = 6 x + 3 y$
$(2, 72)$ $12 + 216 = 228$
$(15, 20)$ $90 + 60 = 150$
$(40, 15)$ $240 + 45 = 285$ (maximum)
Hence the maximum value of $z$ is 285, at $x = 40, y = 15$ .
1 mark

Corner point	$z = 6 x + 3 y$
$(2, 72)$	$12 + 216 = 228$
$(15, 20)$	$90 + 60 = 150$
$(40, 15)$	$240 + 45 = 285$ (maximum)

Answer: Maximum $z = 285$ at $(40, 15)$

Where the marks are

Correct graph with the bounded feasible region and corner points $(2, 72), (15, 20), (40, 15)$ (2 marks)
Evaluates $z$ at the corners and states the maximum 285 at $(40, 15)$ (1 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 paper2023 board paper, set 65/1/1: 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.

CBSE Class 12 Mathematics 2023: question 29

Hints

Step-by-step solution

Revise the chapter

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