NSAA Guides

# NSAA Mathematics - Your Guide to Part A of the NSAA

Written by: Matt Amalfitano-Stroud

The NSAA has four whole subjects that you’re going to need to revise, the three core sciences as well as mathematics. You’ll have a choice between which sciences you want to pick, but mathematics is not only the only questions set you won’t have a choice on but is also a subject that looms over each of the three other subjects. Essentially, you’re going to need to know your maths to succeed at any part of the NSAA. That’s where this guide will come in handy, we’re going to be focusing solely on the pure mathematics that you’ll be expected to know when sitting the exam. Let’s get started!

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## BASICS OF THE NSAA

Before we get our maths hats on, let’s take a moment to refresh our memory on what the NSAA actually is!

### What is the NSAA?

The **Natural Sciences Admissions Assessment (NSAA)** is an exam used by the **University of Cambridge** as an admissions test for their courses in **Natural Sciences** and **Veterinary Medicine**. The NSAA is a **pre-interview assessment **and in 2023, it will be sat on **October 18th**, a few weeks earlier than previous years.

The NSAA is designed to test your** mathematical and scientific knowledge** and to let the admissions team determine your suitability for the university. Both courses are demanding and require an in-depth understanding of everything covered within the assessment, so starting your revision now is a great way to ensure you’ll succeed when you’re actually studying at the university.

The assessment format is **two sections**, **Section 1** and **Section 2**. Both sections involve a **choice between the three core sciences**, **Physics**, Chemistry and Biology, each of which is split into its own “**part**” that contains **20 questions**. Remember, you **only pick one of these parts** so don’t answer more questions than is necessary!

There are two major differences between Section 1 and Section 2. Firstly, Section 2 will be covering **advanced topics**. While Section 1 covers knowledge of each science at around a **GCSE (or equivalent) level**, Section 2 dives deeper into more **unfamiliar territory**, leading to **much harder questions** that you will need to apply to more recent studies to.

Secondly, Section 1 includes an **additional part** before the science questions, **Mathematics**. Here, you will need to answer **20 pure maths questions **which will be to a** standard level**. You will, however, need advanced maths in Section 2, as many of the questions here will require a solid knowledge of **advanced mathematics**.

Every part of the NSAA features **20** **questions**, all of which are **multiple-choice**. Each section lasts **60 minutes**, leading to a time limit of **two hours** for the assessment. This chart gives a quick rundown of the test format:

#### NSAA Format Chart

### How is the NSAA Scored?

The NSAA is scored on a **scale of 1.0 to 9.0**, with each part being scored separately. This score is determined by the raw marks from each part you completed, with every **correct answer increasing your score** on the scale. The conversion between raw marks and the **final score differs between each part**, which you can see below:

#### NSAA Section 1 Score Conversion

#### NSAA Section 2 Score Conversion

There’s **not an official passing or failing score **for the exam, but different colleges within Cambridge will have **different scoring standards**, so be aware of what they may be when choosing a college. Our **NSAA Scoring Guide** goes further into how the exam is scored and what your results mean.

For the NSAA, the only equipment you will need to bring is a **soft pencil and eraser**. **Calculators are not permitted** for either Section, but a periodic table is provided in Section 2.

If you want to find out more about the NSAA, you can check out our **Def****initive NSAA Guide**, which covers everything you need to know about the exam!

**Boost your NSAA skills with Exams.Ninja!**

With the **NSAA Preparation Platform**, you’ll be able to use **over 100 guides** and **over 1,200 practice questions** to get yourself ready for **every part of the NSAA**. Start your preparation now by setting up a **free account!**

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## THE MATHEMATICS OF NSAA SECTION 1

Section 1 of the NSAA doesn’t just require a good knowledge of maths to get through the science questions. It has its own set of pure mathematics questions that you’ll need to get through. What you need to know isn’t as advanced as what we’ll look at later, so. let’s start off with the more standard stuff!

Beyond the **assumed mathematics knowledge** that is required for **Biology, Chemistry and Physics**, you’re going to need to know how to answer some **pure maths questions** in Section 1. Let’s go through the **Official NSAA Specification** to see exactly what you’re going to need to know for this section!

