Download Free Higher Maths Past Papers to Help You Ace Your Exams


  1. Why should I do higher maths past papers?
  2. When to do higher maths past papers
  3. How do to higher maths past papers
  4. What do I need to know about the SQA higher maths past paper?
  5. What skills are tested in a higher maths past paper?
  6. Where can I find SQA higher maths past papers?


Whether you’ve just started prepping for your exam or you’re knee-deep in revision, here at GoStudent, we like to help you out however we can! So, we’ve done the legwork and put together everything you need to know about SQA higher maths past papers.Exam With Higher Maths Past Papers

Why should I do higher maths past papers?


In addition to your own detailed revision plan, going through as many higher maths past papers as you can get your hands on will undoubtedly help you prepare for the exam itself. Knowing what each paper might include can help you keep your cool in that stuffy exam room.

It’s not enough to memorise everything you can from your notes. Exams don’t just test knowledge, they also test your ability to put what you know into action. This means showing the examiner how you’ve solved a problem or analysed information to get your answer.

Doing a few higher maths past papers before you take your seat on exam day can help you in lots of different ways.

Get familiar with exam structure

Doing past papers can help you get to grips with the different parts of the exam. The more familiar with the layout of the paper you are, the easier it will be to sit down and get started on the day. Simple things like knowing how many questions there are and what the paper will look like can really help you stay in control.🧘

Become familiar with the different types of question

Not all higher maths questions are created equal! Some are worth a mark or two while others may be big hitters! Knowing what to expect from the higher maths questions can help you plan your time and think about where to start. 

Know the lingo

When you’re revising using your own notes or revision cards, it all makes perfect sense – but seeing things written in a different way can throw you. The language of exams and exam questions can be topic-specific. So, doing as many higher maths past papers as possible will help. The more familiar you are with exam language, the more comfortable you’ll feel getting stuck in on exam day.

Time yourself

You don’t have to treat every higher maths past paper like a mock, but knowing where you need to speed up and where it’s essential to take your time can make the whole experience less daunting.

Doing timed papers helps you practice staying focused and will help you plan your exam strategy. Perhaps you need a 5-minute breather after a certain point. If you can factor it in and make it part of your plan, you’ll feel in control when it comes to the crunch.⏲️


When to do higher maths past papers


Don’t wait until just before the exam to have a go at higher maths past papers. Do them early as it’ll give you time to get used to the exam format and will help you identify what you need to go over when revising. If you’ve been revising by reading and writing your own notes, then doing past papers gives you the chance to solve problems in real time and will encourage you to practice showing your working out in a clear and logical way.⚙️

You don’t have to approach every higher maths past paper as a mini-mock. You can mix it up and do some as practice exercises and others under exam conditions.


How do to higher maths past papers


Get the lingo

Firstly, check you know what they’re on about. The language used in higher maths questions is pretty specific so make sure you’re fully clued up about the symbols and terms that might come up.

Check you know the symbols and terms used in the question paper (you may be expected to understand these but don’t necessarily have to use them in your answers):



conventions used for representing sets

{ }



empty set



— ℕ, the set of natural numbers,{ } 1, 2, 3, ...
— 𝕎, the set of whole numbers,{ } 0, 1, 2, 3, ...
— ℤ, the set of integers
— ℚ, the set of rational numbers
— ℙ, the set of real numbers


You’ll also need to make sure you’re comfortable with the language used in the question. These are often called ‘command words’ and can be words as follows:

Command word

What it means


examine something in detail


consider a situation and give your answer based on the information you have


use numbers to work out the value of something


give the meaning something


talk about what you can see (no need for reasons)


give reasons for something


provide evidence to explain something


provide a summary of something

Use colour!

You can use highlighters to highlight key/command words in questions to help you stay on track while answering the question. 🌈

Once you’ve finished your higher maths past paper, use different colours to highlight mistakes so that you can go back and look at them in more detail.

Learn about mark schemes

The good thing about doing higher maths past papers with answers is that you’ll get to know how each question affects your final grade. Mark schemes can be a little scary at first, but the more you look at them, the more you’ll be able to work out a strategy to get the best grade for yourself.

