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

Easy Medium Hard

Question 1

Paper 1Medium6 marks

A laboratory receives microscopic particles in sealed deliveries. Figure 1 shows the mass of one particle and the numbers of particles and cartridges in a cartridge and a delivery, respectively. The diagram is not to scale.

(a) Calculate the mass of particles in one cartridge. Give your answer in standard form, correct to 2 significant figures. [2]

(b) Calculate the total mass of particles in one delivery. Give your answer in standard form, correct to 2 significant figures. [2]

(c) A laboratory requires at least $1.0\,\text{g}$ of particles. Find the minimum whole number of deliveries required. [2]

(a)

Calculate the mass of particles in one cartridge. Give your answer in standard form, correct to 2 significant figures.

[2]

(b)

Calculate the total mass of particles in one delivery. Give your answer in standard form, correct to 2 significant figures.

[2]

(c)

A laboratory requires at least $1.0\,\text{g}$ of particles. Find the minimum whole number of deliveries required.

[2]

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 1Medium7 marks

A town records the annual grants it gives to a park. The recorded grants are not exactly arithmetic, so an arithmetic model is to be used for estimating future grants. Use the values for years 1 and 5 to estimate the common difference.

(a)

Write down the first term of the arithmetic model.

[1]

(b)

Estimate the common difference and hence write an expression for the estimated grant, $G_n$ in year $n$

[2]

(c)

Write, in sigma notation, an expression for the total estimated grants in years 1 to 6, and calculate this total.

[2]

(d)

Using the model, determine the first year in which the estimated grant exceeds $7000.

[2]

Question 4

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 5

Paper 1Medium6 marks

A fountain jet is modelled, for $0\leq x\leq 6$ by

h=ax^2+bx+c,

where $x$ is the horizontal distance from the nozzle, in metres, and $h$ is the height above the ground, in metres. The model passes through the points shown.

$x$	0	3	6
$h$	1.20	4.80	5.10

(a)

Use technology to determine the values of $a$ $b$ and $c$ giving each to at least three significant figures.

[3]

(b)

Hence determine the horizontal distance(s), in the interval $0\leq x\leq 6$ at which the model gives $h=3.00$ Give your answer to the nearest $0.01$ m.

[3]

Question 6

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 7

Paper 1Medium6 marks

The equation $x^2-8x+25=0$ has roots $z$ and $\bar z$ where $\operatorname{Im}(z)>0$

(a)

Find $z$ in Cartesian form.

[2]

(b)

Let

q=\frac{z-2}{\bar z-2}.

Find $q$ in Cartesian form.

[2]

(c)

Find the modulus and principal argument of $q$ Give the argument in radians, correct to three significant figures, in the range $-\pi<\arg(q)\leq\pi$

[2]

Question 8

Paper 1Medium5 marks

A shop records the weekly sales of two products. The sales are shown in Figure 1. The selling prices remain constant: product A costs $4.80 per unit and product B costs $6.25 per unit.

(a)

Write down the $2\times3$ matrix $S$ of weekly sales, with rows corresponding to products A and B, in that order, and columns corresponding to Weeks 1, 2 and 3, in that order.

[2]

(b)

Let $\mathbf p=\begin{bmatrix}4.80&6.25\end{bmatrix}$ the row matrix of prices in dollars per unit.

Using your matrix from part (a), show that the revenue in Weeks 1, 2 and 3 is given by

\mathbf pS=\begin{bmatrix}215.20&219.00&261.40\end{bmatrix}.

[2]

(c)

Hence, state the week with the greatest revenue.

[1]

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