IB Maths AI SL Topic 1: Number and algebra 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 2Easy6 marks

At the beginning of year 1, a savings account contains $6000. At the beginning of each following year,$ 500 is added to the account. The account pays simple interest at 4% per year on the amount deposited. Interest is withdrawn at the end of each year and is not added to the amount on which interest is calculated.

(a)

Calculate the interest earned in year 2.

[1]

(b)

State the first term and the common difference of the sequence of annual interest amounts.

[2]

(c)

Hence, write the total interest earned over the first 6 years using sigma notation, and calculate this total.

[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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