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 1Hard16 marks

A workshop makes a circular cover. Its radius is measured as $6.4\text{ cm}$ correct to the nearest $0.1\text{ cm}$ The workshop estimates the area using this displayed radius and rounds the estimate to three significant figures for its label.

For all parts, use unrounded calculator values in subsequent calculations unless a specified rounding is requested. Percentage error is defined as

\frac{|\text{estimate}-\text{true value}|}{\text{true value}}\times 100\%.

(a)

Calculate the estimated area of the cover, giving your answer to three significant figures.

[2]

(b)

Write down the lower and upper bounds for the radius.

[2]

(c)

Let $A\text{ cm}^2$ be the true area of the cover. Show that a guaranteed interval for $A$ is

126.67\leq A<130.70.

[3]

(d)

Calculate the maximum percentage error in the area estimate.

[3]

(e)

The workshop prints $129\text{ cm}^2$ on the cover’s label. Determine whether this label is guaranteed to have a percentage error of less than $2\%$ Justify your answer.

[2]

(f)

A new gauge displays the radius as $6.42\text{ cm}$ correct to the nearest $q\text{ cm}$ Determine the greatest possible value of $q$ for which the estimated area is guaranteed to have a percentage error of at most $1\%$ State whether rounding to the nearest $0.1\text{ cm}$ and to the nearest $0.01\text{ cm}$ meet this requirement.

[4]

Question 3

Paper 1Hard15 marks

A workshop is producing identical circular badges for a display. Their common radius is measured as $3.4$ cm, correct to the nearest $0.1$ cm. The combined face area of a batch is the sum of the areas of its badges. The display requires a combined face area of at least $680\text{ cm}^2$

(a)

State the lower and upper bounds for the actual radius $r$

[2]

(b)

Let $A\text{ cm}^2$ be the area of one badge. Show that

11.2225\pi \leq A < 11.9025\pi.

[3]

(c)

Calculate the estimated area of one badge, based on the reported radius, giving your answer to 3 significant figures. Find a conservative upper bound for the percentage error in this estimate, using the estimated area as the denominator. Give the percentage bound to 3 significant figures, rounding upwards.

[4]

(d)

Using the estimated area from part (c), determine the minimum number of badges indicated by the estimate to meet the display’s area requirement.

[2]

(e)

The workshop orders the number of badges found in part (d). Assess the supervisor’s claim that this order is guaranteed to meet the display’s area requirement, and determine the smallest order that would guarantee it.

[4]

Question 4

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 5

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 6

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 7

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 8

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 9

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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