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

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

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

Q30(a) · 3 marks · Short answerLinear Programming

Solve the following Linear Programming Problem graphically:
Minimize: $z = x + 2 y$ ,
subject to the constraints: $x + 2 y \geq 100, 2 x - y \leq 0, 2 x + y \leq 200, x, y \geq 0$ .

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

Plot each constraint line and shade the side that satisfies the inequality.

Hint 2

Find the corner points by solving pairs of lines. Notice that two corners give the same value of $Z$ .

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 the lines $x + 2 y = 100$ (through $(100, 0), (0, 50)$ ), $2 x - y = 0$ (through $(0, 0), (50, 100)$ ) and $2 x + y = 200$ (through $(100, 0), (0, 200)$ ) in the first quadrant. The feasible region lies above $x + 2 y = 100$ , on the side of $2 x - y = 0$ towards the $y$ -axis, and below $2 x + y = 200$ . It is the bounded quadrilateral with vertices $A (0, 50)$ , $B (20, 40)$ , $C (50, 100)$ , $D (0, 200)$ . Here $B$ comes from solving $x + 2 y = 100, y = 2 x$ , and $C$ from solving $y = 2 x, 2 x + y = 200$ .
1½ marks
Values of $Z = x + 2 y$ at the corners:
Corner point $Z = x + 2 y$
$A (0, 50)$ 100 (minimum)
$B (20, 40)$ 100 (minimum)
$C (50, 100)$ 250
$D (0, 200)$ 400
1 mark
The minimum value of $Z$ is 100. It occurs at every point on the segment joining $(0, 50)$ and $(20, 40)$ .
½ mark

Corner point	$Z = x + 2 y$
$A (0, 50)$	100 (minimum)
$B (20, 40)$	100 (minimum)
$C (50, 100)$	250
$D (0, 200)$	400

Answer: Minimum $Z = 100$ at all points on the segment joining $(0, 50)$ and $(20, 40)$

Where the marks are

Correct graph and feasible region with corner points (1½ marks)
Evaluates $Z$ at the corner points (1 mark)
States the minimum 100 on segment AB (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 2023-24: 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 sample paper 2023-24: question 30(a)

Hints

Step-by-step solution

Revise the chapter

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