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Summary MATHEMATICS FORMULAS AND SUMMARIES
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MATHEMATICS FORMULAS AND SUMMARIES
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MATHEMATICSFORMULAS
AND SUMMARIES
, SYMBOLS
S.I units are used in the K.C.S.E four year course including the symbols +, −, ×, ÷, ± could also be used. The
following rational symbols are important both in mathematics and physics:
= Is equal to
≠ Is not equal to
> Is greater than
≥ Is greater than or equal to
< Is less than
≤ Is less than or equal to
a: b Ratio of a to b
∝ Varies as
≡ Is congruent to or identical to
≃ Approximately equal to
- Is equivalent
⇒ Implies
∴ Therefore
Table of Contents
SYMBOLS .........................................................................................................................................................................................................
2. 00 NUMBERS................................................................................................................................................
Addition and Subtraction Rules: .....................................................................................................................................................................
37 – 45 = 37 + −45 = −(45 −37 ) = −8 ............................................................................................................................................................
(B) BRACKETS (E) EXPONENT (D) DIVISION (M) MULTIPLICATION or of (A) ADDITION (S)SUBTRACTION – B.E.D.M.A.S ......
2.02 Prime numbers /Factors ........................................................................................................................................................................
2.03 G.C.D OR H.C.F /LCM ...........................................................................................................................................................................
TO FIND THE L.C.M OF A SET OF NUMBERS: ...........................................................................................................................................
The L.C.M is the product of ALL the common prime factors and the remaining prime factors.(G.C.D = product of common prime
factors only) ............................................................................................................................................................................................
2.04 FRACTIONS AND DECIMALS ..............................................................................................................................................................
,2.05 SQUARES, SQUARE ROOTS ................................................................................................................................................................
2.06 CUBES AND CUBE ROOTS ...................................................................................................................................................................
2.07 RECIPROCALS ......................................................................................................................................................................................
2.08 RATIOS, RATES, PERCENTAGES AND PROPOTIONS ......................................................................................................................
2.09 COMPOUND PROPORTIONS ..............................................................................................................................................................
2.10 RATES OF WORK ..................................................................................................................................................................................
2.11 MIXTURES ............................................................................................................................................................................................
2.12 LINEAR MOTION ..................................................................................................................................................................................
2.13 INDICES, LOGARITHMS AND FURTHER LOGARITHMSINDICES ................................................................................................
2.14 LOGARITHMS USING MATHEMATICAL TABLES ............................................................................................................................
NOTE:...............................................................................................................................................................................................................
2.15 FURTHER LOGARITHMS ....................................................................................................................................................................
2.16 APPROXIMATIONS AND ERRORS .....................................................................................................................................................
3.00 MEASUREMENTS I ......................................................................................................................................
3.01 LENGTH AND PERIMETERS ...............................................................................................................................................................
3.02 PYTHAGORUS THEOREM ...................................................................................................................................................................
3.06 TIME:......................................................................................................................................................................................................
2) TRAVEL TIME TABLES ........................................................................................................................................................................
4.00 MEASUREMENTS II ......................................................................................................................................
4.11 AREA OF TRIANGLES ..........................................................................................................................................................................
4.12 AREA OF QUADRILATERALS .............................................................................................................................................................
4.13 AREA OF PARTY OF A CIRCLESECTOR .............................................................................................................................................
4.14 SEGEMENT ..............................................................................................................................................................................................
4.15 AREA OF A REGULAR POLYGON .......................................................................................................................................................
4.16 COMMON REGIONS BETWEEN TWO CIRCLES ...............................................................................................................................
4.20 SURFACE AREA OF SOLIDS ................................................................................................................................................................
4.22 CONE ......................................................................................................................................................................................................
4.23 PYRAMIDS .............................................................................................................................................................................................
4.24 FRUSTUM ..............................................................................................................................................................................................
4.25 SPHERES ...............................................................................................................................................................................................
4.27 SURFACE AREAS OF COMBINED SOLIDS ........................................................................................................................................
4.31 PRISM ....................................................................................................................................................................................................
4.32 PYRAMID ...............................................................................................................................................................................................
4.33 CONE ......................................................................................................................................................................................................
,4.34 FRUSTUM ..............................................................................................................................................................................................
4.35 SPHERE .................................................................................................................................................................................................
4.36 HEMISPHERE .......................................................................................................................................................................................
4.37 VOLUMES OF COMBINED SOLIDS ....................................................................................................................................................
5.00 ALGEBRA ...................................................................................................................................................... 2
5.11 SIMPLIFICATION OF ALGEBRAIC EXPRESSIONS ...........................................................................................................................
