MathsCBSE Class 12 · Board exam 2024, set 65/1/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2024: question 30

Set in the CBSE Class 12 board exam of 2024, paper 65/1/1, 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 the following linear programming problem graphically :
Maximise $z = 500 x + 300 y$ ,
subject to constraints
$x + 2 y \leq 12$
$2 x + y \leq 12$
$4 x + 5 y \geq 20$
$x \geq 0, y \geq 0$

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 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 find the side that satisfies the inequality.

Hint 2

Solve $x + 2 y = 12$ and $2 x + y = 12$ together to get one corner.

Hint 3

Evaluate z at every corner of the bounded region and pick the largest.

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 $x + 2 y = 12$ through $(12, 0), (0, 6)$ ; $2 x + y = 12$ through $(6, 0), (0, 12)$ ; $4 x + 5 y = 20$ through $(5, 0), (0, 4)$ . The feasible region (in the first quadrant, below both $x + 2 y = 12$ and $2 x + y = 12$ , above $4 x + 5 y = 20$ ) is the bounded pentagon with corners $A (0, 4)$ , $B (0, 6)$ , $C (4, 4)$ [intersection of $x + 2 y = 12$ and $2 x + y = 12$ ], $D (6, 0)$ , $E (5, 0)$ .
1½ marks
Values of $z = 500 x + 300 y$ at corners:
Corner point $z$
A(0, 4) 1200
B(0, 6) 1800
C(4, 4) 3200
D(6, 0) 3000
E(5, 0) 2500
1 mark
Maximum $z = 3200$ at $x = 4, y = 4$ .
½ mark

Corner point	$z$
A(0, 4)	1200
B(0, 6)	1800
C(4, 4)	3200
D(6, 0)	3000
E(5, 0)	2500

Answer: Max z = 3200 at x = 4, y = 4

Where the marks are

Correct graph with shaded feasible region and corner points (1.5 marks)
Correct corner-point table of z values (1 mark)
States maximum 3200 at (4, 4) (0.5 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 paper2024 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2024: question 30

Hints

Step-by-step solution

Revise the chapter

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