QUANTITATIVE APTITUDE

PERMUTATION AND COMBINATION

Permutation is used where arrangements of things is relevant for e.g. arrangement in a queue, words, nos, sitting arrangements etc. Permutation of n things taken at a time is has r factors. For eg : has 3 factors i.e. has 2 factors i.e. Combination is used where selection is relevant for eg : Selection of a team, committee etc. Combination of n things taken r at a time is Note : Fundamental Principles of countinga. Multiplication Rule [AND] When things are within same arrangement and all things are required then choices are multiplied.b. Addition Rule [OR] When there are different arrangements and either of the arrangement can work then choices are added. Factorial 1 ! = 1 2 ! = 2 3 ! = 6 4 ! = 24 5 ! = 120 6 ! = 720 7 ! = 5040 8 ! = 40320 Note : 1) Factorial cannot be calculated for a ve no. 2) Factorial cannot be calculated for a fractional no. Example 7 Note : 1) Sum of all 4 digit no. which can be formed using a,b,c,d is (a+b+c+d) X 6666 2) Sum of all 3 digit no. which can be formed using a,b,c is (a+b+c) X 222 Circular Permutation1) n things can be arranged in a line in n ! ways.2) n things can be arranged in a circle in (n 1) ! ways.3) n persons can be arranged in a circle so that no person has same 2 neighbors in ways.4) OR no, of necklaces that can be formed with n beads of different colors in ways. Permutation with Restriction : In case of restriction in an arrangement first of all position with restriction should be considered. Note : 1) No. of ways in which n things can be arranged so that 2 particular things are never together = total ways ways in which 2 particular things are always together. 2) Above formula cannot be applied for more than 2 things. Properties of Combination :1) 2) 3) Note : Permutation = Selection with arrangement Combination = Selection without arrangement Properties of Permutation

SEQUENCE AND SERIES

Sequence 1, 3, 5, 7, 9, . . . . . . . 2, 4, 8, 16, 32, . . . . . .Series 1 + 3 + 5 + 7 + 9 .. . . . . . . 2 + 4 + 8 + 16 + 32 . . . . . . . .

ARITHMETIC PROGRESSION (A.P.)

a, (a+d), (a+2d), (a+3d), . . . . . . . . nth term of A.P. Sum of n terms Where = = Arithmetic Mean (A.M.) A.M. between 2 no.s a & b is and if n A.M. are inserted between 2 no.s a & b then a, A, A, A, A. . . . . . . . , b are in A.P. Note : 1) In case of A.P., T - T = T - T = d 2) T + T = 2T or T + T = 2T and so on. 3) If 3 terms are in A.P., then terms are (a-d), a, (a+d) 4) If 4 terms are in A.P., then terms are (a-3d), (a-d), (a+d), (a+3d) 5) If 4 terms are in A.P., then terms are (a-2d), (a-d), a, (a+d), (a+2d)Note : 1) Sum of first n natural no.s 1 + 2 + 3 + 4 . . . . . . . . . n = 2) Sum of squares of first n natural no.s 1+2+3+4 . . . . . . . . . . . n = 3) Sum of cubes of first n natural no.s 1+2+3+4+ . . . . . . . . . . . n= Note : When d= -ve, we get two answers for n. In such case, first value of n better option. GEOMETRIC PROGRESSION (G.P.)

nth term of G.P. Sum of n term of G.P. Sum of Infinity Geometric Mean (G.M.) G.M. of 2 no.s a & b is If n G.M. are inserted between a & b then a, G, G, G, . . . . . . . . . . , b are in G.P. Note : 1) In G.P. 2) T + T = (T) 3) 3 terms in a G.P. are 4) 4 terms in G.P. are 5) 5 terms in a G.P. are Note : 1) For 2 equal no.s A.M. = G.M. = H.M. 2) For 2 unequal no.s A.M. > G.M. > H.M. 3) For 2 no.s A.M. G.M. H.M

RATIO, PROPORTION, INDICES & LOGARITHMS

RATIO 3 : 4 4 : 3 Ratio has no unit1) Inverse Ratio of 2) Compound Ratio of 3) Duplicate ratio of 4) Triplicate ratio of 5) Sub duplicate ratio of 6) Sub triplicate ratio of 7) Continued ratio is ratio of more than 2 things, 2 : 3 : 48) Ratio of 2 integers is said to be commensurable and ratio of 2 non-integers is incommensurable. :. is commensurable and is incommensurable. Proportion An equality of 2 ratios is called proportion. E.g. : then a, b, c, d are proportionate i.e. a : b = c : d or a : b : : c : d Product of Extreme = Product of mean Mean proportion of 2 no.s a, b is Properties of Proportion :1) 2) 3) 4) 5) 6) 7)

