IGCSE Maths

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Prime factorisation

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-3²=-9(-3)²=9Exponential growthUse same formula as compound interest∅The empty set (proper subset)ℰUniversal set∈is an element of {set}∉is not an element of {set}A ⊂ BA is a proper subset of BA ⊆ BA is a subset of BProper subsetAll subsets except the original set (P)Venn diagrams - A'Complement of A (all items not in A)A ∩ BIntersection of A and B (items in both)A ∪ BUnion of A and B (items in A or B or both)2-²1/4 (The reciprocal of the number without the negative)Vulgar fractionA fraction represented by a numerator and denominator and not by decimals.Multiplying with indicesAdd indices, numbers stayDividing with indicesSubtract indices, numbers stayAdding and subtracting indicesPowers and variables must be the samen(A)Number of elements in set AVulgar fractionsAll fractions excluding mixed fractions0.7 recurring as a fraction10f = 7.777..., 10f-f=7, 9f=7, f=7/90.3181818 recurring as a fraction100f =31.818181..., 100f-f=31.5, 99f=31.5, f=315/990=7/22Upper and lower boundsMinimum and maximum any rounded number could possibly bex to the power of ½the square root of xx to the power of ¾(the fourth root of x)cubedx to the power of -¼1/the fourth root of xx to the power of -¾1/(the fourth root of x)cubedx to the power of -5The reciprocal of x to the power of 5 (1/x to the 5)ax to the power of -5a/x to the power of 5Solve Inverse proportion (a ∝ the reciprocal of b)Find total by multiplying values (a=k/b, k=constant)Solve Direct proportion (a ∝ b)Divide to find the constant of proportionality (a=kb, k=constant)Average speedDistance/timeIncrease or decrease by a percentageFind the multiplier (24%=0.24)Currency conversionsUse value that is not one for both exchanges (either times by for currency represented by 1 or divide by for currency represented by point something)Compound interest formulaP(1+r/100)^n where r is the percentage, n is the time and P is the amount invested.Exponential growth and decayUse compound interest principlesthe quadratic formulaLinear sequencesLook for: difference between each number in sequence, difference between that and the first termQuadratic sequencesCompare to square numbers first. Difference between the difference between each number (2nd difference) is the same. 1/2 of the 2nd difference will be the coeffecient of n squared. Take that away from original sequence and find the rule for the remaining numbers.Cubic sequencesCompare to cube number first. Difference between the second difference is the same (3rd difference). 1/6 of the 3rd difference will be the coefficient of n cubed. Take that away from the original sequence and find the rule for the remaining numbers.a/b + c/d(ad + bc)/bda/b - c/d(ad - bc)/bda/b x c/dac/bdUpper and lower bounds e.g. 27.5≥x>28.5The upper bound is the maximum possible the rounded number could have been before rounding and the lower bound the opposite. Upper bound will be the number with 5 after the last decimal or as a unit depending on how it is rounded. The lower bound is the number 5 after one less than the last unit.RatioRatio is a way of comparing the sizes of two or more quantities. and will be set out in the form x:y. It can also be given in the form x/y. All the units in a ratio must be the sameA ratio as a fractionYou can express particular numbers in the ratio as fractions by finding what the whole ratio adds up to and using that as the denominator. e.g. 3/5 is the first part of the ratio 3:2Dividing a quantity in a ratioTo divide a quantity in a ratio, you must first find the total number of parts. Divide the quantity by the total number of parts to find one part and then multiply this to find parts of the ratio.Ratios when only some information is knownFind one part of the ratio. If 30 is 2 parts of the total ratio, divide it by 2 to find one part.Increases in ratiosIncrease 450 in the ratio 5:3. 450 = 3 parts, need to find 5 parts. Increased amount = (5/3) x 450Decreases in ratiosDecrease 450 in the ratio 5:3. 450=5 parts, need to find 3 parts. Decreased amount = (3/5) x 450RatesSpeed applied more generally to show changes in quantities. e.g. temperature goes down by 5 degrees every minuteUsing ratesTemperature lowers 5 degrees every minute. After 6 minutes, temperature lowers (5x6=30) 30 degrees. Temperature lowers by 1 degree in (1/5=0.2) 0.2 minutes. Temperature lowers by 15 degrees in (15/5=3) 3 minutesMap scales1:900000 (1cm on map = 900000cm on ground). 1cm = 9km. Actual distance km = distance on map cm x 9kmReverse percentages - unitary method110%=x. Find the original amount: 1%=x/110 original price = 1% x 100 (Used for decreases as well)Reverse percentages - multiplier method110%=x. Find the original amount: x/1.1 (the multiplier) = the original amount. 