Practice · Maths AI SL · 2021 syllabus · 2.5
IB Maths AI SL 2.5 Modelling with functions: practice questions
3 exam-style questions on 2.5 Modelling with functions, SL: 2 Paper 1 short and 1 Paper 2 long. Each one comes with the mark scheme, a model answer and the reasoning behind every mark. Free with an account.
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2.5-02 · Paper 1 style · short question · 8 marks
Exponential depreciation
Tamsin buys a new electric bicycle. Its value, \(V\) euros, is modelled by\[V(t) = 400 + 2600e^{-kt}, \quad t \ge 0,\]where \(t\) is the number of months since she bought it and \(k\) is a positive constant.
(a) Calculate the value of the bicycle on the day Tamsin bought it. [2]
Twelve months after Tamsin bought it, the bicycle has lost 35% of its value.
(b) Find the value of \(k\). [4]
Tamsin believes that the bicycle will always be worth more than 400 euros.
(c(i)) State a property of the graph of \(V\) that supports Tamsin's belief. [1]
(c(ii)) State one reason, in context, why Tamsin's belief might be wrong. [1]
Show the answer and the reasoning
(a) Mark scheme
- (A1) \(400 + 2600e^{0}\) OR \(400 + 2600 \times 1\) seen;
- A1 3000 euros;
A model answer: \(V(0) = 400 + 2600e^{0} = 400 + 2600 \times 1 = 3000\) euros.
(b) Mark scheme
- (M1) \(3000 \times 0.65\) OR \(3000 - 0.35 \times 3000\) OR \(0.35 \times 3000\);
- (A1) 1950;
- (M1) \(1950 = 400 + 2600e^{-12k}\), an equation with 12 and their 1950;
- A1 \(k = 0.0431\) (0.0431047...);
A model answer: After 12 months the bicycle keeps 65% of its value: \(3000 \times 0.65 = 1950\) euros. So \(400 + 2600e^{-12k} = 1950\), which gives \(e^{-12k} = \frac{1550}{2600}\). The GDC solver gives \(k = 0.0431\) (0.0431047...).
(c(i)) Mark scheme
- A1 the graph has a horizontal asymptote at \(V = 400\), so \(V\) approaches 400 but never reaches it;
A model answer: The graph of \(V\) has a horizontal asymptote, \(V = 400\): as \(t\) increases, \(V\) gets closer and closer to 400 but never reaches it.
(c(ii)) Mark scheme
- R1 the bicycle could be damaged, broken or stolen, OR its battery could wear out, so its real value could fall below 400 euros;
A model answer: If the bicycle is wrecked in an accident, or its battery no longer holds a charge, it could be worth less than 400 euros, or nothing at all.
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The 3 questions in 2.5
- 2.5-01 A tidal pool
- 2.5-02 Exponential depreciation
- 2.5-03 Path of a kicked ball
More of 2 Functions
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Every Maths question here is original, each one modelled on a question from the AI SL or AA SL papers set since 2021, with a new context and new numbers. Developed independently from the International Baccalaureate Organization, which has not endorsed it. Found a mistake? Email vytis@limeadmissions.com with the question id.