1) The document is a formula sheet for quantitative ability and entrance exams that provides formulas and properties for arithmetic, algebra, geometry, trigonometry, and statistics.
2) It includes tables, rules, and definitions for topics like percentages, fractions, logarithms, progressions, roots, factoring, binomials, counting principles, and probability.
3) The formula sheet is intended to be a one-stop reference for various mathematical concepts tested in exams like the CAT.
FellowBuddy.com is an innovative platform that brings students together to share notes, exam papers, study guides, project reports and presentation for upcoming exams.
We connect Students who have an understanding of course material with Students who need help.
Benefits:-
# Students can catch up on notes they missed because of an absence.
# Underachievers can find peer developed notes that break down lecture and study material in a way that they can understand
# Students can earn better grades, save time and study effectively
Our Vision & Mission – Simplifying Students Life
Our Belief – “The great breakthrough in your life comes when you realize it, that you can learn anything you need to learn; to accomplish any goal that you have set for yourself. This means there are no limits on what you can be, have or do.”
Like Us - https://www.facebook.com/FellowBuddycom
FellowBuddy.com is an innovative platform that brings students together to share notes, exam papers, study guides, project reports and presentation for upcoming exams.
We connect Students who have an understanding of course material with Students who need help.
Benefits:-
# Students can catch up on notes they missed because of an absence.
# Underachievers can find peer developed notes that break down lecture and study material in a way that they can understand
# Students can earn better grades, save time and study effectively
Our Vision & Mission – Simplifying Students Life
Our Belief – “The great breakthrough in your life comes when you realize it, that you can learn anything you need to learn; to accomplish any goal that you have set for yourself. This means there are no limits on what you can be, have or do.”
Like Us - https://www.facebook.com/FellowBuddycom
Direct solution of sparse network equations by optimally ordered triangular f...Dimas Ruliandi
Triangular factorization method of a power network problem (in form of matrix). Direct solution can be found without calculating inverse matrix which usually considered an exhaustive method, especially in large scale network.
Special webinar on tips for perfect score in sat mathCareerGOD
Math is the language of logic and is therefore tested in all the major examinations where SAT is no exception.
Scoring well in Math can do wonders to your career and college candidature. Conversely, any complacency in Math affect your score and thus prove dangerous.
In this webinar, “Tips for perfect Math score in SAT and SAT- Math subject test” from the 5-day webinar series ‘Experts’ Speak: Demystifying US Admissions’, seasoned math trainers and subject experts with decades of experience in the industry share important insights on maximising your Math scores and minimising mistakes to lose out on Math scores.
Visit www.careergod.com for more info.
Main obstacles of Bayesian statistics or Bayesian machine learning is computing posterior distribution. In many contexts, computing posterior distribution is intractable. Today, there are two main stream to detour directly computing posterior distribution. One is using sampling method(ex. MCMC) and another is Variational inference. Compared to Variational inference, MCMC takes more time and vulnerable to high-dimensional parameters. However, MCMC has strength in simplicity and guarantees of convergence. I'll briefly introduce several methods people using in application.
Key Trends Shaping the Future of Infrastructure.pdfCheryl Hung
Keynote at DIGIT West Expo, Glasgow on 29 May 2024.
Cheryl Hung, ochery.com
Sr Director, Infrastructure Ecosystem, Arm.
The key trends across hardware, cloud and open-source; exploring how these areas are likely to mature and develop over the short and long-term, and then considering how organisations can position themselves to adapt and thrive.
Software Delivery At the Speed of AI: Inflectra Invests In AI-Powered QualityInflectra
In this insightful webinar, Inflectra explores how artificial intelligence (AI) is transforming software development and testing. Discover how AI-powered tools are revolutionizing every stage of the software development lifecycle (SDLC), from design and prototyping to testing, deployment, and monitoring.
Learn about:
• The Future of Testing: How AI is shifting testing towards verification, analysis, and higher-level skills, while reducing repetitive tasks.
• Test Automation: How AI-powered test case generation, optimization, and self-healing tests are making testing more efficient and effective.