### Units

We’re starting off with the **absolute basics** here! You will of course be required to know how to use **various units in a mathematical and scientific context**. These units range from your basic measures of time, distance and weight, currency, area and volume.

Given that this is a **science-focused exam**, you will also need to have a good understanding of your **various scientific units,** be it those relating to forces, electricity and more! Those aren’t specific to mathematics however so we’ll need to cover those another time.

### Numbers

We can only hope that you’ve learnt how to use **standard number functions** at this stage in your education! This topic essentially covers your understanding of integers, place values, basic mathematical symbols, exponents, parentheses and the four operations. Part of that will include your understanding of the **relationships between operations**, particularly the **order of operations**, otherwise known as **PEMDAS**.

As a basic example, solve this problem using the **correct order of operations**:

**(17 – 6 ÷ 2)+ 4 x 3**

This topic also covers your understanding of fractions, decimals and percentages, as well as your use of the **above concepts** with them. All of this simply boils down to **knowing how to do basic maths!**

There are a few terms and **slightly more advanced principles** to bear in mind too, including negatives, squaring, roots, Pi, surds and rounding. There’s too much to cover everything in detail, but these are all concepts that should be i**ngrained in your brain** at this point!

(FYI: the answer was 26!)

### Ratio and Proportion

This topic is all about **scales and quantities**. Ratios in their **most basic form** are essentially the following:

Here are **five blocks**, two orange and three blue. **The ratio here is 2:3**. Of course, this can easily get more complex by adding **greater variation** and **higher quantities**. This will then pave the way for concepts such as **scale factors, scale diagrams** and **interchangeable ratios**. You’ll also need to be conformable with using ratios alongside **fractions, decimals and percentages**, plus, of course, the use of ratios in **practical, real-world scenarios**.

Proportionality could be seen as an even simpler topic. It’s essentially the r**elationship between different values when manipulated**. A linear example could be the increase of a shopping cost as more of an item at the same price is added. The cost of the item and the cost of all the shopping **are linked and increase in a proportional manner**. Non-linear proportion is much more advanced and goes hand in hand with a topic like **geometric manipulation**, being used in relation to lengths and areas.

Overall, the concepts at play here aren’t too tricky, the challenge comes more from the **values and contexts you’ll be working with**.

### Algebra

Algebra is where we really start to sink our teeth into the kind of knowledge that is really going to be tested in the exam.

It’s essentially the use of **symbols to represent values** in a manner that is easier to work with. The questions in the NSAA for algebra can range from finding missing values represented by letters, solving simplifying extended equations into a more understandable format or even representing algebraic equations in a graphic format.

The most essential thing to know for algebra is **how values are represented**. Here are a few common examples:

*ab *= *a* x *b*

**3 a = a + a + a**

*a²* = *a* x *a* * *

*a*/*b *= *a *÷ *b*

From here, your job is going to be to solve one of the following:

**Quadratic Equations **

First of all, you’ll definitely need to know the **quadratic formula**: ** ax2 + bx + c = 0** (

*a*,

*b*and

*c*represent actual values). You’ll also want to know the formula

**.**

*x*= (*b*± √*b*² − 4*ac*)/2*a*Using this is one of the **three ways you can solve a quadratic equation**, with the other two being **Factorisation and Completing the Square**. No matter what method you use, your goal here is usually to work out the **missing value in the equation**.

#### Simultaneous Equations

This typically involves manipulating **two separate equations** to find the **common values **featured within both, typically represented by ** x and y**. Sometimes you’ll be given one of these values, sometimes you won’t! Take these equations as an example:

**3 x+y=10**

**4 x−3y=9**

There are **four steps** we need to take to solve this problem. We need to **pick a variable to eliminate** (for this example we’ll take away *y*), **eliminate said variable**, **solve the remaining variable** and **substitute the now known variable back into the equations** so we can find *y*. By taking these steps, we can find that ** x = 3** and

**.**

*y*= 1**Graphic Questions**

As we mentioned, any algebraic equation can be **represented in a graphic form**, which is something you’re likely going to have to do in some form for the NSAA. All of the questions are multiple-choice, so you **won’t need to actually sketch a graph **for a question, but you may need to **identify a correct graph** or **find points on a line**. It’s also a handy skill for when you are working out more advanced questions.

