LOGIC Maths.pdf

8/20/2019 LOGIC Maths.pdf

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RADIAN ACADEMY ANNA NAGAR & NSK NAGAR-ARUMBAKKAM [email protected] Ph: 98404-00825, 30025003 Visit OUR ACADEMY’s FREE APTITUDE Yahoo Group: http://groups.yahoo.com/group/APTITUDEandATTITUDE

RADIAN ACADEMY MATHS for GROUP-I EXAM E-Copy MATERIALS

NATURAL NUMBERS:The set of the natural numbers is denoted by N, thus.N = {1, 2, 3, 4 . . . . }

Remember : 0 (zero) is not a natural number and set ofnatural numbers is infinite.

WHOLE NUMBERS:The set of whole numbers is denoted by W, thus.W = {0, 1, 2, 3, 4 . . . . }

INTEGERS : Natural numbers, along with their negativesincluding 0 (zero) are called Integers.The set of integers is denoted by I or Z thusI = { …., -4, -3, -2, -1, 0, 1, 2, 3, 4, ….}

RATIONAL NUMBERS : A number of the form p/q. wherep and q are integers and q ≠ 0 is called a RationalNumber.

The set of rational numbers is denoted by Q thus,Q = { p/q : p, q are Integers and q ≠ 0}

IRRATIONAL NUMBERS: A number which can’t beexpressed in the form p/q is called an Irrational Number.

Thus, √2. √3, √7, 4√2, 6√18 are irrational numbers.

REAL NUMBERS: The rational and irrational numberstaken together constitute Real Numbers.

The set of real numbers is denoted by R.

ABSOLUTE VALUE: The Absolute Value of a realnumber is that number, which is obtained by droppingthe sign of the real number if any and is denoted byplacing the real number with in the symbol | | .Thus, |-7 | =7 , |-9.64 | = 9.64, |25| = 25

Note: In general an even number is represented as 2n,

n € N, and an odd number as (2n-1) where n € N

PRIME NUMBERS: A natural number that is divisible by 1and itself only is called a Prime Number.

Thus the numbers 2, 3, 5, 7, 11, 13 … are prime numbers.

COMPOSITE NUMBERS: A natural number that isneither 1 nor a prime number is called a Compositenumber.

Thus the numbers 4, 6, 8, 10,. 12, 14 . . . . are compositenumbers.

NOTE: Number 1 is neither a prime number nor acomposite number.

RECURRING OR REPEATING DECIMALS: Inrepeating decimals a digit or a block of digits repeatsitself again and again. We represent such decimals byputting a bar on repeated digit or digits.

i) PURE RECURRING: Decimal in which all thefigures after the decimal point are repeated, isknown as a pure recurring decimal such as0.666666……., 0.2626262626…… etc, are purerecurring decimals.

ii) MIXED RECURRING: A decimal in which at leastone figure after the decimal point is repeated isknown as a mixed recurring decimal.0.17777777……., 0.2959595959595……. etc, arecalled mixed recurring decimals.

RATIO & PROPORTIONRATIO

The ratio of two quantities a and b in the same

units, is the fractionb

a and we write it as a : b.

In the ratio a : b, we call a as the first term orantecedent and b, the second term or consequent.

Example: The ratio 5 : 9 represents95 with antecedent

5, consequent 9.

INCOMMENSURABLE: If the ratio of two quantitiescan not be expressed as the ratio of two integers it issaid to be incommensurable. As an example the ratio

of the side of a square to its diagonal is 1 : 2 .

PROPERTIES:a) If both the quantities x and y of a ratio are

multiplied or divided by the same quantity, theresult does not change.

b) Two or more ratios can be compared by makingtheir denominator same.

EXAMPLE: 4 : 5 = 8 : 10 = 12 : 15 = 4/7 : 5/7 etc.

1. Compound Ratio: Ratios are compounded bymultiplying together the antecedents for a newantecedent, and the consequents or a newconsequent. The compounded ratio of the ratios(a: b) , (c : d) & (e : f) is (ace : bdf).

2. If a : b is the given ratio, thena

1 :b

1 or b : a is

called its inverse or reciprocal ratio.

3. Comparison of Ratios: ( a : b) > (c : d ) ifb

a >d

c

4. If the antecedent = the consequent, the ratio is

called the ratio of equality. Ex. 3 : 3.5. If the antecedent > the consequent, the ratio is

called the ratio of greater inequality. Ex. 4:3.

6. If the antecedent < the consequent, the ratio iscalled the ratio of less inequality. Ex. 3:4.

7. Duplicate ratio of a : b is (a2 : b2)

8. Sub-duplicate ratio of a : b is ( a : b )

9. Triplicate ratio of a:b is (a3 : b3)


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10. Sub-triplicate ratio of (a : b) is (a1/3 : b1/3)

11. If sum of two numbers is A and their difference is athen the ratio of the two numbers is (A+a):(A–a).

12. The ratio between two numbers is a:b. If eachnumber is increased by x, the ratio becomes c:d,

then the two numbers arebc ad

d c Xa

−− )( and

bc ad

d c Xb

−− )( .

13. A number which when added to the terms of the

ratio a:b makes it equal to c:d isd c

bc ad

−−

14. The incomes of persons are in the ratio a:b andtheir expenditures are in the ratio c:d. If each of themsaves Rs. X, then their incomes are given by

bc ad

c d Xa

−− )( and

bc ad

c d Xb

−− )( .

15. If in x litres mixture of milk and water, the ratio ofmilk and water is a:b, the quantity of water added to

be added in order to make it equal to c:d isd c

bc ad

−−

PROPORTIONThe equality of two ratios is called Proportion.If a/b = c/d, then a, b, c, d are proportional. This

can be expressed as a : b = c : d or a : b :: c : d. Herea and d are called extremes, while b and c are calledmean terms.

1. Product of means = Product of extremes.Thus if, a : b :: c : d, then bc = ad.

2. Fourth Proportional If a:b = c:d, then d is called the fourth proportionalto a, b, c.

3. Third Proportional If a : b = b : c, then c is called the third proportionalto a and b.

4. Mean Proportional Mean proportional between a and b is ab .

5. Invertendo

Ifb

a =

d

c, then

a

b =

c

d

6. Alternendo

Ifb

a =

d

c, then

c

a =

d

b

7. Componendo

Ifba =

d c , then

bba + = d

d c +

8. Dividendo

Ifb

a =

d

c, then

b

ba − =

d

d c −

9. Componendo-Dividendo

Ifb

a =d

c , thenba

ba

−+ =

d c

d c

−+

VARIATIONIf x is Directly Proportional to y, then x = ky for

some constant k and we write it asx α y

If x is Inversely Proportional to y then xy = k forsome constant k and we write ,

x α1y

CONTINUED PROPORTION: When the first is to thesecond as the second is to the third, as the third is to

the fourth, and so on, are equal they are said to be incontinued proportion i.e.