Note : Third proportion of a X b is x such that :. X =

INDICES/INDEX Laws :1) 2) 3) 4) 5) = 1 6) 7) LOGARITHM1) If, then 2) 3) 4) 5) 6) 7) Base changing formula common base means base = 10. EQUATION

Quadratic Equation 8 Nature of Roots :1) If 2) If 3) If 4) If Note : Irrational roots occur in pairs ,. Sum of 2 roots = Products of 2 roots = Quadratic Equation Formation Cubic Equation : In case of cubic equation, factors should be made if possible. If factorization is not possible then only verification of answer should be preferred. Co ordinate Geometry 1. Distance between 2 points A (x, y) and B () 2. Equation of straight line y = mx + c , m = slope of line Note : 1) Slope of a parallel lines is same. 2) Product of slope of 2lines = -1 3) Equation of line is 4) Intercept form of Equation Cost slope = Variable Cost per unit = Limits and Continuity

Some Important Limits1) L Hospital Rule For Limit = 2) 3) = 14) 5) 6) Types of functions1) Even function : If f(-x) = f(x)2) Odd function : If f(-x) = -f(x)3) Neither even nor odd : f(-x) f(x) -f(x) Notes : A polynomial function is always continuous.

STATISTICAL DESCRIPTION OF DATA

Latin word StatusItalian word StastitaGerman word StastistikFrench word StatistiqueDefinitition of Statistics can be in plural sense or singular sense. When satistic is used as a plural noun then it may be defined as data qualitative and quantitative.When used as singular noun then it is defined as scientific method of drawing conclusions about some important characteristic It means Science of counting or Science of averages.Application of Statistics1) Economics2) Business management3) Commerce and Industry4) Health careLimitations of Statistics :1) It deals with Aggregates (Total)2) Statistics is concerned with quantitative data.3) Future projections are valid under specific set of conditions.4) Theory of Statistics is based on Random SamplingCOLLECTION OF DATADATA: is quantitative information about particular characteristic.VARIABLE : A quantitative characteristic is called variable.ATTRIBUTE : A qualitative characteristic is called Attribute. For e.g. : gender, nationality, color etc.VARIABLES are of 2 types :A) DISCRETE VARIABLE : When a variable can assume finite no, of isolated valued then it is discrete. E.g. : Petals in a flower no. of road accidents, marks in paper etc.B) CONTINUOUS VARIABLE : When a variable can assume any value from a given interval. E.g. : Weight, height, sale etc.

DATA can be of 2 types :A) PRIMARY DATA when data is collected by the person using the data.B) SECONDARY DATA When data is used by the person other than collecting the data sources of secondary data are : 1) International sources WHO (World Health Organisation) ILO (International Labour Organisation) IMF (International Monetary Fund), World Bank, etc. 2) Govt. sources CSO (Central Statistical Organisation), ministers, etc. 3) Private and Quasi Govt. organisation RBI, NCERT 4) Research Institute and Researcher

Collection of Primary Data : There are 4 methods of collection of primary data.a) Interview Method 1) Personal Interview Natural calamities 2) Indirect Interview Accidents 3) Telephonic Interview Maximum non response b) Mailed Questionnaire Maximum non response Widest coveragec) Observation Method Time consuming, expensive and applicable only for small population.d) Questionnaire filled by enumeratorNOTE: Scrutiny of data means checking data for any possible error. Two or more series of figures which are related to each other can be compared for internal consistency. PRESENTATION OF DATA CLASSIFICATION OF DATAa) Chronological or Temporal or Time series datab) Geographical or spatial data.c) Qualitative or ordinald) Quantitative or cardinalMODE OF PRESENTATIONa) Textual presentation Least preferred because it is monotonous.b) Tabular presentation (Page 10.7) 1) Table has a serial no, 2) Table can be divided into 4 parts Caption is upper part of table describing columns and sub columns. Boxhead is entire upper part including column and sub column numbers, units of measurement alongwith caption. Stub is left part of table Body is main part of table.

