Paper 2 Section C Practice

Linear Programming • 10-Mark Mastery

SPM Paper 2 Section C Specialty

Form 5 Chapter 7: Linear Programming (Pengaturcaraan Linear)

Total Marks: 40 Marks • Time Allowed: 50 Minutes

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A Bahagian A: Mathematical Modeling Fundamentals [10 Marks]

Answer all questions
Soalan 1 [5 Marks]

A bakery bakes $x$ chocolate cakes and $y$ fruit cakes per day. Write down an inequality in $x$ and/or $y$ for each of the following conditions:

(a) The maximum daily production of chocolate cakes is 50. [1 mark]

(b) The number of fruit cakes baked is at least half the number of chocolate cakes. [1 mark]

(c) Baking a chocolate cake requires 200g of butter, and a fruit cake requires 300g of butter. The bakery has a maximum daily allocation of 18 kg of butter. [2 marks]

(d) The total number of cakes baked daily is more than 40. [1 mark]

Soalan 2 [5 Marks]

A feasible region $R$ in the Cartesian plane has vertices at $(10, 20)$, $(10, 50)$, $(30, 40)$, and $(40, 10)$. If an objective profit function is defined as $P = 150x + 200y$:

(a) State the gradient of the objective search line. [1 mark]

(b) Determine which vertex maximizes the profit $P$, and calculate the maximum profit value. [4 marks]

B Bahagian B: SPM Paper 2 Section C Full Blueprint [20 Marks]

Answer both 10-mark questions
Soalan 3 (Paper 2 Section C Logistics Model) [10 Marks]
A courier company intends to deploy $x$ delivery vans and $y$ electric cargo motorcycles to service an urban district. The daily fleet operations are governed by the following constraints:
  • I: The total number of vehicles deployed must not exceed $80$.
  • II: The number of motorcycles deployed is at least three times the number of delivery vans.
  • III: The minimum daily total parcels delivered must be at least $1,200$. A van delivers $30$ parcels and a motorcycle delivers $20$ parcels daily.

    • (a) Write three linear inequalities, other than $x \ge 0$ and $y \ge 0$, which satisfy all the above constraints. [3 marks]

    (b) Using a scale of $2\text{ cm to } 10\text{ units}$ on both axes, construct and shade the feasible region $R$. [3 marks]

    (c) Using the graph constructed in (b), find:

    (i) The range of the number of motorcycles deployed if $15$ vans are utilized. [2 marks]

    (ii) The minimum daily operating cost if the operational cost of a van is RM 120 and of a motorcycle is RM 40. [2 marks]

    Soalan 4 (KBAT High-Tech Semiconductor Model) [10 Marks]
    A microchip fabrication plant manufactures two types of silicon microprocessors: Standard AI Core ($x$ batches) and Ultra Quantum Core ($y$ batches) per month. The production is subject to three critical clean-room constraints:
  • I: The ratio of Ultra Quantum batches to Standard AI batches must not exceed $2 : 1$.
  • II: Standard AI cores require $3$ hours of photolithography while Ultra Quantum cores require $6$ hours. The plant has at most $360$ hours of photolithography machine time available monthly.
  • III: The total energy consumed by the plant must not exceed $800$ megawatt-hours (MWh). Each Standard AI batch consumes $8\text{ MWh}$ and each Ultra Quantum batch consumes $10\text{ MWh}$.

    • (a) Formulate three linear inequalities, other than $x \ge 0$ and $y \ge 0$, representing the monthly production constraints. [3 marks]

    (b) Using a scale of $2\text{ cm to } 10\text{ batches}$ on both axes, draw the graph and shade the feasible region $R$. [3 marks]

    (c) Each batch of Standard AI Cores yields a net revenue of RM 50,000, while each batch of Ultra Quantum Cores yields RM 80,000.

    (i) Construct an objective revenue line on your graph to determine the optimal production quantity of each microprocessor model to maximize monthly revenue. [2 marks]

    (ii) Calculate the maximum total monthly revenue. [2 marks]