# CGP Mathematics Facts

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Maths Exam Questions
Physics: Unit One
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Maths
Physics
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Prime Numbers
Prime numbers don't divide by anything: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 ...
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How to find prime numbers: 1) It must end in 1, 3, 7, or 9. 2) It won't divide by any of the primes below the value of its own square root.
Multiples, Factors and Prime Factors
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Multiples of a number are simply its 'times tables'.
Factors of a number are all the numbers that divide into it.
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To find the prime factors, you must use a factor tree: Start at the top and split your number off into factors. Each time you get a prime number you ring it and finally end up with all the prime factors, which you can then arrange in order.
LCM and HCF
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LCM (Lowest Common Multiple) - Method: List the multiples of all the numbers. Find the smallest one that's in all the lists
HCF (Highest Common Factor) - Method: List all the factors of the numbers. Find the biggest one that's in all the lists
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Fractions
Terminating decimals are finite.
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Fractions where the denominator has prime factors of only 2 or 5 will give terminating decimals.
Recurring decimals have a pattern of numbers which repeats forever. All other fractions will give recurring decimals.
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How to turn recurring decimals into fractions: 1) Indentify the repeating unit, which will now become the numerator. 2) The same number of numbers will be on the denominator, and they will all be 9s.
Fractions (2)
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Multiplying: multiply top and bottom separately.
Dividing: first turn the second fraction upside down and then multiply.
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Cancelling down: divide top and bottom by the same number until they won't.
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Finding a fraction of something: multiply the 'something' by the top of the fraction, then divide by the bottom.
Ratios
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Formula triangle:
Proportional Division: 'a total amount is to be split in a certain ratio'. 1) Add up the parts. 2) Find the amount for one part, divide total amount by number of parts. 3) Hence find the three amounts.
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Percentages
Type 1 of question: 'Find X% of Y' -- 0.X x Y
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Type 2 of question: 'Express X as a percentage of Y' -- (X / Y) x 100
Percentage Change - 'change' / original x100
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'change' = profit, loss, appreciation, depreciation, increase, decrease, error, dicsount, etc.
Manipulating Surds and Use of Pi
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Rational Numbers include whole numbers, negative numbers, fractions, terminating decimals, and recurring decimals.
Irrational Numbers incude never-ending non-repeating decimals, square roots, and cube roots.
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1) √a x √b = √ab
2) √a / √b = √a/b
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3) √a + √b = √a + √b
4) (a + √b)^ = a^ + 2a√b + b
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5) Express 3/√5 in the form a√5/5 where a and b are whole numbers = rationalise the denominator.
Bounds and Reciprocals
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Estimating Square Roots: 1) Identify the square numbers either side of the number. 2) Find the square roots and pick an number between.
Upper and Lower bounds of a measurement: The real value can be as much as half the rounded unit above and below the rounded-off value.
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Reciprocals: 1) The reciprocal of a number is '1/' the number. 2) You can find the reciprocal of a fraction by turning it upside down. 3) A number multiplied by its reciprocal gives 1. 4) 0 has no reciprocal as you cannot divide anything by 0.
Metric and Imperial Units
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1 foot = 12 inches. 1 yard = 3 feet. 1 gallon = 8 pints. 1 stone = 14 pounds (lbs). 1 pound = 16 ounces (Oz).
1kg = 2 1/4 lbs; 1 gallon = 4.5 litres; 1m = 1 yard + 10%; 1 foot = 30cm; 1 litre = 1 3/4 pints; 1 metric tonne = 1 imperial tonne; 1 inch = 2.5cm; 1 mile = 1.6km (or 5 miles = 8km).
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Finding the nth term
Common Difference Type: nth term = 'dn + (a - d)'. 1) 'a' is the first term in the sequence. 2) 'd' is the common difference between the terms.
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Changing Difference Type: 'a + (n - 1)d + 1/2(n - 1)(n - 2)C'. 1) 'a' is the first term. 2) 'd' is the first difference. 3) 'C' is the change difference between one difference and another.
Areas
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Area of triangle = 1/2 x base x vertical height
Area of triangle = 1/2 ab SIN C
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Area of parallelogram = base x vertical height
Area of trapezium = average of parallel sides x distance between them
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Areas (2)
Area of circle = π x (radius)^
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Circumference = π x diameter
Area of Sector = θ/360 x area of full circle
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Length of Arc = θ/360 x circumference of full circle
Finding the area of a segment: 1) Find the area of the sector using the above formula. 2)Find the area of the triangle, then subtract it from the sector's area. You can do this using the '1/2 ab SIN C' formula for area of the triangle.
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Volume
Volume of Cuboid = length x width x height
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Volume of Prism = cross-sectional area x length
Volume of Sphere = 4/3 π r^
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Volume of Pyramid = 1/3 x base area x height
Volume of Cone = 1/3 x πr^ x height
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Volume of Frustum = volume of origina cone - volume of removed cone. = 1/3 πR^H - 1/3 πr^h.
Geometry Rules
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1) Angles in a triangle add up to 180 degrees.
2) Angles on a straight line add up to 180 degrees.
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4) Angles round a point add up to 360 degrees.