.......=====m

u

u

t

t

z

z

y

y

x

The quantities x, y, z, t, u, m are said to be in continuedproportion.RESULTS:1. Four quantities are in proportion if and only if,

product of the extreme terms is equal to theproduct of middle terms and conversely.

2. If three quantities are in continued proportion thenthe product of the extreme terms is equal to the

square of the middle terms.

3. FUNDAMENTAL THEOREM: If three quantitiesare in continued proportion then the ratio of first tothird is the squared ratio of the first to second.

PERCENTAGE, PROFIT, LOSS AND DISCOUNT

a) Gain % = (Gain x 100)CP

b) Loss % = (Loss x 100)CP

c) SP = (100 x Gain %) x CP100

d) SP = (100 – Loss %) x CP100e) CP = 100 x SP

(100 + Gain %)f) CP = 100 x SP

(100 – Loss %)TRADE DISCOUNT: The discount is always given onthe marked price. Successive discounts are attractiveto the buyer but profitable to the seller e.g. twodiscounts of 20% and l10% come out to be only 28% tothe purchaser. As a matter of fact purchaser thinks it30% discount.

SIMPLE & COMPOUND INTERESTCOMPOUND INTEREST: Compound interest isdefined as the interest which is every time added to theprincipal whenever it is due. Addition is done after afixed period, usually after a year. After the interest isadded to the principal, the total amount acts asprincipal. Thus the difference between the originalprincipal and final amount is called compound interest.


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PRINCIPAL: The money lended on interest is calledprincipal or sum.

SIMPLE INTEREST: The extra money paid by theborrower is called interest.

AMOUNT: Amount – Principal – Interest.

Formulae for Simple Interest: If P, R and T areprincipal, rate and time then S.I. is given by

S.I. =100

T R P ××

P =T R

I S ×× ..100

R =T P

I S

×× .100

T = R P

I xS

×.100

COMPOUND INTEREST: CI = Amount – P If P = principal, R = rate % p.a. and T = time (years)then(a) Amount after T years (compounded annually)

= P

r R

+ 1001

(b) Amount after T year (compounded half yearly)

= P

T R

2

10021

+

In this case rate becomes half and time becomesdouble.(c) If the rate be p% , q%, and r% during first year,

second year and third year, then amount after 3years.

=P

+

+

+

1001

1001

1001 r q p

POPULATION GROWTH FORMULAE:a) If P is the population and R % is the growth ratethen in n years population will be

= P xn

R

+100

1

b) If p% is the growth rate during first year and q%during second year then the population after 2 years isgiven by.

= p

+

+100

1

100

1 q p

This formula can be used for more than two years.c) If R % per annum is the decrease in population

then after n years.

= p xn

R

−

1001

DEPRECIATION: It is a well known fact that the valueof a machine or car or any other article decreases withtime due to wear and tear. The decrease in value iscalled depreciation value.

Thus , if V is the value at a time t and R% p.a is therate of depreciation, then the value of machine after nyears is given by

= V xn

R

−

1001

Amount after T years is given by

A = PT

R

−

1001

NOTE: (a) For 2 years the difference between the compoundinterest and the simple interest is equal to simpleinterest for 1 year on 1st year’s interest.(b) The amount of the previous year is the principal forthe successive year.(c) The difference between the amount due at the endof two consecutive years = simple interest for one yearon the lesser amount.(d) When the interest is payable half yearly, divide therate by 2 and multiply the time by 2.(e) When the interest is payable quarterly or once in1/4th year divide the rate by 4 and multiply the time by4.

(f) There is no difference between simple interest andcompound interest on the principal for first year. C.I , ismore that S.I. after one year.

REMAINDER THEOREM: Let f(x) be a polynomial ofdegree greater than or equal to one and ‘a’ be any realnumber. If f(x) is divisible by (x-a) , then the remainderis equal to f(a).Example: Determine the remainder when thepolynomial f(x) = x3 - 3x2 + 2x + 1 is divided by (x-1).

Solution: By remainder theorem, the requiredremainder is equal to f(1).Now, f(x) = x3 – 3x2 + 2x + 1=> f(1) = 1 – 3 + 2 + 1 = 1.Hence , the required remainder is equal to 1.

FACTOR THEOREM: Let f(x) be a polynomial ofdegree greater than or equal to one and a be a realnumber such that f(a) = 0, then (x-a) is a factor of f(x),Conversely, if (x+a) is a factor of f(x), then f(-a) =0.

REMARK:i) (x+a) is a factor of a polynomial f(x) if f(-a) =0.ii) (ax-b) is a factor of a polynomial f(x) if f(b/a) = oiii) ax + b is a factor of a polynomial if f(-b/a) = oiv) (x-a) (x-b) is a factor of a polynomial f(x) if f(a) = 0

and f(b) = 0.TIME, SPEED & DISTANCE

SPEED: Distance covered per unit time is called speed.

Speed =Distance

Time

Distance = Speed × Time (or) Time = Distance/Speed

If the speed of a body is changed in the ratio a : b then the ratioof the time taken changes in the ratio b : a


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NOTE: Distance is normally measured in kilometres, metres ormiles; Time in hours or seconds and Speed in km/hr (kmph),miles/hr (mph) or metres/second (m/s).

To convert speed in kmph to m/sec, multiply it with 5/18.To convert speed in m/sec to kmph, multiply it with 18/5.

AVERAGE SPEED: Average speed of a body travelling atdifferent speeds is defined as follows.

Average Speed =takentimeTotal

travelled distanceTotal

NOTE: The average speed of a moving body is NOT EQUAL tothe average of the speeds.

A body travels from point A to another point B with a speed of xkmph and back to point A (from point B) with a speed of y kmph.

x kmph

A y kmph B

Let AB = d, the time taken by the body to travel from A to B bet1 and that from B to A be t2.

Then t1 = d/x and t2 = d/y. The total distance travelled is 2d.

Average Speed =21 tt

2

+

d =

y d x d

d

+

2 =

+ y x

d

d

11

2

=

+

y x

11

2 =

yx

2+xy

Average Speed =yx

2xy+

kmph

NOTE: This formula does not depend on the distance between A and B. This formula can be used only if the distancestravelled in each case are equal.