• Visual Testing: Explore the emerging capabilities of AI in visual testing and how it's set to revolutionize UI verification.
• Inflectra's AI Solutions: See demonstrations of Inflectra's cutting-edge AI tools like the ChatGPT plugin and Azure Open AI platform, designed to streamline your testing process.
Whether you're a developer, tester, or QA professional, this webinar will give you valuable insights into how AI is shaping the future of software delivery.
Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
Paper presented at SYNERGY workshop at AVI 2024, Genoa, Italy. 3rd June 2024
https://alandix.com/academic/papers/synergy2024-epistemic/
As machine learning integrates deeper into human-computer interactions, the concept of epistemic interaction emerges, aiming to refine these interactions to enhance system adaptability. This approach encourages minor, intentional adjustments in user behaviour to enrich the data available for system learning. This paper introduces epistemic interaction within the context of human-system communication, illustrating how deliberate interaction design can improve system understanding and adaptation. Through concrete examples, we demonstrate the potential of epistemic interaction to significantly advance human-computer interaction by leveraging intuitive human communication strategies to inform system design and functionality, offering a novel pathway for enriching user-system engagements.
UiPath Test Automation using UiPath Test Suite series, part 3DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 3. In this session, we will cover desktop automation along with UI automation.
Topics covered:
UI automation Introduction,
UI automation Sample
Desktop automation flow
Pradeep Chinnala, Senior Consultant Automation Developer @WonderBotz and UiPath MVP
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
- For advanced developers: master the skills to efficiently apply PowSyBl functionalities to your real-world scenarios.
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. What’s changed? What has stayed the same? Where do we go from here? Join a group of leading sales peers from across the industry for a conversation about the lessons learned since the popularization of digital books, best practices, digital book supply chain management, and more.
Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
JMeter webinar - integration with InfluxDB and GrafanaRTTS
Watch this recorded webinar about real-time monitoring of application performance. See how to integrate Apache JMeter, the open-source leader in performance testing, with InfluxDB, the open-source time-series database, and Grafana, the open-source analytics and visualization application.
In this webinar, we will review the benefits of leveraging InfluxDB and Grafana when executing load tests and demonstrate how these tools are used to visualize performance metrics.
Length: 30 minutes
Session Overview
-------------------------------------------
During this webinar, we will cover the following topics while demonstrating the integrations of JMeter, InfluxDB and Grafana:
- What out-of-the-box solutions are available for real-time monitoring JMeter tests?
- What are the benefits of integrating InfluxDB and Grafana into the load testing stack?
- Which features are provided by Grafana?
- Demonstration of InfluxDB and Grafana using a practice web application
To view the webinar recording, go to:
https://www.rttsweb.com/jmeter-integration-webinar
Slack (or Teams) Automation for Bonterra Impact Management (fka Social Soluti...Jeffrey Haguewood
Sidekick Solutions uses Bonterra Impact Management (fka Social Solutions Apricot) and automation solutions to integrate data for business workflows.
We believe integration and automation are essential to user experience and the promise of efficient work through technology. Automation is the critical ingredient to realizing that full vision. We develop integration products and services for Bonterra Case Management software to support the deployment of automations for a variety of use cases.
This video focuses on the notifications, alerts, and approval requests using Slack for Bonterra Impact Management. The solutions covered in this webinar can also be deployed for Microsoft Teams.
Interested in deploying notification automations for Bonterra Impact Management? Contact us at sales@sidekicksolutionsllc.com to discuss next steps.
Connector Corner: Automate dynamic content and events by pushing a buttonDianaGray10
Here is something new! In our next Connector Corner webinar, we will demonstrate how you can use a single workflow to:
Create a campaign using Mailchimp with merge tags/fields
Send an interactive Slack channel message (using buttons)
Have the message received by managers and peers along with a test email for review
But there’s more:
In a second workflow supporting the same use case, you’ll see:
Your campaign sent to target colleagues for approval
If the “Approve” button is clicked, a Jira/Zendesk ticket is created for the marketing design team
But—if the “Reject” button is pushed, colleagues will be alerted via Slack message
Join us to learn more about this new, human-in-the-loop capability, brought to you by Integration Service connectors.
