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Year 1 Pure Mathematics Cheat Sheet
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Trigonometry and Vector Mathematics
Trigonometric equations follow the same principles as regular algebra, but you need to consider multiple solutions within the given interval. Use the CAST diagram to determine which quadrants give positive values for sin, cos, and tan.
The unit circle and special angle values are essential knowledge. Learn that sin(30°) = 1/2, cos(60°) = 1/2, and tan(45°) = 1 - these pop up constantly in exam questions.
Vectors represent both magnitude and direction. Column vectors like (3,4) can be written as 3i + 4j. The magnitude of vector a is |a| = √, and the unit vector in direction a is a/|a|.
Vector addition follows the triangle law: to get from A to C via B, use AC = AB + BC. Position vectors always start from the origin, making calculations more straightforward.
Key Identity: Remember that sin²θ + cos²θ = 1 - this trigonometric identity solves many complex problems!
Fundamental Algebraic Skills
The laws of indices are the building blocks of algebra - master these and you'll breeze through exponential equations. Key rules include aᵐ × aⁿ = aᵐ⁺ⁿ and (aᵐ)ⁿ = aᵐⁿ.
Surds and rationalising denominators often trip students up, but the technique is straightforward. To rationalise 1/√a, multiply by √a/√a. For more complex expressions like a/, multiply by the conjugate /.
Quadratic equations can be solved using the formula x = /2a, but completing the square gives you more insight. The completed square form a² + q immediately shows the turning point at .
The discriminant tells you everything about the roots: positive means two real roots, zero means one repeated root, and negative means no real roots.
Pro Tip: When solving inequalities, remember to flip the inequality sign if you divide by a negative number!
Exponential and Logarithmic Equations
Ever wondered how to solve equations where x appears as an exponent? Logarithms are your secret weapon for bringing those tricky powers down to earth.
When solving exponential equations like 3^x = 2x + 1, take the logarithm of both sides. This transforms the equation into something manageable: x log 3 = log 2. From here, it's just algebra - collect like terms, factorise, and solve for x.
Natural logarithms (ln) work hand-in-hand with the exponential function e^x. They're inverse functions, so ln = x and e^(ln x) = x. This relationship is crucial for solving more complex exponential equations.
Remember that e^x is always positive - it can never equal a negative number. If you get e^x = -2, something's gone wrong in your working!
Quick Tip: When you see e^(2x), think of it as ² - it's a quadratic function in disguise!
Circle Geometry and Algebraic Techniques
Circle intersections with straight lines follow predictable patterns - a line can intersect a circle twice, once (tangent), or not at all. Understanding this helps you predict how many solutions your equations will have.
To find a circle's equation, remember the three-step process: find the centre coordinates (a,b), calculate the radius using any given point, then substitute into ² + ² = r². For right-angled triangles inscribed in circles, the hypotenuse is always the diameter.
Algebraic fractions and polynomial division might look intimidating, but they follow the same rules as arithmetic division. The factor theorem is particularly useful - if is a factor of f(x), then f(a) = 0.
Binomial expansion using Pascal's triangle gives you the coefficients for expressions like ⁿ. The index n+1 tells you which row of Pascal's triangle to use. For binomial estimation, you can approximate values like 0.975⁵ by treating it as ⁵ where x = 0.025.
Remember: Tangents to circles are always perpendicular to the radius at the point of contact.
Advanced Trigonometry and Differentiation
The sine and cosine rules are your go-to tools for non-right-angled triangles. Use the cosine rule when you have two sides and the included angle, or three sides. The sine rule works when you have matching pairs of sides and angles.
Trigonometric graphs follow predictable patterns. Understanding transformations like y = sin (horizontal shift) and y = k sin x (vertical stretch) helps you sketch complex trig functions quickly.
Differentiation gives you the gradient of curves at any point. The power rule is your foundation: xⁿ becomes nx^. Always rearrange expressions into the form axⁿ before differentiating.
Stationary points occur where dy/dx = 0. Use the second derivative test: if f''(x) > 0, you've found a minimum; if f''(x) < 0, it's a maximum. This is much quicker than testing points either side.
