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 1 Paper 2

Difficulty

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Question 1

Paper 1Hard6 marks

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

(a)

Show that the data from Instrument A require $1.296\times10^{16}$ 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 2

Paper 1Easy5 marks

A neighbourhood repair fund plans to increase its annual contribution by the same amount each year. Figure 1 shows the planned contributions for the first three years.

The planned contributions form an arithmetic sequence.

(a)

Write down the first term and the common difference of the sequence.

[2]

(b)

Find the planned contribution for year 8.

[1]

(c)

Calculate the total planned contributions for the first eight years.

[2]

Question 3

Paper 1Easy6 marks

A laboratory compares the hydrogen ion concentrations of two aqueous solutions. Figure 1 shows the measured concentrations. For these solutions, the one with the greater hydrogen ion concentration is more acidic.

For the pH calculations, use

\mathrm{pH}=-\log_{10}(C),

where $C$ is the numerical value of the hydrogen ion concentration in $\mathrm{mol\,dm^{-3}}$

(a)

Write down the hydrogen ion concentration of solution B in standard form, including its unit.

[1]

(b)

Calculate the pH of solution A, giving your answer to 2 decimal places.

[2]

(c)

Determine the ratio of the hydrogen ion concentration of solution A to that of solution B, giving your answer in standard form.

[2]

(d)

Hence state which solution is more acidic.

[1]

Question 4

Paper 1Hard6 marks

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

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

where $I$ is the sound intensity and $I_0$ is a fixed reference intensity. When machines operate together, their sound intensities add.

Machine A alone produces a sound level of $67.4$ dB and machine B alone produces a sound level of $72.1$ dB.

(a)

Show that if the sound levels of A and B are $L_A$ and $L_B$ respectively, then

\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 $0.1$ 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 1Easy6 marks

A point $P$ on an Argand diagram is represented by $z=1+2i$

(a)

Write down the real part and the imaginary part of $z$

[2]

(b)

Write down the complex conjugate of $z$

[1]

(c)

Determine the distance of $P$ from the origin, in units.

[1]

(d)

Use your GDC to calculate $z^3$ giving your answer in Cartesian form.

[2]

Question 6

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 $\mathbf{x}$ and is encoded as $\mathbf{y}=K\mathbf{x}$

The transmitted pairs, in order, are

\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=\begin{pmatrix}2&1\\1&1\end{pmatrix} \quad\text{or}\quad Q=\begin{pmatrix}1&2\\1&3\end{pmatrix}.

(a)

Show that $P^{-1}=\begin{pmatrix}1&-1\\-1&2\end{pmatrix}$

[2]

(b)

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

[4]

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