1
THE SHOMATO
HANDBOOK
OF K.C.S.E
MATHEMATICS
FORMULAS
AND SUMMARIES
YAA K.W.S (B.ED (SCI) - EGERTON UNIV. CERT. EDUC
LEADERSHIP AND MANAGEMENT- AGHAN UNIV. DIPLOMA-
EDUCATION LEADERSHIP- K.E.M.I))
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A compilation of mathematical formulas and concepts relevant to high school students. It covers a range of topics, including algebra, geometry, trigonometry, and statistics. Examples and explanations to aid in understanding and application. It is designed to serve as a quick reference guide for students studying mathematics at the high school level, offering a concise overview of key principles and techniques. Topics such as bedmas, prime numbers, fractions, indices, measurements, spheres, lines, series, binomial expansion, trigonometry, matrices and transformations.
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THE SHOMATO
HANDBOOK
OF K.C.S.E
MATHEMATICS
FORMULAS
AND SUMMARIES
YAA K.W.S ( B.ED (SCI) - EGERTON UNIV. CERT. EDUC
LEADERSHIP AND MANAGEMENT- AGHAN UNIV. DIPLOMA-
EDUCATION LEADERSHIP- K.E.M.I))
PREFACE
The Shomato Handbook of Mathematics formulas is a quick reference aid for K.C.S.E students. This complete revision course for K.C.S.E contains more than ------------ of the most useful formulas and hints and equations found and necessary in the K.C.S.E four years course, covering mathematics. The exhaustive contents pages, allows the required formulas to be located swiftly and simply by the students. All variables involved in formulas have been accurately identified. The Handbook is designed to be a compact, portable reference book suitable for everyday work, problem solving, or examination revision for all students from form one to form four. All students and teachers in mathematics and physics will want to have this essential reference book within reach. Yaa K.W.S is a seasoned teacher of Mathematics and Physics, having taught these two subjects for over twenty years now. He is also a K.N.E.C examiner for K.C.S.E Physics paper three for many years. His valuable experience will go a long way to helping students in Kenya. Yaa K.W.S was the H.O.D Physical sciences in Khamis High school, Mombasa County. He became the Deputy Principal of the same school years back and is currently the Principal of Mwakirunge secondary school, teaching Mathematics and Physics. Mwakirunge secondary school is in Kisauni Sub- County, in Mombasa County.
CONTENTS/INDEX PAGE
- Preface
- 1.00 Symbols
- 2.00 NUMBERS
- 2.01 Integers
- 2.02 Prime factors
- 2.03 GCD /LCM
- 2.04 Fractions and Decimals
- 2.05 Squares and square roots
- 2.06 Cubes and Cube roots
- 2.07 Reciprocals
- 2.08 Ratios, rates, percentages and proportions
- 2.09 Compound proportions
- 2.10 Rates of work
- 2.11 Mixtures
- 2.12 Linear
- 2.13 Indices
- 2.14 Logarithms
- 2.15 Further logarithms
- 2.16 Approximations and errors
- 3.00 MEASUREMENTS I
- 3.01 Lengths and perimeters
- 3.02 Pythagoras theorem
- 3.03 Areas and Volumes of simple prisms
- 3.04 Volume and Capacity relationship
- 3.05 Mass, density and weight
- 3.06 Time
- 4.00 MEASUREMENTS II
- 4.10 FURTHER AREAS
- 4.11 Areas of Triangles
- 4.12 Areas of quadrilaterals
- 4.13 Areas of part of a circle
- 4.14 Area of a segment
- 4.15 Areas of regular polygons
- 4.16 Area of common region between two circles
- 4.20 SURFACE AREA OF SOLIDS
- 4.21 Surface area of prisms
- 4.22 Surface area of a cone
- 4.23 Surface area of a pyramid
- 4.24 Surface area of a frustum
- 4.25 Surface area of sphere
- 4.26 Surface area of a hemisphere
- 4.27 Surface area of combined solids
- 4.30 VOLUMES OF SOLIDS
- 4.31 Volumes of prisms
- 4.32 Volumes of pyramids
- 4.33 Volumes of cones
- 4.34 Volumes of frustums
- 4.35 Volumes of spheres
- 4.36 Volumes of hemisphere
- 4.37 Volumes of combined solids
- 5.00 ALGEBRA
- 5.10 Algebraic expressions
- 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
- 5.21 Linear Equations in one unknown
- 5.22 Linear Equations from real life situations
- 5.23 Simultaneous linear equations
- 5.30 EQUATIONS OF STRAIGHT LINES
- 5.31 Gradient of a straight line
- 5.32 Equations of straight lines
