RADIAN ACADEMY
MATHS for GROUP-I EXAM E-Copy
MATERIALS
i) PURE RECURRING: RECURRING: Decimal in which all the figures after the decimal point are repeated, is known as a pure recurring decimal such as 0.666666……., 0.2626262626…… etc, are pure recurring decimals. Remember Remember : 0 (zero) is not a natural number and set of
NATURAL NUMBERS: NUMBERS: The set of the natural numbers is denoted by N, thus. N = {1, 2, 3, 4 . . . . } natural numbers is infinite. WHOLE NUMBERS: NUMBERS: The set of whole numbers is denoted by W, thus. W = {0, 1, 2, 3, 4 . . . . }
ii) MIXED RECURRING: RECURRING: A decimal in which at least one figure after the decimal decimal point point is repeated is known as a mixed recurring decimal. 0.17777777……., 0.2959595959595……. etc, are called mixed recurring decimals.
INTEGERS : Natural numbers, along with their negatives including 0 (zero) are called Integers. The set of integers is denoted by I or Z thus RATIO I = { …., -4, -3, -2, -1, 0, 1, 2, 3, 4, ….}
RATIO & PROPORTION
The ratio of two quantities a and b in the same RATIONAL NUMBERS : A number of the form p/q. where units, is the fraction a and we write it as a : b. p and q are integers and q ≠ 0 is called a Rational Number.
The set of rational numbers is denoted by Q thus, Q = { p/q : p, q are Integers and q ≠ 0}
b
In the ratio a : b, we call a as the first term or antecedent and b, the second term or consequent or consequent.. Example: Example: The ratio 5 : 9 represents 5 with antecedent 9
IRRATIONAL NUMBERS: NUMBERS: A number which can’t be 5, consequent 9. expressed in the form p/q is called an Irrational Number. Thus, √2. √3, √7, 4√2, 6√18 are irrational numbers.
INCOMMENSURABLE: INCOMMENSURABLE: If the ratio of two quantities can not be expressed as the ratio of two integers it is REAL NUMBERS: NUMBERS: The rational and irrational numbers said to be incommensurable. As an example the ratio taken together constitute Real Numbers. of the side of a square to its diagonal diagonal is 1 : 2 . The set of real numbers is denoted by R.
PROPERTIES: PROPERTIES: a) If both the quantities x and y of a ratio are ABSOLUTE VALUE: VALUE: The Absolute Value of a real multiplied or divided by the same quantity, the number is that number, which is obtained by dropping result does not change. the sign of the real number if any and is denoted by b) Two or more ratios can be compared compared by making placing the real number with in the symbol | | . their denominator same. Thus, |-7 | =7 , |-9.64 | = 9.64, |25| = 25
Note: Note: In general an even number is represented as 2n, n € N, and an odd number as (2n-1) (2n-1) where n € N
EXAMPLE: EXAMPLE: 4 : 5 = 8 : 10 = 12 : 15 = 4/7 : 5/7 etc.
1. Compound Ratio: Ratios are compounded by PRIME NUMBERS: A natural number that is divisible by 1 multiplying together the antecedents for a new antecedent, and the consequents or a new and itself only is called a Prime Number. consequent. The compounded compounded ratio of the ratios (a: b) , (c : d) & (e : f) is (ace : bdf). Thus the numbers 2, 3, 5, 7, 11, 13 … are prime numbers. 1 : 1 COMPOSITE NUMBERS: NUMBERS: A natural number that is 2. If a : b is the given ratio, then a b neither 1 nor a prime number is called a Composite called its inverse or reciprocal or reciprocal ratio. number.
or b : a is
a Thus the numbers 4, 6, 8, 10,. 12, 14 . . . . are composite 3. Comparison of Ratios: ( a : b) > (c : d ) if b > numbers.
c d
4. If the antecedent = the consequent, consequent, the ratio is is equality. Ex. 3 : 3. NOTE: Number 1 is neither a prime number nor a called the ratio of equality. composite number. 5. If the antecedent > the consequent, consequent, the ratio is is called the ratio of greater inequality. Ex. 4:3. RECURRING OR REPEATING DECIMALS: DECIMALS: In consequent, the ratio is is repeating decimals a digit or a block of digits repeats 6. If the antecedent < the consequent, called the ratio of less inequality. inequality . Ex. 3:4. itself again and again. We represent such decimals by putting a bar on repeated digit or digits. 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) RADIAN ACADEMY
ANNA NAGAR & NSK NAGAR-ARUMBAKKAM
[email protected] Ph: 98404-00825, 30025003
10. Sub-triplicate ratio of (a : b) is (a1/3 : b1/3)
VARIATION If x is Directly Proportional to y, then x = ky for 11. If sum of two numbers is A and their difference is a some constant k and we write it as xαy then the ratio of the two numbers is (A+a):(A–a). If x is Inversely Proportional to y then xy = k for 12. The ratio between two numbers is a:b. If each some constant k and we write , number is increased by x, the ratio becomes c:d, 1 xα Xa(c − d ) Xb(c − d ) y then the two numbers are and . ad − bc ad − bc CONTINUED PROPORTION: When the first is to the second as the second is to the third, as the third is to 13. A number which when added to the terms of the the fourth, and so on, are equal they are said to be in continued proportion i.e. ratio a:b makes it equal to c:d is ad − bc c − d
x
=
y
=
z
=
t
=
u
= .......