You should expect to find **linear and quadratic graphs** in these questions, which will typically look something like this:

##### Linear Graph

##### Quadratic Graph

### Geomerty

Geometry is basically the **mathematics of shapes**, a favourite for the visual learners out there! You’ll be needing to find angles, perimeters, areas, volumes and more. The most basic rule to remember is that **angles of any complete shape add up to 360º**.

Different shapes have different rules, the most basic of these being **squares and rectangles**. The angles here are simple, at **90º in each corner**. The perimeter is just a matter of **adding all the sides together**, while the **area = length x width**. The same applies to a **cuboid**, except **height will also be included in the equation**.

Other shapes such as trapeziums and circles have different rules when it comes to area:

**Trapezium – A = 1/2( a + b)h** (

*a*and

*b*represent the lengths of the parallel sides of the shape)

**Circle – A = π r²** (

*r*is the radius of the shape, which is found by dividing the circumference by 2π)

Circles in particular have a **very different set of rules**, known as **Circle Theorem**. These relate to the angles, radius and chords that are found within a circle and are essential to know.

**Basic trigonometry** is also covered in the part of the specification, including knowledge of **Pythagoras’s theorem** (a² + b² = c²), **trigonometric ratios** and **basic congruence criteria**.

### Statisitics

Statistics is essentially the **mathematics of data**. The skills required for this subject is a mixture of **interpreting data and representing it in different formats**. You’re going to need to be able to read through **potentially large amounts of numbers and related data **and use what you have available to you to solve problems.

In terms of graphic interpretation of the data, you can expect to see anything from** bar charts, line graphs and pie charts**. It’s important that you know how to read these and are able to **extract pure data** from them. You may also need to **compare data between two or more different formats** in order to come to a single conclusion.

Lastly, you’re going to need to know how to **calculate the averages** of grouped and ungrouped data:

**Mean** – **m = sum of terms/number of term** (essentially the overall value of the data divided by the number of individual pieces of data).

**Median **– This is the **value that fits in the exact centre of the data set**, it will likely need to be found by hand at this level as you won’t have too much data to go through.

**Range** – This is the **difference between the highest and lowest values **and is simply found by **subtracting the lowest value from the highest value**.

**Mode **– the least common of the averages, this is the **most frequent value to appear in a data set** and will again need to be found by hand.

### Probability

This is basically about chance. The **chance of an event happening under certain circumstances** and the **chances of multiple events happening together**. A key term to understand here is uncertainty, which is what probability seeks to resolve.

Questions on probability will almost always revolve around **real-world scenarios**, even if it’s something as simple as tossing a coin. Therefore, you’ll need to **read the question carefully** to ensure you understand every detail that you need to consider.

There are **4 major rules to probability** that you should learn:

**Equally Likely Events:**If events are equally likely, that means that the probability is a**simple split between the different events**(for example picking out a red or blue ball from a bag when there are 3 of each colour). A question with this kind of event will likely involve an event that makes things unequal.**Probabilities must sum to 1:**This one’s simple, the**values of all your probabilities need to equate to 1**, as there is always a 100% chance of one of the potential events happening (at least in context of an NSAA question).**Multiple Events:**Sometimes you may be asked the probability of**multiple events happening**. That doesn’t mean two events happening at the same time or one after the other, but rather**combining the chances of two potential events into one**(for example, the probability of rolling a 1 or 2 on a dice). These would initially be considered as 2 separate chances (e.g. 1/6 chance for each). But with this rule, we would combine the probabilities to 2/6, or simplify to 1/3.**Conditional Probability:**Here, we must consider if a probability is**reliant on a separate event**. Going back to the red and blue balls example, if 1 red ball were to be taken out of the 6, the probability of then picking out a red ball would be 2/5, and blue 3/5. These probabilities are conditional on the fact that a red ball was initially picked.

You will also need to understand various** graphic representations of probability**, including **Venn diagrams** and **Tree diagrams** (one of which is shown below).

If you’re looking to learn more about the other parts of Section 1, then our **NSAA Section 1 Guide** will be perfect for you!

