IB Maths AI SL AI 1.3 Exponents, logarithms and equations 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

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Question 1

Paper 1Medium7 marks

A sensor records the height $h$ in metres, of a test rocket $t$ seconds after launch. For $0\leq t\leq 6$ its height is modelled by

h(t)=at^2+bt+c,

where $a$ $b$ and $c$ are constants.

$t$ (s)	0	2	5
$h$ (m)	1.2	7.8	6.7

(a)

Use a graphic display calculator to find $a$ $b$ and $c$ Show the three equations used.

[4]

(b)

Find the time or times, to three significant figures, when the rocket is at a height of $6$ m. Give your answer in seconds.

[3]

Question 2

Paper 2Medium6 marks

An engineer models the height $h$ in metres, of the lower edge of an arch above a level path by

h(x)=ax^2+bx+c,

where $x$ is the horizontal distance, in metres, from a marker. The model is assumed to pass exactly through the observations below.

$x$	0	2	4
$h(x)$	1.10	4.10	4.30

(a)

Write down a system of linear equations in $a$ $b$ and $c$ from these observations.

[2]

(b)

Using your system from part (a), use a GDC to solve it and hence show that

h(x)=-0.35x^2+2.20x+1.10.

[2]

(c)

The arch meets the path when $h(x)=0$ Use the model to determine the positive value of $x$ to three significant figures, and state what it represents.

[2]

Question 3

Paper 2Medium13 marks

At a rehearsal venue, the intensity $I$ in $\text{W m}^{-2}$ of a short electronic tone $t$ seconds after it ends is modelled by

I(t)=I_0e^{-kt},\qquad t\geq 0,

where $I_0=8.00\times10^{-5}$ A sensor reading gives $I(5)=2.40\times10^{-5}$

The sound level $L$ in decibels (dB), is modelled by

L=10\log_{10}\left(\frac{I}{I_{\mathrm{ref}}}\right), \qquad I_{\mathrm{ref}}=1.00\times10^{-12}\text{ W m}^{-2}.

The venue’s policy is to begin recording only when the predicted sound level is at most $70.0$ dB.

(a)

For the initial intensity:

(i) Write $\dfrac{I_0}{I_{\mathrm{ref}}}$ in standard form.

(ii) Calculate $L(0)$ giving your answer to three significant figures.

[3]

(b)

Determine $k$ giving your answer to three significant figures and including its unit.

[3]

(c)

Show that

L(t)=L(0)-10kt\log_{10}(e).

[2]

(d)

Hence or otherwise, determine the first time at which the predicted sound level is at most $70.0$ dB. Give your answer to the nearest $0.1$ s.

[3]

(e)

A manager claims that recording can begin $8.0$ seconds after the tone ends. Assess this claim using the model, and state one assumption or limitation relevant to the decision.

[2]

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