14. The incomes of persons are in the ratio a:b and y z t u m their expenditures are in the ratio c:d. If each of them The quantities x, y, z, t, u, m are said to be in continued saves Rs. X, then their incomes are given by proportion. Xa(d − c ) RESULTS: and Xb(d − c ) . 1. Four quantities are in proportion if and only if, ad − bc ad − bc product of the extreme terms is equal to the product of middle terms and conversely. 15. If in x litres mixture of milk and water, the ratio of milk and water is a:b, the quantity of water added to 2. If three quantities are in continued proportion then be added in order to make it equal to c:d is ad − bc the product of the extreme terms is equal to the c − d square of the middle terms. PROPORTION The equality of two ratios is called Proportion. 3. FUNDAMENTAL THEOREM: If three quantities If a/b = c/d, then a, b, c, d are proportional. This are in continued proportion then the ratio of first to can be expressed as a : b = c : d or a : b :: c : d. Here third is the squared ratio of the first to second. a and d are called extremes, while b and c are called mean terms. PERCENTAGE, PROFIT, LOSS AND DISCOUNT 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 proportional to a, b, c. 3. Third Proportional If a : b = b : c, then c is called the third proportional to a and b. 4. Mean Proportional Mean proportional between a and b is ab . 5. Invertendo If
a b
=
c d
, then
6. Alternendo If
a b
=
c d
, then
7. Componendo If
a b
=
c d
, then
8. Dividendo If
a b
=
c d
, then
b
=
a a
=
c
d c b d
a+b b a −b b
=
=
d
a−b
RADIAN ACADEMY
Gain % = (Gain x 100) CP b) Loss % = (Loss x 100) CP c) SP = (100 x Gain %) x CP 100 d) SP = (100 – Loss %) x CP 100 e) CP = 100 x SP (100 + Gain %) f) CP = 100 x SP (100 – Loss %) TRADE DISCOUNT: The discount is always given on the marked price. Successive discounts are attractive to the buyer but profitable to the seller e.g. two discounts of 20% and l10% come out to be only 28% to the purchaser. As a matter of fact purchaser thinks it 30% discount.
c + d
SIMPLE & COMPOUND INTEREST
d
COMPOUND INTEREST: Compound interest is defined as the interest which is every time added to the principal whenever it is due. Addition is done after a fixed period, usually after a year. After the interest is added to the principal, the total amount acts as principal. Thus the difference between the original principal and final amount is called compound interest.
c − d d
9. Componendo-Dividendo If a = c , then a + b = c + d b
a)
c − d
ANNA NAGAR & NSK NAGAR-ARUMBAKKAM
[email protected] Ph: 98404-00825, 30025003
PRINCIPAL: The money lended on interest is called principal or sum. Thus , if V is the value at a time t and R% p.a is the rate of depreciation, then the value of machine after n SIMPLE INTEREST: The extra money paid by the years is given by borrower is called interest. n R = V x 1 − AMOUNT: Amount – Principal – Interest.
100
Formulae for Simple Interest: If P, R and T are Amount after T years is given by principal, rate and time then S.I. is given by T R P × R × T A = P 1 − S.I. =
100 100 × S . I .
100
NOTE: (a) For 2 years the difference between the compound P= R × T interest and the simple interest is equal to simple interest for 1 year on 1st year’s interest. 100 × S . I R= (b) The amount of the previous year is the principal for P × T the successive year. 100 xS . I (c) The difference between the amount due at the end T= P × R of two consecutive years = simple interest for one year on the lesser amount. COMPOUND INTEREST: CI = Amount – P (d) When the interest is payable half yearly, divide the If P = principal, R = rate % p.a. and T = time (years) rate by 2 and multiply the time by 2. then (e) When the interest is payable quarterly or once in (a) Amount after T years (compounded annually) 1/4th year divide the rate by 4 and multiply the time by r 4. R (f) There is no difference between simple interest and = P 1 + 100 compound interest on the principal for first year. C.I , is more that S.I. after one year. (b) Amount after T year (compounded half yearly)
R 2 = P 1 + 100
2T
REMAINDER THEOREM: Let f(x) be a polynomial of degree greater than or equal to one and ‘a’ be any real number. If f(x) is divisible by (x-a) , then the remainder is equal to f(a). In this case rate becomes half and time becomes Example: Determine the remainder when the polynomial f(x) = x3 - 3x2 + 2x + 1 is divided by (x-1). double. (c) If the rate be p% , q%, and r% during first year, second year and third year, then amount after 3 Solution: By remainder theorem, the required remainder is equal to f(1). years. Now, f(x) = x3 – 3x2 + 2x + 1 p q r => f(1) = 1 – 3 + 2 + 1 = 1. =P 1 + 1 + 1 + Hence , the required remainder is equal to 1. 100 100 100 POPULATION GROWTH FORMULAE: a) If P is the population and R % is the growth rate FACTOR THEOREM: Let f(x) be a polynomial of degree greater than or equal to one and a be a real then in n years population will be number such that f(a) = 0, then (x-a) is a factor of f(x), n R Conversely, if (x+a) is a factor of f(x), then f(-a) =0. = P x 1 +
100
b) If p% is the growth rate during first year and q% REMARK: during second year then the population after 2 years is 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) = o given by. iii) ax + b is a factor of a polynomial if f(-b/a) = o p q iv) (x-a) (x-b) is a factor of a polynomial f(x) if f(a) = 0 = p 1 + 1 + 100 100 and f(b) = 0. TIME, SPEED & DISTANCE This formula can be used for more than two years. c) If R % per annum is the decrease in population SPEED: Distance covered per unit time is called speed. then after n years. =px
1 − R 100
n
Speed =
Distance Time
Distance = Speed × Time (or) Time = Distance/Speed
DEPRECIATION: It is a well known fact that the value speed of a body is changed in the ratio a : b then the ratio of a machine or car or any other article decreases with Ifofthe the time taken changes in the ratio b : a time due to wear and tear. The decrease in value is called depreciation value. RADIAN ACADEMY
ANNA NAGAR & NSK NAGAR-ARUMBAKKAM
[email protected] Ph: 98404-00825, 30025003
NOTE: Distance is normally measured in kilometres, metres or If a body covers part of the journey at speed x and the remaining miles; Time in hours or seconds and Speed in km/hr (kmph), part of the journey at speed y and the distances of the two parts miles/hr (mph) or metres/second (m/s). of the journey are in the ratio m : n, then 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 at different speeds is defined as follows. Average Speed =
Total distance travelled Total time taken
The average speed for the entire journey is
( m + n ) xy xn+ ym
TRAINS Time taken by a train of length “d” metres to pass a pole or a standing man or a signal post is equal to the time taken by the train to cover “d” metres. 1.