And...
Speakers:
Akshay Agnihotri, Product Manager
Charlie Greenberg, Host
Dev Dives: Train smarter, not harder – active learning and UiPath LLMs for do...UiPathCommunity
💥 Speed, accuracy, and scaling – discover the superpowers of GenAI in action with UiPath Document Understanding and Communications Mining™:
See how to accelerate model training and optimize model performance with active learning
Learn about the latest enhancements to out-of-the-box document processing – with little to no training required
Get an exclusive demo of the new family of UiPath LLMs – GenAI models specialized for processing different types of documents and messages
This is a hands-on session specifically designed for automation developers and AI enthusiasts seeking to enhance their knowledge in leveraging the latest intelligent document processing capabilities offered by UiPath.
Speakers:
👨🏫 Andras Palfi, Senior Product Manager, UiPath
👩🏫 Lenka Dulovicova, Product Program Manager, UiPath
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
Guy Korland, CEO and Co-founder of FalkorDB, will review two articles on the integration of language models with knowledge graphs.
1. Unifying Large Language Models and Knowledge Graphs: A Roadmap.
https://arxiv.org/abs/2306.08302
2. Microsoft Research's GraphRAG paper and a review paper on various uses of knowledge graphs:
https://www.microsoft.com/en-us/research/blog/graphrag-unlocking-llm-discovery-on-narrative-private-data/
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Divisibility Rule
Number Rule Example
2 If last digit is 0,2,4,6, or 8 22, 30, 50, 68, 1024
3 If sum of digits is divisible by 3 123 is divisible by 3 since 1 + 2 + 3 = 6 (and 6
is divisible by 3)
4 If number created by the last 2
digits is divisible by 4
864 is divisible by 4 since 64 is divisible by 4
5 If last digit is 0 or 5 5, 10, 15, 20, 25, 30, 35, 2335
6 If divisible by 2 & 3 522 is divisible by 6 since it is divisible by 2 & 3
9 If sum of digits is divisible by 9 621 is divisible by 9 since 6 + 2 + 1 = 9 (and 9
is divisible by 9)
10 If last digit is 0 10, 20, 30, 40, 50, 5550
Logarithms
Definition: 𝒚 = 𝒍𝒐𝒈 𝒃 𝒙 → 𝒙 = 𝒃 𝒚
Example: 𝒍𝒐𝒈 𝟐 𝟑𝟐 = 𝟓 → 𝟑𝟐 = 𝟐 𝟓
Properties
𝒍𝒐𝒈 𝒙 𝒙 = 𝟏 𝒍𝒐𝒈 𝒙 𝟏 = 𝟎
𝒍𝒐𝒈 𝒙 𝒙 𝒏
= 𝒏 𝒙𝒍𝒐𝒈 𝒙 𝒚
= 𝒚
𝒍𝒐𝒈 𝒙 𝒚 𝒏
= 𝒏𝒍𝒐𝒈 𝒙 𝒚
𝒍𝒐𝒈 𝒙 𝒂 × 𝒃 = 𝒍𝒐𝒈 𝒙 𝒂 + 𝒍𝒐𝒈 𝒙 𝒃
𝒍𝒐𝒈 𝒙
𝒂
𝒃
= 𝒍𝒐𝒈 𝒙 𝒂 − 𝒍𝒐𝒈 𝒙 𝒃
Special Exponents
Reciprocal 𝟏
𝒂 𝒎
= 𝒂−𝒎
Power 0 𝒂 𝟎
= 𝟏
Power 1 𝒂 𝟏
= 𝒂
Operations Involving Exponents
Multiplication 𝒂 𝒎
× 𝒂 𝒏
= 𝒂(𝒎+𝒏)
Division 𝒂 𝒎
÷ 𝒂 𝒏
= 𝒂(𝒎−𝒏)
Power (𝒂 𝒎
) 𝒏
= 𝒂 𝒎×𝒏
Roots
𝒂
𝒎
𝒏 = 𝒂 𝒎𝒏
= ( 𝒂
𝒏
) 𝒎
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Progressions
Arithmetic Progression: 𝑛 𝑡
term of an
Arithmetic Progression
𝒂 𝒏 = 𝒂 𝟏 + 𝒏 − 𝟏 𝒅
Sum of n terms of an arithmetic expression
𝒔 𝒏 =
𝒏 𝒂 𝟏 + 𝒂 𝒏
𝟐
=
𝒏 𝟐𝒂 𝟏 + 𝒏 − 𝟏 𝒅
𝟐
The first term is 𝑎1, the common difference
is 𝑑, and the number of terms is 𝑛.