5.12 REMOVAL OF BRACKETS ....................................................................................................................................................................
5.13 FACTORIZATION BY GROUPING .......................................................................................................................................................
5.14 SUBSTITUTION AND EVALUATION ..................................................................................................................................................
5.20 LINEAR EQUATIONS ...........................................................................................................................................................................
5.22 LINEAR EQUATIONS FROM REAL LIFE SITUATIONS ....................................................................................................................
5.23 SIMULTANEOUS EQUATIONS ...........................................................................................................................................................
a) ELIMINATION METHOD .....................................................................................................................................................................
b) SUBSTITUTION METHOD ...................................................................................................................................................................
5.30 EQUATIONS OF STRAIGHT LINES ....................................................................................................................................................
5.32 EQUATION OF A STRAIGHT LINE .....................................................................................................................................................
5.33 PERPENDICULAR LINES AND THEIR GRADIENTS ........................................................................................................................
5.34 PARALLEL LNES AND THEIR GRADIENTS ......................................................................................................................................
5.35 THE X – AND Y – INTERCEPTS OF A LINE .......................................................................................................................................
5.40 QUADRATIC EXPRESSIONS AND EQUATIONS I .............................................................................................................................
5.42 QUADRATIC IDENTITIES....................................................................................................................................................................
5.43 FACTORIZING QUADRATIC EXPRESSIONS .....................................................................................................................................
5.50 QUADRATIC EXPRESSIONS AND EQUATIONS II ............................................................................................................................
5.52 COMPLETING THE SQUARE ...............................................................................................................................................................
5.53 QUADRATIC FORMULA.......................................................................................................................................................................
5.54 TABLE OF VALUES FOR A GIVEN QUADRATIC RELATION ...........................................................................................................
5.55 GRAPHS OF QUADRATIC EQUATIONS AND SOLUTIONS ..............................................................................................................
5.56 SIMULTANEOUS EQUATIONS, ONE LINEAR AND THE OTHER QUADRATIC............................................................................
5.60 LINEAR INEQUALITIES ......................................................................................................................................................................
5.62 SIMPLE LINEAR INEQUALITIES IN ONE UNKNOWN ....................................................................................................................
5.63 COMPOUND INEQUALITY STATEMENTS IN ONE UNKNOWN .....................................................................................................
5.64 GRAPHICAL REPRESNTATION OF LINEAR INEQUALITIES .........................................................................................................
5.65 SOLUTIONS OF LINEAR INEQUALITIES ..........................................................................................................................................
5.66 FORMATION OF SIMPLE LINEAR INEQUALITIES FROM GRAPHS..............................................................................................
,5.70 LINEAR PROGRAMING – LINEAR INEQUALITIES II .....................................................................................................................
5.72 GRAPHICAL SOLUTIONS OF LNEAR INEQUALITIES AND OPTIMIZATION ...............................................................................
6.00 SURDS ......................................................................................................................................................... 3
1. ADDITION AND SUBTRACTION OF SURDS: ....................................................................................................................................
6.02 RATIONALIZATION OF DENOMINATORS .........................................................................................................................................
2. FOR DIVISION: SURDS ARE SIMPLIFIED VIA RATIONALIZING THE DENOMINATOR: ...........................................................
7.00 SEQUENCES AND SERIES .............................................................................................................................
7.02 G.P SERIES ............................................................................................................................................................................................
7.03 SUM OF A.P SERIES .............................................................................................................................................................................
7.04 SUM OF G.P SERIES .............................................................................................................................................................................
8.00 BINOMIAL EXPANSION ................................................................................................................................
8.02 COEFFICIENTS OF TERMS IN A BINOMIAL EXPANSION ..............................................................................................................
8.03 COMPUTATION USING BINOMIAL EXPANSION .............................................................................................................................
8.04 EVALUATION OF NUMERICAL CASES USING BINOMIAL EXPANSION. .....................................................................................
9.00 FORMULAE AND VARIATION ......................................................................................................................
9.01 DIRECT VARIATION ............................................................................................................................................................................
9.02 INVERSE VARIATION: .........................................................................................................................................................................
9.03 JOINT VARIATION ...............................................................................................................................................................................
9.04 PARTIAL VARIATION: .........................................................................................................................................................................
9.05 FORMULAE: ..........................................................................................................................................................................................
10.00 GEOMETRY ................................................................................................................................................. 3
10.10 ANGLES AND PLANE FIGURES ..........................................................................................................................................................
10.12 ANGLE PROPERTIES OF POLYGONS.................................................................................................................................................
b) REGULAR POLYGONS .........................................................................................................................................................................
c) ALL POLYGONS ....................................................................................................................................................................................