Essential Skill: Area of any triangle = ½ab sin C, where C is the angle between sides a and b.
Graph Sketching and Coordinate Geometry
Cubic and quartic graphs follow predictable shapes based on their leading coefficient. Positive coefficients start low and end high, while negative coefficients do the opposite. Reciprocal graphs like y = 1/x have asymptotes at x = 0 and y = 0.
Graph transformations follow clear rules: f shifts left by k units, f(x) + k shifts up by k units. Reflections use f for the y-axis and -f(x) for the x-axis.
The gradient formula / and distance formula √ are fundamental tools. Parallel lines have the same gradient, while perpendicular lines have gradients that multiply to give -1.
Circle equations come in two forms: x² + y² = r² for circles centred at the origin, and ² + ² = r² for circles with centre (a, b). The expanded form x² + y² + 2fx + 2gy + c = 0 has centre and radius √.
Memory Aid: For perpendicular gradients, if one is m, the other is -1/m - they're negative reciprocals!
Calculus and Exponential Functions
Integration is the reverse of differentiation. The power rule becomes: xⁿ integrates to x^/ + C. Never forget the constant of integration C - it's essential for indefinite integrals.
Definite integrals calculate the area under curves between two limits. Use the format ᵇₐ = F(b) - F(a). If the area lies below the x-axis, your answer will be negative.
Exponential functions like eˣ have unique properties. When differentiating e^(kx), you get ke^(kx). Exponential models using e^(kt) represent exponential growth, while e^ represents exponential decay.
Logarithm laws are crucial for simplifying complex expressions. Key rules include: log(ab) = log a + log b, log = log a - log b, and log(aⁿ) = n log a. These laws transform multiplication into addition and powers into products.
Integration Tip: When finding the constant C, substitute known coordinates into your integrated function and solve!
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The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
Year 1 Pure Mathematics Cheat Sheet
UKRevisionResources
@ukrevisionresources
This comprehensive guide covers Chapters 1-14 of Pure Maths, from basic algebra through to advanced calculus and exponential functions. Whether you're tackling quadratics, trigonometry, or differentiation, these key concepts will help you master your A-level maths with confidence.
Trigonometry and Vector Mathematics
Trigonometric equations follow the same principles as regular algebra, but you need to consider multiple solutions within the given interval. Use the CAST diagram to determine which quadrants give positive values for sin, cos, and tan.
The unit circle and special angle values are essential knowledge. Learn that sin(30°) = 1/2, cos(60°) = 1/2, and tan(45°) = 1 - these pop up constantly in exam questions.
Vectors represent both magnitude and direction. Column vectors like (3,4) can be written as 3i + 4j. The magnitude of vector a is |a| = √, and the unit vector in direction a is a/|a|.
Vector addition follows the triangle law: to get from A to C via B, use AC = AB + BC. Position vectors always start from the origin, making calculations more straightforward.
Key Identity: Remember that sin²θ + cos²θ = 1 - this trigonometric identity solves many complex problems!
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By signing up you accept Terms of Service and Privacy Policy
Fundamental Algebraic Skills
The laws of indices are the building blocks of algebra - master these and you'll breeze through exponential equations. Key rules include aᵐ × aⁿ = aᵐ⁺ⁿ and (aᵐ)ⁿ = aᵐⁿ.
Surds and rationalising denominators often trip students up, but the technique is straightforward. To rationalise 1/√a, multiply by √a/√a. For more complex expressions like a/, multiply by the conjugate /.
Quadratic equations can be solved using the formula x = /2a, but completing the square gives you more insight. The completed square form a² + q immediately shows the turning point at .
The discriminant tells you everything about the roots: positive means two real roots, zero means one repeated root, and negative means no real roots.
Pro Tip: When solving inequalities, remember to flip the inequality sign if you divide by a negative number!
Sign up to see the contentIt's free!
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Improve your grades
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Exponential and Logarithmic Equations
Ever wondered how to solve equations where x appears as an exponent? Logarithms are your secret weapon for bringing those tricky powers down to earth.