If the entire journey AD is travelled with the different speeds, Ato B with a uniform speed of x kmph, B to C with a uniform

speed of y kmph and C to D with a uniform speed of z kmphsuch that AB = BC = CD.

x kmph y kmph z kmph

A B C DThe average speed from A to D is given by the formula

Average Speed =

z y x

111

3

++ =

xyz

xy zx yz ++3

Average Speed =

zx yz xy

xyz

++

3

In general the ‘n’ equal distances are travelled with the speedsof x1 kmph, x2 kmph, ...., xn kmph, then the average speed isgiven by

Average Speed =

1 2

1 1 1....

n

n

x x+ + +

kmph

NOTE: The above is the harmonic mean of n numbers.

If a body covers part of the journey at speed x and the remainingpart of the journey at speed y and the distances of the two partsof the journey are in the ratio m : n, then

The average speed for the entire journey is( )m n xy xn ym

+

+

TRAINS

1. Time taken by a train of length “d” metres to pass a pole ora standing man or a signal post is equal to the time taken bythe train to cover “d” metres.

2. Time taken by a train of length “d1” metres to pass a

stationary object of length “d2” metres is the time taken by thetrain to cover (d1 + d2) metres.

3. If two trains or two bodies are moving in the same directionat u m/s and v m/s, where u > v, then their relatives speed = (u– v) m/s.

4. If two trains are moving in opposite directions atu m/s and v m/s then the relative speed is = (u + v) m/s.

5. If two trains of length “a” metres and “b” metres aremoving in opposite directions at u m/s and v m/s, then time

taken by the trains to cross each other isa b

u v

++

sec.

6. If two trains of length “a” metres & “b” metres are movingin the same direction at u m/s and v m/s, then the time taken by

the faster train to cross the slower train isa b

u v

++++−−−−

sec.

7. If two trains “A” & “B” start at the same time from points“P” and “Q” towards each other and after crossing they take“a” secand “b” sec in reaching B and A respectively, then

(A’s speed): (B’s speed) = ( √b : √a )

BOATS AND STREAMS

1. In river, the direction along the stream is calleddownstream and, the direction against the stream is calledupstream.

2. If the speed of a boat in still water is u km/hr and thespeed of the stream is v km/hr , then:

Speed of boat in downstream = (u + v) km/hr.Speed of the boat in upstream = ( u – v) km/hr.

3. If the speed downstream is “x” km/hr and the speedupstream is “y” km/hr, then:

Speed in still water = (x + y)/2 km/hrRate of stream = (x – y)/2 km/hr.

RACES AND CIRCULAR TRACK

Let the two persons “A” and “B” with respective speeds of a and b (a > b) be running around a circular track (of length L)

starting at the same point at the same time.Running in theSAME direction

Running in theOPPOSITE dir.

Time taken to meetfor the FIRSTTIME some whereon the track. ba

L

−

ba

L

+

Time taken to meetfor the first time atthe sameSTARTINGPOINT.

LCM

b

L

a

L, LCM

b

L

a

L,


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THREE PERSONS

Let the three people A, B and C with respective speeds of a , band c (a > b > c) be running around a circular track (of length L)starting at the same point at the same time in the samedirection.

CLOCKS

How many times the hands of a clock Coincide or makingan angle 0o or lie in a straight line facing same direction in a day?

Note: Day in a problem means 24 hours not 12 hours.

In 12 hrs, the two hands of the clock coincide once in every 1hour. Between 11 and 12, the coincidence is at 12 O' clock.Between 12 and 1, there is no further coincidence, because itcoincides at 12. In 12 hrs, the two hands of the clock coincide11 times only. In a day, the two hands coincide 22 times.

How many times the hands of a clock are at 180° or lie in astraight line but facing opposite direction in a day?

In 12 hrs, the two hands of the clock at straight angle once inevery 1 hr.Between 5 and 6, the angle between them is 180° at 6 O' clockonly. Also, between 6 O' clock and 7 O' clock, they will not be

at 180° as it start from 180°.In 12 hrs, 11 times. In 24 hrs, 22 times, they are at 180°.

How many times the hands of a clock are at right angles ina day?

Every one hour, the two hands are at right angles twice, exceptbetween 3 & 4 and 9 & 10.

Considering 2 to 3 they are at right angles for first timebetween 2:25 to 2:30. For the second time they are at rightangles at 3. Between 3 and 4, they are at right angles onlyonce. (ie) between 3.30 and 3.35.Similar argument holds for 9 & 10.The hands of a clock are at right angles 22 times in 12 hrs.In a day, 44 times they are at right angles.

How many times the hands of a clock lie on the samestraight line in a day?

The two hands lie on the same straight line, when they coincideand when they are at straight angle.In 12 hrs. the hands of the clock lie on the same straight line 22times.In a day, they lie on the same straight line 44 times.

The following table sum up the above discussions:

Number of timesAngle b/w the hands12 hrs 24hrs (Day)

0° (Coincidence) 11 22180° (Straight Angle) 11 22

0° or 180°(Straight line)

22 44

90° (Right angle) 22 44

MINUTE HANDIn 1 hour, the minute hand makes a complete rotation of 360°.In 1 minute it rotates about 360/60 = 6°.

HOUR HANDIn 1 hour, the hour hand makes a complete rotation of 30°. In 1minute it rotates about 30/60 = ½ °.

QUADRATIC EQUATIONSA general quadratic equation is expressed asax2 + bx + c = 0, where a≠0; a, b and c are constants.

Roots of the quadratic equation:A quadratic equation has two roots α and β given by

α =a

acbb

2

42 −+−

and β =a

acbb

2

42 −−−

The quantity D = b2 – 4ac is known as the discriminant.

I. If D = b2 – 4ac > 0 the roots are real and distinct.II. If D = b2 – 4ac = 0 the roots are real and equal.III. If D = b2 – 4ac < 0 the roots are imaginary.

RELATION BETWEEN ROOTS AND COEFFICIENTS If α and β are the roots of the equation ax2 + bx + c = 0

then α + β =-b

a and α β =

c

a

Hence x2 – ( α + β) x + α β = 0 (or) (x – α) (x – β) = 0

HIGHER DEGREE EQUATION: P(x) = a0x

n + a1xn-1 + …. + a n-1 x + an = 0

Where the coefficients a0, a1, …. an and a0 ≠ 0 is calledan equation of nth degree, which has exactly ‘n’ rootsα1, α2, … αn.

