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Multiple Forces Acting on an Object: Analysis & Applications

AS-Level Maths: Master Core Concepts & Skills

Set Theory

auto-gen

Set notation

Understanding How to Use Set Notation

Set builder notation

Intersection and union of 2 sets

Intersection and Union of 2 Sets

Intersection and union of 3 sets

Intersection and Union of 3 Sets

Number System

Understanding the number systems

Prime factorization

Number properties: Prime or Composite

Number properties: Prime?

Prime factors – Word Problem

Greatest Common Factors (GCF)

Greatest Common Factor 2 Numbers

Greatest common factors (GCF)

Greatest Common Factor 2 Numbers – Word Problem

Least Common Multiple (LCM)

Least common multiple (LCM)

Lowest Common Multiple - 2 Numbers – Word Problem

Lowest Common Multiple 2 Numbers

Rational vs. Irrational numbers

Converting repeating decimals to fractions

Surds

Evaluating and simplifying radicals

Square and square roots

Square Root of a Perfect Square

Cubic and cube roots

Cube root of a perfect cube

Converting radicals to mixed radicals

Converting radicals to entire radicals

Adding and subtracting radicals

Adding radicals

Subtracting radicals

Multiplying and dividing radicals

Dividing radicals

Multiplying radicals

Rationalize the denominator 

Laws of Indices

Indices: Product rule <i>(a^x)(a^y) = a^(x+y)</i>

Exponents: Product rule (a^x)(a^y) = a^(x+y)

Indices: Division rule (a^x / a^y) = a^(x-y)

Exponents: Division rule: a^x / a^y = a^(x-y)

Indices: Power rule <i>(a^x)^y = a^(x * y)</i>

Exponents: Power rule: (a^x)^y = a^(xy)

Exponents: Power rule

Indices: Negative indices

Exponents: Negative exponents

Zero index: <i>a^0 = 1</i>

Exponents: Zero exponent: a^0 = 1

Rational indices

Exponents: Rational exponents

Combining laws of indices

Combining the exponent rules

Standard form

Scientific notation

Solving for indices

Solving for exponents

Linear Functions

Distance formula: \(d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)

Distance formula: <katex>d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}</katex>

Distance formula

Midpoint formula: \(M = ( \frac{x_1+x_2}2  ,\frac{y_1+y_2}2)\)

Midpoint formula

Midpoint formula: <katex>M = ( \frac{x_1+x_2}2  ,\frac{y_1+y_2}2)</katex>

Midpoint formula: <katex>M = ( \frac{x_1+x_2}2 ,\frac{y_1+y_2}2)</katex>

Gradient equation: \(m = \frac{y_2-y_1}{x_2- x_1}\)

Slope equation: <katex>m = \frac{y_2-y_1}{x_2- x_1}</katex>

Gradient intercept form: y = mx + b

Slope intercept form: y = mx + b

General form: Ax + By + C = 0

Gradient-point form: \(y - y_1 = m (x - x_1)\)