Geometric Progression: 𝑛 𝑡
term of a
Geometric Progression
𝒂 𝒏 = 𝒂 𝟏 𝒓 𝒏−𝟏
Sum of n terms of a geometric progression:
𝒔 𝒏 =
𝒂(𝒓 𝒏+𝟏
− 𝟏)
𝒓 − 𝟏
The first term is 𝑎1, the common ratio is 𝑟, and
the number of terms is 𝑛.
Infinite Geometric Progression
Sum of all terms in an infinite geometric series =
𝒂 𝟏
(𝟏−𝒓)
𝑤𝑒𝑟𝑒 − 𝟏 < 𝒓 < 𝟏
Roots of a Quadratic Equation
A quadratic equation of type 𝐚𝐱 𝟐
+ 𝐛𝐱 + 𝐜 has two solutions, called roots. These two solutions
may or may not be distinct. The roots are given by the quadratic formula:
−b± b2−4ac
2a
where
the ± sign indicates that both
–b− b2−4ac
2a
and
–b+ b2−4ac
2a
are solutions
Common Factoring Formulas
1. 𝒙 𝟐
− 𝒚 𝟐
= 𝒙 − 𝒚 × 𝒙 + 𝒚
2. 𝒙 𝟐
+ 𝟐𝒙𝒚 + 𝒚 𝟐
= (𝒙 + 𝒚) 𝟐
3. 𝒙 𝟐
− 𝟐𝒙𝒚 + 𝒚 𝟐
= (𝒙 − 𝒚) 𝟐
4. 𝒙 𝟑
+ 𝟑𝒚𝒙 𝟐
+ 𝟑𝒚 𝟐
𝒙 + 𝒚 𝟑
= (𝒙 + 𝒚) 𝟑
5. 𝒙 𝟑
− 𝟑𝒚𝒙 𝟐
+ 𝟑𝒚 𝟐
𝒙 − 𝒚 𝟑
= (𝒙 − 𝒚) 𝟑
Binomial Theorem
The coefficient of 𝑥(𝑛−𝑘)
𝑦 𝑘
in (𝑥 + 𝑦) 𝑛
is:
𝐶𝑘
𝑛
=
𝑛!
𝑘! (𝑛 − 𝑘)!
Applies for any real or complex numbers x and y, and any non-negative integer n.
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Summary of counting methods
Order matters Order doesn’t matter
With
Replacement If 𝒓 objects are taken from a set of 𝒏
objects, in a specific order with
replacement, how many different
samples are possible? 𝒏 𝒓
N/A
Without
Replacement
Permutation Rule: If 𝒓 objects are
taken from a set of 𝒏 objects without
replacement, in a specific order, how
many different samples are
possible?
𝑃𝑟
𝑛
=
𝑛!
(𝑛 − 𝑟)!
Combination Rule: If 𝒓 objects are
taken from a set of 𝒏 objects without
replacement and disregarding order,
how many different samples are
possible?
𝐶𝑟
𝑛
=
𝑛!
𝑟!(𝑛−𝑟)!