Example : Status of workers Caption Status Member of TU Non members Total M F T M F T M F T

Body

Source FootnoteC) Diagrammatic representation of Data1) Line Diagram or Historiagram Line diagrams are used when data vary over time. Logarithmic or ratio chart is used where there is wide fluctuation. Multiple line chart is used for 2 or more related time series with same units. Multiple axis line chart are used for 2 or more series with different units.

2) Bar Diagram Horizontal Bar diagram used for qualitative data or data varing over space (Geographical)

Vertical Bar diagram used for quantitative data or data varing over time.

Multiple or group Bar diagram used to compare related series.

Component or subdivided Bar diagram (Percentage)

3) Pie Chart

FREQUENCY DISTRIBUTIONa) Discrete or ungrouped frequency distribution (Simple) This is used for discrete variable E.g. : Marks No. of students (f) 5 20 6 30 7 50 8 10 9 15 10 20B) Grouped frequency Distribution used for continuous variables.E.g. :1) Marks / Weight f 0 10 10 10 20 5 Mutually exclusive classification used for 20 30 15 continuous variable 30 40 20 50

2) Marks / Weight f 0 9 10 10 19 5 Mutually inclusive classification used for 20 29 15 discrete variable 30 39 20

CLASS LIMIT Lower Class limit (LCL), Upper Class Limit (UCL) LCL UCL LCB UCB 10 20 10 20 10 20 20 30 20 30 20 30 LCL = LCB 30 40 30 40 30 40 UCL = UCB CLASS BOUNDARIES are real class limits Lower class boundary (LCB), Upper class boundary (UCB) LCL UCL LCB UCB 10 19 10 19 9.5 19.5 20 29 20 29 19.5 29.5 LCL LCB 30 39 30 39 29.5 39.5 UCL UCB Note : 29.5 39.5 includes 29.5 but excludes 39.5 CLASS MARK Class width/ Interval

Class Mark Class width / Interval 10 19 14.5 9.5 19.5 10 20 29 24.5 19.5 29.5 10 30 39 34.5 29.5 39.5 10CLASS WIDTH = CLASS SIZE = CLASS INTERVAL Class Cumulative Frequency Relative Percentage frequency interval frequency density frequency frequency10 20 8 10 8 20 30 12 10 20 30 40 20 10 40

* FREQUENCY DENSITY = * RELATIVE FREQUENCY = * PERCENTAGE FREQUENCY =

GRAPHICAL REPRESENTATION OF FREQUENCY DISTRIBUTION1) Histogram

Mode Most frequently occurring item.2) Frequency polygon is used for simple frequency distribution. It can be used for grouped frequency distribution provided width of class interval is same.

Note : Frequency curve is limiting form of frequency polygon3) Cumulative frequency curve or ogive (Two types)

Less than ogive can be used to calculate median, quartile deciles etc.Frequency curve It is limiting form of frequency polygon or histogram for which total area is 1.a) Bell shaped Most common

b) U shaped

MEASURE OF CENTRAL TENDENCY AND DISPERSION

CENTRAL TENDENCIES ARITHMETIC MEAN (A.M.) = Note : For calculation of A.M., mutually inclusive series need not be converted into mutually exclusive series. Properties of Arithmetic Mean :1) A.M. is neither free of origin (affected by addition or subtraction of a constant) Nor free of scale (multiplication and division of a constant).2) Combined A.M. = GEOMETRY MEAN (G.M.) G.M. = G.M. = A.L. Properties of G.M.1) If all the observations are k then G.M. = k2) If z = xy then G.M. of z = 3) If z = HARMONIC MEAN (H.M.) = Combined H.M. of two series = E.g.: H.M. of 4, 6, 10 H.M. =

E.g. : x 2 4 6 f 3 2 5 H.M =

E.g. : Distance (f) 100 km 200km Speed (x) 60km/h 80km/h Average of Speed = Note : For average of speed, H.M. is used E.g.: Home College 40 km/h College Home 30 km/h Average speed = Relationship between A.M., G.M., H.M.- A.M. G.M. H.M.a) For unequal no. A.M. > G.M. > H.M.b) For equal no. A.M. = G.M. = H.M.* For 2 numbers : (G.M.) = A.M. x H.M.* G.M. cannot be calculated for negative items.* G.M. is used where compound growth is there.* A.M. is most popular central tendency. WEIGHTED MEAN =

E.g. : x w xw food 10 10 100 Rent 20 6 120 Education 30 4 120 20 340 Weighted Average = MEDIAN If all the items are arranged in increasing order then middle item is median. Median ( in case of discrete items) = Median (for grouped items) = L +

