IB Maths AI HL 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 1Paper 2
Difficulty
EasyMediumHard

Question 1

Paper 2Hard6 marks

A comet lander collects dust during a 4.04.0-minute sampling pulse. Figure 1 shows the rate RR at which particles enter the collection chamber. Assume that RR varies linearly between consecutive readings. The mean mass of a particle is 2.4×10−13 kg2.4\times10^{-13}\,\text{kg}

0123400.20.40.60.811.21.41.61.82t (min)R (108 particles s-1)
(a)

Determine the total mass of dust collected during the pulse. Give your answer in scientific notation to 2 significant figures.

[4]
(b)

A return canister has a maximum dust payload of 7.5×10−3 kg7.5\times10^{-3}\,\text{kg} Is it sufficient for the dust collected during the full pulse? Justify your answer with a numerical comparison.

[2]

Question 2

Paper 1Hard6 marks

A remote-sensing archive receives data from two instruments. Instrument A records 4.8×10124.8\times10^{12} observations, each requiring 2.7×1032.7\times10^3 bytes. Instrument B produces 2.24×10152.24\times10^{15} bytes of data. The archive has 4.1×10154.1\times10^{15} bytes of free space. Each additional storage module can hold at most 3.6×10143.6\times10^{14} bytes.

(a)

Show that the data from Instrument A require 1.296×10161.296\times10^{16} bytes. Give your answer in scientific notation.

[2]
(b)

Determine the least number of additional storage modules required to store all the data from both instruments. Justify that this number is least.

[4]

Question 3

Paper 2Hard6 marks

Two accounts each receive a deposit of $10 000. The principal is returned in full by either account, so only interest and fees are compared.

Account A pays simple interest at 3%3\% per year and has no fees.

The annual interest payments observed for account B are shown below.

Year1234
Interest payment ($)246267284306

For a projection, an adviser models account B’s annual interest payments as an arithmetic sequence. The first term is the payment in year 1, and the common difference is the mean of the three consecutive changes shown in the table. Assume this model continues for nn years. Account B has a one-time opening fee of $200.

(a)

State the first term and common difference of the arithmetic model for account B.

[2]
(b)

Let SnS_n be the total interest paid by account B during the first nn years. Show that

Sn=n(10n+236).S_n=n(10n+236).

Hence determine the first whole number of years for which account B’s cumulative interest, after its opening fee, exceeds account A’s cumulative interest. Justify that this is the first such year.

[4]

Question 4

Paper 1Hard6 marks

During an acoustic test, sound level LL in decibels (dB), is defined by

L=10log⁡10(II0),L=10\log_{10}\left(\frac{I}{I_0}\right),

where II is the sound intensity and I0I_0 is a fixed reference intensity. When machines operate together, their sound intensities add.

Machine A alone produces a sound level of 67.467.4 dB and machine B alone produces a sound level of 72.172.1 dB.

(a)

Show that if the sound levels of A and B are LAL_A and LBL_B respectively, then

IBIA=10(LB−LA)/10.\frac{I_B}{I_A}=10^{(L_B-L_A)/10}.
[2]
(b)

Determine the sound level when A and B operate together. Give your answer to the nearest 0.10.1 dB.

[2]
(c)

Identical copies of machine A are used instead. Determine the least whole number of copies needed for their combined sound level to be at least the level found in part (b).

[2]

Question 5

Paper 1Hard6 marks

A field station sends four-letter locker identifiers using a linear code. The letters A to Z are represented by the integers 1 to 26, respectively. Each successive pair of letters is written as a column vector x\mathbf{x} and is encoded as y=Kx\mathbf{y}=K\mathbf{x}

The transmitted pairs, in order, are

(2714)and(4828).\begin{pmatrix}27\\14\end{pmatrix} \quad\text{and}\quad \begin{pmatrix}48\\28\end{pmatrix}.

The same encoding matrix was used for both pairs. It was one of

P=(2111)orQ=(1213).P=\begin{pmatrix}2&1\\1&1\end{pmatrix} \quad\text{or}\quad Q=\begin{pmatrix}1&2\\1&3\end{pmatrix}.
(a)

Show that P−1=(1−1−12)P^{-1}=\begin{pmatrix}1&-1\\-1&2\end{pmatrix}

[2]
(b)

Determine which matrix was used and hence decode the locker identifier. Justify your answer.

[4]

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