Maths questions

Maths questions often start with the command words 'Calculate...' or 'Determine...'. They will then have a blank space for you to show your working. It is important that you show your working, don't just write the answer down. Calculation errors carried forward are worked through to give credit for later working.

In some maths questions you will be required to give the units. This may earn you an additional mark. Don't forget to check whether you need to do this. Take extra care when converting between units.

Maths questions might include graphs and tables as well as calculations. Don't forget to take a ruler and scientific calculator into the exam.

If drawing graphs, make sure you:

  1. put the independent variable on the x-axis and the dependent variable on the y-axis
  2. construct regular scales for the axes
  3. label the axes appropriate
  4. plot each point accurately
  5. draw a straight or curved line of best fit

If you are asked to calculate an answer and it has lots of decimal places, don't forget to only use one more than the data in the question. For example, if whole numbers are given in the question, then your answer would be to one decimal place. Don't forget to check your rounding.

These questions have been written by Bitesize consultants as suggestions to the types of questions that may appear in an exam paper.

Sample question 1 - Foundation

A study looked at people's sodium intake - mostly in the form of salt - and rates of stomach cancer in populations of different countries.

A scatter diagram of the results of the study are shown below.

A scatter diagram showing the number od deaths from stomach cancer.
Question

Discuss the results of the study. [1 mark]

There is a correlation between the intake of sodium (ions) and the death rate from stomach cancer - as the intake of sodium increases, the rate of stomach cancer increases also.

Question

Suggest what might affect the validity of the study in investigating the effects of sodium intake. [4 marks]

  • not all the data fits the pattern - rates of stomach cancer in Japan and South Korea are higher than expected from the trend, while the rate in Canada is lower than expected by the trend
  • other factors may be involved, such as genetic factors, or some other factor that promotes the cancer or is protective against it
  • salt intake may not have been monitored accurately - by individuals or across the population
  • the reliability of recording rates of stomach cancer may vary from country to country
Question

Suggest what the scientists should do before drawing a firm conclusion about sodium ion intake, or salt intake, and stomach cancer. [1 mark]

  • investigate a possible mechanism for the link, ie a mechanism by which salt intake could cause cancer
  • in fact, we know that salt promotes the growth of a bacterium called Helicobacter pylori in the stomach, which is linked with the development of stomach cancer

Sample question 2 - Foundation

Question

A study in Switzerland investigated the relationship between the number of cigarettes smoked per day by mothers and the number of premature births.

Some of the results of the study are shown below:

Number of cigarettes smoked by the mother per dayNumber of babies born prematurely (%)
05.4
1 - 96.2
10 - 198.4
≥ 2012.6

Plot a histogram of the data. [4 marks]

Your histogram should look like this:

A graph showing the number of babies born prematurely.
  • axes and scales correct [1]
  • all bars plotted correctly [2] or two to three bars plotted correctly [1]
  • no gaps between bars [1]

In an exam, the wording of the axis labels does not have to match precisely that of the graph above, but the labels must represent accurately the data plotted. Use the table headings as a guide.

As this is a histogram, the bars must not have gaps between them, as the x-axis is continuous.

Sample question 3 - Higher

Question

The diameter of an alveolus can range in size from 200 to 400 μm.

Alveoli are roughly spherical.

The surface area of a sphere can be calculated using the formula:

 \text{surface~area~of~a~sphere} = {4}~{\pi} {r}^{2}

The volume of a sphere can be calculated using the formula:

 \text{volume of a sphere} = \frac{4}{3}~{\pi}{r}^{3}

Where  \pi = pi use the value of  \pi as 3.14 for your calculations.

r = the radius of the sphere.

In this example, assume that an alveolus is a perfect sphere.

Calculate the surface area to volume ratios of:

  • an alveolus 200 μm in diameter
  • an alveolus 400 μm in diameter

Show all your working. [6 marks]

For:

  • an alveolus 200 μm in diameter - surface area to volume ratio is 0.030.
  • an alveolus 400 μm in diameter - surface area to volume ratio is 0.015.

Award:

  • [1 mark] each for the calculation of surface areas
  • [1 mark] each for the calculation of volumes
  • [1 mark] each for the calculation of surface area to volume ratios

Note that if the question says that you must show all the stages of your working out, then you must do that.

Calculations:

The radius of a circle is half the diameter.

For an alveolus 200 μm in diameter - radius 100 μm - the surface area to volume ratio is:

 \text {Surface area of the sphere} = {4}~{\pi} {r}^{2}

 = {4}\times {3.14} \times {(100 \times 100)}= {125~600}~μm^2

 \text{Volume~of~the~sphere} = \frac {4}{3}~{\pi}{r}^{3}

 = \frac{4}{3} \times {3.14} \times {(100 \times 100 \times 100)} = {4~186~667}~μm^3

The surface area to volume ratio =  \frac{125~600}{4~186~667} = {0.030}

For an alveolus 400 μm in diameter:

\text{Surface area of the sphere} = {4}~{\pi}{r}^{2}

= {4}\times{3.14} \times{(200 \times 200)} = {502~400~μm}^{2}

\text{Volume of the sphere} = \frac{4}{3}~{\pi}{r}^{3}

 = \frac{4}{3}\times{3.14}\times{(200\times200\times200)} ={33~493~333}~μm^{3}

The surface area to volume ratio =  \frac{502~400}{33~493~333} = 0.015

Sample question 4 - Higher

Question

The number of new cases of malignant skin cancer in the UK increased at a steady rate between 1975 and 2010.

YearNumber of new cases per 100,000 of population
19753.20
201017.50

Calculate the rate of increase in the number of new cases of malignant skin cancer. [3 marks]

The rate of increase is 0.41 cases per 100,000 of the UK population every year.

Calculation:

Increase in cases = Number of new cases per 100,000 of population in 2010 - Number of new cases per 100,000 of population in 1975

= 17.5 - 3.2 = 14.3

Rate of increase = \frac{increase~per~100~000~of~population}{number~of~years}

 = \frac{14.3}{35} = {0.41}