When solving exponential equations like 3^x = 2x + 1, take the logarithm of both sides. This transforms the equation into something manageable: x log 3 = log 2. From here, it's just algebra - collect like terms, factorise, and solve for x.
Natural logarithms (ln) work hand-in-hand with the exponential function e^x. They're inverse functions, so ln = x and e^(ln x) = x. This relationship is crucial for solving more complex exponential equations.
Remember that e^x is always positive - it can never equal a negative number. If you get e^x = -2, something's gone wrong in your working!
Quick Tip: When you see e^(2x), think of it as ² - it's a quadratic function in disguise!
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Circle Geometry and Algebraic Techniques
Circle intersections with straight lines follow predictable patterns - a line can intersect a circle twice, once (tangent), or not at all. Understanding this helps you predict how many solutions your equations will have.
To find a circle's equation, remember the three-step process: find the centre coordinates (a,b), calculate the radius using any given point, then substitute into ² + ² = r². For right-angled triangles inscribed in circles, the hypotenuse is always the diameter.
Algebraic fractions and polynomial division might look intimidating, but they follow the same rules as arithmetic division. The factor theorem is particularly useful - if is a factor of f(x), then f(a) = 0.
Binomial expansion using Pascal's triangle gives you the coefficients for expressions like ⁿ. The index n+1 tells you which row of Pascal's triangle to use. For binomial estimation, you can approximate values like 0.975⁵ by treating it as ⁵ where x = 0.025.
Remember: Tangents to circles are always perpendicular to the radius at the point of contact.
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Improve your grades
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Advanced Trigonometry and Differentiation
The sine and cosine rules are your go-to tools for non-right-angled triangles. Use the cosine rule when you have two sides and the included angle, or three sides. The sine rule works when you have matching pairs of sides and angles.
Trigonometric graphs follow predictable patterns. Understanding transformations like y = sin (horizontal shift) and y = k sin x (vertical stretch) helps you sketch complex trig functions quickly.
Differentiation gives you the gradient of curves at any point. The power rule is your foundation: xⁿ becomes nx^. Always rearrange expressions into the form axⁿ before differentiating.
Stationary points occur where dy/dx = 0. Use the second derivative test: if f''(x) > 0, you've found a minimum; if f''(x) < 0, it's a maximum. This is much quicker than testing points either side.
Essential Skill: Area of any triangle = ½ab sin C, where C is the angle between sides a and b.
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By signing up you accept Terms of Service and Privacy Policy
Graph Sketching and Coordinate Geometry
Cubic and quartic graphs follow predictable shapes based on their leading coefficient. Positive coefficients start low and end high, while negative coefficients do the opposite. Reciprocal graphs like y = 1/x have asymptotes at x = 0 and y = 0.
Graph transformations follow clear rules: f shifts left by k units, f(x) + k shifts up by k units. Reflections use f for the y-axis and -f(x) for the x-axis.
The gradient formula / and distance formula √ are fundamental tools. Parallel lines have the same gradient, while perpendicular lines have gradients that multiply to give -1.
Circle equations come in two forms: x² + y² = r² for circles centred at the origin, and ² + ² = r² for circles with centre (a, b). The expanded form x² + y² + 2fx + 2gy + c = 0 has centre and radius √.
Memory Aid: For perpendicular gradients, if one is m, the other is -1/m - they're negative reciprocals!
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Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Calculus and Exponential Functions
Integration is the reverse of differentiation. The power rule becomes: xⁿ integrates to x^/ + C. Never forget the constant of integration C - it's essential for indefinite integrals.
Definite integrals calculate the area under curves between two limits. Use the format ᵇₐ = F(b) - F(a). If the area lies below the x-axis, your answer will be negative.
Exponential functions like eˣ have unique properties. When differentiating e^(kx), you get ke^(kx). Exponential models using e^(kt) represent exponential growth, while e^ represents exponential decay.
Logarithm laws are crucial for simplifying complex expressions. Key rules include: log(ab) = log a + log b, log = log a - log b, and log(aⁿ) = n log a. These laws transform multiplication into addition and powers into products.
Integration Tip: When finding the constant C, substitute known coordinates into your integrated function and solve!
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
Where can I download the Knowunity app?
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The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user