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## ADVANCED MATHEMATICS IN THE NSAA

Moving onto Section 2 of the NSAA, this is where things will really begin to heat up. As well as the maths you’ve just gone through, you’re going to need. to be confident in the use of advanced mathematics. Bear in mind, you won’t be answering any pure maths questions, but all of this stuff is going to be needed in whichever advanced science part you decide to take on. Let’s get straight to it!

### Algebra and Functions

We looked at this topic already, but there’s loads more that we need to consider for advanced maths. Let’s start off with something a bit simpler. We discussed before about the functions within algebra, but in advanced maths, we’re going to need to take into account the **Laws of Indices**. These essentially show us the **functions that can be performed with all rational exponents and intergers**, which are represented by *n* and *m*:

*a ^{n}* x

*a*=

^{m}*a*

^{n + m}**( a^{n})^{m} = a^{nm}**

*a ^{n}*/

*a*=

^{m}*a*

^{n – m}*a*^{0} = 1

*a*^{1} = *a*

*a*^{-1} = 1/*a*

*a*^{1/n} = ^{n}*√a*

Knowing these will be vital for succeeding in any algebra-based question.

Following on from this, you are also going to need to understand various techniques, including **manipulation of surds, solving quadratic inequalities and simplifying polynomials.**

it is also important that you understand various **basic functions **and how they may be used graphically. This is explained below:

- A function can be
**one-to-one or many-to-one**(they cannot be one-to-many). When trying to figure out which way round these are it does help to think of functions as a ‘**black box**‘ which takes an input (or several) and, as if by magic, gives an**output for each of them**. - If a function is one-to-one,
**one value of**(for example, the line*x*only gives one value of*y*gives*y*= 3*x*+ 2*y*= 2 when*x*= 0, and*y*doesn’t equal 2 anywhere else on the line. - If a function is many-to-one,
**lots of values of**(for example, in quadratic graphs like*x*can give the same value of*y**y*=*x*² ‘inputting’ both 2 and -2 as*x*would give the same output of 4) - Things like
**circles**would be examples of**one-to-many situations**(in a circle centred on the origin with radius 5, the input*x*= 0 gives an output of both*y*= 5 and*y*= -5) – this**is not a function**as a result – if you were inputting numbers into a computer and it had loads of outputs and ‘didn’t know which it wanted’ that’d usually be pretty unhelpful!

### Sequence and Series

This area of maths is, in simple terms, about **pattern recognition**. A sequence in this context is defined as an ordered list of numbers, worked out by a given formula or relation.

There are two types of sequences that you will need to know about:

**Arithmetic Sequence: **These are progressions in which the term increases by a **fixed amount each time**, a, a+d, a+d+d, a+d+d+d, and so on… things like 1, 3, 5, 7, 9…

**Geometric Sequence: **These are progressions in which the **term is multiplied (or divided) by a fixed amount each time**, a, 2a, 4a, 8a, and so on… this looks something like 5, 15, 45, 135…

Questions involving either of these will likely be asking you to either find the *n*th term in the sequence or find the next term.

The other major aspect of this topic is **Binomial Expansion**, which is an extremely helpful way of finding **statistics or approximate expressions without a calculator** (which you’re going to need to know considering you won’t have a calculator).

### Coordinate Geometry

This is another topic we already had a look at, but there’s still plenty more to be aware of. When we say coordinate geometry, this essentially means **two-dimensional**, as you will only be working with the ** x and y planes**.

First of all, let’s remind ourselves of some **key equations** that you’ll need to remember when it comes to graphing:

**The Equation of a Straight Line –** *y* = *mx* + *c*

Here are a few examples of this equation in action:

We can also use the equations:

**y – y _{1} = m(x – x_{1}) **

**b. ax + by + c = 0 **

**Circle Equation – ****(x − a)²**** ****+ ( y − b)²**

**= r²**

**The circle of this equation can be seen below:**

We can also use the equation: *x²* + *y*² + *cx* + *dy* + *e* = 0

There are various **circle properties** to understand, including:

a. The perpendicular from the centre to a chord bisects the chord.

b. The tangent at any point on a circle is perpendicular to the radius at that point.

c. The angle subtended by an arc at the centre of a circle is twice the angle subtended by the arc at any point on the circumference.

d. The angle in a semicircle is a right angle.

e. Angles in the same segment are equal.

f. The opposite angles in a cyclic quadrilateral add to 180°.

g. The angle between the tangent and chord at the point of contact is equal to the angle in the alternate segment.