Σαi = α1 + α2 + ….αn =0

1

a

a

Σαiα j = α1α2 + … + α n-1αn = -0

2

aa

∏ αi = α1 × α2 × …× αn = (-1)n

0

n

a

a

FUNCTION

A function from X to Y is defined as a relation X x Ysuch that no two different ordered pairs of the relationhave the same first component and every element of Xhas an image in Y.

It is denoted by f : X → Y or X x Y

DOMAIN: Domain of a function is the set of values ofa, when (a, b) belongs to the function.

RANGE: Range of a function is the set of value of b,when (a, b) belongs to the function.

CO-DOMAIN: If (a, b) belong to a function f: A -> Bthen b is called co-domain of the function. Range is a

Time taken to meetfor the FIRST TIME on the track.

LCM

−− cb L

ba

L,

Time taken to meet forthe first time at the

STARTING POINT.

LCM

c

L

b

L

a

L,,


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subset of co-domain, sometimes the range and co-domain have the same elements.

FUNCTION DOMAINSin-1 x [-1.1]Cos-1 x [-1.1]Tan-1 x ] [∞−∞, Cot-1x ] [∞−∞, Sec-1 x (- ∞ , -1] U [ 1, ∞ )Cosec-1 x (- ∞ , -1] U [ 1, ∞ )

1. The function is called an onto function if everyelement of set Y has at least one pre-image in setX.

. X Y 1 a2 b3 c4

2. The function is called one-one if distinct elementshave distinct images.

X Y1 a

2 b3 c

3. The function is called many-to-one, if one or moreelements of set X there correspond only oneelement of set Y.

X Yab 1

c

NOTE: 1. One-one is also written as 1 – 1.2. An onto function is also called ‘surjection’3. An into function is also called ‘Injection’4. Both Injective & Surjective in called Bijective

LOGARITHMSCOMMON LOGARITHMS: Logarithms calculated tothe base 10. These consists of two parts:

1) Characteristic (the integral value)2) Mantissa (the positive fraction)

CHARACTERISTIC:1) To find the characteristic of a number greater thanone.

“Characteristic is one less than the number of digits tothe left of the decimal point in the given number”.

Ex. characterstic of 514.34 is 2 and 3125.875 is 3.

2) To find the characteristic of a number less than one.

“Characteristic is one more than the number of zerosbetween the decimal point and the first significant digitof the number and is negative”.

Ex. characterstic of 0.34 is 1 and 0.00075 is 4 .

MANTISSA: Mantissa of a number is found with thehelp of logarithmic tables.1. The mantissa is the same for the logarithms of all

numbers having the same significant digits.2. The logarithm of one digit number, say 2, is to be

see in the table, opposite to 20.3. The mantissa is always taken positive.

ANTILOGARITHM: If log a = m, then a = antilog of m,i.e., The number corresponding to a given logarithm iscalled antilogarithm.

1. If the characteristic of the logarithm is positive,then: “put the decimal point after ( n+1)th digit,where n is equal to characteristic.

2. If the characteristic of the logarithm is negative,the:”put the decimal point so that the first significantdigit is at ‘n’th place, where n = characteristic’.

Properties of Logarithms.1. Log 1 = 0 , irrespective of the base2. Log a a = 1, logarithm of any number to its own

base is always 1.3. Logarithm of product

Log a (mn) = Log a m + Log a n

4. Logarithm of ratioLog a (m/n) = Log a m - Log a n5. Logarithm of a Power

Log a mn = nLog a m

6. Base changing formulaLog a m = Log a m x Log a b

7. Log a q(np) = Log n p / Log a q irrespective of

the base.8. Particular case

log a an = n

9. a log a n = nIn particular e In n = n

SOME IMPORTANT POINTS: Those logarithmswhose base is 10 are known as Common (decimal) logarithms while which has base e (e = 2.71828….) areknown as natural or Napierian logarithms. Naturallogarithm is changed to decimal logarithm as

PERMUTATIONS AND COMBINATIONS

PERMUTATIONS: It is defined as the ways ofarranging object. Here the order i.e. position isimportant.

The number of permutations of objects taken r at atime isnPr = n (n-1)(n-2)(n-3)…(n-r+1) =

)!(

!

r n

n

−

nPn = n!;nP0 = 1;

nP1 = n

NOTE: n! = n×(n-1)×(n-2) ……. 3×2×1 =1

RESULTS:

i) The total number of permutation of n items taken alltogether, when ‘p’ items are of one type, ‘q’ are ofsecond type and ‘r’ of then third kind and the remaining

are of different type is!!!

!r q p

n


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ii) The number of circular permutations of n differentobjects is n-1!.

COMBINATIONSWhen r objects taken out of n objects thencombination of n objects taken r at a time, we writenCr = C (n, r) =

!)!(

!r r n

n

−

Note: nCr =nC n-r

STATISTICS

STATISTICS is concerned with scientific methods forcollecting, organizing, summarizing, presenting andanalyzing data, as well as drawing valid conclusions andmaking reasonable decisions on the basis of suchanalysis.

LIMITATIONS OF STATISTICS1. Statistics is not suited to the study of qualitative

phenomenon.2. Statistics does not study individuals but is used only

to analyse an aggregate of objects. We study groupcharacteristics through statistical analysis.

3. Statistical decisions are true only on an average andalso the average is to be taken for a large number ofobservations. For a few cases in succession thedecision may not be true.

4. Statistical decisions are to be made carefully byexperts. Untrained persons using statistical tools, maylead to false conclusions.

CHARACTERISTICS OF STATISTICAL ANALYSIS.

1. In statistics all information are to be expressed inquantitative terms. Even in the study of quality likeintelligence of a group of students we require scoresor marks secured in a test.

2. Statistics deals with a collection of facts not anindividual happening.

3. Statistical data are collected with a definite object inmind. i.e. there must be a definite field of enquiry.

4. In every field of enquiry there are large number offactors, each of which contributes to the final datacollected. So statistics may be affected by amultiplicity of causes.

5. Statistics is not an exact science.6. Statistics should be so related that cause and effect

relationship can be established.7. A statistical enquiry passes through four stages,

Collection of data, Classification & tabulation of data,Analysis of data and Interpretation of data.

COMMONLY USED TERMS:

1. Data: A collection of observations expressed innumerical figures, obtained by measuring or counting.

2. Population: A population or a universe consists ofthe totality of the set of objects, with which we areconcerned, e.g. all workers working in a plant, alltimes produced by a machine in a particular periodetc.