- 5.33 Perpendicular lines and their gradients
- 5.34 Parallel lines and their gradients
- 5.35 The x- and y- intercepts of a line or a curve
- 5.40 QUADRATIC EXPRESSIONS AND EQUATIONS I
- 5.41 Expanding algebraic expressions
- 5.42 Quadratic Identities
- 5.43 Factorizing quadratic expressions
- 5.44 Solving quadratic equations by factorization
- 5.50 QUADRATIC EXPRESSIONS AND EQUATIONS II
- 5.51 Further factorization
- 5.52 Completing the square
- 5.53 Quadratic formula
- 5.54 Tables of values for quadratic equations
- 5.55 Graphs of quadratic equations and solutions of equations
- 5.56 Simultaneous equations, one linear and the other quadratic
- 5.60 LINEAR INEQUALITIES I
- 5.61 Linear inequalities in one unknown on a number line
- 5.62 Simple inequalities in one unknown
- 5.63 Compound inequalities in one unknown
- 5.64 Graphical solutions of linear inequalities
- 5.65 Solutions of simultaneous inequalities
- 5.66 Formation of simple linear inequalities from graphs
- 5.70 LINEAR INEQUALITIES II
- 5.71 Formation of linear inequalities in linear programming
- 5.72 Graphical solutions of linear inequalities and optimization
- 6.00 SURDS
- 6.01Simplification of Surds
- 6.02 Rationalization of denominators of Surds
- 7.00 SEQUENCES AND SERIES
- 7.01 A.P series
- 7.02 G.P series
- 7.03 Sum of A.P series
- 7.04 Sum of G.P series
- 8.00 BINOMIAL EXPANSION
- 8.01 Pascal’s Triangle
- 8.02 Coefficients of terms of Binomial expansion
- 8.03 Computation using Binomial expansion
- 8.04 Evaluation of numerical values 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
- 10.10 ANGLES AND PLANE FIGURES
- 10.11 Angle on a transversal
- 10.12 Angle properties of a polygon
- 10.13 Angles at a point
- 10.20 GEOMETRICAL CONSTRUCTIONS
- 10.21 Bisection of a line
- 10.22 Bisection of an angle
- 10.23 Construction of an angle of
- 10.24 Construction of angles of 60^0
- 10.25 Construction of angle multiples of 7
- 10.26 Perpendicular bisector of a line
- 10.27 Perpendicular bisector from a point to a line
- 10.28 Parallel line through a given point to a line
- 10.29 Construction of polygons
- 10.30 LOCI
- 10.31 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 locus
- 10.38 Circumscribed circle
- 10.39 Inscribed circle
- 10.40 SCALE DRAWING
- 10.41 Bearings
- 10.42 Scale drawing
- 10.43 Angles of elevation and depression
- 10.44 Surveying techniques
- 10.50 COMMON SOLIDS
- 10.51 Nets and solids
- 10.52 Distance between two points on the surface of a solid
- 10.60THREE DIMENSIONAL GEOMETRY
- 10.61 Angles between lines
- 10.62 Angles between a line and a plane
- 10.63 Angles between a plane and a plane
- 10.64 Angles between skew lines
- 11.00 ANGLE PROPERTIES OF A CIRCLE
- 11.01 Angles subtended by the same arc at the circumference
- 11.02 Angles in a semi-circle
- 11.03 Angles at the centre and at the circumference by the same arc
- 11.04 Angles in a cyclic quadrilateral
- 12.00 CIRCLES: CHORDS AND TANGENTS
- 12.01 Lengths of Tangents and chords
- 12.02 Properties of chords and tangents
- 12.03 Intersecting chords
- 12.04 Escribed circle
- 12.05 Construction on Tangents to a circle from an exterior point
- 12.06 Construction of Direct common Tangents to two circles
- 12.07 Construction of Inverse common Tangents to two circles
- 12.08 Angles in the alternate segment
- 13.00 GRAPHS
- 13.10 COORDINATES AND GRAPHS
- 13.11 Choice of scale
- 13.12 Tabulation of values for linear relation
- 13.13 Graphical solutions of linear equations
- 13.20 GRAPHICAL METHODS
- 13.21 Table of values for graphs of a given relation
- 13.22 Graphs of quadratic and cubic relation
- 13.23 Solutions of quadratic and cubic equations from graphs
- 13.24 Quadratic curves
- 13.25 Cubic curves
- 13.26 Average and instantaneous rates of change
- 13.27 Changing non-linear laws to linear laws
- 13.28 Best line of fit and Empirical data
- 13.29 Equations of a circle
- 14.00 TRIGONOMETRY
- 14.10 TRIGONOMETRIC RATIOS I
- 14.11 Sine, Cosine and Tangent of angles
- 14.12 Sine and Cosine of Complementary angles
- 14.13 Special angles of 30^0 , 45^0 and 60^0
- 14.20 TRIGONOMETRIC RATIOS II