Where L = lower limit of median class N = Total items N = Cumulative frequency of preceeding class f = frequency of median class c = class interval E.g. : f c.f. 10 19 10 10 20 29 15 25 30 39 25 50 Median class 40 49 20 70 Median = 29.5 + QUARTILE Divide total items in 4 parts Q and Q DECILE (10 parts) PERCENTILE (100 parts)

MODE is most frequently occurring item. For grouped data, Mode = L

Where L = lower limit of modal class = frequency of modal class = frequency of preceeding class = frequency of succeeding class c = class intervalNote : For moderately skewed data Mode = 3Median 2Mean Note : All central tendencies are neither free of origin nor free of scale

MEASURE OF DISPERSION Absolute measure Relative measure 1) Range 1) Coefficient of range 2) Mean Deviation 2) Coefficient of mean deviation 3) Standard Deviation 3) Coefficient of variation 4) Quartile deviation 4) Quartile deviation (Coefficient)Note : All measures of dispersion are free of origin but not free of scale. RANGE = L S L = Largest S= Smallest Coefficient of Range = MEAN DEVIATION about mean = Coefficient of mean deviation + STANDARD DEVIATION S.D. = Coefficient of variation = S.D. of 2 numbers a and b = S.D. of first n natural numbers = Combined S.D. of 2 groups

d = d = = mean of first group = mean of second group = combined mean

QUARTILE DEVIATION = Coefficient of quartile deviation = Note : Range is not free of units (Rs., Kg)Note : Measure of dispersion is free of origin but not free of scale.Note : S.D. is of range of two numbers. CORRELATION AND REGRESSION

Correlation and Regression Extent of Relationship Prediction -1 to 11) MARGINAL DISTRIBUTION There are two marginal distribution Marks in Maths (f) Marks in Stats (f) 0 10 7 8 10 20 19 24 20 30 32 262) CONDITIONAL DISTRIBUTION There are m + n conditional distribution where m is no. of rows and n is no. of columns. Marks in Maths [If marks in stats is 10 20] f 0 10 2 10 20 7 20 30 15Note : No. of cells = m X n CORRELATION Correlation analysis is establishing relation between two variable (positive, negative or zero) and measuring the extent of relationship between two variables (-1 to 1). Correlation can be measured by four methods :a) Scatter Diagramb) Karl Pearsons product moment correlation coefficient.c) Spearmans rank correlation coefficient.d) Coefficient of concurrent deviations. Scatter Diagram : can be used for linear as well as curvilinear variables. Exact measure of correlation cannot be measured by this method.

KARL PEARSONS PRODUCT MOMENT COEFFICIENT OF CORRELATION is the best method for finding correlation between two variable having linear relationship. = Where Covariance = Or = Note : Coefficient of correlation is free of origin as well as free of scale.Note : r is free of unit. SPEARMANS RANK CORRELATION : Rank correlation is used to study relationship between Qualitative variables.

Where D = Difference of Ranks t = no. of tied ranksCOEFFICIENT OF CONCURRENT DEVIATION where m = no. of pairs of deviations c = No. of +ve signs = No. of concurrent deviationsREGRESSION ANALYSIS There are two regression lines. Regression is concerned with predicting dependent variable when independent variable is known. In simple, Regression model, if y depends on x then line y on x is given by y = a + bx dependent IndependentRegression line Y on X where Regression coefficient of y on x. Regression line X on Y Properties of Regression 1) 2) 3) 4) Arithmetic mean of x and y are solution of two regression equations.5) Because

PROBABILITY AND EXPECTED VALUE

Probability can be divided into two categories :a) Subjective Probabilityb) Objective Probability Random Experiment Experiment means performance of an Act. Random means that Probability of all outcomes is equal. Event is result of and experimenta) Simple or Elementary eventsb) Composite or Compound events An event is simple if it cannot be decomposed into further event. Compound event are made of two or more simple event,MUTUALLY EXCLUSIVE EVENTS When happening of an event makes happening of another event impossible then two events are mutually exclusive.EXHAUSTIVE EVENTS : are set of all possible events i.e. events have to be from set of Exhaustive events. Sum of probability of Exhaustive events is always equal to 1. EQUALLY LIKELY or Equi Probable Events having same probability. CLASSICAL DEFINITION OF PROBABILITY P (A) = Note : 1) It is applicable when total no. of events is finite. 2) It can be used only when events are equally likely. 3) Above definition is also termed as a priori and is useful only in coin tossing, dice throwing etc. Note : E.g. : In one coin H T In two coins 0H 1H 2H In three coins 0H 1H 2H 3H