88%=x. Find the original amount: x/0.88 (the multiplier) = the original amount.Finding the multiplierA multiplier is the percentage divided by 100 to find the decimal. However to decrease a quantity by 7%, the multiplier used will be 0.93. If a quantity has been decreases by 7%, the amount left will be 93% of the totalFactorising simple quadratic equations (x²+bx+c)Factorise in the morning - AM stands for add, multiply. Find two numbers that add to the second number and multiply to the third number.Factorising quadratic equations with coefficients (ax²+bx+c)Write out the factors of a and c. Find the pair of a factors that multiply with the pair of c factors to make two numbers that add to b.Factorising tipsFactor out ALL common factors even if these are brackets, numbers and multiple variables.Simplifying algebraic fractionsAlways fully factorise and cancel common brackets. Brackets that are the same with opposite sides are equal to -1 on the numerator. Two negatives over each other cancel each other out. Variables should be in brackets before they are cancelled.Adding and subtracting fractionsFor fractions to be added or subtracted, the denominator MUST be the same.Multiplying fractionsPut the product orf the numerators over the product of the denomiators and then simplify if needed.Dividing fractionsTo divide fractions, times the first fraction by the reciprocal (turned upside down) of the second.A power to a power (x²)³If a power is raised a variable that is already being squared, you multiply the indices. The example will equal x to the power of 6.Negative indicesIf something is raised to a negative indices, find the reciprocal.Multiplying fractional indicesFractional indices can be added in multiplication in the same way as normal indices. They should be left as improper fractions not mixed numbers.(2x⁵)³ =8x to the power of 15Solving quadratic equations by completing the squareSolves quadratic equations by rewriting x squared + px +q in the form (x+a)^2 +b. Found by putting it into the form (x+(p/2))^2 -(p/2)^2 + q and rearranging to solve.Solve x²+6x-7 by completing the square(x+(6/2))^2 - (6/2)^2 - 7 = (x+3)^2 -3^2 - 7 = (x+3)^2 - 16. x=-2 +or- the square root of 16Sketching an inequalities graph - linear programmingSketch a broken line if the symbol is not equal to. Sketch a solid line if it is. Shade the unwanted region, e.g. if the wanted is y is less than x, shade the region where y is more than x.y=mx + cm is the gradient of the line, c is where the line intercepts the y axis.Parallel lines on a graphFor a line to be parallel it must have the same gradient, so variable m must be the same in both equations in y=mx + cUsing coordinates - gradientdifference between y-coordinates/difference between x-coordinatesUsing coordinates - midpointsAdd end values and divide by two for each x and y separatelyUsing coordinates -the distance between two pointsUse pythagoras' thereom to find the distance by using the gradients as the sides of a right angled trianglePerpendicular lines on a graphIf a line is perpendicular to the other line, the gradient will be the negative reciprocal of it. Also, the product of both gradients will be 1Quadratic graphs - parabolasThis will have either a u or upside down u shape and will be a smooth, continuous curve with no straight linesFinding the points of a quadratic graphDraw out a table and in each row put a separate component of the equation and the answer, then add these all together to find the points.Reciprocal graphs - y=a/xValues should be rounded to two decimal points. x will never equal zero. It will be symmetrical. The graph never touches either axes.Exponential graphsThis graph will have a changing gradientEstimating gradientsYou can estimate the gradient of a curved line by drawing a tangent to it and finding the gradient of that line.Function notationf(x)=3x-6Inverse functions - f-¹(x)To find the inverse of a function, replace f(x) with y and rearrange the equation to make x the subject. In the final answer, change the y to x.Composite functionsWhen functions are put together, read them backwards. e.g. fg(2) means start with 2, apply g then apply fComposite function expressionsTo make an expression for the composite function fg(x), put the expression for g, in the place of x in the expression for f. "Take x, apply g, then apply f"Different form of function notation - f:x→3x-5This is the same as f(x)=3x-5F angles on parallel linesCalled corresponding angles. They are equalU angles on parallel linesCalled interior angles. They add up to 180 degreesZ angles on parallel linesCalled alternate angles. They are the same.Congruent shapesShapes with the same size and shape as each other are congruent, even if they are placed in different positions.Similar shapesShapes are similar if one is an enlargement or reduction of the other. The corresponding angles of similar shapes are equal. Their sides can be found using ratios or linear scale factorsLinear scale factorUsed in similar shapes. To find it, find two corresponding sides and put the bigger side as the numerator of a fraction. Simplify the fraction.Area and volume scale factorUsed in similar shapes. This applies the same principles as the linear scale factor but is squared or cubed. To find the linear scale factor from these, find the square or cube root.Angles in a triangle...add to 180 degreesAngles in a quadrilateral...add to 360 degreesParallelogramOpposite sides are parallel and equal. Opposite angles are equal.RhombusAll sides are equal, opposite sides are parallel. Diagonals bisect at right angles. Opposite angles are equalKiteTwo pairs of equal adjacent sides. Has one long diagonal and one