- 14.21 Trigonometric ratios of negative angles and angles greater than 90^0
- 14.22 Derivation of Sine rule
- 14.23 Derivation of Cosine rule
- 14.30 TRIGONOMETRIC RATIOS III
- 14.31 Trigonometric graphs
- 14.32 Trigonometric simple equations
- 14.33 Trigonometry Amplitude, period, wavelength and Phase angle or shift
- 15.00 COMMERCIAL ARITHMETIC
- 15.10 COMMERCIAL ARITHMETIC I
- 15.11Profit and loss
- 15.12 Exchange rates
- 15.13 Discounts
- 15.14 Commissions
- 15.20 COMMERCIAL ARITHMETIC II
- 15.21 Simple interest
- 15.22 Compound interest
- 15.23 Compound interest formula for constant multiple investments
- 15.24 Appreciation and Depreciation
- 15.25 Hire purchase
- 15.26 Income tax
- 16.00 STATISTICS AND PROBABILITY
- 16.10 STATISTICS I
- 16.11 Measures of ungrouped data
- 16.12 Measures of grouped data
- 16.13 Pie charts
- 16.14 Histograms
- 16.15 Frequency polygons
- 16.20 STATISTICS II
- 16.21 Mean using assumed mean
- 16.22 Mean using further coding
- 16.23 Ogive
- 16.24 Median and quartiles by calculation
- 16.25 Range, quartiles and inter-quartile range
- 16.26 Quartile deviations and Variance
- 16.27 Standard deviation
- 16.30 PROBABILITY
- 16.31 Sample space
- 16.32 Independent events
- 16.33 Mutually exclusive events
- 16.34 Probability tree diagram
- 17.00 VECTORS
- 17.10 VECTORS I
- 17.11 Addition of vectors
- 17.12 Column vectors
- 17.13 Magnitude of a column vector in two dimensions
- 17.14 Addition and subtraction of column vectors
- 17.15 Multiplication of a column 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 vector in three dimensions
- 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 co linearity
- 17.25 Ratio theorem
- 18.00 TRANSFORMATIONS
- 18.10 ROTATION
- 18.11 Properties of rotation
- 18.12 Centre of rotation
- 18.20 REFLECTION AND CONGRUENCE
- 18.30 SIMILARITY AND ENLARGEMENT
- 18.31 Similarity
- 18.32 Enlargement
- 18.33 Scale factors and their relations
- 19.00 MATRICES
- 19.01 Matrix addition and subtraction
- 19.02 Matrix multiplication
- 19.03 Determinant of a matrix
- 19.04 Inverse of a matrix
- 1905 Matrix solution of simultaneous linear equations
- 20.00 MATRIX OF TRANSFORMATION
- 20.01 Matrix of transformation
- 20.02 Successive transformation matrix
- 20.03 Inverse of a transformation matrix
- 20.04 Area scale factor and the determinant of a transformation matrix
- 20.05 Areas of triangles and quadrilaterals from the coordinates of their vertices
- 20.06 Stretch transformation and its matrix
- 20.07 Stretch scale factor
- 20.08 Stretch invariant line
- 20.09 Shear transformation and its matrix
- 20.10 Shear scale factor
- 20.11 Shear invariant line
- 21.00 NAVIGATION
- 21.10 LONGITUDES AND LATITUDES
- 21.11 Distance in nautical miles and kilometers along a great circle
- 21.12 Distance in nautical miles and kilometers along a small circle
- 21.13 Time and longitudes
- 21.14 Speed in knots and kilometers per hour
- 22.00 AREA APPROXIMATION
- 22.01 Area approximation by counting technique
- 22.02 Area approximation by Trapezium rule
- 22.03 Area approximation by Mid-ordinate rule
- 23.00 ELEMENTARY CALCULUS
- 23.10 DIFFERENTIATION
- 23.11 Gradient of a curve at a point
- 23.12 Equations of tangents and of normal lines
- 23.13 Stationary points
- 23.14 Curve sketching
- 23.15 Application of differentiation to velocity and acceleration
- 23.16 Maxima and Minima
- 23.20 INTEGRATION
- 23.21 Indefinite integrals
- 23.22 Definite integrals
- 23.23 Exact Area under a curve
- 23.24 Applications of integration to velocity and acceleration
2. 00 NUMBERS
2.01 INTEGERS
Addition and Subtraction Rules: 1. Same signs: add and keep the sign (Same sign + same sign = same sign sum) 3+5 = 8; − + − = − 21
- Different signs: Subtract and keep the sign of the larger absolute number (Different sign + different sign = subtract, and keep the sign of the largest absolute value). The absolute value of a whole number is the same number but always positive. − + = −( − )= − 13 REMEMBER: Subtraction is same as “Adding the Opposite.” a – b = a + (-b) To subtract numbers:
- Change subtraction sign to addition.2. Change the sign of the second number.