Sum of no. on two dice 2 3 4 5 6 7 8 9 10 11 12

STATISTICAL DEFINITION OF PROBABILITYIf a random experiment is repeated a very good no. of times say n under identical conditions and an event occurs times then ratio of and n when n tends to infinity is defined as statistical definition of probability. P =

1) 2) For Independent events 3) 4) 5) * Only A + only B + A or B + A and B = 16) P (A) = 1 P(A) Mutually exclusive events Exhaustive Equally/likely

Conditional Probability EXPECTED VALUE is sum of product of different values and its probability. Variance = (S.D.) = = Properties of Expected value :1)2) 3) Mean = S.D. = or V = V =

THEORETICAL DISTRIBUTION

Theoretical Distribution exists only in theory. It is theoretical probability distribution of experiments. BINOMIAL DISTRIBUTION : Features are1) There are 2 possible outcomes.2) Trials are independent.3) n is small4) It is Biparametric Distribution (n, p)5) Probability of successes out of n trials is 6) Mean = np, variance = npq7) It can be unimodal or bimodal Mode = Largest integer in (n+1) P if (n+1) p is a non integer. = (n+1) P and (n+1)p 1 if (n+1)p is integer.8) Variance is maximum when p = q = .5 Maximum variance = Additive Property If x and y are two independent variables such that X + Y * {p = success , q = failure}

POISION DISTRIBUTION : Properties1) Probability of success in very small time interval (t, t+ dt) is kt where k is constant.2) Probability of success is independent.3) Probability of success in this time interval is very small.4) This distribution is called distribution of rare events.5) n is large and p is very small such that n X p is finite.6) Mean = Variance = n X p = m7) It is uniparametric distribution (m).8) There are two possible outcomes.9) Probability of r success out of n 10) Mode is largest integer in m if m is non integer Mode is m and m 1 if m is an integer. Additive Property : If x and y are two independent poisson variables and z = x + y x y z = x + y NORMAL OR GAUSSIAN DISTRIBUTIONP(x) = f(x) = Probability density functionf(x) = m = mean = S.D. Properties of Normal Distribution1) Mean = median = mode2) Mean deviation = .8 S.D.3) First Quartile = Q = Mean .675 S.D. Third Quartile = Q = Mean + .675 S.D.4) Point of inflexion are a) Mean S.D.b) Mean + S.D.5) In standard normal variate, Mean = 0 , S.D. = 16) Area under normal curve z = CHI-SQUARE DISTRIBUTION (X-DISTRIBUTION) Properties :1) It is continuous, positively skewed probability distribution.2) Mean = n3) S.D. = 4) When n is large it follows normal distribution.

STUDENTS T DISTRIBUTION 1) It is a continuous symmetrical distribution.2) Mean = 03) S.D. = 4) For large n (n>30) t distribution is identical to z distribution. F Distribution1) F Distribution is positively skewed.2) It is continuous.Note : z Area under the normal curve 1 .6826 1.64 .90 (90%) 1.96 .95 (95%) 2.33 .98 (98%) 2.58 .99 (99%) 3 .9973 (99.73%)