NOTE: The average speed of a moving body is NOT EQUAL to 2. Time taken by a train of length “d1” metres to pass a stationary object of length “d2” metres is the time taken by the the average of the speeds. train to cover (d1 + d2) metres. A body travels from point A to another point B with a speed of x kmph and back to point A (from point B) with a speed of y kmph. 3. If two trains or two bodies are moving in the same direction at u m/s and v m/s, where u > v, then their relatives speed = (u – v) m/s. x kmph A
y kmph
B
Let AB = d, the time taken by the body to travel from A to B be t1 and that from B to A be t2. Then t1 = d/x and t2 = d/y. The total distance travelled is 2d.
2d
Average Speed =
=
t1 + t 2
=
2d d d + x y
2d = 1 1 d + x y
2 2 xy = 1 1 x+y + x y
If two trains are moving in opposite directions at u m/s and v m/s then the relative speed is = (u + v) m/s. 4.
If two trains of length “a” metres and “b” metres are moving in opposite directions at u m/s and v m/s, then time 5.
taken by the trains to cross each other is
a +b u+v
sec.
If two trains of length “a” metres & “b” metres are moving in the same direction at u m/s and v m/s, then the time taken by 6.
the faster train to cross the slower train is
a+b sec. u − v
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 7.
2xy kmph x+y NOTE: This formula does not depend on the distance between A and B. This formula can be used only if the distances travelled in each case are equal. Average Speed =
(A’s speed): (B’s speed) = ( √b : √a )
BOATS AND STREAMS
1. In river, the direction along the stream is called If the entire journey AD is travelled with the different speeds, A downstream and, the direction against the stream is called to B with a uniform speed of x kmph, B to C with a uniform upstream. speed of y kmph and C to D with a uniform speed of z kmph such that AB = BC = CD. 2. If the speed of a boat in still water is u km/hr and the speed of the stream is v km/hr , then: x kmph y kmph z kmph Speed of boat in downstream = (u + v) km/hr. Speed of the boat in upstream = ( u – v) km/hr. A BC D The average speed from A to D is given by the formula 3. If the speed downstream is “x” km/hr and the speed upstream is “y” km/hr, then: 3 3 Speed in still water = (x + y)/2 km/hr Average Speed = = 1 1 1 yz + zx + xy Rate of stream = (x – y)/2 km/hr.
x
+ + y
z
xyz
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) xy + yz + zx starting at the same point at the same time. Running in the Running in the In general the ‘n’ equal distances are travelled with the speeds SAME direction OPPOSITE dir. of x1 kmph, x2 kmph, ...., xn kmph, then the average speed is Time taken to meet given by for the FIRST L L TIME some where n Average Speed = kmph a −b a+b on the track. 1 1 1 Time taken to meet + + .... + for the first time at x1 x2 xn L L LCM L , L the same LCM , STARTING a b a b NOTE: The above is the harmonic mean of n numbers. POINT. Average Speed
=
RADIAN ACADEMY
3 xyz
ANNA NAGAR & NSK NAGAR-ARUMBAKKAM
[email protected] Ph: 98404-00825, 30025003
QUADRATIC EQUATIONS
THREE PERSONS
A general quadratic equation is expressed as 2 Let the three people A, B and C with respective speeds of a , b ax + bx + c = 0, where a≠0; a, b and c are constants. and c (a > b > c) be running around a circular track (of length L) Roots of the quadratic equation: starting at the same point at the same time in the same A quadratic equation has two roots α and β given by direction. CLOCKS
How many times the hands of a clock Coincide or making an 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 1 hour. Between 11 and 12, the coincidence is at 12 O' clock. α = Between 12 and 1, there is no further coincidence, because it coincides at 12. In 12 hrs, the two hands of the clock coincide 11 times only. In a day, the two hands coincide 22 times.
L , L a − b b − c
Time taken to meet for the first time at the STARTING POINT.
LCM
L , L , L a b c
− b + b 2 − 4ac 2a
and β =
− b − b 2 − 4ac 2a
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 -b c How many times the hands of a clock are at right angles in then α + β = and α β = a day? a a
Every one hour, the two hands are at right angles twice, except between 3 & 4 and 9 & 10. Considering 2 to 3 they are at right angles for first time between 2:25 to 2:30. For the second time they are at right angles at 3. Between 3 and 4, they are at right angles only once. (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.
LCM
How many times the hands of a clock are at 180° or lie in a The quantity D = b2 – 4ac is known as the discriminant. straight line but facing opposite direction in a day?
In 12 hrs, the two hands of the clock at straight angle once in every 1 hr. Between 5 and 6, the angle between them is 180° at 6 O' clock only. 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°.
Time taken to meet for the FIRST TIME on the track.
How many times the hands of a clock lie on the same straight line in a day?