- Follow rules for addition. 37 – 45 = 37 + −45 = −(45 −37 ) = − 8 Multiplication and Division Rules
- Multiplying or dividing numbers with the same signs gives a positive answer. 4 ×12 = 48; 32 ÷2 = 16; − ÷ −4 = 3; − × − = 30
- Multiplying or dividing numbers with different signs gives a negative answer 4 × −12 = −48; 32 ÷ −2 = − 16
- Definition of division: = × = 12×
When several operations are applied on integers please use: BEDMAS. (B) BRACKETS (E) EXPONENT (D) DIVISION (M) MULTIPLICATION or of (A) ADDITION (S) SUBTRACTION – B.E.D.M.A.S ( ) ÷ × × 2010 p1 no 1. ( ) ( ) √
√ = −2√
2.02 Prime numbers /Factors These are numbers having exactly two factors: one and itself. These include 2, 3, 5, 7, 11, 13, 17, 19, 23 etc. All composite numbers can be written as products of prime factors using a factor ladder or a factor tree. Begin with the least prime number that is a factor. Repeat until the quotient is prime or one. Express 10500 in terms of prime factors. 2011 p1 no 14.
To express a mixed number as an improper fraction, first multiply the whole number by the denominator and add the numerator. Then write this sum over the denominator. . = × =
To express an improper fraction as a mixed number, divide the denominator into the numerator. . =
To add or subtract a set of fractions, you must make sure they have a Common Denominator which will be the L.C.M of the set of denominators. With the L.C.M rewrite the fractions as equivalent fractions and then add or subtract fractions by adding or subtracting numerators together and reduce answer to lowest terms. +
To add or subtract mixed fractions, Whole numbers are added together first. Then determine LCM for fractions. Reduce fractions to their LCM. Add numerators together and reduce answer to lowest terms. Add sum of fractions to the sum of whole numbers.
5 + 2 − 3 = (5 + 2 − 3)+ + − = (4)+ + (^) − = 4 + (^) = 4 (^)
To multiply fractions change any mixed fractions to improper fractions before multiplying. Then multiply numerator to numerator and denominator to denominator. × =
To divide by a fraction, multiply by its inverse or reciprocal. ÷ = × =
Mixed fractions should be changed into improper fractions first before dividing or multiplying them. To divide fractions, multiply the first fraction with the reciprocal of the second fraction. When several operations are applied on fractions please use: BEDMAS. 2.05 SQUARES, SQUARE ROOTS Some K.C.S.E questions could require simplification using the rules on integers and fractions. To solve these types of problems change the numbers involved into products of prime factors. If decimals are involved, write them as suitable whole numbers that are multiples of powers of ten first. Remember √ ^ = ^. Then use the factor method to quickly sort out the questions with great accuracy. In almost all the questions in this area do not require the use of a four figure table or a calculator. A calculator and four figure mathematical tables MUST not be used. 2.06 CUBES AND CUBE ROOTS The cube root of a number can be found using factor method, the four figure table of the calculator.
- To use factor method, the number is written as a product of prime factors as indicated above for square roots.
- To use a four figure table, some numbers must be written as suitable powers of ten before using the tables to find the answer.
2.07 RECIPROCALS
The reciprocals section of the four figure table can be used to find the reciprocals of numbers. Some numbers must be suitably written as powers of ten first before checking for their reciprocals. Do not forget to correctly use the reciprocal of the power of ten. 2.08 RATIOS, RATES, PERCENTAGES AND PROPOTIONS
To divide a number in a stated ratio, first find the total ratio and then multiply the number by (^) for each ratio or as is required. Percent increase describes an amount that has grown and percent decrease describes an amount that has reduced.
Percent of Change = or , expressed as a percent.