Graphing linear functions using table of values

Graphing linear functions using x- and y-intercepts

Graphing from slope-intercept form y=mx+b

Graphing linear functions using a single point and gradient

Word problems of graphing linear functions

Rate of change

Parallel and perpendicular lines in linear functions

Applications of linear relations

Quadratic Functions

Introduction to quadratic functions

Transformations of quadratic functions

Quadratic function in general form: <i>y = ax^2 + bx + c</i>

Converting from general to vertex form by completing the square

Shortcut: Vertex formula

Converting general form into vertex form by applying the vertex formula

Vertex Formula Application

Graphing quadratic functions: General form VS. Vertex form

Finding the quadratic functions for given parabolas

Applications of quadratic functions

Quadratic function in vertex form: <i>y = a(x-p)^2 + q</i>

Completing the square

Completing the Square

Quadratic Equations

Solving quadratic equations by factorising

Solving quadratic equations by factoring

Solving quadratic equations by completing the square

Using quadratic formula to solve quadratic equations

Solving a quadratic equation with TWO REAL SOLUTIONS

Solving a quadratic equation with TWO COMPLEX SOLUTIONS

The discriminant: Nature of roots of quadratic equations

Applications of quadratic equations

Simultaneous Equations

Simultaneous linear equations

System of linear equations

Simultaneous linear-quadratic equations

System of linear-quadratic equations

Simultaneous quadratic-quadratic equations

Inequalities

Express linear inequalities graphically and algebraically

Graph the Inequality

Linear Inequalities – Word Problem 

Linear inequalities

Solving one-step linear inequalities

One-step Linear Inequalities

One-step linear inequalities

One-step Linear Inequalities Decimal

One-step Linear Inequalities Decimal with Fraction

One-step Linear Inequalities Fraction

Solve One Step Linear Inequalities Find Solution– Word Problem

One-step Linear Inequalities — Word Problem

Solving multi-step linear inequalities

Multi-step Linear Inequalities

Two-step Linear Inequalities

Multi-step linear inequalities

Two-step linear inequalities

Solve Multi-step Linear Inequalities – Word Problem 

Solve Multi Step Linear Inequalities Solve Equation – Word Problem

Solve Multi Step Linear Inequalities Equation – Word Problem

Solving quadratic inequalities

Factorisation

Factorise difference of cubes

Factoring difference of cubes

Factorise sum of cubes

Factoring sum of cubes

Factorise by taking out the greatest common factor

Factor by taking out the greatest common factor

Factor by taking out GCF

Factorise by grouping

Factor by grouping

Factorise difference of squares: x^2 - y^2

Factoring difference of squares: <katex>x^2 - y^2</katex>

Factoring difference of squares: x^2 - y^2

Factorise trinomials

Factoring trinomials

Factoring Trinomials

Operations with Algebraic Fractions

Partial fraction decomposition

Solving algebraic fraction equations

Solving rational equations

Simplifying algebraic fractions and restrictions

Simplifying rational expressions and restrictions

Adding and subtracting algebraic fractions

Adding and subtracting rational expressions

Adding and Subtracting Rational Expressions

Multiplying algebraic fractions

Multiplying rational expressions

Multiplying Rational Expressions

Multiplying Rational Expressions in Factored Form

Dividing algebraic fractions

Dividing rational expressions

Dividing Rational Expressions

<p>Get better marks with our comprehensive AS level Maths help ? whether it&apos;s for <a href="https://qualifications.pearson.com/en/qualifications/edexcel-a-levels/mathematics-2017.html">Edexcel</a> (Pearson), <a href="https://www.aqa.org.uk/subjects/mathematics/as-and-a-level">AQA</a>, <a href="https://www.ocr.org.uk/qualifications/by-subject/mathematics/">OCR</a> or <a href="http://www.wjec.co.uk/qualifications/mathematics/mathematics-gce-a-as/">WJEC</a>. Our AS Maths tutors have you all covered!</p>
<p>Keeping with the specifications of all the exam boards, our thorough help for the exam includes topics such as Trigonometry, Factorisation, Quadratic functions, Exponential functions, Logarithms, Calculus, Statistics, and a lot more. Learn the concepts with our online video tutorials that show you step-by-step solutions to even the hardest maths problems. Then, strengthen your understanding with tons of as maths practice.</p>
<p>All our lessons are taught by experienced AS Maths teachers. Let&apos;s finish your revision in no time, and ACE that maths exam.</p>

AS-Level Maths

math

<h2>Is AS level Maths Hard?</h2>
<p>As the name suggests, advanced subsidiary (AS) level maths, expands beyond the basic concepts of maths and into more complex or "advanced" areas of mathematics. If the thought of this seems a little daunting to you, you&apos;re not alone. There are many students across the UK that are put off by the complex sounding maths problems and terms like simultaneous equations, algebraic expressions, and more.</p>
<p>From the outside looking in, AS level maths can seem like a hard course of study, but if you have a firm grasp of the basics and you paid attention in secondary school maths, you should be well equipped to tackle even the trickiest of problems within this course.</p>
<p>Even if you struggled in <a href="[[region_textbook.html|{"textbook_key":"uk-year10"}]]">year 10</a> or <a href="[[region_textbook.html|{"textbook_key":"uk-year11"}]]">year 11</a> maths, you could still earn a good grade within AS level maths. All it would take is some organization, studying and an effective revision tool like <a href="[[region_textbook.html|{"textbook_key":"uk-as-level-maths"}]]">StudyPug</a>.</p>
<p>Our platform has a vast collection of video tutorials and revision guides for each topic of AS level maths. Our content will walk you through all the relevant topics that are bound to come up during your course of study, providing AS level maths step-by-step with videos on the following:</p>
<ul type="disc">
<li>Simultaneous Equations</li>
<li>Differentiation</li>
<li>Circle Theorems</li>
<li>Operations with Surds</li>
<li>Law of Indices</li>
<li>Binomial Expansion</li>
<li>And more.</li>
</ul>
<p>If you want to familiarize yourself with the topics you studied in secondary school before you take on an A level course, we have an extensive collection of materials for revision in trigonometry, statistics, integration vectors, and all other topic studied across the GCSE curriculum.</p>
<p>Our AS level Maths tutoring service will show you easy to understand solutions to even the hardest AS level maths questions. Furthermore, the information that we deliver, reflects everything you&apos;d expect to find in modern maths textbooks and covers the content most likely to appear on your AQA, Edexcel, MEI, WJEC or OCR maths exams.</p>
<p>We understand that each student will have their own unique learning style and not every student will come to us at the same skill level. That&apos;s why we offer an "AS level maths for dummies" approach that starts with the basics and assumes no prior knowledge. The beauty of our on-demand video service is in is flexibility. You don&apos;t need to sit through lessons you already know. You can skip them entirely or simply fast-forward to the section you need.</p>
<p>If you <a href="[[signup.html]]">join StudyPug today</a> as a subscriber, you&apos;ll have direct access to all of our GCSE and AS level maths help, giving you unprecedented access to a wealth of quality videos and resources that can supplement your studies and help you find your way to the right maths solutions.</p>

