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Math

Mathematics involves numbers, shapes, algebra, and many more topics, but it’s much more than that. It is the study of the ‘language of the universe’. The logic and examples that underpin many mathematical concepts existed long before mathematics was created, and we can use them to help us understand how things work. Maths is essential for understanding the sciences. For example, physics can tell us how objects should behave due to gravity, which is supported by the diverse tools that mathematics provides.

Mathematics is one of the subjects many students find most challenging during their studies. The key to mastering mathematics is to learn where topics come from and how they are linked and complete lots of practice questions to know how to approach all kinds of problems.
At StudySmarter, you will find mathematics revision notes covering every aspect needed for your exams, complete with flashcards to help you practice and gain a solid grasp of the key concepts!

We can split Mathematics into many subtopics; however, there are three main areas: Pure Maths, Mechanics, and Statistics.

Topics and learning goals of pure maths

Pure Maths is dedicated to the study of maths independent of their application but can still be implemented in everyday life.

Mathematics, especially Pure Mathematics, tries to extract the main concepts behind operations and numbers and utilizes them using symbols and relationships. The work done with these concepts can then be applied to different fields such as social sciences, logic, engineering, biology, chemistry, or physics. For instance, calculus, which is in the field of Pure Maths, is the base of many engineering courses. Many of the topics within Pure Maths, such as Vectors, Geometry, Algebra, and Coordinate Spaces, relate to each other.

Within Pure Maths, you can find the following topics on StudySmarter:

    • Proof – is the area of maths that includes the verification of math theorems and laws.
    • Algebra – is the topic comprising the study of the types of numbers and mathematical operations.
    • Functions – comprises the understanding of the relationships between two groups of numbers, the function input (independent variable) and the function output (the dependent variable). The relationship is created by applying operations on the input to obtain the output.
    • Coordinate Geometry – is the study of geometrical objects using a system to locate them in space. Cartesian geometry is an excellent example of this, this is the coordinate system you use to know the positions of objects in a 3D space when studying triangles and other shapes.
    • Sequences and Series – sequences are lists of numbers such as n,n2,n3,n4,n5… where all n numbers relate to each other. While series are the sum of sequences.
    • Trigonometry – examines geometrical objects in coordinate space, strongly related to the topic of coordinate geometry.
    • Exponentials and Logarithms – are special functions in maths, different from the ones you will study in the ‘Function’ topic; as it introduces something called a limit. When this limit is reached, the changes to the function input (independent variable) will produce little to almost zero changes in the function output (dependent variable). These functions are widely used to model different mechanisms in biology, physics, and other areas of science.
    • Differentiation – is the area of pure maths used to measure the rate of change of one function. The modern basis of these two areas of maths was created by Isaac Newton and Wilhelm Leibniz. Differentiation is widely used in many areas of science and engineering, providing solutions for many practical problems such as heat transfer, fluid mechanics, and dynamics of systems in movement.
    • Integration is the inverse of differentiation used to find the area under a graph. It is utilized to obtain many physical properties such as energy, work, area, etc.
    • Numerical Methods – this area uses mathematical approximations to solve mathematical problems that can not usually be solved using the standard mathematical tools. It is used in conjunction with differentiation and integration to solve many practical problems in engineering and science.
    • Vectors – are representations used to model quantities that have a direction. A common example is the representation of wind. The vector will have two main properties: a magnitude (value) and a direction. In terms of the wind example, the magnitude is its speed, and the direction where it is blowing is the vector direction.

Topics and learning goals of statistics

Statistics involves the collection, analysis, organization, and presentation of data. It allows you to obtain meaningful results and conclusions from data collected through experiments, surveys, interviews or observations. As a student, statistics provides other important areas of application for your knowledge of mathematics.

Statistics is widely used in social sciences, economics, the natural sciences and engineering. There are many different techniques and models in statistics, and you’ll need to learn how and when to apply them.
StudySmarter will help you understand statistics and covers sampling, data interpretation, distributions, and hypothesis testing.

Sampling

The only way of ensuring data is accurate is to test every item or individual in a population. Sampling is the method of extracting a subset of the population to make general statements or predictions about the dataset. StudySmarter will help you to understand the basics of sampling and its methods.

Data interpretation

Once data has been collected, it needs to be interpreted to analyze it effectively and draw conclusions. This process uses key information from your data analysis. It’s also helpful to represent the data in a graph or chart, so it’s easier to understand and identify patterns. At StudySmarter, we will teach you how to extract key information such as the measures of central tendency. You will learn how to present this data using plots, graphs, histograms, etc.