To change a decimal to a %, move decimal point two places to right and write percent sign. 2.09 COMPOUND PROPORTIONS Combining ratios a:b and c:d , the numbers in the bold positions must have the same value for the ratio to combine and become the ratio x:y:z. Hence, multiply the ratio as c (a: b) and b(c: d) to give ac: bc and bc: bd. So x:y:z = ac: bc : bd Remember: a: b = x: y means x/y = a/b Compound proportions questions related to work could be sorted out a table as shown below: quantity name Q1 Q2 Q Value 1value 2 ad b ce
To find the missing number for Q2, check on how the variations in Q1 and Q3 will affect b, in terms of whether the ratio will be an increasing one or a decreasing one and multiplying accordingly. For example :1993 p2 no 5 It takes 30 workers 6 days working 8 hours a day to harvest maize in a farm. How many days would 50 workers working 6 hours a day take to harvest the maize? quantity workers days hours value 1 30 6 8 value 2 50 x 6 Increasing workers reduces days of working → , reducing working hours increases days of working → , hence x = 6 × × = 4 days.
2.10 RATES OF WORK
If B can complete work in b hours, then B does work at the rate of per hour. If C can complete the work in c hours, the rate of work is per hour. If B and C work together the rate of work will be the sum of their rates ( + ) per hour. Time to complete the work when working together will be the reciprocal of the Sum. For Taps, the rates of the taps bringing in water are summed up. If a drain pipe or tap is involved, the difference of the rates of the drain tap to the filling taps gives the rate at which the tank accumulates water. The time to fill the tank equals the reciprocal of the difference.
2.14 LOGARITHMS USING MATHEMATICAL TABLES
For all v, w>0; 1.log( )= log + log 2.log( (^) )= log − log
3.log( ) ^ = klog 4. log 1 = 0
- log = 1 6. Change of base, (^) a = (^ ^ )
- log = (^) 8. log √ ^ = log
If n = m x 10 ^ where 1≤m˂10, then log = log + log 10 = c + log , c is the characteristic of n. log is the mantisaa of n gotten from the four figure mathematical tables. Hence log = mantissa + characteristic. To use the logarithms four figure table:
- Write the number in standard form: n = m x 10
- Identify the first four numbers of m say – abcd
- Select row ‘ab’ from the logs tables ; 4. locate number at column ‘c’ from the row ‘ab’ say x.
- If d≠0, locate number at column ‘d’ of mean difference from the row ‘ab’ say y. ( if d = 0 , then y=0 ) log = c + .( + ) = .( + ) Antilog can be found using the following procedure:
- If = . , and n≥0, the Antilog ( ) = .( = ) × 10 ( )^ ; If = . , and ˂0, the Antilog ( ) = .( + )x 10 (^ )
- Select row ‘ab’ from the antilog table.
- Select column ‘c’ of row ‘ab’ from the antilog table to locate the number say x.
- Select the column ‘d’ of mean difference of row ‘ab’ from the antilog table to locate the number say y. (if d=o then y=0). If 4 ^ + 0.0867 = 4 . NOTE:
- Logarithms add up if their numbers multiplied. logarithms subtract if their numbers divided. for these two the normal rules of addition and subtraction applies.
- A logarithm of the form . , can be divided by a whole numbers like any decimal. Extra care should be taken for a logarithm of the form ( . ).
For . , express . as ( + . )/ . If m is exactly divisible by n proceed with division normally, if not make the following adjustments. Go for the next number greater than m that can be divided exactly by n say p. Then rewrite the expression as ̅ ( ). and proceed with division normally. For example . , becomes ( ). = . = 2 (^) + .8559 = 2 .
2.15 FURTHER LOGARITHMS
If (^) y = x , then y = ^ which can be used to solve logarithmic equations. Laws of logarithms are used to simplify the equations that lead to their solutions. Normal laws of solving equations also apply.
- (^) x + (^) y = (^) xy 2. (^) x - (^) y = (^) 3. x (^) y = (^) ( ^ )
- (^) ( )^ = 1 5. (^) 1 = 0 Resulting equations are solved using the normal laws. For equations of the form c( ^ ) + b( ^ ) + d = 0, let y = ^ and reduce the equation to a quadratic equation, i.e cy^2
- by + d = 0. 2.16 APPROXIMATIONS AND ERRORS True value – estimated value = error absolute error = ( ℎ ℎ ) Relative error r. e = (^) Percentage error p. e = (^) x 100 When numbers add or subtract, the absolute errors of each value are summed up. When numbers multiply or divide, their percentage or relative errors are summed up. The absolute error is = ( ℎ ℎ ℎ .) If a number p, is raised to power n i. e ( ) ^ the percentage error or relative error in( ) ^ will be ( )( )