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Mathematical Modeling · Form 5

Mathematical Modeling: Practice Questions

Original SPM-style practice questions for Mathematical Modeling, each with full worked solutions, Paper 1 multiple-choice and Paper 2 structured.

Written by the spmmath.com.my editorial team.

Original practice questions for Mathematical Modeling, in the style of Mathematics Paper 1 (Objective) and Mathematics Paper 2 (Subjective). Each answer is worked so you can check your method, not just the result.

Multiple-choice (Paper 1 style)

Question 1

A taxi company charges a flag-fall of RM4 plus RM1.50 for each kilometre travelled. If the total charge is RM C for a journey of x km, which of the following is the correct model?

  1. A. C = 4 + 1.5x
  2. B. C = 1.5 + 4x
  3. C. C = 4x + 1.5
  4. D. C = 5.5x

Question 2

Using the model C = 4 + 1.5x, what is the total charge for a journey of 10 km?

  1. A. RM15
  2. B. RM19
  3. C. RM17.50
  4. D. RM55

Question 3

A rectangular flower bed has a length that is 3 m longer than its width. If the width is x m, which model gives the area A in m²?

  1. A. x² + 3x
  2. B. x² + 3
  3. C. 3x²
  4. D. x² + 9x

Question 4

The height of a ball, h metres, t seconds after it is kicked upward is modelled by h = 20t − 5t². What is the maximum height reached?

  1. A. 15 m
  2. B. 25 m
  3. C. 20 m
  4. D. 40 m

Question 5

For the same model h = 20t − 5t², after how many seconds does the ball return to the ground (h = 0)?

  1. A. 2 s
  2. B. 4 s
  3. C. 5 s
  4. D. 8 s

Question 6

A monthly phone plan is modelled by C = 30 + 0.10m, where C is the cost in RM and m is the number of call minutes. If a bill is RM55, how many minutes were used?

  1. A. 250 minutes
  2. B. 220 minutes
  3. C. 350 minutes
  4. D. 850 minutes

Question 7

To make the flower bed in Question 3 have an area of 40 m², the equation x² + 3x − 40 = 0 gives x = 5 or x = −8. Which solution is rejected and why?

  1. A. x = −8, because width cannot be negative
  2. B. x = 5, because it is too small
  3. C. x = −8, because the area becomes negative
  4. D. x = 5, because it is an odd number

Question 8

A water tank drains at a steady rate. Its volume is modelled by V = 500 − 25t, where V is in litres and t in minutes.

After how many minutes is the tank empty?

  1. A. 12.5 minutes
  2. B. 20 minutes
  3. C. 25 minutes
  4. D. 475 minutes

Question 9

A stall's daily profit is modelled by P = −2x² + 40x − 150, where P is in RM and x is the selling price in RM. Which selling price gives the maximum profit?

  1. A. RM5
  2. B. RM10
  3. C. RM20
  4. D. RM40

Question 10

For the profit model P = −2x² + 40x − 150, at which selling prices does the stall break even (P = 0)?

  1. A. RM5 and RM15
  2. B. RM10 and RM15
  3. C. RM3 and RM25
  4. D. RM5 and RM30

Structured (Paper 2 style)

Question 1 (6 marks)

A farmer builds a rectangular chicken enclosure against a long straight wall. The wall forms one side, and 24 m of fencing is used for the other three sides.

Let the width (each side perpendicular to the wall) be x m. (a) Show that the area is modelled by A = 24x − 2x².

[2] (b) Find the values of x when the area is 70 m². [3] (c) State the length of the enclosure for each value of x.

[1]

  1. The side parallel to the wall uses the remaining fencing: 24 − 2x metres.
  2. Area = width × length = x(24 − 2x) = 24x − 2x², as required.
  3. Set 24x − 2x² = 70, which rearranges to 2x² − 24x + 70 = 0.
  4. Divide throughout by 2: x² − 12x + 35 = 0, then factorise: (x − 5)(x − 7) = 0.
  5. So x = 5 or x = 7; both are positive lengths, so both are accepted.
  6. When x = 5, length = 24 − 2(5) = 14 m; when x = 7, length = 24 − 2(7) = 10 m.

Question 2 (5 marks)

Two mobile data plans are charged on the extra data used beyond the quota. For x GB of extra data, Plan A costs RM CA = 40 + 5x and Plan B costs RM CB = 55 + 2x.

(a) Find the amount of extra data for which the two plans cost the same. [3] (b) Determine which plan is cheaper when 8 GB of extra data is used, and by how much.

[2]

  1. Set the two models equal: 40 + 5x = 55 + 2x.
  2. Collect like terms: 5x − 2x = 55 − 40, so 3x = 15.
  3. Solve: x = 5, so the plans cost the same at 5 GB of extra data.
  4. At x = 8: CA = 40 + 5(8) = RM80 and CB = 55 + 2(8) = RM71.
  5. Since RM71 < RM80, Plan B is cheaper by RM80 − RM71 = RM9.

Question 3 (7 marks)

A stone is thrown upward from the top of a 30 m building. Its height above the ground, h metres, t seconds after being thrown is modelled by h = 30 + 5t − 5t².

(a) Find the height of the stone after 2 seconds. [1] (b) Find the time taken to reach maximum height, and the maximum height.

[3] (c) Find the time when the stone hits the ground. [3]

  1. (a) Substitute t = 2: h = 30 + 5(2) − 5(2²) = 30 + 10 − 20 = 20 m.
  2. (b) Maximum height occurs at t = −b/(2a) = −5/(2×−5) = 0.5 s.
  3. Substitute t = 0.5: h = 30 + 5(0.5) − 5(0.5²) = 30 + 2.5 − 1.25 = 31.25 m.
  4. (c) The stone hits the ground when h = 0: 30 + 5t − 5t² = 0.
  5. Divide throughout by −5: t² − t − 6 = 0, then factorise: (t − 3)(t + 2) = 0.
  6. So t = 3 or t = −2; reject t = −2 because time cannot be negative, giving t = 3 s.

Question 4 (6 marks)

A bakery finds that when the price of a cake is RM p, the number of cakes sold in a day is modelled by n = 120 − 4p. (a) Show that the daily sales revenue is R = 120p − 4p².

[2] (b) Find the price that gives the maximum revenue. [2] (c) Find the maximum revenue, and verify it using the number of cakes sold at that price.

[2]

  1. (a) Revenue = price × quantity = p × n = p(120 − 4p) = 120p − 4p².
  2. (b) The maximum of the quadratic is at p = −b/(2a) = −120/(2×−4) = RM15.
  3. (c) Substitute p = 15: R = 120(15) − 4(15²) = 1800 − 900 = RM900.
  4. Check: at p = 15, n = 120 − 4(15) = 60 cakes, so R = 15 × 60 = RM900, which matches.

Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)

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Frequently asked questions

What is the difference between the Paper 1 MCQ and Paper 2 structured practice for mathematical modelling?

Paper 1 MCQ questions check whether you can set up a model quickly and pick the correct equation or graph from options, matching the objective format of 1449/1. Paper 2 structured questions ask you to build the full model step by step and show working, as required in 1449/2.

Why should I attempt each mathematical modelling question myself before checking the worked solution?

Working through the problem first shows you exactly where your model breaks down, wrong assumption, missed variable, or a calculation slip. Checking the solution too early skips this diagnosis, so you keep repeating the same mistake in later mathematical modelling questions instead of fixing it.

What are common pitfalls students make in mathematical modelling questions?

Students often misread which quantity is the independent variable, forget to state assumptions, or round too early and carry the error through the model. Some also stop once they get an equation without checking it fits the given data, which is where marks are usually lost.

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