3. A sample: A sample is a sub-set of the population i.e.it is a selected number of individuals each of which isa member of the population.

4. Characteristic : A quality possessed by an individualperson, object or item of a population, e.g. heights ofindividuals, nationality of a group of passengers on aflight etc.

5. Variable and attribute: A measurable characteristicis called a variable or a variate. A non-measurablecharacteristic is called an attribute. It may be notedhere that by measurable characteristics we meanthose characteristics which are expressible in terms ofsome numerical units, e.g. age, height, income etc.

CONTINUOUS AND DISCRETE VARIABLE.

A variable which can theoretically assume any valuebetween two given values is called a Continuous variableotherwise it is a discrete variable; heights, weights ,agricultural holding are some examples of continuousvariables whereas number of workers in a factory, numberof defectives produced, readings on a Taxi meter areexamples of discrete variables.

Data which can be described by a discrete or continuousvariable are called discrete data or continuous datarespectively.

The first and foremost task of a Statistician is to collectand assemble his data. When he himself prepares thedata, it is called a primary data but when he borrows themfrom other sources (Government, semi-Government ornon-official records) the data is called a secondary one.

MEASURES OF CENTRAL TENDENCY

The term of ‘Central Tendency of a given statistical data’we mean that central value of the data about which theobservations are concentrated. A central value whichenables us to comprehend in a single effort thesignificance of the whole is know as Statistical Average orsimply average.The three common measures of Central Tendency arei) Mean ii) Median iii) Mode

THE MOST COMMON AND USEFUL MEASURE IS THEMEAN.

ARITHMETIC MEAN

Advantages:1. This is the widely used measure of Central Tendency.2. It is simple to understand and easy to Calculate.3. It is rigidly defined4. Calculations depend on all the values5. It is suitable for algebraic treatment.6. It is least affected by sampling fluctuations.

Disadvantages:i) Cannot be determined by inspectionii) It is very much affected by the presence of a few

extremely large or small values of the variableiii) Mean cannot be calculated if a single term is

missing.iv) A.M. cannot be calculated for grouped frequency

distribution with open end classes, unless someassumptions are made.


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GEOMETRIC MEANAdvantages:i) G.M. is not widely used. It is particularly suitable

for averaging rates of changes.ii) It is rigidly defined and depends on all values of

the series.iii) It is suitable for algebraic treatmentiv) G.M. is not affected by the presence of very large

or small values of the variable.Disadvantages:i) Unlike A.M, G.M. is neither simple to understand

nor simple to calculate.

ii) If any value of the series is Zero. G.M. cannot becalculated.iii) Calculation of G.M. is impossible unless all the

values are positive.

HARMONIC MEAN: Advantages:i) It is useful in averaging rates ratios and prices.ii) It is suitable for algebraic treatmentsiii) Its calculation is based on all values of the series.Disadvantages:i) It is very limited use and not easy to understandii) H.M. cannot be calculated if any value is Zero.

RELATION BETWEEN A.M., G.M. and H.M.For any set of positive values of a variable, we can writeA.M. ≥ G.M. ≥ H.M. equality occurring only when thevalues are equal.

For a pair of observations only, AM x HM = (GM)2

MEDIAN:Advantages:i) It is easily understood.ii) Not affected by extreme values.iii) Can be determined by inspection in case of a

simple frequency distribution.iv) It can be calculated from a grouped frequency

distribution with open-end classes, provided byclosed classes are of equal width.

Disadvantages:i) It is not well-defined and also it is not possible to

find a well defined mode.ii) It is not suitable for algebraic treatmentiii) It is not based on all values of the variableiv) It is affected by sampling fluctuations.

MEDIAN: The value of the item which divides the datainto two equal parts is called median.

Median of ungrouped data: If the n items in the dataare arranged in ascending or descending order and

if n is ODD then ,1

2

n ++++ th item;

if n is EVEN, then the average of2

nth,

2

n +1 th

items is called median.

QUARTILE DEVIATION: The items which divide the datainto four parts are called quartiles. They are denoted byQ1 , Q2, Q3

Quartile deviation = Q3 – Q1 2

QUARTILES OF UNGROUPED DATA: Write the nitems of the data in ascending order. Then LowerQuartile Q1 = (n +1)/4

th item

Middle Quartile Q2 (Median) =1

2n ++++

th item

Upper Quartile Q3 = 3 (n+1) /4th item.

DISPERSION: The variation or scattering or deviation ofthe different values of a variable from their average isknown as Dispersion.

ABSOLUTE MEASURES: The three absolute measures

arei) Rangeii) Mean deviationiii) Standard deviation.

Range: Range is the simplest measure of dispersion. It isthe difference between the largest and the smallestvalues of a variable. This is not the widely used measureas it lacks in accuracy.

Coefficient of Mean Dispersion:The coefficient of mean dispersion is defined by theformula.

Coefficient of Mean Dispersion

=eanDeviationfrommean

ean

Or =eanDeviationfromMedian

edian

STANDARD DEVIATION: This is most important absolutemeasure of dispersion. Standard deviation (S.D.) for a setof values of a variable is defined as the positive squareroot of the arithmetic mean o the squares of all thedeviations of the values from their arithmetic mean. In

short, it may be defined as the square root of the Meansquares of deviation from mean.S.D is usually denoted by a greek small letter σ (pronounced Sigma)

If x1, x2 . . . . xn be a series of values of a variable and x their A.M. : then S.D. is defined by

σ =( ) ( ) ( )

2 2 2

1 2 .....− + − + + −n x x x x x x

n

For a frequency distributionThis square of S.D . is known as VARIANCE

σ =( )

2

∑ −i f x x

N , where N = Σf

i.e. variance = σ 2 = (S.D.) 2

i) Coefficient of range = Max.value – min. valuemax.value + min. value

ii) Coefficient of Q.D. = Q3 – Q1 Q3 + Q1


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RELATIVE MEASURES OF DISPERSION

x =1 21 2

1 2

n x n x

n n

++

σ2 =2 2 2 2

1 1 2 2 1 1 2 2

1 2

n n n d n d

n n

σ σ + + ++

where d1 = - , d2 = 2 - x

The relative measures of dispersion are pure numbersand are mainly employed in comparing the dispersions oftwo or more distributions. There are two relativemeasures:

i) Coefficient of Variation

(as percentage) =. .S D

Mean x 100

ii) Coefficient of Mean Deviation (as percentage) = MeanDeviation

MeanorMedianx 100

MEASURES OF SKEWNESS:

The degree of skew ness is measured by its coefficient.The very common measures are:

1. Pearson’s first measure:

Skewness =tan

Mean Mode

S darddeviation

−

2. Pearson’s second measures:

Skewness =( )3tan

ean Mode

S darddeviation

−

BINOMIAL DISTRIBUTION

A random variable X is said to follow binomial distributionif its probability mass function is given by

P(X ) = { ncx px q n-x when x = 0, 1, 2, 3, …. n

{ 0 Otherwise

X denotes the number of successes.n denotes the total number of trials.p is the probability of success in each trial.q is the probability of failure in each trial.We have q = 1-p.

n and p are known as the parameters of the binomialdistribution .