<h2>What is AS level Maths Like?</h2>
<p>A level maths is separated into two variants. An AS Level maths course is one year long, whereas the A level Maths course, is a 2 year program with the AS acting as its first year.</p>
<p>Within the structure of the course, students are given a choice as to what modules they study. The modules are as follows:</p>
<ul type="disc">
<li>C1-4 - Core Maths</li>
<li>S1 & S2 - Statistics</li>
<li>M1 & M2 - Mechanics</li>
<li>D1 & D2 - Decisions</li>
</ul>
<p>Regardless of whether you&apos;re studying AS or A level maths, you will be required to study the C1-4 modules. AS students then pick one further module to study (statistics, mechanics, or decisions) and A level students, will pick two.</p>
<p>As an example, <a href="https://nrich.maths.org">nrich.maths.org</a> has drafted what a typical AS student studying statistics might encounter on their course (see below).</p>
<ol type="1">
<li>Laws of indices</li>
<li>Integration and differentiation of xn</li>
<li>Integration gives areas under curves</li>
<li>Laws of logarithms (log(ab)=log(a)+log(b) etc)</li>
<li>Solution of quadratic equations</li>
<li>Linear and quadratic graphs</li>
<li>
Simple use of sin, cos and tan functions and graphs, including
<ol type="1">
<li>cos2(x)+sin2(x)=1</li>
<li>Radians</li>
<li>Sin rule</li>
</ol>
</li>
<li>Expanding brackets and geometric series.</li>
<li>Basic ideas of statistics: Mean, standard deviation, variance, outliers</li>
<li>Various methods of plotting data and linear regression.</li>
<li>The shape of the normal distribution and use of tables</li>
</ol>

<h2>How to Get an A in AS Level Maths?</h2>
<p>To stand a chance of getting an A grade in your AS level exams, you&apos;ll need to be prepared to get yourself into a good study routine, which should include homework, worksheets, and AS level maths past papers. Using these past papers to sit mock exams, you will be better prepared for you actual tests, and it could help with your confidence and time management.  Work within the time limits for each paper and review your performance to single out any areas of weakness.</p>
<p>If you get a few questions wrong within a specific topic, revisit the whole topic on StudyPug and review your class notes. Our content is broken down into individual topics, so you&apos;ll be able to find the videos that are relevant to you with ease. Watch the videos, resit mock papers, and chart your progress. Eventually, you&apos;ll see an improvement in your scores, and by addressing your weaker areas, you&apos;ll be better prepared for whatever arises in your end of year exam.</p>
<p>When the examiners are marking papers, they&apos;re not just looking for the right answers. Examiners are looking to see if you arrived at the solution through an understanding of the problem or via simple memorization. This is why its key to always show you working out! Even if its a simple question don&apos;t just answer it and move on, show how you arrived at the solution. There&apos;s always the potential to receive extra marks for showing your thought process, and that could mean the difference between a B and an A grade.</p>

<h2>How to Revise for AS level Maths</h2>
<p>When it comes to maths revision, as mentioned above, you should highlight your areas of weakness so that you know what to focus on. Once you know the topics of AS level maths you need to work on, you can visit StudyPug to find video tutorials that cover each topic with handy step-by-step examples.</p>
<p>You, much like many of students, may find that revising via our video format is a lot more engaging and easier to follow than the traditional textbook revision methods. To help get you started, we&apos;re offering a collection of free AS Level Maths lessons across the following subject areas:</p>
<ul type="disc">
<li>Understanding the Number Systems</li>
<li>Least Common Multiple (LCM)</li>
<li>Converting Radicals to Entire Radicals</li>
<li>Slope Intercept Form: y = mx + b</li>
<li>Graphing Linear Functions Using x- and y- Intercepts</li>
<li>And Many More</li>
</ul>
<p>Think of StudyPug as your own personal AS level maths tutor that's available 24/7.</p>