10.20 GEOMETRIC CONSTRUCTIONS OF ANGLES AND TRIANGLES CONSTRUCTION OF LINE AND ANGLES USING RULER
AND COMPASSES ONLY ......................................................................................................................................................................
10.22 BISECTNG AN ANGLE ..........................................................................................................................................................................
10.23. AN ANGLE OF 450: ..............................................................................................................................................................................
10.24 AN ANGLE OF 600: ...............................................................................................................................................................................
10.25 CONSTRUCTION OF ANGLES OF MULTIPLES OF ...........................................................................................................................
CONSTRUCTION OF PERPENDICULAR AND PARALLEL LINES USING RULER AND COMPASSESOR SET SQUARE ONLY .........
10.27 PERPENDICULAR FROM A POINT TO A LINE: ................................................................................................................................
10.28 PARALLEL LINE THROUGH A GIVEN POINT ..................................................................................................................................
,10.29 CONSTRUCTIONOF REGULAR AND IRREGULAR POLYGONS UPTO A HEXAGON ....................................................................
10.30 LOCI........................................................................................................................................................................................................
10.31 THE PERPENDICULAR BISECTOR LOCUS: ......................................................................................................................................
10.32 PARALLEL LINES LOCUS: ...................................................................................................................................................................
10.33 CIRCLE LOCUS: ....................................................................................................................................................................................
10.34 ANGLE BISECTOR LOCUS: ..................................................................................................................................................................
10.35 CONSTANT ANGLE LOCUS: ................................................................................................................................................................
10.36 INTERSECTING LOCUS: ......................................................................................................................................................................
10.37 INEQUALITIES LOCI: ..........................................................................................................................................................................
10.38 CONSTRUCTION OF CIRCUMSCRIBED OR CIRCUMCIRCLE (CIRCLE ROUND A TRIANGLE) .................................................
10.39 CONSTRUCTION OF THE INSCRIBED CIRCLE (CIRCLE INSIDE THE TRIANGLE) ....................................................................
10.40 BEARINGS AND SCALE DRAWING ....................................................................................................................................................
180 + ϴ for ϴ˂180 and ϴ - 180 if ϴ > 180. ......................................................................................................................................................
10.43 ANGLES OF ELEVATION AND DEPRESSION ...................................................................................................................................
10.44 SURVEYING TECHNIQUES .................................................................................................................................................................
10.50 COMMON SOLIDS ................................................................................................................................................................................
DISTANCE BETWEEN TWO POINTS ON THE SURFACE OF A SOLID .....................................................................................................
10.60 THREE DIMENSIONAL GEOMETRY ..................................................................................................................................................
10.61 ANGLE BETWEEN A LINE AND A LINE.............................................................................................................................................
10.64 ANGLE BETWEEN SKEWLINES .........................................................................................................................................................
11.00 ANGLE PROPERTIES OF A CIRCLE ..............................................................................................................
11.02 ANGLES IN A SEMI-CIRCLE ................................................................................................................................................................
11.03 RELATIONSHIP BETWEEN ANGLE SUBTENDED AT THE CENTRE OF A CIRCLE AND ANGLESUBTENEDED ON THE
CIRCUMFERENCE................................................................................................................................................................................
11.04 ANGLE PROPETIES OF A CYCLIC QUADRILATERAL ......................................................................................................................
12.01 LENGTHS OF TANGENTS AND CHORDS ..........................................................................................................................................
12.02 PROPERTIES OF CHORDS AND TANGENTS.....................................................................................................................................
12.03 INTERSECTING CHORDS:...................................................................................................................................................................
12.04 CONTRUCTION OF AN ESCRIBED CIRCLE.......................................................................................................................................
12.05 CONSTRUCTION OF TANGENTS TO A CIRCLE ................................................................................................................................
12.06 DIRECT COMMON TANGENTS TO TWO CIRCLES...........................................................................................................................
12.07 INVERSE COMMON TANGENTS TO TWO CIRCLES ........................................................................................................................
12.08 ANGLES IN THE ALTERNATE SEGMENT .........................................................................................................................................
13.00 GRAPHS ......................................................................................................................................................
,13.10 COORDINATES AND GRAPHS ............................................................................................................................................................
13.12 TABULATION OF VALUES FOR A LINEAR RELATION ....................................................................................................................
13.13 GRAPHICAL SOLUTION OF LINEAR SIMULTANEOUS EQUATIONS............................................................................................
13.20 GRAPHICAL METHODS .......................................................................................................................................................................