Mean = npVariance = npqStandard deviation = √(npq)

Binomial distribution is a discrete distribution. Abinomial distribution can be used when

a) The number of trials is finiteb) The trials are independent of each otherc) The probability of success is constant for each

trial.An experiment which has two mutually disjoint outcomes,usually called “success” and “failure” is called a Bernouilli trial.

An experiment consisting of a repeated number ofBernoulli trials is called a binomial experiment.

POISSON DISTRIBUTION:A random variable X is said to follow Poisson distributionif its probability mass function is given by

P (X=x) =x e

x

λ λ

!

−−−−

when x = 0, 1, 2, 3 , . . . .

0 otherwise

λ is known as the parameter of the Poisson distribution.

Mean = λVariance = λStandard deviation = √λ

NORMAL DISTRIBUTION:A continuous random variable X is said to follow normaldistribution with mean µ and standard deviation σ if itsprobability density function is given by

2

2

1

21

2

( )

( )x

f x e σ

σ π

µ−−−−

==== -

0

x

σ

µ

−∞ < < ∞−∞ < < ∞−∞ < < ∞−∞ < < ∞

−∞ < < ∞−∞ < < ∞−∞ < < ∞−∞ < < ∞

>>>>

µ an σ are called the parameters of the normaldistributionMean = µ

Variance = σ 2

Standard deviation = σ

Properties of normal distribution:The total area under the normal curve is UNITY. Mean, Median and mode of the distribution are all equal.Mean = Median – Mode = µ

The maximum probability density (i.e. the maximumordinate) occurs at x = µ

Maximum ordinate =σ π

1

2

It has only one mode at x = µ , Therefore it is unimodal

Curve is symmetrical about x - µ , so that skewness =0.

NOTE:The cube roots of unity, ie., the values of 11/3 are

1, -1

2 + i

3

2, -

1

2 - I

3

2

These are denoted by 1, ω , ω 2. We havea) 1 + ω + ω 2 = 0 b) ω 3=1


10/13


1. (1 +x)m =1+1

m

!x +

( 1)m m −2!

x2 +

( 1)( 2)m m m− −3!

x3 + . . . .

2 (1+x) -1 = 1 – x + x2 –. . . x3 + x4 – x5 + . . .

3. (1-x)-1 = 1 + x + x2 + . x3 + x4 + x5 + . . .

4. (1 + x) -2 = 1 – 2x + 3x2 – 4x3 + . . .. .

5. (1 - x) -2 = 1 + 2x + 3x2 + 4x3 + . . .. .

6. ex = 1 +1

x

! +

2

2! +

3 x

3! +

4 x

4! + ……

e = 1 +1

1! +

1

2! +

1

3! +

1

4! + ……

7. e –x = 1 -1!

+2 x

2! +

3 x

3! +

4

4!

8. ex – e-x = 2 . ..... x x x + + + 1! 3! 5!

e -1

e

=21 1 1

1 ...... + + + 1! 3! 5!

e -1 or1

e= 1 +

1

1! +

1

2! +

1

3! + ……

8. e x + e -x = 22 4 6

1 ......2 4 6

+ + + ! ! !

x x x

e +1

e = 2

1 1 11 ......

2 4 6

+ + + ! ! !

10. log (1-x) = -x -2 x

2

+3

3

-4

4

- . . . .. .

11.log (1-x) = -x -2

2 -

3

3 -

4 x

4 - . .. .. ..

12.log1 x

x

+1 −

= 23 5

......5

x x x

+ + + 3

13. sin x = x -3

3! +

5

5! x

-7

7! + .. . . . .

14. cos x = 1 -2 4 6

.....2 4 6

x x x+ + +

! ! !

15. tan x = x +3

3 + 2

15 x5 + . . . . .

CONIC : The locus of a point P which moves such that itsdistance from a fixed point S bears a constant ratio to itsdistance from a fixed l is called a conic.

The fixed point S is called the focus.

The fixed line l is called the directrix.

The constant ratioSP

PM is called the eccentricity,

denoted by e.If e = 1 , the conic is called a parabolaIf e < 1 the conic is called an ellipseIf e > 1 the conic is called a hyperbola

The general equation of a conic will be an equation ofsecond degree in x and y, in the formax2+ 2hxy + by2+ 2gx + 2fy + e = 0Conversely, the general equation of second degree in xand y, i.e.,ax2 + 2hxy + by2+ 2gx + 2fy + e = 0 will represent a conic

if abc + 2fgh – af 2- bg2- ch2 ≠ 0 andi) h2 - ab for a parabolaii) h2 < ab for an ellipseiii) h2 > ab for a hyperbolaiv) h2 > ab and a+b =0 for a rectangular

hyperbola.COORDINATE GEOMETRYDistance Formulae: The distance between the pointsA(x1,y1) and B(x2,y2) is given by

AB = 2 1 2 12

2( ) ( )x x y y − + −− + −− + −− + − The distance of the point P(x,y) from the origin O is given

byOP = 2 2+ x y

SECTION FORMULAE:(a) The coordinates (x,y) of a point R which divides the

join of two points P(x1,Y1) and Q(x2,y2) in the ratio m1:m2 internally are given byx = m1x2 + m2x1 , y= m1y2 + m2y1

m1 + m2 m1 + m2 (b) If (x, y) divides the line segment PQ in the ratio k :1

(internally), thenx = kx2 + x1 , y = ky2 + y1

k + 1 k + 1(c) If M(x, y) is a midpoint of PQ, then

X = 1 (x1 + x2), y = 1 (y1 + y2)2 2

(d) If R (x,y) divides PQ externally in the ratio m1:m2 ,then

X = m1 x2 - m2 x1 m1 – m2

Y = m1y2 – m2 y1 m1 – m2

e) If R(x, y) divides PQ externally in the ratio K:1 ,then

X = Kx2 - X1 . y = ky2 – y1 k-1 k-1

CENTROID : It is the point where the three medians of atriangle meet. Centroid divides each median in the ratio

2:1 . The coordinates (x,y ) of the centroid of the trianglewhose vertices are (x1,y1) (x2, y2) (x3+ y3)are given by

X = 1/3 (x1 + x2 +x3) = 1/3 (y1+y2+y3)

INCENTRE: It is the point where the internal bisectors ofa triangle intersect. The coordinates k (x, y) of theincentre are given by:

x = ax1 + bx2 + mcx3 y = ay1 + by2 + cy3

a + b + c a + b + c


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ORTHOCENTRE: The three altitudes (the lines throughthe vertices and perpendicular to the opposite sides) of atriangle interest in a common point called orthocenter ofa triangle.