Hence x2 – ( α + β) x + α β = 0 (or) (x – α) (x – β) = 0 HIGHER DEGREE EQUATION: P(x) = a0xn + a1x n-1 + …. + a n-1 x + an = 0 Where the coefficients a0, a1, …. an and a0 ≠ 0 is called an equation of nth degree, which has exactly ‘n’ roots α1, α2, … αn. Σαi = α1 + α2 + ….αn = a1 a0
Σαiα j = α1α2 + … + α n-1αn = - a2
a0 The two hands lie on the same straight line, when they coincide and when they are at straight angle. ∏ αi = α1 × α2 × …× αn = (-1)n an a0 In 12 hrs. the hands of the clock lie on the same straight line 22 times. In a day, they lie on the same straight line 44 times. FUNCTION The following table sum up the above discussions: Angle b/w the hands
Number of times 12 hrs 24hrs (Day)
0° (Coincidence) 180° (Straight Angle) 0° or 180° (Straight line) 90° (Right angle)
11 11 22
22 22 44
22
44
A function from X to Y is defined as a relation X x Y such that no two different ordered pairs of the relation have the same first component and every element of X has 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 of a, when (a, b) belongs to the function.
MINUTE HAND In 1 hour, the minute hand makes a complete rotation of 360°. RANGE: Range of a function is the set of value of b, In 1 minute it rotates about 360/60 = 6°.
when (a, b) belongs to the function.
HOUR HAND In 1 hour, the hour hand makes a complete rotation of 30°. In 1 CO-DOMAIN: If (a, b) belong to a function f: A -> B minute it rotates about 30/60 = ½ °. then b is called co-domain of the function. Range is a RADIAN ACADEMY
ANNA NAGAR & NSK NAGAR-ARUMBAKKAM
[email protected] Ph: 98404-00825, 30025003
subset of co-domain, sometimes the range and co- MANTISSA: Mantissa of a number is found with the domain have the same elements. help of logarithmic tables. 1. The mantissa is the same for the logarithms of all FUNCTION DOMAIN numbers having the same significant digits. Sin-1 x [-1.1] 2. The logarithm of one digit number, say 2, is to be Cos-1 x [-1.1] see in the table, opposite to 20. 3. The mantissa is always taken positive. ]−∞, ∞[ Tan-1 x ]−∞, ∞[ Cot-1x -1 ANTILOGARITHM: If log a = m, then a = antilog of m, Sec x (- ∞ , -1] U [ 1, ∞ ) -1 i.e., The number corresponding to a given logarithm is Cosec x (- ∞ , -1] U [ 1, ∞ ) called antilogarithm. 1. The function is called an onto function if every element of set Y has at least one pre-image in set 1. If the characteristic of the logarithm is positive, then: “put the decimal point after ( n+1)th digit, X. where n is equal to characteristic. . X Y 2. If the characteristic of the logarithm is negative, 1 a the:”put the decimal point so that the first significant 2 b digit is at ‘n’th place, where n = characteristic’. 3 c 4 Properties of Logarithms. 1. Log 1 = 0 , irrespective of the base 2. The function is called one-one if distinct elements 2. Log a a = 1, logarithm of any number to its own have distinct images. base is always 1. 3. Logarithm of product X Y Log a (mn) = Log a m + Log a n 1 a 4. Logarithm of ratio 2 b Log a (m/n) = Log a m - Log a n 3 c 5. Logarithm of a Power Log a m n = nLog a m 6. Base changing formula 3. The function is called many-to-one, if one or more Log a m = Log a m x Log a b elements of set X there correspond only one 7. Log a q(n p) = Log n p / Log a q irrespective of element of set Y. the base. X Y 8. Particular case a log a a n = n b 1 9. a log a n = n In particular e In n = n c NOTE: 1. 2. 3. 4.
One-one is also written as 1 – 1. An onto function is also called ‘surjection’ An into function is also called ‘Injection’ Both Injective & Surjective in called Bijective
LOGARITHMS COMMON LOGARITHMS: Logarithms calculated to the 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 than one. “Characteristic is one less than the number of digits to the 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 zeros between the decimal point and the first significant digit of the number and is negative”. Ex. characterstic of 0.34 is 1 and 0.00075 is 4 . RADIAN ACADEMY
SOME IMPORTANT POINTS: Those logarithms whose base is 10 are known as Common (decimal) logarithms while which has base e (e = 2.71828….) are known as natural or Napierian logarithms. Natural logarithm is changed to decimal logarithm as PERMUTATIONS AND COMBINATIONS PERMUTATIONS: It is defined as the ways of arranging object. Here the order i.e. position is important. The number of permutations of objects taken r at a time is n! n Pr = n (n-1)(n-2)(n-3)…(n-r+1) = (n − r )!
n
Pn = n!;
n
P0 = 1;
n
P1 = n
NOTE: n! = n×(n-1)×(n-2) ……. 3×2×1 =1
RESULTS: i) The total number of permutation of n items taken all together, when ‘p’ items are of one type, ‘q’ are of second type and ‘r’ of then third kind and the remaining are of different type is n! p! q! r !
ANNA NAGAR & NSK NAGAR-ARUMBAKKAM
[email protected] Ph: 98404-00825, 30025003
ii) The number of circular permutations of n different 4. Characteristic : A quality possessed by an individual person, object or item of a population, e.g. heights of objects is n-1!. individuals, nationality of a group of passengers on a flight etc. COMBINATIONS 5. Variable and attribute: A measurable characteristic When r objects taken out of n objects then is called a variable or a variate. A non-measurable combination of n objects taken r at a time, we write characteristic is called an attribute. It may be noted n! n Cr = C (n, r) = here that by measurable characteristics we mean (n − r )! r ! those characteristics which are expressible in terms of Note: nCr = nC n-r some numerical units, e.g. age, height, income etc.
STATISTICS STATISTICS is concerned with scientific methods for collecting, organizing, summarizing, presenting and analyzing data, as well as drawing valid conclusions and making reasonable decisions on the basis of such analysis. LIMITATIONS OF STATISTICS 1. 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 group characteristics through statistical analysis. 3. Statistical decisions are true only on an average and also the average is to be taken for a large number of observations. For a few cases in succession the decision may not be true. 4. Statistical decisions are to be made carefully by experts. Untrained persons using statistical tools, may lead to false conclusions. CHARACTERISTICS OF STATISTICAL ANALYSIS. 1.