Probability
The probability of an event A, 𝑃 𝐴 is defined as
𝑃 𝐴 =
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑡𝑎𝑡 𝑜𝑐𝑐𝑢𝑟 𝑖𝑛 𝑒𝑣𝑒𝑛𝑡 𝐴
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑙𝑖𝑘𝑒𝑙𝑦 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
Independent Events: If A and B are independent events, then the probability of A happening
and the probability of B happening is:
𝑃 𝐴 𝑎𝑛𝑑 𝐵 = 𝑃(𝐴) × 𝑃(𝐵)
Dependent Events: If A and B are dependent events, then the probability of A happening and
the probability of B happening, given A, is:
𝑃 𝐴 𝑎𝑛𝑑 𝐵 = 𝑃(𝐴) × 𝑃(𝐵 | 𝐴)
Conditional Probability: The probability of an event occurring given that another event has
already occurred e.g. what is the probability that B will occur after A
𝑃 𝐵 𝐴 =
𝑃(𝐴 𝑎𝑛𝑑 𝐵)
𝑃(𝐴)
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Geometry Fundamentals
The sum of angles around a point will always
be 360 degrees.
In the adjacent figure
𝒂 + 𝒃 + 𝒄 + 𝒅 = 𝟑𝟔𝟎°
Vertical angles are equal to each other.
In the adjacent
figure 𝒂 = 𝒄 and
𝒃 = 𝒅
The sum of angles on a straight line is 180˚.
In the adjacent
figure sum of
angles and b is
180 i.e. 𝒂 + 𝒃 =
𝟏𝟖𝟎°
When a line intersects a pair of parallel lines,
the corresponding angles are formed are
equal to each other
In the figure above
𝒄 = 𝒅
When a line intersects a pair of parallel lines,
the alternate interior and exterior angles
formed are equal to each other
In the adjacent
figure alternate
interior angles
a=b and
alternate exterior
angles c=d
Any point on the perpendicular bisector of a
line is equidistant from both ends of the line.
In the figure, k
is the
perpendicular
bisector of
segment AB
and c=d, e=f
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Triangle Properties
1. Sum of all internal angles of a
triangle is 180 degrees i.e.
𝒑 + 𝒒 + 𝒓 = 𝟏𝟖𝟎°
2. Sum of any two sides of a triangle
is greater than the third i.e.
𝐚 + 𝐛 > 𝐜 or 𝐜 + 𝐛 > 𝐚 or 𝐚 + 𝐜 > 𝐛
3. The largest interior angle is
opposite the largest side; the
smallest interior angle is opposite
the smallest side i.e. if 𝐩 > 𝐪 →
𝐚 > 𝐜
4. The exterior angle is supplemental
to the adjoining interior angle i.e.
𝒑 + 𝒛 = 𝟏𝟖𝟎°. Since 𝒑 + 𝒒 + 𝒓 =
𝟏𝟖𝟎° it follows that 𝒛 = 𝒒 + 𝒓
5. The internal bisector of an angle
bisects the opposite side in the
ratio of the other two sides. In the
adjoining figure
BD
DC
=
AB
BC
Pythagoras Theorem
𝒄 𝟐
= 𝒂 𝟐
+ 𝒃 𝟐
Commonly Used Pythagorean Triples
Height, Base Hypotenuse
3,4 or 4,3 5
6,8 or 8,6 10
5, 12 or 12,5 13
7, 24 or 24,7 25
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Special Right Triangles
The lengths of the
sides of a 45- 45-
90 triangle are in
the ratio of 𝟏: 𝟏: 𝟐
The lengths of the sides
of a 30- 60- 90
triangle are in the ratio
of 𝟏: 𝟑: 𝟐
Test of Acute and Obtuse Triangles
If 𝒄 𝟐
< 𝒂 𝟐
+ 𝒃 𝟐
then it is an acute-angled
triangle, i.e. the angle facing side c is an acute
angle.
If 𝒄 𝟐
> 𝒂 𝟐
+ 𝒃 𝟐
then it is an obtuse-angled
triangle, i.e. the angle facing side c is an
obtuse angle.
Polygons
Sum of interior angles of a quadrilateral is 360˚
𝑎 + 𝑏 + 𝑐 + 𝑑 = 360°
Sum of interior angles of a pentagon is 540˚
𝑎 + 𝑏 + 𝑐 + 𝑑 + 𝑒 = 540°
If n is the number of sides of the polygon then, sum of interior angles = (n - 2)180°
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Properties of a Circle
The angle at the centre of a circle is twice any
angle at the circumference subtended by the
same arc.