CIRCUM-CENTRE: This is a point which is equidistantfrom three vertices of a triangle. Thus it is the centre ofthe circle that passes through the vertices of triangle. It isalso the point of intersection of the right bisectors of thesides.

AREA OF A TRIANGLE: The area of a triangle whose

vertices are A(x1, y1), B(x2, y2), C (x3, y3) is given by= 1/2 (x1(y2-y3)+x2(y3-y1) + x3(y2 – y2)}

If there pints A, B, C are collinear (lie on the samestraight line). Then Area = 0.

LOCUS OF A POINT: It is the path traced by a pilotmoving under certain conditions. Thus the locus of apoint which moves such that it is always at a constantdistance from a given point in a plane, is a circle.

EQUATION OF A LOCUS: The equation of the locus of amoving point P(x,y) is an algebraic relation between x andy satisfying the given conditions, under which P moves.Thus, if P(x,y) moves along the circle of radius r havingkits centre at the origin, then equation of the locus is

X2 + y2 = r 2

STRAIGHT LINE;EQUATIONS O A STRAIGHT LINE:(a ) Equations of coordinate axes: Sine at every pointon the x-axis, y=0, hence the equation of the axis of x is y= 0 . Similarly, the equation of the y=axis is x=0.

Equations of straight lines in various forms:(a) Slope Intercept form

y= mx + C

(b) slope-point formy – y1 = m (x -x1)

(c) Intercept formx + y = 1a b

(d) Two point formy – y1 = y2 – y1 (x – x1)

x2 – x1

(e) Parametric form,:

x-x1 = y-y1 = rcos θ sin θ

any point on this line(x1 = r cos θ , y1 = r sin θ)

(f) Normal formX cos θ + y sin θ = p

(g) General equation:Ax + By + C = 0

Angle between he two straight lines:Y = m1 x + c1, y = m2 x + c2

m1 – m2 tan θ = 1 + m1 m2

a) The above two straight lines are perpendicular if,θ = 90o tan 90o = Not defined , i.e. if1 + m1m2 = o or m1 x m2 = 1

b) The above two straight lines are parallel ifθ = 0 => tan θ = 0, i.e. m1 = m2

ANGLE BETWEEN THE TWO STRAIGHT LINES:

a1 x + b1 y + c1 = 0a2 x + b2 y + c2 = 0

tan θ = a1b2 – a2b1 a1a2 + b1b2

a) The above lines are perpendicular ifA1b2 – a2b1 = 0

i.e. a1 = b1 a2 b2

The equations of two parallel lines differ in constant termonly.

Equation of a straight line parallel to the straight lineax + by + c= 0 , is ax + by + k = 0

Equation of a straight lien perpendicular to the straight lineax + by + c = 0, is bx – ay + k = 0

Equation of a straight line through the point ofintersection of the straight lines

a1 x + b1 y + c1 = 0 anda2 x + b2 y + c2 = 0 is

a1x + b1y + c1 + k (a2x + b2y + c2) = 0

m = tan θ ± tan 1 + tan θ tan

Length p of the perpendicular fromP (x1 , y1) to the lien ax + by + c = 0

P = ax1 + by1 + c√a2 + b2

Perpendicular distance p between and parallel straightlines ax + by + c1 = 0 and ax + by + c2 = 0, are

P = c1 - c2 a2 + b2

Equation of angle bisectors between the straight lines,a1x + b1y + c1 = 0 and a2 x + b2y + c2 = 0 , are

a1 x + b1y + c1 = a2x + b2y + c2 √ a21 + b

21 √a

22 + b

22


12/13


Concurrency of the three straight lines, The straight lines:a1x + b1y + c1 = 0a2x + b2y + c2 = 0a3x + b3y + c3 = 0

are concurrent if

a1 b1 c1 a2 b2 c2 = 0a3 b3 c3

TRIGONOMETRY

1. sin ø = p/h = perpendicular / hypotenuse2. cos ø = b / h = base / hypotenuse3. tan ø = p/b = perpendicular/base4. cosec ø = h/p = hypotenuse / perpendicular5. sec ø = h/b = hypotenuse / base6. cot ø = b/ = base / perpendicular

TRIGONEMETRIC RELATIONS:1. sin ø = 1/cosec ø2. cos ø = 1/sec ø3. tan ø = 1/ cot ø4. tan ø = sin ø / cos ø5. cot ø = cos ø / sin ø

QUADRANTS

The two axes Xn OX and Y n OY divides the plane intoFour Quadrants.i.In first quadrant, all trigonometric ratios are positive.

ii. In second quadrant, only sin ø and cosec ø are positive.iii. In third quadrant, only tan ø and cot ø are positive.iv. In fourth quadrant, only cos ø and sec ø are positive.