2. 3. 4.
5. 6. 7.
CONTINUOUS AND DISCRETE VARIABLE. A variable which can theoretically assume any value between two given values is called a Continuous variable otherwise it is a discrete variable; heights, weights , agricultural holding are some examples of continuous variables whereas number of workers in a factory, number of defectives produced, readings on a Taxi meter are examples of discrete variables. Data which can be described by a discrete or continuous variable are called discrete data or continuous data respectively. The first and foremost task of a Statistician is to collect and assemble his data. When he himself prepares the data, it is called a primary data but when he borrows them from other sources (Government, semi-Government or non-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 the observations are concentrated. A central value which enables us to comprehend in a single effort the significance of the whole is know as Statistical Average or simply average. The three common measures of Central Tendency are i) Mean ii) Median iii) Mode
In statistics all information are to be expressed in quantitative terms. Even in the study of quality like intelligence of a group of students we require scores or marks secured in a test. Statistics deals with a collection of facts not an individual happening. Statistical data are collected with a definite object in THE MOST COMMON AND USEFUL MEASURE IS THE mind. i.e. there must be a definite field of enquiry. MEAN. In every field of enquiry there are large number of factors, each of which contributes to the final data ARITHMETIC MEAN collected. So statistics may be affected by a multiplicity of causes. Advantages: Statistics is not an exact science. 1. This is the widely used measure of Central Tendency. Statistics should be so related that cause and effect 2. It is simple to understand and easy to Calculate. relationship can be established. 3. It is rigidly defined A statistical enquiry passes through four stages, 4. Calculations depend on all the values Collection of data, Classification & tabulation of data, 5. It is suitable for algebraic treatment. Analysis of data and Interpretation of data. 6. It is least affected by sampling fluctuations.
COMMONLY USED TERMS: 1. Data: A collection of observations expressed in numerical figures, obtained by measuring or counting. 2. Population: A population or a universe consists of the totality of the set of objects, with which we are concerned, e.g. all workers working in a plant, all times produced by a machine in a particular period etc. 3. A sample: A sample is a sub-set of the population i.e. it is a selected number of individuals each of which is a member of the population. RADIAN ACADEMY
Disadvantages: i) Cannot be determined by inspection ii) It is very much affected by the presence of a few extremely large or small values of the variable iii) 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 some assumptions are made.
ANNA NAGAR & NSK NAGAR-ARUMBAKKAM
[email protected] Ph: 98404-00825, 30025003
GEOMETRIC MEAN QUARTILES OF UNGROUPED DATA: Write the n Advantages: items of the data in ascending order. Then Lower i) G.M. is not widely used. It is particularly suitable Quartile Q1 = (n +1)/4th item for averaging rates of changes. n +1 ii) It is rigidly defined and depends on all values of Middle Quartile Q2 (Median) = 2 th item the series. Upper Quartile Q3 = 3 (n+1) /4th item. iii) It is suitable for algebraic treatment iv) G.M. is not affected by the presence of very large DISPERSION: The variation or scattering or deviation of or small values of the variable. the different values of a variable from their average is Disadvantages: known as Dispersion. i) Unlike A.M, G.M. is neither simple to understand nor simple to calculate. ABSOLUTE MEASURES: The three absolute measures ii) If any value of the series is Zero. G.M. cannot be are calculated. i) Range iii) Calculation of G.M. is impossible unless all the ii) Mean deviation values are positive. iii) Standard deviation. HARMONIC MEAN: Advantages: i) It is useful in averaging rates ratios and prices. ii) It is suitable for algebraic treatments iii) Its calculation is based on all values of the series. Disadvantages: i) It is very limited use and not easy to understand ii) H.M. cannot be calculated if any value is Zero.
Range: Range is the simplest measure of dispersion. It is the difference between the largest and the smallest values of a variable. This is not the widely used measure as it lacks in accuracy. Coefficient of Mean Dispersion: The coefficient of mean dispersion is defined by the formula.
RELATION BETWEEN A.M., G.M. and H.M. Coefficient of Mean Dispersion For any set of positive values of a variable, we can write A.M. ≥ G.M. ≥ H.M. equality occurring only when the MeanDeviationfrommean values are equal. = Mean For a pair of observations only, AM x HM = (GM)2 MeanDeviationfromMedian Or = MEDIAN: Median Advantages: i) It is easily understood. STANDARD DEVIATION: This is most important absolute ii) Not affected by extreme values. iii) Can be determined by inspection in case of a measure of dispersion. Standard deviation (S.D.) for a set of values of a variable is defined as the positive square simple frequency distribution. iv) It can be calculated from a grouped frequency root of the arithmetic mean o the squares of all the distribution with open-end classes, provided by deviations of the values from their arithmetic mean. In short, it may be defined as the square root of the Mean closed classes are of equal width. squares of deviation from mean. Disadvantages: i) It is not well-defined and also it is not possible to S.D is usually denoted by a greek small letter σ (pronounced Sigma) find a well defined mode. ii) It is not suitable for algebraic treatment If x1, x2 . . . . xn be a series of values of a variable and x iii) It is not based on all values of the variable their A.M. : then S.D. is defined by iv) It is affected by sampling fluctuations. MEDIAN: The value of the item which divides the data into two equal parts is called median. Median of ungrouped data: If the n items in the data are arranged in ascending or descending order and n +1 if n is ODD then , th item; 2 if n is EVEN, then the average of items is called median.
