IB Mathematics Applications & Interpretation Topic AHL 2.9—HL Modelling Functions Questionbank (HL)

IB Mathematics Applications & Interpretation Topic AHL 2.9—HL Modelling Functions Questionbank (HL) | RevisionDojo

Practice IB Mathematics Applications & Interpretation (AI) Topic AHL 2.9—HL Modelling Functions with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 2.9—HL Modelling Functions and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

HLPaper 2

A pendulum's displacement, $x (t)$ , in meters, from its equilibrium position at time $t$ seconds is modeled by

$x (t) = 0.5 e^{- 0.1 t} cos (\frac{π}{2} t)$

The pendulum is released from rest at $t = 0$ .

The energy $E (t)$ is measured in joules and is proportional to $x (t)^{2}$ .

Find the initial displacement.

[1]

Determine the period of oscillation.

[2]

Show that the maximum displacement occurs at $t = 0$ .

[3]

Find the time when the displacement is first at its equilibrium position.

[3]

The energy $E (t)$ of the pendulum is proportional to $x (t)^{2}$ . Given $E (0) = 2$ , find the constant of proportionality and express $E (t)$ .

[3]

Determine the rate of change of $E (t)$ at $t = 1$ , correct to three significant figures.

[4]

Question 2

HLPaper 2

A drone's altitude, $A (t)$ , in meters, above a field at time $t$ seconds is modeled by

$A (t) = 50 + 30 sin (\frac{π}{20} t) + 20 e^{- 0.02 t} cos (\frac{π}{20} t + \frac{π}{3}) .$

The drone's battery life depends on its altitude, with power consumption rate $B (t) = 0.1 A (t)$ watts.

Find the period of the drone's altitude oscillations.

[2]

Express the oscillatory part as a single trigonometric function with time-dependent amplitude.

[5]

Find the total energy consumed by the drone over the first 40 seconds, correct to two decimal places.

[4]

Determine the time when the drone first reaches an altitude of 60 meters.

[4]

Question 3

HLPaper 2

A wind turbine's power output, $P (t)$ , in kilowatts, varies over time $t$ hours according to the function

$P (t) = 500 + 300 sin (\frac{π}{12} t + \frac{π}{6}) + 200 e^{- 0.05 t} cos (\frac{π}{12} t)$

where $t \geq 0$ and angles are measured in radians. The turbine operates efficiently when $P (t) \geq 600$ . The wind conditions (and hence the oscillatory components of $P$ ) cycle every 24 hours.

Determine the period of the oscillatory components of $P (t)$ .

[2]

Express the oscillatory part of $P (t), 300 sin (\frac{π}{12} t + \frac{π}{6}) + 200 e^{- 0.05 t} cos (\frac{π}{12} t)$ , in the form $A (t) sin (\frac{π}{12} t + ϕ (t))$ , where $A (t)$ and $ϕ (t)$ are functions of $t$ .

[5]

Sketch the graph of $P (t)$ for $0 \leq t \leq 24$ , indicating where $P (t) = 600$ .

[3]

Find the time intervals in the first 24 hours when the turbine operates efficiently.

[4]

The cost of maintenance, $C (t)$ , in dollars per hour, is modeled as $C (t) = 10 + 0.02 P (t)$ . Find the total maintenance cost over the first 24 hours, correct to two decimal places, using numerical integration.

[4]

Question 4

HLPaper 2

The population of a rare species, $N (t)$ , in thousands, after $t$ years is modeled by

$N (t) = \frac{50}{1 + 49 e ^{- 0.4 t}} + 5 cos (\frac{π}{5} t + \frac{π}{6})$

(where angles are measured in radians)

The species is at risk when $N (t) < 10$ .

Find the initial population.

[1]

Determine the carrying capacity of the logistic component.

[2]

Find the first time $t \geq 0$ for which $N (t) < 10$ .

[4]

The growth rate is modeled by $N^{'} (t)$ . Find the maximum value of $N^{'} (t)$ for $0 \leq t \leq 10$ , correct to three significant figures.

[4]

Propose a modified model $M (t) = \frac{L}{1 + C e ^{- k t}} + 5 cos (\frac{π}{5} t + \frac{π}{6})$ to stabilize the population above $10$ thousand, and determine $L$ .

[5]

Question 5

HLPaper 2

A tide's height, $H (t)$ , in meters, at a coastal station is modeled by

$H (t) = 3 + 2 sin (\frac{2 π}{12.4} t) + 1.5 cos (\frac{2 π}{12.4} t) + 0.5 sin (\frac{4 π}{12.4} t)$

The time $t$ is in hours since midnight.

Find the period of the tide.

[2]

Express $H (t)$ in the form $H (t) = 3 + A sin (\frac{2 π}{12.4} t + ϕ) + 0.5 sin (\frac{4 π}{12.4} t)$ .

[4]

Sketch the graph of $H (t)$ for $0 \leq t \leq 24.8$ .

[3]

Find the times in the first 24.8 hours when $H (t) = 3$ .

[4]

Determine the maximum height of the tide and the time it first occurs.

[4]

IB Mathematics Applications & Interpretation AHL 2.9—HL Modelling Functions Questionbank

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IB Mathematics Applications & Interpretation AHL 2.9—HL Modelling Functions Questionbank

IB Mathematics Applications & Interpretation (AI) Topic AHL 2.9—HL Modelling Functions Questionbank

Unlock all 16 questions with a free account