short diagonal that bisect each other at right angles. The opposite angles between the sides of different lengths are equal.TrapeziumTwo parallel sides. The sum of the interior angles at the ends of each non-parallel side is 180 degrees.A polygon is regular if...all interior angles are equal and all its sides are the same length. e.g. square, pentagon, hexagon, octagon.Interior anglesThe sum of the interior angles of any polygon can be found using 180 x (n-2) where n is the number of sidesExterior anglesThe sum of the exterior angles of any polygon is 360 degreesConstructing a triangleDraw the longest side as the base. Draw the next longest side with a compass as an arc and do the same with the shortest side. The draw the straight line to where the two arcs intersect.TangentA tangent is a straight line that touches a circle or curve as one point only. It is drawn perpendicular to the radiusAngles in a circle - two semi-circlesIf a diameter is drawn and an angle in the semi-circle uses it as its base (with its third point touching the circumference) then the angle at the circumference is 90 degrees.ArcAn arc is a section of the circumference of a circleAngles in a circle - centre, circumferenceIf an angle is extended to the centre of a circle from arc AB and another angle is extended to the circumference from arc AB then the angle at the circumference will be half the angle at the centre.Angles in a circle - same arc at circumferenceAll angles extending out to the circumference from arc AB will have the same angle.Angles in a circle - cyclic quadrilateralsOpposite angles in a cyclic quad add to 180 degrees.Constructing an accurate trapeziumTo construct a trapezium accurately, use first a protractor and a ruler and then a set square to find the parallel side.Constructing a line bisectorOpen compass to abut three quarters the length of the line and draw a semi-circle arc from either point on the line. The bisector is the point where the two arcs intersect.Constructing an angle bisectorWith the vertex of the angle as the centre, draw an arc through both lines and then, with centres where the arc intersects both lines, draw two more arcs. The angle bisector extends from the vertex to the point where the two arcs intersect.Construct a perpendicular from a point to a lineWith the point as the centre, draw an arc that intersects the line on either side. Using these points as the centre, follow the same directions as constructing a perpendicular.Locus (loci)A locus is a shape or a line which is always the same distance from a given point or line. You may be also asked to draw a locus that is equidistant from two points. This will usually be a straight line.Line of symmetry (mirror line)A line that can be drawn through a shape so that what can be seen on one side is a mirror image of what can be seen on the other side.Rotational symmetryA 2D shape has rotational symmetry if it can be rotated about a point to look exactly the same in a new position. It can be represented by the order of rotational symmetry or number of times it can be rotated to look exactly the same until it is back to its original position. e.g. an equilateral triangle has a rotational symmetry of threeSymmetry of 3D shapes3D shapes have planes of symmetry. One is a reflection of the other half. A cuboid has 3 planes of symmetryRotational symmetry of 3D shapes3D shapes have axes of symmetry. A cubiod has three axes of symmetry. Each axis has its own order of rotational symmetry.Equilateral triangular prismIt has four planes of symmetry. One from each of the points lengthways and one widthways halfway down. It also has two axis of rotational symmetry.ChordsA chord is a line in a circle where both ends touch the circumference, cutting a section of the circle.Symmetry in circles - chordsTwo chords of equal length are the same distance from the centre of the circle.Symmetry in circles - bisector of chordsThe perpendicular bisector of a chord passes through the centre of the circleSymmetry in circles - tangentsTwo tangents extending from a point to touch the circle at any two points will be the same lengthMensurationThe part of geometry concerning lengths, areas and volumes1000 kg =1 tonne1000 Litres =1 metre cubed1000 cm cubed =1L1 cm cubed =1 ml1 centiliter =10 ml1000 milliliters =1L1/2(a+b)hArea of a trapeziumbase x heightArea of a square, rectangle and parallelogrampi x diameterCircumference of a circlepi x radius squaredarea of a circlelength x width x heightvolume of a cuboidarea of cross-section x lengthvolume of a prismSectorPart of a circle bounded by two radii and one of the arcs.