<h2>What Calculators are Allowed in AS level Maths?</h2>
<p>All examining boards will follow the regulations of the <a href="https://www.jcq.org.uk/exams-office/ice---instructions-for-conducting-examinations">JCQ</a> (Joint Council for Qualifications) with regards to calculators in exams.</p>
<p>The regulations are as follows:</p>
<p><u>Calculators must be:</u></p>
<ul type="disc">
<li>Of a size suitable for use on the desk</li>
<li>Either battery or solar powered</li>
<li>Free of lids, cases and covers which have printed instructions or formulas</li>
</ul>
<p><u>Calculators must not:</u></p>
<ul type="disc">
<li>Be designed or adapted to offer any of these facilities:
<ul type="circle">
<li>translators</li>
<li>Symbolic algebra manipulation</li>
<li>Symbolic differentiation or integration</li>
<li>Communication with other machines or the internet</li>
</ul>
</li>
<li>Be borrowed from another candidate during an examination for any reason</li>
<li>
Have retrievable information stored in them - this includes:
<ul type="circle">
<li>Databanks</li>
<li>Dictionaries</li>
<li>Mathematical formulas</li>
<li>Text</li>
</ul>
</li>
</ul>

<h2>Is it Worth Getting my AS Level Maths Paper Remarked?</h2>
<p>If you don&apos;t receive the marks you were expecting, having your paper remarked could help. Firstly, you&apos;ll need to consult with your teacher and see if they also expected a higher score from your test paper (based on previous tests and in-class performance). Accessing the papers may be free for the school, but it can cost anywhere from &pound;11 - &pound;15, depending on the exam board.</p>
<p>There will also be an additional cost for the actual remarking and this will vary between exam boards and the nature of the remark (please see below).</p>
<figure class="table">
<table>
<thead>
<tr><th></th><th>AQA</th><th>Edexcel</th><th>OCR</th><th>WJEC</th><th>Deadline</th></tr>
</thead>
<tbody>
<tr><th class="text-nowrap text-right">Access to Marked Paper</th><td>&pound;13.95</td><td>Free</td><td>&pound;11.40</td><td>&pound;11.40</td><td>Aug 24</td></tr>
<tr><th class="text-nowrap text-right">Priority Remark</th><td>&pound;50.30</td><td>&pound;49.70</td><td>&pound;56.30</td><td>&pound;46</td><td>Aug 24</td></tr>
<tr><th class="text-nowrap text-right">Full Re-mark</th><td>&pound;42.25</td><td>&pound;41.70</td><td>&pound;45.60</td><td>&pound;36</td><td>Sept 21</td></tr>
<tr><th class="text-nowrap text-right">Clerical Re-check</th><td>&pound;16.10</td><td>&pound;11.20</td><td>&pound;16.40</td><td>&pound;10</td><td>Sept 21</td></tr>
</tbody>
</table>
<figcaption>Pricing as of 2017</figcaption>
</figure>
<p>It's not a cheap process and in most cases, remarking will only add a few extra marks, it won't change a D grade to an A. With that in mind, check how far away you are from the next possible grade, if you&apos;re 2-3 marks away, it may be worth the cost of a remark, specially it its the difference between a failing and passing grade.</p>

https://dmn92m25mtw4z.cloudfront.net/img/textbook/uk/textbook-AS-maths.png

Understanding Multiple Forces Acting on an Object

What You'll Learn

Apply Newton's second law to analyze objects with multiple forces acting simultaneously

Draw free body diagrams to identify and represent every force acting on an object

Calculate normal force as the contact force between surfaces resisting compression

Determine net force by summing all forces acting on an object using vector addition

Separate force analysis into x and y components for easier calculation

What You'll Practice

Drawing free body diagrams with gravity, normal force, friction, and applied forces

Calculating thrust force needed to launch objects upward against gravity

Finding normal force on objects resting on surfaces or being lifted

Solving for friction force when objects slide across surfaces with acceleration

Why This Matters

This Unit Includes

4 Video lessons

Learning resources

Skills

Newton's Second Law

Free Body Diagrams

Normal Force

Net Force

Vector Addition

Force of Friction

Contact Forces

Force Components