In the adjacent figure
angle 𝒃 = 𝟐𝒂
Every angle subtended at the circumference
by the diameter of a circle is a right angle
(90˚).
In a cyclic quadrilateral, the opposite angles are
supplementary i.e. they add up to 180˚.
In the figure
above angle
𝒂 + 𝒄 =
𝟏𝟖𝟎° and
𝒃 + 𝒅 = 𝟏𝟖𝟎°
The angles at the circumference subtended
by the same arc are equal.
In the
adjacent
figure angle
𝒃 = 𝒂
A chord is a straight line joining 2 points on the
circumference of a circle.
A radius that
is
perpendicular
to a chord
bisects the
chord into
two equal
parts and
vice versa.
In the adjacent figure PW=PZ
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Area and Perimeter of Common Geometrical Figures
Rectangle
Area 𝑨 = 𝒍 × 𝒘
Perimeter 𝑷 = 𝟐 × (𝒍 + 𝒘)
Square
Area 𝑨 = 𝒔 𝟐
Perimeter 𝑷 = 𝟒𝒔
Triangle
Area 𝑨 =
𝟏
𝟐
× 𝒃 × 𝒉
Circle
Area 𝑨 = 𝝅𝒓 𝟐
Perimeter 𝑷 = 𝟐𝝅𝒓
Equilateral Triangle
Area 𝑨 =
𝟑
𝟒
× 𝒔 𝟐
Perimeter 𝑷 = 𝟑𝒔
Altitude 𝒉 = 𝒂
𝟑
𝟐
Trapezoid
Area 𝑨 =
𝟏
𝟐
𝒉 (𝒃 𝟏 + 𝒃 𝟐)
Parallelogram
Area 𝑨 = 𝒉 × 𝒃
Perimeter 𝑷 = 𝟐 × (𝒉 + 𝒃)
Ring
Area 𝑨 = 𝝅(𝑹 𝟐
− 𝒓 𝟐
)
Sector
𝛉 in degrees
Area 𝑨 = 𝝅𝒓 𝟐
×
𝛉
𝟑𝟔𝟎
Length of Arc 𝑳 = 𝝅𝐫 ×
𝛉
𝟏𝟖𝟎
Perimeter 𝑷 = 𝑳 + 𝟐𝒓
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Volume and Surface Area of 3 Dimensional Figures
Cube
Volume 𝒗 = 𝒍 × 𝒘 × 𝒉
Surface Area
𝑨 = 𝟐(𝒍𝒘 + 𝒘𝒉 + 𝒉𝒍)
Sphere
Volume 𝒗 =
𝟒
𝟑
𝝅𝒓 𝟑
Surface Area 𝑨 = 𝟒𝝅𝒓 𝟐
Right Circular Cylinder
Volume 𝒗 = 𝝅𝒓 𝟐
𝐡
Surface Area 𝑨 = 𝟐𝝅𝐫𝐡
Pyramid
Volume 𝒗 =
𝟏
𝟑
𝑩𝒉
Surface Area = 𝑨 = 𝑩 +
𝒑𝒍
𝟐
B is the area of the base
𝒍 is the slant height
Right Circular Cone
Volume 𝒗 =
𝟏
𝟑
𝝅𝒓 𝟐
𝐡
Surface Area 𝑨 = 𝝅𝒓(𝒍 + 𝒓)
Frustum
Volume
𝒗 =
𝟏
𝟑
𝝅𝐡(𝐫 𝟐
+ 𝐫𝐑 + 𝐑 𝟐
)
Coordinate Geometry
Line: Equation of a line 𝒚 = 𝒎𝒙 + 𝒄
In the adjoining
figure, c is the
intercept of the
line on Y-axis i.e.
c=2, m is the
slope
Slope of a line 𝒎 =
𝒚 𝟐−𝒚 𝟏
𝒙 𝟐−𝒙 𝟏
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The slopes of parallel lines are equal
In the
adjoining
figure, the two
lines are
parallel to
each other i.e.
𝒎 𝟏 = 𝒎 𝟐
The slopes of perpendicular lines are opposite
reciprocals of one another.