n
2
th,
n
2
+1 th
σ =
(
2
)
x1 − x
2
+ ( x2 − x) + ..... + ( xn − x)
2
n
For a frequency distribution This square of S.D . is known as VARIANCE σ =
∑ f ( xi − x) N
2
, where N = Σf
QUARTILE DEVIATION: The items which divide the data i.e. variance = σ 2 = (S.D.) 2 into four parts are called quartiles. They are denoted by Q1 , Q2, Q3 i) Coefficient of range = Max.value – min. value max.value + min. value Quartile deviation = Q3 – Q1 ii) Coefficient of Q.D. = Q3 – Q1 2 Q3 + Q1 RADIAN ACADEMY
ANNA NAGAR & NSK NAGAR-ARUMBAKKAM
[email protected] Ph: 98404-00825, 30025003
RELATIVE MEASURES OF DISPERSION x =
n1 x1 + n2 x 2 n1 + n2 n1σ 12
σ2 =
+ n2σ 22 + n1d12 + n2 d 22 n1 + n2
where d1 = x - x , d2 = x 2 - x
Binomial distribution is a discrete distribution. A binomial distribution can be used when a) The number of trials is finite b) The trials are independent of each other c) 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 of Bernoulli trials is called a binomial experiment.
The relative measures of dispersion are pure numbers POISSON DISTRIBUTION: and are mainly employed in comparing the dispersions of A random variable X is said to follow Poisson distribution two or more distributions. There are two relative if its probability mass function is given by measures:
e − λ x when x = 0, 1, 2, 3 , . . . . P (X=x) = x ! 0 otherwise λ
i)
Coefficient of Variation S . D. (as percentage) = x 100 Mean
ii) Coefficient of Mean Deviation (as percentage) = MeanDeviation x 100 MeanorMedian
λ is known as the parameter of the Poisson distribution.
Mean = λ Variance = λ Standard deviation = √λ NORMAL DISTRIBUTION: MEASURES OF SKEWNESS: A continuous random variable X is said to follow normal The degree of skew ness is measured by its coefficient. distribution with mean µ and standard deviation σ if its The very common measures are: probability density function is given by 1. Pearson’s first measure: Mean − Mode Skewness = S tan darddeviation 2. Pearson’s second measures: 3 ( Mean − Mode) Skewness = S tan darddeviation BINOMIAL DISTRIBUTION A random variable X is said to follow binomial distribution if 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 binomial distribution . Mean = np Variance = npq Standard deviation = √(npq)
RADIAN ACADEMY
f (x) =
σ
1 e 2π
1( x − µ )2 2σ 2
−∞ < x < ∞ - −∞ < µ < ∞ σ > 0
µ an σ are called the parameters of the normal distribution Mean = 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 maximum ordinate) occurs at x = µ Maximum ordinate =
σ
1 2π
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 3 3 1 1 1, +i , - -I 2 2 2 2 These are denoted by 1, ω , ω 2. We have a) 1 + ω + ω 2 = 0 b) ω 3=1
ANNA NAGAR & NSK NAGAR-ARUMBAKKAM
[email protected] Ph: 98404-00825, 30025003
m
1. (1 +x)m =1+
m(m − 1)
x+
1! m(m − 1)( m − 2)
x2 +
2!
The constant ratio
x3 + . . . .
3!
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 + . . .. . x x 2 x 3 x 4 x 6. e = 1 + + + + + …… 1! 2! 3! 4! 1 1 1 1 e =1+ + + + + ……
1!
7. e –x = 1 -
2!
x
+
1!
3!
x
4!
2
x
+
2!
3
x 4
+
3!
4!
x + x + x . + ..... 1! 3! 5! 1 1 1 1 e=2 1 + + + ...... e 1! 3! 5!
8. ex – e-x = 2
e
-1
or
1
e
=1+
1
1!
1
+
2!
e+
+
3!
……
-x
1
e
=2
1 + 1 + 1 + 1 ...... 2! 4! 6!
10. log (1-x) = -x -
11.log (1-x) = -x -
x 2
+
2
x 2
x 3
x3
-
x 4
-
3 -
- . . . .. .
4
x 4
- . .. .. ..
2 3 4 x3 x5 1+ x 12.log = 2 x + + + ...... 1− x 3 5 13. sin x = x 14. cos x = 1 15. tan x = x +
x3
3! x2 2! x 3
3
+
+ +
x5
5! x4 4! 2
+
15
x6 6!
x 7
7!
is called the eccentricity,
denoted by e. If e = 1 , the conic is called a parabola If e < 1 the conic is called an ellipse If e > 1 the conic is called a hyperbola The general equation of a conic will be an equation of second degree in x and y, in the form ax2+ 2hxy + by2+ 2gx + 2fy + e = 0 Conversely, the general equation of second degree in x and y, i.e., ax2 + 2hxy + by2+ 2gx + 2fy + e = 0 will represent a conic if abc + 2fgh – af 2- bg2- ch2 ≠ 0 and i) h2 - ab for a parabola ii) h2 < ab for an ellipse iii) h2 > ab for a hyperbola iv) h2 > ab and a+b =0 for a rectangular hyperbola. COORDINATE GEOMETRY Distance Formulae: The distance between the points A(x1,y1) and B(x2,y2) is given by 2
x2 x4 x6 + + ...... 8. e + e = 2 1 + 2! 4! 6! x
PM
AB = ( x2 − x1)2 + ( y2 − y1) The distance of the point P(x,y) from the origin O is given by
1
+
SP
+ .. . . . .
+ .....
x5 + . . . . .
