IB Maths AI SL Topic 2: Functions questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

Paper

Paper 1 Paper 2

Difficulty

Easy Medium Hard

Question 1

Paper 1Easy6 marks

The upper edge of an entrance arch is modelled by $y=-0.4(x-5)^2+10$ where $x$ is the horizontal distance in metres from the left end of the arch and $y$ is the height in metres. The model is valid for $0\leq x\leq10$

Use graphing technology for each part. Give coordinates in metres.

(a)

State the coordinates of the maximum point of the graph.

[2]

(b)

Determine the coordinates of the $x$ -intercepts of the graph.

[2]

(c)

A horizontal crossbar is placed at a height of $6$ metres. Determine the coordinates where it meets the arch. Give your answers to two decimal places.

[2]

Question 2

Paper 1Hard8 marks

An urban greenhouse produces and sells seedlings in complete batches of 1000. Let $q$ be the number of batches produced in a year, where $q$ is an integer and $0\leq q\leq80$

The selling price per seedling is modelled by

p(q)=96-1.2q

dollars. The annual operating cost is modelled by

C(q)=0.4q^2+12q+500

thousand dollars. The greenhouse also pays an annual permit fee of $F$ thousand dollars. Assume that all seedlings produced are sold.

Let $P_F(q)$ denote the annual profit, in thousands of dollars. The greenhouse considers a year acceptable if its profit is at least $350 000.

(a)

Write an expression for $P_F(q)$

[1]

(b)

When $F=200$ use your graphic display calculator to determine all allowed values of $q$ for which the year is acceptable.

[2]

(c)

Determine the greatest annual permit fee that would still allow the greenhouse to have an acceptable year, and the corresponding production.

[4]

(d)

State one limitation of this model when it is used to make the decision in part (c).

[1]

Question 3

Paper 1Easy7 marks

A technician records the volume, $V$ of water in a container as it is filled from a tap.

$t$ (minutes)	0	2	4	6
$V$ (litres)	2.6	4.0	5.5	7.0

(a)

State one feature of the measurements that supports using a linear model for $V$

[1]

(b)

Using the measurements at $t=0$ and $t=6$ determine the gradient $m$ and intercept $c$ of the linear model $V(t)=mt+c$ Hence write down the model.

[3]

(c)

Use your model to estimate the volume at $t=5$ minutes, giving your answer to the nearest $0.1$ litre. State whether this prediction involves extrapolation, giving a reason.

[3]

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