θ (theta)the angle at the centre of a sectorθ/360 x pi x diameterArc lengthθ/360 x pi x radius squaredSector areaWhich ratio to useSOHCAHTOAsine/cosine/tangent of known angle = side/side is used tocalculate the length of a side in a right-angled triangleThe inverse sine/cosine/tangent of known side/known side is used toFind the angles in a right-angled triangleSolving problems of angles of elevation and depressionUse the trigonometric ratios (sine/cosine/tangent)a/sinA=b/sinB=c/sinCThe sine rule (can be inverted so sines are on top)Sine rule notesWhen you are calculating a side, use the rule with sides on top. When you are calculating an angle, use the rule with sines on top. Is not restricted to right angled trianglesCosA= b squared+c squared - a squared/2bcThe cosine ruleCosine rule notesb and c in the rule are the sides making the angle being calculated and a is the opposite side. Is not restricted to right-angled trianglesChoosing the right rule - two sides and included angleCosine rule followed by sine ruleChoosing the right rule - two angles and a sideSine ruleThree sidesCosine rule followed by sine ruleFinding the area of a triangle using sine1/2 ab sinC (two sides with included angle)Obtuse sineThe sine of an obtuse angle is equal to that of its supplementary angle (sin100 = sin80)Obtuse cosineThe cosine of an obtuse angle is equal to the negative of its supplementary angle (cos100 = -cos80)VectorsVectors have both a magnitude and a direction. They can be represented by the start and end points with an arrow over top or as a lower case letter printed in bold OR underlined. On graphs they are represented by lines with arrows indicating directionVectors on a coordinate gridOn a coordinate grid, vectors can be represented by two numbers in a bracket as a fraction without the line. The top number is the amount moved across (negative left) and the bottom number is the amount moved vertically (negative down)The magnitude of a vectorthe square root of x square plus y squared - written with two vertical lines on either side of the vectorDescribing translations with vectorsAs shapes that have been translated are congruent, all points have moved the same amount. The translation is how any one point moved from its original place to its new place with its movement across on top and up and down on the bottom of the fraction without a line.Reflecting a shape over an axisThe easiest and most effective methods is counting how many squares away each individual point is from the mirror line and then making the point that many squares away on the opposite side. This should be done using the most direct route (even diagonally through squares)Rotating a shapeThe easiest and most effective way to rotate a shape about a point is to count how many squares across and up and down each individual point is away from the axis, then turn your paper the amount required and count the squares using the new axes.Enlarging a shapeAn enlargment will occur about a point and the distance away from this point should be taken into consideration as well as the size. The lengths of each side can be easily found by multiplying the original length by the scale factor.Enlarging a shape - ray methodIf you need to find the point from which the shape was enlarged, the ray method is helpful. Draw lines through corresponding points on the shape. The point where these lines meet is the point from which the shape was enlarged.Order of the matrixThe number of rows and columns in a matrix. Written as rows x columnsAdding and subtracting matricesFor matrices to be added or subtracted they must be of the same order.Multiplying matrices togetherWhen you multiply two matrices together, each row in the first matrix combines with each column of the second to give a single number. (The first to the left will combine with the uppermost, the second to the left will combine with the second down)Order of multiplied matricesYou can only multiply matrices if the number of columns in the first is equal to the number of rows in the second. The other two values will combine to give the order of he product.The zero matrix (Z)The zero matrix is a 2 x 2 matrix which is just zeros and acts like a normal zero in equations.The identity matrixA 2 x 2 matrix with 1 0 as its top numbers and 0 1 as its bottom numbers. This matrix is the same as multiplying it by 1.The determinant of a matrixThe determinant helps find the inverse. It can be represented by two lines around the letter representing the matrix. It is found using the formula ad - bc of matrix abcdFinding the inverse of a 2 x 2 matrixOf matrix abcd, swap the positions of a and d and change the signs of b and c. Then divide all the numbers by the determinant.If the determinant of a matrix is 0...It is a singular matrix and the inverse does not existTransformations using matricesYou can transform a shape using matrices, the matrix for a shape is the vector for each point put together e.g. a triangle will form a 2 x 3 matrix. You can then multiply it by a specific 2 x 2 matrix to find the vectors of the transformed shape.Finding a matrix transformationTo do this, you only need to find the vector for two points. Find where 1 0 (over each other) and 0 1 (over each other) will end up and put the vectors for these together in a matrix. Put the image vector of the vector that started on the x-axis first.Frequency tableA tally chart of the all the options. The first column is the options or numbers you are tallying, the second is the tallys and the third is the numerical value of the number of tallys in the second column.PictogramsPictograms are frequency tables which use pictures to represent frequency. Each picture represents a number of items. It helps people to understand it more quickly but doesn't allow for fractions of pictures.Bar chartsAll bars are the same width, the heights represent frequency. A dual chart is sometimes