In the
adjoining the
two lines are
perpendicular
to each other
i.e. 𝒎 𝟏 =
−𝟏
𝒎 𝟐
Midpoint between two points (𝒙 𝟏, 𝒚 𝟏) and
(𝒙 𝟐, 𝒚 𝟐) in a x-y plane:
𝒙 𝒎, 𝒚 𝒎 =
𝒙 𝟐 − 𝒙 𝟏
𝟐
,
𝒚 𝟐 − 𝒚 𝟏
𝟐
Distance between two points (𝒙 𝟏, 𝒚 𝟏) and
(𝒙 𝟐, 𝒚 𝟐) in a x-y plane:
𝒅 = (𝒙 𝟐 − 𝒙 𝟏) 𝟐 + (𝒚 𝟐 − 𝒚 𝟏) 𝟐
Horizontal and Vertical line
Equation of line parallel to x-axis (line p) with
intercept on y-axis at (0, b) is 𝒚 = 𝒃. Equation
of line parallel to y-axis (line q) with intercept
on x-axis at (a,0) is 𝒙 = 𝒂
Equation of a circle with center (h,k) and
radius r
(𝒙 − 𝒉) 𝟐
+ (𝒚 − 𝒌) 𝟐
= 𝒓 𝟐
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Trigonometry Basics
Definition: Right Triangle definition for angle θ
such that 0 < θ < 90°
𝑠𝑖𝑛𝜃 =
𝑒𝑖𝑔𝑡
𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
, 𝑐𝑠𝑐𝜃 =
1
𝑠𝑖𝑛𝜃
𝑐𝑜𝑠𝜃 =
𝑏𝑎𝑠𝑒
𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
, 𝑠𝑒𝑐𝜃 =
1
𝑐𝑜𝑠𝜃
𝑡𝑎𝑛𝜃 =
𝑒𝑖𝑔𝑡
𝑏𝑎𝑠𝑒
, 𝑐𝑜𝑡𝜃 =
1
𝑡𝑎𝑛𝜃
Pythagorean Relationships
𝑠𝑖𝑛2
𝜃 + 𝑐𝑜𝑠2
𝜃 = 1
𝑡𝑎𝑛2
𝜃 + 1 = 𝑠𝑒𝑐2
𝜃
𝑐𝑜𝑡2
𝜃 + 1 = 𝑐𝑠𝑐2
𝜃
Tan and Cot
𝑡𝑎𝑛𝜃 =
𝑠𝑖𝑛𝜃
𝑐𝑜𝑠𝜃
,
𝑐𝑜𝑡𝜃 =
𝑐𝑜𝑠𝜃
𝑠𝑖𝑛𝜃
Half Angle Formulas
𝑠𝑖𝑛2
𝜃 =
1
2
(1 − cos 2𝜃 )
𝑐𝑜𝑠2
𝜃 =
1
2
1 + cos 2𝜃
𝑡𝑎𝑛2
𝜃 =
(1 − cos 2𝜃 )
(1 + cos 2𝜃 )
Even/Odd
sin −𝜃 = −𝑠𝑖𝑛𝜃
cos −𝜃 = 𝑐𝑜𝑠𝜃
tan −𝜃 = −𝑡𝑎𝑛𝜃
sin 2𝜃 = 2𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃, cos 2𝜃 = 𝑐𝑜𝑠2
𝜃 − 𝑠𝑖𝑛2
𝜃
Periodic Formula: Where n is an integer
sin 𝜃 + 2𝜋𝑛 = sin 𝜃 → sin 𝜃 + 180° = 𝑠𝑖𝑛𝜃
cos(𝜃 + 2𝜋𝑛) = cos 𝜃 → cos(θ + 180°) = 𝑐𝑜𝑠𝜃
tan 𝜃 + 𝜋𝑛 = tan 𝜃 → tan θ + 90° = tanθ
0° 30° 45° 60° 90°
Sin 0 1
2
1
2
3
2
1
Cos 1 3
2
1
2
1
2
0
Tan 0 1
3
1 3 ∞