When you’ve finished a past paper, use the mark scheme to work out your result. Seeing the marks tot up can be really motivating and can help build your confidence. Besides, it’s always rewarding to see your hard work paying off. By using the marking instructions, you’ll also be able to look at your work from an examiner’s point of view and find out more about what they are looking for in terms of showing your working out and justifying your answers.🧮

Identify gaps in your knowledge

If you’re scoring badly on a particular type of question – don’t get disheartened. Use it to help you plan your next revision session! When revising, it’s tempting to focus on getting really good at what we already know. 

Real learning is often about allowing yourself to stumble so that you can learn from your mistakes. Doing higher maths past papers is no different. You don’t need to pass with flying colours every time for it to be a useful learning tool – in fact, embracing failure while revising higher maths questions can help you plan your revision more effectively.🙌🏽


What do I need to know about the SQA higher maths past paper?


The SQA higher maths exam is divided into two papers. In Paper 1, you’re allowed to use a calculator but in Paper 2, it’s all down to you and your giant maths brain.🧠

Which paper?

Can I use a calculator?

How many marks?

How long do I have?

Paper 1 



1 hour and 30 minutes

Paper 2



1 hour and 45 minutes



What skills are tested in a higher maths past paper?


Maths is a broad subject and there’s a lot of ground to cover when doing SQA higher maths past papers. Here’s a handy breakdown of the different areas that you might be asked questions on.

(Skills tables and information adapted from SQA course specifications.)

Algebra and trigonometry

Can you…

What you might be asked to do in the exam

manipulate algebraic expressions

  • factorise a cubic/quartic polynomial
  • simplify a numerical expression using
    laws of logarithms and exponents

manipulate trigonometric expressions

  • apply the addition formulae and/or double
    angle formulae
  • apply trigonometric identities
  • convert ɑ cos x + b sin x to k cos (x ± α) or 

k sin (x  ± α), k > 0

identify and sketch related functions

  • identify a function from a graph, or sketch
    a function after a transformation of the form kf (x ), f (kx), f (x) + k , f (x + k) 

or a combination of these

  • sketch y = f ‘(x) given the graph of 

y = f (x)

  • sketch the inverse of a logarithmic or an
    exponential function
  • complete the square in a quadratic expression where the coefficient of x2  is non-unitary

determine composite and inverse

  • show knowledge and use of the terms ‘domain’ and ‘range’
  • determine a composite function given f (x)
    and g (x) , where f (x) and g (x)  can be
    trigonometric, logarithmic, exponential or
    algebraic functions
  • determine f -1(x)of functions

solve algebraic equations

  • solve a cubic or quartic polynomial equation
  • use the discriminant to find an unknown, given the nature of the roots of an equation
  • solve quadratic inequalities, 

ɑx2 + bx + c  ≥ 0  (or ≤ 0)

  • solve logarithmic and exponential equations
  • use the laws of logarithms and exponents
  • solve equations of the following forms for
    ɑ and b, given two pairs of corresponding
    values of x and y :

log y = b log x + log ɑ, y - ɑxb


log y = x log b + log ɑ, y - ɑbx

  • use a straight-line graph to confirm
    relationships of the form y = ɑxb , y = ɑbx 
  • mathematically model situations involving the logarithmic or exponential function
  • find the coordinates of the point(s) of
    intersection of a straight line and a curve or of two curves

solve trigonometric equations

  • solve trigonometric equations in degrees or
    radians, including those involving the wave
    function or trigonometric formulae or identities, in a given interval


Can you…

What you might be asked to do in the exam

determine vector connections

  • determine the resultant of vector pathways in three dimensions
  • work with collinearity
  • determine the coordinates of an internal
    division point of a line

work with vectors

  • evaluate a scalar product given suitable
    information and determine the angle between
    two vectors
  • apply properties of the scalar product
  • use and find unit vectors including i, j, k as
    a basis


