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 1Medium6 marks

A laboratory sample contains $3.6 \times 10^{13}$ identical nanoparticles. Each nanoparticle has a mass of $4.7 \times 10^{-19}$ kg.

(a)

Calculate the total mass of the nanoparticles in the sample, in kg. Give your answer in scientific notation.

[2]

(b)

A filter removes nanoparticles with a combined mass of $5.64 \times 10^{-6}$ kg. Using your answer to part (a), determine the number of nanoparticles remaining. Give your answer in scientific notation.

[4]

Question 2

Paper 1Medium6 marks

Maya borrows $3,800 to repay a credit-card balance. Interest is charged at a nominal annual rate of 18.6%, compounded monthly. Maya pays$ 240 at the end of each month unless this would exceed the amount then owed, in which case she pays only the amount needed to clear the balance. There are no fees or further charges.

Give monetary amounts to the nearest cent.

(a)

Use the financial package on a graphic display calculator to determine the number of monthly payments Maya makes before the loan is cleared.

[2]

(b)

Using your answer to part (a),

(i)

find the amount of Maya’s final payment.

[2]

(ii)

hence calculate the total interest Maya pays over the loan.

[2]

Question 3

Paper 1Medium7 marks

The Northgate Credit Union offers simple interest at 2.7% per year on the original deposit. Leila deposits $2400 and makes no further deposits or withdrawals. Let $B_n$ be the balance, in dollars, at the end of $n$ years.

(a)

Calculate the common difference between consecutive year-end balances.

[1]

(b)

Show that $B_n=2400+64.80n$

[2]

(c)

Hence, determine the first whole year at the end of which the balance exceeds $3000.

[1]

(d)

A report summarizes the mean of the balances at the ends of years 1 to 10. Write the sum of these balances in sigma notation and hence determine the mean.

[3]

Question 4

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 5

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 6

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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