Lastly, be sure to understand how **parallel and perpendicular lines** work in a geometric context.

### Trigonometry

This could be lumped into the same boat as geometry, and in a broad sense they are the same, but trigonometry has so many **unique rules and methods** that it really needs to be looked at as its own separate thing.

Starting with some basics, you need to be aware of the **sine and cosine rules**.

**Sine Rule – ***a*/*sin A *= *b*/*sin B *= *c*/*sin C*

**Cosine Rule – ***a²* = *b²* + *c²* – 2*bc cosA*

We also need to be able to find the area of a triangle using the form 1/2*ab sinC. *As an example, **The area of a triangle ABC shown below is given by **1/2*ab sinC *= 1/2*bc sin A *= 1/2*ac sin B* ** **

You’re also going to need to be aware of **Radian Measure**, which is the **ratio of the circular arc (a) to the radius of the arc (r)** in any given part of a circle’s circumference.

Lastly, you need to know about the **Sine, Cosine and Tangent functions**. This will include their graphs, symmetries and periodicity. The basics of these functions in regard to a right-angled triangle are as follows:

**sin θ = opposite/hypotenuse**

**cos θ = adjacent/hypotenuse**

**tan θ = opposite/adjacent**

You’ll also need to be aware of the following equations:

**a. tan θ = sinθ/cosθ **

**b. sin²θ + cos²θ = 1 **

### Exponentials and Logarithms

An exponential function is a **function of the form y = ax**, where ‘a’ can be

**any positive real number**. These can be fairly quickly put into a graph, which you will need to know how to do for simple positive numbers in this case. Take

**as an example:**

*a*= 2As for logarithms, you going to need to be familiar with their laws:

** a^{b} = c **⇔

*b*=*log*_{a}c**Product Rule: log_{a }x + log_{a} y = log_{a}(xy) **

**Quotient Rule: log_{a}x – log_{a} y = log_{a} (y/x)**

**Power Rule: x log_{a }(y) = log_{a }y^{x}**

*klog _{a} x* =

*log*(

_{a}*xk*)

The are also two special cases to consider:

*log _{a}*(1/

*x*) = –

*log*

_{a}*x*

*log _{a} a* = 1

You’ll mostly be using all of this to **solve equations**, such as 3^{2x} = 4 and 25^{x} – 3 × 5^{x} + 2 = 0

### Differentiation

Differentiation simply means ‘find the gradient/slope of a curve’ in its most basic sense. It could be described as finding the rate of change in one thing with respect to another.

If we differentiate *y *with respect to *x*, the function we get is written as *dy*/*dx. *

Often, we have a function written as *y* = *f*(*x*) in which case we also sometimes write dydx=f′(x)

“>*d**y*/*dx *= *f*′(*x*).

One key area you will need to revise is differentiating* x ^{n}* for rational

*n.*

### Graphs of Functions

Graphs of functions have been a theme throughout a lot of what we’ve spoken about, but we’ll quickly refresh our memory on what they look like to finish things off.

#### Line Graphs

#### Quadratic Graphs

#### Polynomials Graphs

This graph represents the polynomial* **y *= (*x*−1)2(*x*+2).

*y *= *sinx*

*y *= *cosx*

*y *= *tanx*

*y *= log²*x*

*y *= 2*x*

*y *= 1/*x*

Remember the four major types of graphic transformations as well:

*y*=*af*(*x*) – stretch of scale factor aa in the*y*direction.*y*=*f*(*x*) +*a*– translation of aa in the positive*y*direction.*y*=*f*(*ax*) – stretch of scale factor 1/*a*in the*x*direction.*y*=*f*(*x*+*a*) – translation of scale factor −*a*in the positive*x*direction.

That covers **everything that you’re going to need** to know for the NSAA, at least in terms of pure mathematics.