Distributions

Statistical outcomes can be modeled in different ways depending on the event. For example, some events are independent, meaning the likelihood of it occurring is unaffected by previous results, while some are dependent – one result affects another. For this reason, different probability distributions can be used in different scenarios, such as normal, binomial, and Poisson distributions.

StudySmarter will guide you through learning how to analyze data and derive information from its distribution.

Hypothesis testing

Before collecting and analyzing data, you often predict the expected result. This is known as a hypothesis. The process of hypothesis testing allows you to quantify the accuracy of your prediction. You will need to know how to test hypotheses for different probability distributions, but the general method is the same.

On StudySmarter, you will learn all about the procedure of hypothesis testing and when to apply them.

Topics and learning goals of mechanics

Mechanics studies the relationships between force, matter and motion. To do so, we use the tools of calculus, which was invented for this purpose.

Mechanics involves using examples of systems in movement, using real-life simplified scenarios for more straightforward calculations. Examples include a box on a slope or a weight on a rope pulley.

On StudySmarter, Mechanics is divided as shown below:

  • Quantities, Units and Assumptions
  • Kinematics
  • Forces and Newton’s Laws

Quantities, Units, and Assumptions

This subtopic covers the fundamental knowledge you will need to solve problems involving mechanics. This includes common assumptions you make and their implications for your answer – e.g., if you assume there is no air resistance on a moving object, you will get a higher theoretical velocity than would be the case in practice. You will also learn key units and how to convert between them.

Kinematics

Kinematics is the mechanics of motion, including acceleration (both constant and variable) and parabolic motion (projectiles).

Within this subtopic, you will also learn how to use the equations of uniform acceleration (constant acceleration), a set of equations that link displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (t). We refer to these equations as suvat equations.

Forces and Newton’s laws

A force is a ‘push’ or ‘pull’ acting on an object, which might cause a change in its motion. An example is gravity, which is always acting on Earth at a force of approximately 9.8 N on a unit mass object.

Forces appear in a variety of situations. On StudySmarter, you will learn how to calculate forces when applied parallel to the object in motion or at an angle to the object in motion (component forces). An example of when forces have components is when a box slides over a slope.

Newton’s Laws are a set of three laws that define the interaction between forces, acceleration and movement. The third law can be summarised as ‘every action has an equal and opposite reaction’. We know that gravity is a force acting downwards, but in order to keep us in equilibrium, there must be some other force acting upwards, too – otherwise, we would sink into the ground!

How can StudySmarter support me through maths?

To gain a solid understanding of mathematics, it’s important to complete lots of practice questions to know which techniques you need to use in different situations and how to approach different types of questions. StudySmarter provides notes, flashcards, quizzes, and other comprehensive revision materials to help you get top grades in your exams – completely free!

In addition, the StudySmarter app:

  • Creates an intelligent learning plan just for you.
  • Tracks your progress and motivates you with badges and awards.
  • It lets you quickly create notes and flashcards and share them with other students in your class. You can also join a wider learning community of other maths students.

By using our resources, you will be well prepared and confident for every exam level.

Final Math Quiz

Question

What is algebra?

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Answer

Algebra is a branch of mathematics that represents problems as mathematical expressions, using letters or variables (ie x, y or z) to represent unknown values ​​that can change. The purpose of algebra is to find out what the unknown values ​​are, by using predefined rules to manipulate each mathematical expression.

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Who invented algebra?

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Answer

Algebra was invented by Abu Ja'far Muhammad ibn Musa al-Khwarizmi

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Question

What is the equation for the coefficient of friction?

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Answer

μ = F / L


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What are pulleys?


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Answer

A pulley comprises a wheel and a fixed axle, with a groove along the edges to guide a rope or a cable

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 Why are Newton's laws important?


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Answer

They tell us why objects move or sit still, why we are not floating, and informs the mechanism in guns and cars

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Why is force considered a vector quantity?


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Answer

They are considered vector quantities because they have a magnitude and a direction.

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What is a resultant force?


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Answer

A resultant force is a single force which is a representation of the vector sum of more than one force

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What are Newton's three laws of motion?


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Answer

  • Newton's first law of motion states that an object continues to be in a state of rest or a state of motion at a constant speed along a straight line unless it is compelled to change that state by a net force acting on it. Net force is the vector sum of all the forces acting on an object. 

  • Newton's second law of motion states that the time rate of change of the momentum of a body is equal in both magnitude and direction to the force imposed on it. 

  • Newton's third law states that when two bodies interact, they apply forces to one another that are equal in magnitude and opposite in direction.

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What is the formula for solving moments?


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Answer

moment = force distance (from pivot perpendicular to force's line of action). 


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What is the distributive property of algebra?