useful to compare two bar charts with similar data.Pie chartsThese do not show individual frequencies but instead compare the frequencies. Each is represented as a sector of the circle where the angle of each sector is proportional to the frequency it represents. The sectors should always be labelled.Calculating the angles of a pie chartTo find what angle a sector will be, take the frequency divided by the total number and then times that by 360. These angles do not have to labelled on the chartScatter diagramsCompares two variables by plotting corresponding values on a graph. They can have positive, negative or zero correlation between the points.HistogramsHistograms are similar to bar charts except, there are no gaps between bars, the horizontal axis has a continuous scale, the area of each bar represents the frequency.Histograms with bars of unequal widthFrequency is represented by area. THe vertical axis should read frequency density. To find the height of a bar where class widths are different, use the formula: frequency density=frequency of class interval/width of class interval.Histograms with unequal width - proportionalityAreas are proportional to frequenciesMedianThe middle value when put in order of size. It is not heavily affected by extreme values and is easy to find for ungrouped data. But, it doesn't use all the values and can be hard to understand.ModeThe value that occurs most often in a set of data. It can be used for qualitative data, is easy to find and is not affected by extreme values. But, it doesn't use all the values and may not exist.MeanThe mean is the sum of all the values in a set divided by the total number of values in the set. IT uses all the values although extreme values can affect it and it has to be calculated.RangeThe range is the highest value in the set minus the lowest value in the set. It shows the spread of the data, it can help to compare data and consistency.Using frequency tables to find the medianTo find the median from a frequency table, add all the values to find the total number of cars surveyed then divide this by 2 to find the middle value. Now add the frequencies up cumulatively to find which group contains this middle number.Using frequency tables to find the meanTo find the mean, the frequency must be multiplied by the by the class it is in. These final values should then be added up and divided by the total frequency. If the classes are grouped, use the middle value for the multiplication.Continuous dataData that can have any value within a range of values, for example mass, height, time.Discrete dataThis is data which contains separate numbers like goals scored, number of children or shoe size.Cumulative frequency diagramsA frequency table where the next the value has been added on to the previous value so that the number is always rising. This can then be plotted with the top value of the class group against the cumulative frequency. Plots can be joined with a freehand curve.Inter-quartile rangeBases the measure of spread on the middle 50% of the data removing extreme valuesEstimating the median from a cumulative frequency diagramThe median can be found by looking at the mark on the vertical axis which is half of the total. Follow this along to meet the curve and the value on the horizontal axis down from this is an estimate of the median. The median is also the 50th percentileFinding the interquartile range.The interquartile range can be found by taking the upper quartile (75th percentile, 75%) and taking away the lower quartile (25th percentile, 25%) to give the middle 50%. The upper and lower quartile can be found in the same way as the median but by finding 1/4 and 3/4 of the data instead of 1/2.The probability scaleThe probability scale goes from 0 being impossible to 1 being certain. Any probability values fall in between 0 and 1.Calculating probabilityProbability can be found with a probability fraction. The fraction of P(outcome) is the number of ways the outcome can happen/the total number of possible outcomes. This can then be changed to a decimal or percentage.The probability of an event occurring =1 - the probability of the event not occurring.Theoretical probability vs experimental probabilityThe way of calculating a probability is actually only theoretical and in practice, this probability is only an estimate of the real results. However, the more times you take data, the closer the experimental probability will get to the theoretical probabilityRelative frequencyAlso known as experimental probability, it is given by frequency of the outcome of event/total number of trials. This relative frequency is an estimate of the probabilityCombined frequency of two eventsYou can illustrate the possibilities of two combined events on a possibility diagram where all the possible outcomes for one variable is on one axis and for the other on the other axis. Every possible outcome for both variables is then represented on the diagram and the probability can be found.Tree diagramsProbabilities of sequences of events can be shown on a tree diagram. Each brach represents an outcome which branches into more outcomes. When doing this type of diagram, whether the outcome is put back into the total should be taken into consideration.