SPM 2024 Add Maths Paper 2, Question 13Linear Programming

Solved on video in English and Bahasa Melayu · 1 marks · Form 5, Linear Programming · 5:32 video

The question

(a)

Mus wants to sell two types of crisps. Table 3 shows information relating to the two types of crisps.

Table 3:

Crisp P: x packets, cost price RM4.00/packet, selling price RM10.00/packet
Crisp Q: y packets, cost price RM2.50/packet, selling price RM7.00/packet

(a) Mus has RM125 to buy the two types of crisps. Write an inequality to represent this constraint. [1 mark]

Mus has RM125 to buy the two types of crisps. Write an inequality to represent this constraint. [1 mark]

[1 marks]

(b)

(b) The shaded region on the graph on page 33 (of the exam paper) represents the second constraint. Write an inequality to represent the constraint.

The shaded region on the graph on page 33 represents the second constraint. Write an inequality to represent the constraint.

(c)

(c) The number of packets of crisp P sold is less than 3/2 of the number of packets of crisp Q sold.

On page 33, construct and label the region R which satisfies all the constraints and x > 0, y > 0.

The number of packets of crisp P sold is less than 3/2 of the number of packets of crisp Q sold.

On page 33, construct and label the region R which satisfies all the constraints and x > 0, y > 0.

(d)(i)

(d) Use the graph constructed in (c) to answer the following question:

(i) State the maximum number of packets of crisp P sold if 12 packets of crisp Q were sold.

State the maximum number of packets of crisp P sold if 12 packets of crisp Q were sold.

(d)(ii)

(d)(ii) Mus plans to save 50% of the total sales profit. He also needs to pay a wage of RM40 to a worker from the remainder.

By drawing the objective function line to find the profit obtained, determine whether or not the selling prices are suitable. Give your justification.

[You may assume that he sells all of the crisps that he buys.]

Mus plans to save 50% of the total sales profit. He also needs to pay a wage of RM40 to a worker from the remainder. By drawing the objective function line to find the profit obtained, determine whether or not the selling prices are suitable. Give your justification. [You may assume that he sells all of the crisps that he buys.]

Part (a)

Step 1

Total cost of buying x packets of P and y packets of Q must not exceed RM125.

1 more step, with the reason for it, in the full solution

Final answer

4x + 2.50y ≤ 125 (or 8x + 5y ≤ 250)

Where students lose marks

Watch out: the budget is a limit, not an exact amount, so write ≤ 125, not = 125.

Part (b)

3 more steps, each with the reason for it, in the full solution

Final answer

x + y ≥ 20

Where students lose marks

Watch out: the line is x + y = 20; read the sign from the shading, not something like 20x + 20y > 0.

Part (c)

6 more steps, each with the reason for it, in the full solution

Final answer

x < (3/2)y; region R drawn and labelled (see video)

Where students lose marks

Watch out: 'less than' is <, not ≥, and shade the side that satisfies each inequality. Wrong sign or wrong side makes R wrong.

Part (d)(i)

5 more steps, each with the reason for it, in the full solution

Final answer

17 packets

Where students lose marks

Watch out: stay inside region R. Reading the maximum x from beyond the dashed line x = (3/2)y gives 18 instead of 17.

Part (d)(ii)

9 more steps, each with the reason for it, in the full solution

Final answer

Suitable: min profit RM91.50, half = RM45.75 > RM40

Where students lose marks

Watch out: evaluate profit at the minimum point of R, not any point inside, then compare the 50% remainder with RM40 to justify.

Watch every step of this question solved on video (5:32), with the reason behind each step and a box that checks your own answer.

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