Can you…

What you might be asked to do in the exam

differentiate functions

  • differentiate an algebraic function which is, or can be simplified to, an expression in powers of x
  • differentiate k sin x  and k cos x
  • differentiate a composite function using the
    chain rule

use differentiation to investigate
the nature and properties of

  • determine the equation of a tangent to a curve at a given point by differentiation
  • determine where a function is strictly increasing or decreasing
  • sketch the graph of an algebraic function by
    determining stationary points and their nature as well as intersections with the axes and behaviour of f (x)  for large positive and negative values of x

integrate functions

  • integrate an algebraic function which is, or can be, simplified to an expression of powers of x
  • integrate functions of the form

f (x) = (x  + q)n , n ≠ −1

  • integrate functions of the form

f (x) = p cos and f (x) = p sin x

  • integrate functions of the form

f (x) = (px  + q)n , n ≠ −1

  • integrate functions of the form

f (x) = p cos (qx  + r)  and p sin (qx + r)

  • solve differential equations of the form

dydx = f (x)

use integration to calculate definite integrals

  • calculate definite integrals of functions with
    limits which are integers, radians, surds or

apply differential calculus

  • determine the optimal solution for a given
  • determine the greatest and/or least values of a function on a closed interval
  • solve problems using rate of change

apply integral calculus

  • find the area between a curve and the x-axis
  • find the area between a straight line and a
    curve or two curves
  • determine and use a function from a given
    rate of change and initial conditions

Algebra and geometry

Can you…

What you might be asked to do in the exam

apply algebraic skills to
rectilinear shapes

  • find the equation of a line parallel to and a line perpendicular to a given line
  • use m = tan θ to calculate a gradient or angle
  • use properties of medians, altitudes and
    perpendicular bisectors in problems involving the equation of a line and intersection of lines
  • determine whether or not two lines are

apply algebraic skills to circles
and graphs

  • determine and use the equation of a circle
  • use properties of tangency in the solution of a problem
  • determine the intersection of circles or a line
    and a circle

model situations using

  • determine a recurrence relation from given
    information and use it to calculate a required
  • find and interpret the limit of a sequence,
    where it exists


Can you…

What you might be asked to do in the exam

interpret a situation where
mathematics can be used and
identifying a strategy

  • analyse a situation and identify an
    appropriate use of mathematical skills

explain a solution and, where
appropriate, relate it to context

  • explain why a particular solution is appropriate
    in a given context



Where can I find SQA higher maths past papers?


The SQA have made a few higher maths papers PDF files available.

Higher maths sample papers

Firstly, you can look at the SQA’s higher maths sample papers which have handy marking instructions at the end. Spoiler alert! You may see some of the questions in the sample paper repeated in the past papers.🔁

Sample paper 1

Sample paper 2

Higher maths past papers

You can also find free higher maths past papers for the exam from 2016 – 2019 here (past papers from before 2019 carry a different amount of marks).


Higher maths past papers


Marking instructions


past papers 1 and 2

Paper 1 – 70 marks

Paper 2 – 80 marks

marking instructions for 2019


past papers 1 and 2

Paper 1 – 60 marks

Paper 2 – 70 marks

marking instructions for 2018


past papers 1 and 2

Paper 1 – 60 marks

Paper 2 – 70 marks

marking instructions for 2017


past papers 1 and 2

Paper 1 – 60 marks

Paper 2 – 70 marks

marking instructions for 2016

Learning from others - Real examples of higher maths past paper answers

Another way to make the most of higher maths past papers is to look at an example of a real candidate’s answers and look at how they were marked by reading the examiner’s commentary. ⚖️

Here you can see the answers from a candidate from 2019 and how they scored. (It’s probably better to look at this after you’ve completed the 2019 past paper yourself)



Paper 1 candidate’s answers

Paper 1 commentary

Paper 2 candidate’s answers

Paper 2 commentary


Finally, as good old Will Shakespeare himself wrote,

All things are ready, if our mind be so.

By taking the time to have a bash at higher maths past papers, you’re readying your mind for the challenge of the exam and will give yourself the best possible chance of success.

While you’re revising, you might find that you need a little more help getting to grips with a certain topic or type of question. If that’s the case, then why not think about booking a free trial session with one of our expert tutors.