Don’t forget, there are **three whole sciences that you need to be prepared for**, so be sure to plan out your revision so you have time to look at everything! Not sure how to plan it out? Our **NSAA 6-Month Preparation Timeline** will guide you in the right direction!

**Still unsure about what you need to know for the NSAA?**

The **NSAA Preparation Platform **provides you with **over 100 expertly written tutorials** that cover everything you need to know about NSAA mathematics and science, as well as plenty of **amazing revision and exam tips!**

Start your NSAA revision with Exams.Ninja today and **boost your chances of success!**

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## NSAA MATHS PRACTICE QUESTIONS

Here are a handful of practice questions to try out what you’ve revised so far. These are all from Section 1 of the exam as there aren’t any pure maths questions in Section 2. Therefore, these are mainly going to be using standard maths than advanced. That doesn’t mean they’re easy though, so give them a try!

### NSAA Maths Question 1

**Calculate the following:**

A) 0

B) 1

C) 55

D) 110

E) 1.25 x 10^{8}

F) 5.5 x 10^{7}

G) 5.5 x 10^{8}

**The correct answer is C.**

**Let y = 1.25 x 10^{8}; this is not necessary, but helpful, as the question can then be expressed as:**

**(100 y + 10y)/2y = 110y/2y = 55**

### NSAA Maths Question 2

**The diagram to the right shows a series of regular pentagons. What is the product of angles a and b?**

A) 580º

B) 1,111º

C) 3,888º

D) 7,420º

E) 9,255º

**The correct answer is C.**

**Remember the interior angles of a pentagon add up to 540° (three internal triangles), so each interior angle is 540/5 = 108°. Therefore angle a is 108°. Recalling that angles within a quadrilateral sum to 360°, we can calculate b. The larger angle in the central quadrilateral is 360° – 2 x 108° (angles at a point) = 144°. Therefore the remaining angle, b = (360 – 2(144)]/2 = 36°. The product of 36 and 108 is 3,888°. **

### NSAA Maths Question 3

**What is the equation of the line of best fit for the scatter graph below?**

A) *y* = 0.2*x* + 0.35

B) *y* = 0.2*x* – 0.35

C) *y* = 0.4*x* + 0.35

D) *y* = 0.4*x* – 0.35

E) *y* = 0.6*x* + 0.35

**The correct answer is E.**

**Begin by drawing your line of best fit, remembering not to force it through the origin. Begin fitting the general equation y = mx + c to your line. Calculate the gradient as ∆y/∆x and read the y intercept off your annotated graph. **

### NSAA Maths Question 4

**The table below shows the results of a study investigating antibiotic resistance in staphylococcus populations. **

**A single staphylococcus bacterium is chosen at random from a similar population. Resistance to any one antibiotic is independent of resistance to others. Calculate the probability that the bacterium selected will be resistant to all four drugs.**** **

A) 1 in 10^{12}

B) 1 in 10^{6}

C) 1 in 10^{20}

D) 1 in 10^{25}

E) 1 in 10^{30}

F) 1 in 10^{35}

**The correct answer is D.**

**The key here is to note that the answers are several orders of magnitude apart so you can round the numbers to make your calculations easier: **

**Probability of bacteria being resistant to every antibiotic = P (Res to Antibiotic 1) x P (Res to Antibiotic 2) x P (Res to Antibiotic 3) x P (Res to Antibiotic 4)**

**=100/10 ^{11} x 100/10^{9} x 100/10^{8} x 10/10^{5}**

**=10 ^{8}/10^{33} = 1/10^{25}**

Looking for even tougher practice questions? You can learn more about Section 2 and put your advanced maths skills to the test with our** Definitive Section 2 Guide**!

You can also keep an eye out for **more NSAA practice questions** on **Exams.Ninja** coming soon!

So, that’s pretty much everything covered by the **NSAA specification** for **assumed mathematical knowledge!** The majority of these topics should be pretty well known to you by now but it’s always good to have a general refresh, especially when **considering the context of the NSAA itself.**

Don’t forget to **prepare for the sciences** as well, though all of these topics will be relevant in some way when ploughing through **Section 2** of the exam. How you revise is up to you, but **we wish you the very best of luck!**

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