IMPORTANT RELATIONSI. sin2ø + cos2ø = 1II. 1 + tan2ø = sec2øIII. 1 + cos2ø = cosec2ø

SUM AND DIFFERENCE FORMULAE:1) sin (A±B) = sin A cos B ± cos A sin B2) cos (A±B) = cos A cos B – or + sin A sin B3) tan (A±B) = tan A ± tan B / 1 ± tan A tan B4) sin (A±B) sin (A – B)5) sin (A±B) sin (A ± B)

= sin2 A – Sin2B = Cos2B – Cos2A6) cos ( A +B) Cos (A – B)

= cos2A – sin2 B = cos2b – sin2A

DOUBLE – ANGLE FORMULAE:a) sin2ø = 2sin ø cos ø = 2tanø / 1+ tan2øb) cos2 ø = cos2ø – sin2ø= 2cos2ø – 1= 1 – 2sin2ø= 1-tan2ø / 1 + tan2ø

7) cos2 ø = ½ (1 + cos2ø)8) tan2 ø= 2tan ø / 1 – tan2 ø

Triple-Angle Formulae:a) sin3ø = 3sin ø - 4 sin3 øb) cos3ø = 4cos3 ø - 3 cos øc) tan3ø = 3tanø - tan3ø / 1-3tan2ø

Sum or Differnce nto product:b) sin A + sin B = 2sin A+B/2cos A-B/2c) sinA – SinB = 2cos A+B/2sin A-B/2d) cosA + cos B = 2cosA =B /2cos A-B / 2e) cos A – cos B = -2A+B/2sin A-B/2

Product into sum or difference:a) 2sinA cosB = sin (A +B) + sin (A-B)b) 2cosA cosB = cos(A+B) + cos(A-B)c) 2sinA sinB = cos (A-B) – cos (A+B)

Relations between he sides and angles of a triangle:

In ∆ ABC,a) Sinc formula

sin

a

A=

sin

b

B=sin

c

C = 2R

b) Consine formulae

Cos A =2 2 2

2

b c a

bc

+ −

Cos B =2 2 2

2

c a b

ca

+ −

Cos C =2 2 2

2

a b c

ab+ −

PROJECTION FORMULAE:a) a = b cosC + c cosBb) b = c cosA + a cosCc) c = a cosB + b cosA

General values of Trigonometric Functions:a) If sin ø = SinαThen , θ = n π+ (-1)n α, n €1

b) If cos θ = cos α

Then, θ = n π ± α, n € 1

i) velocity at time ‘t’ is v =ds

dt

ii) acceleration at time ‘t’ is a =dv

dt =

2

2d s

dt


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RADIAN

LOGICAL REASONING for TNPSC EXAMS RADIAN IAS ACADEMY

Logic is the science and art of reasoning correctly, thescience of the necessary laws of thought; Reasoning isthe mind’s power of drawing conclusions and deductinginference from premises. And so, Logical Reasoningimplies the process of drawing logical conclusions fromgiven facts in conformity to what is fairly to be expected orcalled for. It must be noted that logical conclusionsmeans what is derived by reasoning or logic and not thetruth or fact.

PROPOSITION: The logical proposition is an expression

or a statement which affirms or denies something, so thatit can be characterised as true or false, valid or invalid.Like any other grammatical sentence, a proposition has asubject, a predicate and a copula connecting the two.

Example: Philosophers are intelligent.

Here, ‘Philosophers’ the subject, ‘intelligent’ is predicateand ‘are’ is copula.

The propositions can be classified into Four categories.

(I) CATEGORICAL PROPOSITION: Emphasises what isand what is not, i.e., a subject is a predicate or is not

predicate.Example: I. All cats are dogs.

II. No hens are ducks.Logically speaking, all cats must be dogs irrespective ofthe truth that cats can never be dogs. So, also in secondsentence, no hens are ducks leaves no argument thatsome hens may be ducks.

(II) DISJUNCTIVE PROPOSITION: Leave every scope ofconfusion as they have either -------- or --------- in then

Example: Either she is shy or she is cunning.These type of propositions give two alternatives.

I. Antecedent i.e. ‘she is shy’ andII. Consequent i.e. ‘or she is cunning’

The inferences drawn on such statements are probablytrue or probably false. The right inference often dependson one’s own ability to sense and analyse the validity ofthe logic.

(III) HYPOTHETICAL PROPOSITION: Correspond to theconditions, and the conditional part starts with words suchas ‘if’.Example: If I am late, I will miss the train. Here also,proposition has two parts.

I. antecedent – if I am late, and

II. consequeny –I will miss the train

(IV) RELATIONAL PROPOSITION: Denote the relationbetween the subject and the predicate. The relation canbe (I) symmetrical (II) non-symmetrical or (III)asymmetrical.

Example: I. She is as tall as PinkiII. Jai is wiser than RoyIII. Tim is brother of Ria.

For the validity of drawing inference in an argumentthe propositions are also classified on the basis ofquality; as Affirmative (Positive) or Negative, andQuantity; as Universal or Particular

a) UNIVERSAL AFFIRMATIVE – ‘A’ PropositionOnly subjective term is distributed:Example: I. All men are strong.

II. All Birds have beaks.In the above statements, subject is ‘All’ , i.e. ‘All men’and ‘All’ birds;

b) UNIVERSAL NEGATIVE – ‘E’ Proposition:Both subjective and predicative terms are distributedExample I. No man is perfect

II. No fools are wiseIn the above statements, the distributed term is ‘No’,‘No one’. When no man is perfect, then one who isperfect cannot be man. Similarly, when no fools arewise, then one who is wise cannot be a fool.

c) PARTICULAR AFFIRMATIVE – ‘I’ Proposition:Neither of the terms is distributed.Example: I. Some children are very naughty

II. Some politicians are dishonest

In the above statements, the distributed term is notparticular, i.e. ‘some’. When some children arenaughty, then some of those who are naughty may bechildren. Similarly, when some politicians aredishonest, then some dishonest men may bepoliticians. There is no defined certainty.

d) PARTICULAR NEGATIVE: ‘O’ Propositions:Here the predicative term is distributed. ‘Some usedwith a negative sign is a particular negativeproposition.Example: I. Some students are not intelligent

II. All animals are not pets.In the statement ‘All animals’ may mislead it to be a

Universal negative but ‘All’ with ‘not’ is a particularnegative. However, words such as ‘some’ ‘mostly’ ‘allbut one’ etc. are particular Propositions.

PREMISE is a proposition stated or assumed for after-

reasoning especially one of the two propositions in asyllogism, from which the conclusion is drawn. Of the twostatements, the first is major premise and the second isminor premise. Example: All dogs are hens. (major premise)

All pups are dogs. (minor premise)Inference: All pups are hens.

Based on the two premises, the inference is drawn.

TERM is a word used in a specially understood or defined

source which may be subject or predicate of aproposition. The terms in the major premise are calledmajor terms and that in the minor premise are calledminor terms. The middle term occurs in both the premise. In the above example, dogs, hens and pups are threeterms used. Of these ‘hens’ is the major term, ‘pups; is theminor term and ‘dogs’ is the connecting or the middleterm.

INFERENCE is the act of drawing a logical conclusionfrom given premise. This logical deduction followsnecessarily from the reasoning of given premises and notof the truth.

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