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Answer

 a × (b + c) = a × b + b × c


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What is the commutative property of multiplication of algebra?


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Answer

a × b = b × a

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What is the associative property of addition of algebra?

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Answer

a + (b + c) = (a + b) + c

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What is the additive inverse property of algebra?


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Answer

a + (-a) = 0

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What are the steps to solve linear algebraic equations?


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Answer

Step 1: each side of the equation must be simplified by removing parentheses and combining terms

Step 2: add or subtract to isolate the variable on one side of the equation

Step 3: multiply or divide to obtain the value of the unknown variable

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What are the steps to solve word problems in algebra?


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Answer

Step 1: Assign variables to unknown values

Step 2: Construct the equations

Step 3: Solve the equations

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Question

Solve the equation: x + 2 = 5


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Answer

x = 3

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Solve the equation: 2x - 5 = 1


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x = 3

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Solve the equation: x/3 = 9


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x = 27

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Solve the equation: 2(x + 6) = 4(x + 1)


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x = 4

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Solve the equation: 5x - 2 = 3x + 8


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Answer

x = 5

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Question

Mary is 15. If we know that Ben's age is three times Mary's age minus 4, how old is Ben?


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Answer

Ben is 41 years old

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What is a sequence?

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Answer

A sequence is a set of numbers that follow a specific rule

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What is a series?

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Answer

A series is the sum of a sequence

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What are the two types of sequences?

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Answer

Arithmetic and geometric

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When do you use the sigma notation?

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Answer

This notation is used to identify the sum of the series.

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What are the three trigonometric functions?

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Answer

Sine, cosine and tangent

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Question

What are the names of the three sides of a right-angled triangle?

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Answer

Hypotenuse, opposite and adjacent

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How do you find the length of a side in a non-right-angled triangle?

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Answer

You can use the sine or cosine rule.

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When would you use the sine rule?

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Answer

You would use the sine rule when finding the missing side length or angle when you have two opposite sides and angles.

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What is the difference between a vector and scalar?

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Answer

A vector has direction and magnitude, whereas a scalar only has magnitude.

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Question

Find the missing vertex of the parallelogram given by (7, 1), (3, 1), (2, 0), (123)

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Answer

(6, 0)

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What are powers?

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Answer

Power is the exponent that a variable or number is being raised to, which in practice means that the number or variable is multiplied by itself as many times as the value of the power or exponent.

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What are roots?


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Roots are the opposite of powers, they find what number multiplied n times equals the number inside the root, where n is the index of the root.

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How many solutions do odd roots have?

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Answer

One solution

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How many solutions do even roots have?

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Two solutions

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What are surds?

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Answer

Surds are expressions that contain a square root, cube root or other roots, which produce an irrational number as a result, with infinite decimals. They are left in their root form to represent them exactly.

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Question

What do the letters in SUVAT stand for?

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Answer

  • s = Displacement
  • u = initial velocity
  • v = final velocity
  • a = acceleration
  • t = time.


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Question

What is free fall?


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Answer

 It is when an object experiences acceleration due to gravity.

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Question

Give two examples of projectiles.


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Answer

  • A bullet’s movement at the instant it is fired from a gun.

  • A car driven off a cliff.

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What is constant acceleration?

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Answer

It is also called one-dimensional equations of motion for constant acceleration. It deals with all kinematic problems where the acceleration is stable and constant.



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What is a particle in equilibrium?


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Answer

A particle is said to be in equilibrium when its net force is zero.


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What is variable acceleration?


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Answer

When acceleration is different between points along its path, it is considered variable acceleration. 


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What is an example of variable acceleration?


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Answer

A car moving in a crowded environment will slow down till it has enough clear space to move forward.


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What are projectiles?


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Answer

Projectiles deal with objects projected through the air either by being thrown, hit, or fired.


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Question

The path of a projectile.


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Answer

Trajectory.


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What are kinematic equations?


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Answer

 They are the equations used to describe the motion of objects with constant acceleration.


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Question

How do you multiply brackets containing surds?


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Answer

To multiply brackets containing surds, each term in the first bracket must be multiplied by each term in the second bracket.

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Question

How do you simplify surds?


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Answer

The steps to simplify surds are:

  • Write the number inside the root as the multiplication of its factors. One of the factors should be a square number 
  • Split the factors into separate roots 
  • Simplify the terms
  • Take out the multiplication symbol

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Question

How do you rationalise the denominator of surds?

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Answer

  • If the denominator is a surd, then multiply the numerator and denominator by that surd.
  • If the denominator has two terms, one rational and a surd, then multiply the numerator and denominator by the expression conjugate of the denominator.

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Question

What is a function?

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Answer

A function involves an input and an output.

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