13.22 GRAPHS OF QUADRATIC AND CBIC RELATIONS ...........................................................................................................................
13.23 SOLUTIONS OF QUADRATIC AND CUBIC EQUATIONS FROM GRAPHS .....................................................................................
13.24 QUADRATIC CURVES ..........................................................................................................................................................................
13.25 CUBIC GRAPHS .....................................................................................................................................................................................
13.28 BEST LINE OF FIT, EMPIRICAL DATA AND THEIR GRAPHS .........................................................................................................
13.29 EQUATIONS OF A CIRCLE, THE CENTRE AND RADIUS OF A CIRCLE .........................................................................................
14.00 TRIGONOMETRY ......................................................................................................................................... 5
14.11 TANGENT, SINE AND COSINE OF ANGLES ......................................................................................................................................
14.12 SINE AND COSINE OF COMPLEMENTARY ANGLES .......................................................................................................................
14.13 TRIGONOMETRIC RATIOS OF SPECIAL ANGLES ............................................................................................................................
14.20 TRIGONOMETRY II ..............................................................................................................................................................................
14.30 TRIGONOMETRY III.............................................................................................................................................................................
14.32 SIMPLE TRIGONOMETRIC EQUATIONS ..........................................................................................................................................
14.33 AMPLITITUDE, PERIOD, WAVELENGTH AND PHASE ANGLE OF TRIGONOMETREIC ............................................................
15.00 COMMERCIAL ARITHMETIC .......................................................................................................................
15.10 COMMERCIAL ARITHMETIC I............................................................................................................................................................
15.12 EXCHANGE RATES...............................................................................................................................................................................
15.13 DISCOUNTS ...........................................................................................................................................................................................
15.14 COMMISSIONS .....................................................................................................................................................................................
15.20 COMMERCIAL ARITHMETIC II ..........................................................................................................................................................
15.22 COMPOUNG INTEREST FORMULA AND CALCULATION ...............................................................................................................
15.23 COMPOUND INTEREST EQUATION FOR MULTIPLE INVESTMENTS IN EQUAL AMOUNTS ...................................................
15.24 APPRECIATION AND DEPRECIATION (USE OF COMPOUND INTEREST FORMULA) ...............................................................
15.25 HIRE PURCHASE ..................................................................................................................................................................................
15.26 INCOME TAX.........................................................................................................................................................................................
16.00 STATISTICS AND PROPBABILITY ................................................................................................................
16.10 STATISTICS I .........................................................................................................................................................................................
16.12 MEASURES OF GROUPED DATA ........................................................................................................................................................
REPRESENTATION OF DATA .......................................................................................................................................................................
16.14 HISTOGRAM .........................................................................................................................................................................................
,16.15 FREQUENCY POLYGONS.....................................................................................................................................................................
16.20 STATISTICS II .......................................................................................................................................................................................
16.21 MEAN USING ASSUMED MEAN .........................................................................................................................................................
16.22 MEAN BY CODING ................................................................................................................................................................................
16.23 CUMULATIVE FREQUENCY TABLES AND OGIVE ...........................................................................................................................
16.24 MEDIAN AND QUARTILES BY CALCULATION ................................................................................................................................
16.25 RANGE, QUARTILES AND INTERQUARTILE RANGE .....................................................................................................................
16.26 QUARTILE DEVIATIONS AND VARIANCE ........................................................................................................................................
16.27 STANDARD DEVIATION ......................................................................................................................................................................
16.30 PROBABILITY DEFINATION OF PROBABILITY ...............................................................................................................................
16.31 SAMPLE SPACE .....................................................................................................................................................................................
COMBINED EVENTS ......................................................................................................................................................................................
16.32 INDEPENDENT EVENTS .....................................................................................................................................................................
16.33 MUTUALLY EXCLUSIVE EVENTS: .....................................................................................................................................................
16.34 PROBABILITY TREE DIAGRAMS ........................................................................................................................................................
17.00 VECTORS ..................................................................................................................................................... 6
17.10 VECTORS I VECTOR NOTATION ........................................................................................................................................................
17.11 ADDITION AND SUBTRACTION OF VECTORS .................................................................................................................................
17.12 COLUMN VECTORS ..............................................................................................................................................................................
17.13 MAGNITUDE OF A VECTOR ................................................................................................................................................................
EQUIVALENT VECTORS ................................................................................................................................................................................
17.14 ADDITION AND SUBTRACTION OF COLUMN VECTORS................................................................................................................
17.15 MULTIPLICATION OF A VECTOR BY A SCALAR ..............................................................................................................................
17.16 TRANSLATION VECTOR ......................................................................................................................................................................