SAMPLING

BASIC PRINCIPLES OF SAMPLE SURVEY :1) Law of Statistical regularity : States a large sample drawn at random from population possesses characteristics of population at an average.2) Principle of inertia : States that results drawn from samples are likely to be more reliable, accurate and precise as sample size increases, other things remaining same.3) Principle of Optimization : means maximum efficiency at given cost or minimum cost for optimum level of efficiency.4) Principle of Validity : States that sampling designs is valid only if it is possible to obtain valid results.Probabilistic sampling ensures this validity. COMPARISON BETWEEN SAMPLE AND COMPLETE ENUMERATION (CENSUS)1) Speed 2) Cost 3) Reliability 4) Accuracy 5) Necessity ERRORS IN SAMPLING :1) Sampling errorsa) Errors due to defective sampling design.b) Errors arising out due to substitution.c) Errors due to faulty demarcation.d) Errors due to wrong choice of statistic.e) Variability in the population.2) Non-Sampling Errors : Memory lapse, preference for certain digits, ignorance, psychological factors like vanity, non-response etc.SAMPLING DISTRIBUTION AND STANDARD ERROR OF A STATISTICStandard Error (SE) is standard deviation of sample statistic.1) SE of means = SE ((with replacements) where = population S.D. n = Sample size2) SE ( where N = population size3) SE ( s = Sample S.D.4) SE (5) Standard error of proportion = 6) SE (P) (WOR) = INTERVAL ESTIMATION OF MEAN AND PROPORTION1) Interval estimation of mean =2) Interval estimation of mean ifa) Sample size is less than 30.b) Population S.D. is not known. then interval estimation of mean 3) Interval estimation of proportion = Sample size1) Sample size for population mean = 2) Sample size for proportion = TYPES OF SAMPLINGI) PROBABILITY SAMPLINGII) NON-PROBABILITY SAMPLINGIII) MIXED SAMPLING PROBABILITY SAMPLING : When each member of population has equal chance of selection.a) Simple Random Sampling (SRS) should be used when 1) Population is small. 2) Sample size is not large. 3) Population is homogeneous.b) Stratified Sampling (strata = layer) : is used when 1) Population is large. 2) Population is heterogeneous c) Multi stage Sampling : If population is very large then it is cost effective and flexible system of sampling.E.g. : Estimation of foodgrain production in India.PURPOSIVE OR JUDGEMENT OR NON PROBABILISTIC SAMPLING : Probability of selection of each unit is not equal. MIXED SAMPLING : (Systematic Sampling) is partly probabilistic and partly non probabilistic. E.g. : Sampling done by Auditors.Systematic Sampling has a drawback If there is unknown or undetected periodicity in sampling frame and sampling interval is multiple of that period then we get most biased samples. CRITERIA FOR IDEAL ESTIMATION1) Unbiasedness and minimum variance (MVUE) Minimum variance unbiased estimation2) Consistency and Efficiency3) Sufficiency

INDEX NUMBERS

Index Numbers is an average of ratios expressed as percentage. Two or more time periods are involved one of which is base time period. The value at the time of base period serves as standard of comparison.

INDEX NUMBER ARE OF FOLLOWING TYPES :1) PRICE INDEX2) QUANTITY INDEX3) VALUE INDEX4) COST OF LIVING INDEX OR CONSUMER PRICE INDEX METHODS OF CONSTRUCTION OF INDEX NUMBER (PRICE INDEX)1) SIMPLE AGGREGATIVE METHOD (0 is base year, 1 is current year) 2) SIMPLE AVERAGE OF PRICE RELATIVE 3) WEIGHTED METHODSa) Laspeyres Price Index (L) b) Passches Price Index (P) c) Marshall Edge worth d) Fishers ideal Index no. = e) Bowleys Index no. = 4) WEIGHTED AVERAGE OF PRICE RELATIVE METHOD [Similar to Laspeyres Index no.]5) Chain Index = Link Relative =QUANTITY INDEX REPLACE P BY Q AND Q BY P IN PRICE INDEX FORMULA.

VALUE INDEX COST OF LIVING INDEX (CLT) OR CONSUMER PRICE INDEX (CPI) = [is weighted average of indices] or [same as Laspeyres]DEFLATING Deflated Value = or BASE SHIFTING Shifting Price Index = Note : Index no. of base year is always 100. Real wages of Base year = Real wages in Current year = Decrease in Real wages = 295.45 250 = 45.45TEST OF ADEQUACY1) UNIT TEST This test requires index no. to be free of unit. All index no. expect simple aggregative method satisfy this test.2) TIME REVERSAL TEST is cleared if a) Fisher Index no. b) Marshall Edgeworth sastisfy this text.3) FACTOR REVERSAL TEST is satisfied if a) Fisher Index no. b) Simple Aggregative method satisfy this text.

4) CIRCULAR TEST is satisfied if It is extension of time reversal testa) simple Geometric mean of Price relativeb) Weighted Aggregative with fixed weight

c) simple aggregative meet this test.

DIFFERENTIAL CALCULUS

Derivative of y w..t. x is change in y w.r.t. x. when change in x is very small

Standard Results 1) 2) 3) 4) 5)

Product Rule Quotient Rule Gradient = Derivative = * Ordinate = value of x Abscissa = value of y

INTEGRATION

1) 2) 3) 4) 5) 6)

SETS, FUNCTIONS AND RELATIONS

Sets : A set is defined as collection of well defined distinct objects. For e.g. :A = {a,e,i,o,u} = {x : x is a vowel in the alphabets}B = {2, 4, 6, 8, 10} ={ x : x = 2m and m is an integer o