OP = x 2 + y 2 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 by x = m1x2 + m2x1 , y= m1y2 + m2y1 m1 + m2 m1 + m2 (b) If (x, y) divides the line segment PQ in the ratio k :1 (internally), then x = 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 a triangle meet. Centroid divides each median in the ratio 2:1 . The coordinates (x,y ) of the centroid of the triangle whose vertices are (x1,y1) (x2, y2) (x3+ y3)are given by X = 1/3 (x1 + x2 +x3) = 1/3 (y1+y2+y3)
CONIC : The locus of a point P which moves such that its distance from a fixed point S bears a constant ratio to its INCENTRE: It is the point where the internal bisectors of a triangle intersect. The coordinates k (x, y) of the distance from a fixed l is called a conic. incentre are given by: The fixed point S is called the focus. x = ax1 + bx2 + mcx3 y = ay1 + by2 + cy3 a+b+c a+b+c The fixed line l is called the directrix. RADIAN ACADEMY
ANNA NAGAR & NSK NAGAR-ARUMBAKKAM
[email protected] Ph: 98404-00825, 30025003
ORTHOCENTRE: The three altitudes (the lines through the vertices and perpendicular to the opposite sides) of a triangle interest in a common point called orthocenter of a triangle. CIRCUM-CENTRE: This is a point which is equidistant from three vertices of a triangle. Thus it is the centre of the circle that passes through the vertices of triangle. It is also the point of intersection of the right bisectors of the sides.
Angle between he two straight lines: Y = m1 x + c1, y = m2 x + c2 m1 – m2 1 + m1 m2
tan θ =
a) The above two straight lines are perpendicular if, θ = 90o tan 90o = Not defined , i.e. if 1 + m1m2 = o or m1 x m2 = 1 b) The above two straight lines are parallel if θ = 0 => tan θ = 0, i.e. m1 = m2
AREA OF A TRIANGLE: The area of a triangle whose vertices are A(x1, y1), B(x2, y2), C (x3, y3) is given by ANGLE BETWEEN THE TWO STRAIGHT LINES: = 1/2 (x1(y2-y3)+x2(y3-y1) + x3(y2 – y2)} If there pints A, B, C are collinear (lie on the same straight line). Then Area = 0. LOCUS OF A POINT: It is the path traced by a pilot moving under certain conditions. Thus the locus of a point which moves such that it is always at a constant distance from a given point in a plane, is a circle.
a1 x + b1 y + c1 = 0 a2 x + b2 y + c2 = 0 tan θ =
a1b2 – a2b1 a1a2 + b1b2
a) The above lines are perpendicular if A1b2 – a2b1 = 0
EQUATION OF A LOCUS: The equation of the locus of a moving point P(x,y) is an algebraic relation between x and i.e. a1 = b1 y satisfying the given conditions, under which P moves. a2 b2 Thus, if P(x,y) moves along the circle of radius r having kits centre at the origin, then equation of the locus is The equations of two parallel lines differ in constant term X2 + y2 = r 2 only. STRAIGHT LINE; Equation of a straight line parallel to the straight line EQUATIONS O A STRAIGHT LINE: ax + by + c= 0 , is ax + by + k = 0 (a ) Equations of coordinate axes: Sine at every point on the x-axis, y=0, hence the equation of the axis of x is y Equation of a straight lien perpendicular to the straight line = 0 . Similarly, the equation of the y=axis is x=0. ax + by + c = 0, is bx – ay + k = 0 Equations of straight lines in various forms: (a) Slope Intercept form y= mx + C (b) slope-point form y – y1 = m (x -x1) (c) Intercept form x+ y =1 a b (d) Two point form y – y1 = y2 – y1 (x – x1) x2 – x1 (e) Parametric form,: x-x1 = y-y1 = r cos θ sin θ any point on this line (x1 = r cos θ , y1 = r sin θ) (f) Normal form X cos θ + y sin θ = p (g) General equation: Ax + By + C = 0
RADIAN ACADEMY
Equation of a straight line through the point of intersection of the straight lines a1 x + b1 y + c1 = 0 and a2 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 from P (x1 , y1) to the lien ax + by + c = 0 P=
ax1 + by1 + c √a2 + b2
Perpendicular distance p between and parallel straight lines 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 √ a21 + b21
=
a2x + b2y + c2 √a22 + b22
ANNA NAGAR & NSK NAGAR-ARUMBAKKAM
[email protected] Ph: 98404-00825, 30025003
Concurrency of the three straight lines, The straight lines: Sum or Differnce nto product: a1x + b1y + c1 = 0 b) sin A + sin B = 2sin A+B/2cos A-B/2 a2x + b2y + c2 = 0 c) sinA – SinB = 2cos A+B/2sin A-B/2 a3x + b3y + c3 = 0 d) cosA + cos B = 2cosA =B /2cos A-B / 2 e) cos A – cos B = -2A+B/2sin A-B/2 are concurrent if Product into sum or difference: a1 b1 c1 a) 2sinA cosB = sin (A +B) + sin (A-B) a2 b2 c2 = 0 b) 2cosA cosB = cos(A+B) + cos(A-B) a3 b3 c3 c) 2sinA sinB = cos (A-B) – cos (A+B) TRIGONOMETRY 1. 2. 3. 4. 5. 6.
sin ø = p/h = perpendicular / hypotenuse cos ø = b / h = base / hypotenuse tan ø = p/b = perpendicular/base cosec ø = h/p = hypotenuse / perpendicular sec ø = h/b = hypotenuse / base cot ø = b/ = base / perpendicular
TRIGONEMETRIC RELATIONS: 1. sin ø = 1/cosec ø 2. cos ø = 1/sec ø 3. tan ø = 1/ cot ø 4. tan ø = sin ø / cos ø 5. cot ø = cos ø / sin ø QUADRANTS The two axes Xn OX and Y n OY divides the plane into Four Quadrants. i.In first quadrant, all trigonometric ratios are positive.