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5) Exterior angle of a triangle is the sum of opposite interior angles.
6) In an isosceles triangle, two sides are the same and two angles are the same.
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7) Whenever one line crosses two parallel lines, then the two bunches of angles are the same, and a + b = 180 degrees.
Loci and Construction
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Locus is a line that shows all the points which fit in with a given rule.
1) Locus of points that are 'fixed distance from given point'. 2) Locus of points that are 'fixed distance from a given line'. 3) Locus of points that are 'equidistant from two lines'. 4) Locus of points that are 'equidistant from two points'
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The Four Transformations
Translation - one detail. 1) Vector of translation. (rise/run)
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Enlargement - two details. 1) Scale factor. 2) Centre of enlargement.
Rotation - three details. 1) Angle turned. 2) Direction eg. clockwise, etc. 3) Centre of rotation.
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Reflection - one detail. 1) Mirror line
Congruence and Similarity
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If two shapes are congruent, they are simply the same (same size, same shape).
If one of these conditions hold, the triangles are congruent:
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1) SSS - three sides are the same
2) AAS - two angles and a side match
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3) SAS - two sides and the angle between them match
4) RHS - a right angle, the hypotenuse and one other side match
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Length, Area and Volume
For a scale factor n: Sides are -- n times bigger. Areas are -- n^ times bigger. Volumes are -- n^ times bigger.
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Length formulas (such as perimeter) always have lengths occurring singly. Area formulas always have lengths multiplied in pairs. Volume formulas always have lengths multiplied in groups of three.
Pythagoras' Theorum and Bearings
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Pythagoras' Theorum: a^ + b^ = c^
Bearings: 1) Place your pencil on the point in which you are going 'from'. 2) Draw a northline. 3) Draw in the angle clockwise from the northline to the line joining the two points.
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3D Pythagoras and Trigonometry
Angle between line and plane - using a diagram: 1) Make a right-angled triangle using the line, a line in the plane and a line between the two.
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2) Draw this right-angled triangle again so that you can see it clearly. Label the sides. You might have to use Pythagoras to work out the length of one of the sides. 3) Use trigonometry to calculate the angle.
Sine and Cosine Rules
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Sine Rule: a = b = c
Cosine Rule: a^ = b^ + c^ - 2bc COS A. or COS A = b^ + c^ - a^
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Sine and Cosine Rules (2)
1) Two angles given plus any side - sine rule needed
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2) Two sides given plus an angle not enclosed by them - sine rule needed
3) Two sides given plus the angle enclosed by them - cosine rule needed
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4) All three sides given but no angles - cosine rule needed
Graph of Sine
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Graph of Tan
Graph of Cosine
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D/T Graphs and V/T Graphs
Distance-Time Graphs: 1) At any point, gradient = speed. 2) The steeper the graph, the faster it's going. 3) Flat sections are where it is stopped.
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Velocity-Time Graphs: 1) At any point, gradient = acceleration (m/s^). 2) Negative slope is deceleration. 3) Flat sections are steady speed. 4) Area under graph = distance travelled.
Coordinates and Distances
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Finding the midpoint of a line segment: 1) Find the average of the two x-coordinates, then do the same for the y-coordinates. 2) These will be the coordinates of the midpoint.
Use Pythagoras to find the distance between points: 1) Draw a sketch to show the right-angled triangle. 2) Find the lengths of the sides of the triangle. 3) Use Pythagoras to find the length of the diagonal (answer).
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Gradient = change in y / change in x
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Equation of a straight line: y = mx + c.
Parallel lines have the same gradient.
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The gradients of two perpendicular lines multiply to give -1
If the gradient of the first line is m, the gradient of the other will be -1/m, because m x -1/m = -1
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Graph Shapes
Probability
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All probabilities are between 0 and 1. 0 = it will never happen, 1 = it will definitely happen
Tree Diagrams: 1) Always multiply along the branches. 2) On any set of branches which all meet at a point, the numbers must always add up to 1. 3) Add downwards to get your end result.
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Powers and Roots
1) When multiplying, you add the powers.
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2) When dividing, you subtract the powers.
3) When raising one power to another, you multiply them.
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4) Anything to the power of 1 is just itself.
5) Anything to the power of 0 is just 1.
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6) 1 to any power is still just 1.
7) Fractions - Apply power to both top and bottom.
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Compound Growth and Decay & Algebra
N = N. (1 + r/100)^. existing amount at this time, initial amount, percentage change per day/hour/year, number of days/hours/years.
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Difference of two squares: a^ - b^ = (a + b)(a - b).
Quadratic Formula & Completing the Square
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Quadratic Formula: x = -b +(or)- √ b^ - 4ac / 2a
Completing the Square: 1) As always, rearrange the quadratic into the standard format - ax^ + bx + c = 0. 2) If 'a' is not 1, then divide the whole equation by 'a'.
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## Other cards in this set

Maths

Physics

Prime Numbers

### Card 4

#### Front

How to find prime numbers: 1) It must end in 1, 3, 7, or 9. 2) It won't divide by any of the primes below the value of its own square root.

### Card 5

#### Front

Multiples of a number are simply its 'times tables'.