17.17 POSITION VECTOR ..............................................................................................................................................................................
17.18 MID-POINT OF A VECTOR ..................................................................................................................................................................
17.20 VECTORS II ...........................................................................................................................................................................................
17.21 COLUMN AND POSITION VECTORS IN THREE DIMENSION ........................................................................................................
17.22 COLUMN VECTORS IN TERMS OF i,j and k .......................................................................................................................................
17.23 MAGNITUDE OF A VECTOR IN THREE DIMENSIONS....................................................................................................................
17.24 PARALLEL VECTORS AND COLLINEARITY ......................................................................................................................................
17.25 THE RATIO THEOREM ........................................................................................................................................................................
18.00 TRANSFORMATIONS................................................................................................................................... 6
18.11 PROPERTIES OF ROTATION ...............................................................................................................................................................
,CONGRUENCE AND ROTATION ..................................................................................................................................................................
18.12 CENTRE OF ROTATION .......................................................................................................................................................................
18.13 ANGLE OF ROTATION .........................................................................................................................................................................
18.14 ROTATIONAL SYMETRY......................................................................................................................................................................
N.B: SYMMETRY .............................................................................................................................................................................................
18.30 SIMILARITY AND ENLARGEMENT....................................................................................................................................................
PROPERTIESB OF SIMILAR FIGURES.........................................................................................................................................................
18.32 ENLARGEMENT ...................................................................................................................................................................................
18.33 LINEAR, AREA AND VOLUME SCALE FACTORS AND THEIR RELATION ....................................................................................
19.00 MATRICES ...................................................................................................................................................
19.01 MATRIX COMPATIBILITY IN ADDITION AND SUBTRACTION......................................................................................................
19.02 MATRIX COMPATIBILITY IN MULTIPLICATIONMULTIPLICATION OF A MATRIX BY A SCALAR k: ......................................
MULTIPLICATION OF MATRICES................................................................................................................................................................
19.03 DETERMINANT OF A 2X2 MATRIX ...................................................................................................................................................
19.04 INVERSE OF A 2X2 MATRIX ...............................................................................................................................................................
SINGULAR MATRIX .......................................................................................................................................................................................
19.05 SOLUTION OF SIMULTANEOUS EQUATIONS BY MATRIX METHOD ..........................................................................................
20.00MATICES AND TRANSFORMATIONS ........................................................................................................... 6
20.01 IDENTIFICATION OF A TRANSFORMATION MATRIX....................................................................................................................
20.02SUCCESSIVE TRANSFORMATION .....................................................................................................................................................
SINGLE MATRIX OF TRANSFORMATION FOR SUCCESSIVE TRANSFORMATIONS ............................................................................
20.03INVERSE OF A TRANSFORMATION MATRIX ...................................................................................................................................
20.06STRETCH TRANSFORMATION AND ITS MATRIX ...........................................................................................................................
20.07 THE SCALE FACTOR k .........................................................................................................................................................................
20.09THE SHEAR FACTOR k, ......................................................................................................................................................................
21.00 NAVIGATIOIN .............................................................................................................................................
21.10 LONGITUDES AND LATTITUDES.......................................................................................................................................................
21.13 TIME AND LONGITUDES ....................................................................................................................................................................
21.14 SPEEDS IN KNOTS AND KILOMETERS PER HOUR .........................................................................................................................
22 .00 AREA APPROXIMZATION............................................................................................................................
22.01 AREA BY COUNTING TECHNIQUE ....................................................................................................................................................
22.02 TRAPEZIUM RULE AND AREA ...........................................................................................................................................................
22.03 MID-ORDINATE RULE AND AREA ....................................................................................................................................................
23.00 ELEMENTARY CALCULUS...........................................................................................................................
, 23.10 DIFFERENTIATION..............................................................................................................................................................................
23.12 EQUATIONS OF TANGENTS AND NORMALS TO THE CURVE .......................................................................................................
23.13 STATIONARY POINTS ..........................................................................................................................................................................
23.14 CURVE SKETCHING .............................................................................................................................................................................
23.15 APPLICATIONS TO VELOCITY AND ACCELERATION .....................................................................................................................
23.16 MAXIMA AND MINIMA .......................................................................................................................................................................
23.20 INTEGRATION ......................................................................................................................................................................................
23.21 INDEFINITE INTEGRALS ....................................................................................................................................................................
23.22 DEFINATE INTEGRALS .......................................................................................................................................................................
23.23 AREA UNDER A CURVE BY INTEGRATION ......................................................................................................................................
23.24 APPILICATIONS TO VELOCITY AND ACCELERATION ....................................................................................................................
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