Relations between he sides and angles of a triangle: In ∆ ABC, a) Sinc formula a b c = = = 2R sin A sin B sin C b) Consine formulae b2 + c 2 − a2 Cos A = 2bc Cos B =
Cos C =
c2 + a2
− b2
2ca
a2
+ b2 − c2 2ab
PROJECTION FORMULAE: a) a = b cosC + c cosB b) b = c cosA + a cosC ii. In second quadrant, only sin ø and cosec ø are positive. c) c = a cosB + b cosA iii. In third quadrant, only tan ø and cot ø are positive. iv. In fourth quadrant, only cos ø and sec ø are positive. General values of Trigonometric Functions: a) If sin ø = Sinα IMPORTANT RELATIONS Then , θ = n π+ (-1)n α, n €1 I. sin2ø + cos2ø = 1 II. 1 + tan2ø = sec2ø b) If cos θ = cos α III. 1 + cos2ø = cosec2ø Then, θ = n π ± α, n € 1 SUM AND DIFFERENCE FORMULAE: 1) sin (A±B) = sin A cos B ± cos A sin B 2) cos (A±B) = cos A cos B – or + sin A sin B 3) tan (A±B) = tan A ± tan B / 1 ± tan A tan B 4) sin (A±B) sin (A – B) 5) sin (A±B) sin (A ± B) = sin2 A – Sin2B = Cos2B – Cos2A 6) cos ( A +B) Cos (A – B) = cos2A – sin2 B = cos2b – sin2A
i) velocity at time ‘t’ is v =
ds dt
ii) acceleration at time ‘t’ is a =
dv dt
=
d 2s 2
dt
DOUBLE – ANGLE FORMULAE: a) sin2ø = 2sin ø cos ø = 2tanø / 1+ tan2ø b) 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 ø b) cos3ø = 4cos3 ø - 3 cos ø c) tan3ø = 3tanø - tan3ø / 1-3tan2ø RADIAN ACADEMY
ANNA NAGAR & NSK NAGAR-ARUMBAKKAM
[email protected] Ph: 98404-00825, 30025003
LOGICAL REASONING
for
TNPSC EXAMS
Logic is the science and art of reasoning correctly, the science of the necessary laws of thought; Reasoning is the mind’s power of drawing conclusions and deducting inference from premises. And so, Logical Reasoning implies the process of drawing logical conclusions from given facts in conformity to what is fairly to be expected or called for. It must be noted that logical conclusions means what is derived by reasoning or logic and not the truth or fact. PROPOSITION: The logical proposition is an expression or a statement which affirms or denies something, so that it can be characterised as true or false, valid or invalid. Like any other grammatical sentence, a proposition has a subject, a predicate and a copula connecting the two.
RADIAN IAS ACADEMY For the validity of drawing inference in an argument the propositions are also classified on the basis of quality; as Affirmative (Positive) or Negative, and Quantity; as Universal or Particular a) UNIVERSAL AFFIRMATIVE – ‘A’ Proposition
Only 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:
Here, ‘Philosophers’ the subject, ‘intelligent’ is predicate and ‘are’ is copula.
Both subjective and predicative terms are distributed Example I. No man is perfect II. No fools are wise In the above statements, the distributed term is ‘No’, ‘No one’. When no man is perfect, then one who is perfect cannot be man. Similarly, when no fools are wise, then one who is wise cannot be a fool.
The propositions can be classified into Four categories.
c) PARTICULAR AFFIRMATIVE – ‘I’ Proposition:
Example: Philosophers are intelligent.
(I) CATEGORICAL PROPOSITION: Emphasises what is and 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 of the truth that cats can never be dogs. So, also in second sentence, no hens are ducks leaves no argument that some hens may be ducks. (II) DISJUNCTIVE PROPOSITION: Leave every scope of confusion as they have either -------- or --------- in then Example: Either she is shy or she is cunning. These type of propositions give two alternatives.
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 not particular, i.e. ‘some’. When some children are naughty, then some of those who are naughty may be children. Similarly, when some politicians are dishonest, then some dishonest men may be politicians. There is no defined certainty. d) PARTICULAR NEGATIVE: ‘O’ Propositions:
Here the predicative term is distributed. ‘Some used with a negative sign is a particular negative proposition. 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 particular negative. However, words such as ‘some’ ‘mostly’ ‘all but one’ etc. are particular Propositions.
I. Antecedent i.e. ‘she is shy’ and II. Consequent i.e. ‘or she is cunning’ The inferences drawn on such statements are probably true or probably false. The right inference often depends PREMISE is a proposition stated or assumed for afteron one’s own ability to sense and analyse the validity of reasoning especially one of the two propositions in a the logic. syllogism, from which the conclusion is drawn. Of the two statements, the first is major premise and the second is (III) HYPOTHETICAL PROPOSITION: Correspond to the minor premise. conditions, and the conditional part starts with words such Example: All dogs are hens. (major premise) as ‘if’. All pups are dogs. (minor premise) Example: If I am late, I will miss the train. Here also, Inference: All pups are hens. proposition has two parts. 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 a proposition. The terms in the major premise are called (IV) RELATIONAL PROPOSITION: Denote the relation major terms and that in the minor premise are called between the subject and the predicate. The relation can minor terms. The middle term occurs in both the premise. be (I) symmetrical (II) non-symmetrical or (III) In the above example, dogs, hens and pups are three terms used. Of these ‘hens’ is the major term, ‘pups; is the asymmetrical. minor term and ‘dogs’ is the connecting or the middle Example: I. She is as tall as Pinki term. II. Jai is wiser than Roy INFERENCE is the act of drawing a logical conclusion III. Tim is brother of Ria. from given premise. This logical deduction follows necessarily from the reasoning of given premises and not of the truth. I. II.
antecedent – if I am late, and consequeny –I will miss the train