Maths · Linear programming · chapter 12 of 13

Class 12 Maths Chapter 12: Linear Programming case study question with answers

One case-based question of the kind the board asks: a passage and a figure, then 3 parts worth 4 marks in all, with the answer and the marking scheme for each. Written from the NCERT chapter.

Case study · 4 marks

A start-up in Bhopal is launching a study app and plans to advertise it on a local radio station and in a city newspaper. A radio spot costs ₹2000 and reaches about 3000 listeners; a newspaper advertisement costs ₹5000 and reaches about 6000 readers. The advertising budget is ₹40,000. The radio station can give at most 15 spots, and the newspaper at most 6 advertisements. Let x be the number of radio spots and y the number of newspaper advertisements. The reach, in thousands of people, is R = 3x + 6y. The feasible region is shaded in the figure.

Part 1 · 1 mark

The budget constraint is

(a) 5x + 2y ≤ 40(b) 2x + 5y ≥ 40(c) x + y ≤ 40(d) 2x + 5y ≤ 40

Show answer

Answer: (d) 2x + 5y ≤ 40

Cost 2000x + 5000y must not exceed 40000; dividing by 1000, 2x + 5y ≤ 40. 5x + 2y swaps the two prices, and ≥ would make the budget a minimum spend.

Part 2 · 1 mark

The reach at the corner point C(5, 6), in thousands, is

(a) 45(b) 51(c) 57(d) 36

Show answer

Answer: (b) 51

R = 3(5) + 6(6) = 15 + 36 = 51 thousand. 45, 57 and 36 are the values at A, B and D.

Part 3 · 2 marks

Find the plan that gives the greatest reach, and that reach.

Answer and marking scheme

The region is bounded, with corners O(0, 0), A(15, 0), B(15, 2), C(5, 6), D(0, 6). R there: 0, 45, 57, 51, 36. The greatest reach is 57 thousand people, with 15 radio spots and 2 newspaper advertisements (B).

R at all five corners (1)
57 thousand at (15, 2): 15 radio spots, 2 newspaper advertisements (1)

If the radio station could give at most 10 spots instead of 15, find the plan that gives the greatest reach, and that reach.

Answer and marking scheme

With x ≤ 10 the corners are (0, 0), (10, 0), (10, 4) [x = 10 meets 2x + 5y = 40], (5, 6) and (0, 6). R there: 0, 30, 54, 51, 36. The greatest reach is 54 thousand people, with 10 radio spots and 4 newspaper advertisements.

new corners, including (10, 4), and R at each (1)
54 thousand at (10, 4): 10 radio spots, 4 newspaper advertisements (1)

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