quantitative recipe !!! by veera karthik

Veera Karthik G

Quantitative Recipe !!!

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Quantitative Recipe !!! Veera Karthik G

Number Theory

Contents

1. Properties of real numbers2. Number systems3. Tests of divisibility4. HCF ( Highest common

factor )5. LCM ( Least common

multiple )6. Key points on L.C.M &

H.C.F7. Simplification8. Complete Remainder9. Fractions and ordering

fractions10. Surds and Indices

11. Finding the Square root

Properties of real numbers

In Hindu Arabic system we use ten symbols 0,1,2,3,4,5,6,7,8,9 called digits to represent any number. This is the decimal system where we use the digits 0 to 9. Here o is called insignificant digit where as 1,2,3,4,5,6,7,8,9 are significant digits

Number systems

Natural numbers : The numbers 1,2,3,4,5,.......125,126, ........... which we use in counting are known as natural numbers The set of all natural numbers can be represented by N = {1,2,3,4,5 ........... }

Whole Numbers :- If we include 0 among the natural numbers then the numbers 0,1,2,3,4,5 ........... are called whole numbers The set of whole numbers can be represented by W = {0,1,2,3,4,5, ........... .}

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Clearly, every natural number is a whole number but 0 is a whole number which is not a natural number

Integers All counting numbers and their negatives including zero are known as integers. The set of integers can be represented by z or I = {.........-4,-3,-2,-1,0,1,2,3,4..} Clearly every natural number is an integer but not every integer is natural number

Positive Integers The set I + = {1,2,3,4........} is the set of all positive integers. Clearly positive integers and natural numbers are synonyms

Negative Integers The set I - = {-1,-2,-3,.......} is the set of all negative integers 0 is neither positive nor negative.

Non negative integers : The set {0,1,2,3........} is the set of all non negative integers

Rational Numbers The numbers of the form P/q, where p and q are integers and q≠0, are

known as rational numbers. e.g: etc ,e.g is not define

The set of all rational numbers is denoted by Q. i.e Q = { x :x = ; where q ≠ 0} Irrational Numbers Those numbers which when expressed in decimal form are neither terminating nor repeating decimals are known as irrational numbers

e.g = The absolute value of & etc

The absolute value of P is irrational. is rational. Real Numbers The rational and irrational numbers together are called real numbers. e.g: 13/21, 2/5, -3/7, / 3, +4 / 2 etc are real numbers The set of real numbers is denoted by R Even Numbers All those natural numbers which are exactly divisible by 2 are called even numbers e.g: 2,6,8,10 etc are even numbers.

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Odd Numbers All those numbers which are not exactly divisible by 2 are called odd numbers. e.g: 1,3,5,7. Etc are odd numbers Prime Numbers The natural numbers other than 1, is a prime number if it is divisible by 1 and it self only e.g: Each of the numbers 2,3,5,7,11 ,13,17,etc are prime number. 1 Is not a prime number 2 Is the least and only even prime number 3 Is the least odd prime number Prime numbers upto 100 are 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,57,61,67,71,73,79,83,89,97. i.e 25 prime numbers

Composite Number Natural numbers greater than 1 which are not prime, are known as composite numbers The number 1 is neither prime number nor composite number Two numbers which have only '1' the common factor are called co-primes (or) relatively prime to each other e.g. 3 and 5 are co primes

Tests of Divisibility

Divisibility by2: A number is divisible by 2 if the unit's digit is either zero or divisible by2. e.g. : units digit of 76 is 6 which is divisible by 2 hence 76 is divisible by 2 and units digit of 330 is 0 and hence it is divisible by 2 Divisibility by3 A number is divisible by 3 if the sum of all digits in it is divisible by 3. e.g: The number 273 is divisible by 3 since 2+ 7 + 3 = 12 which is divisible by 3.

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Divisibility by 4 A number is divisible by 4, if the number formed by the last two digits in it is divisible by 4, or both the last digits are zeros e.g The number 5004 is divisible by 4 since last two digits 04 is divisible by 4. Divisibility by 5 A number is divisible by 5 if the units digit in the number is either 0 or 5 e.g: 375 is divisible by 5 as 5 is present in units place in the number. Divisibility by 6 A number is divisible by 6 if the number is even and sum of all its digits is divisible by 3 e.g: The number 6492 is divisible as it is even and sum of its digits 6 + 4 + 9 + 2 = 21 is divisible by 3 Divisibility by 7 A number is divisible by 7 if the difference of number' obtained by omitting the unit digit' and 'twice the units digits' of the given number is divisible by 7 e.g: Consider the number 10717 On doubling the unit digit '7' we get 14. On omitting the unit digit of 10717 we get 1057. Then 1071 -14= 1057 is divisible by 7 Therefore 10717 is divisible by 7 Divisibility by 8 A number is divisible by 8,if the number formed by last 3 digits is divisible by 8. e.g: The number 6573392 is divisible by 8 as the last 3 digits '392' is divisible by 8. Divisibility by 9 A number is divisible by 9 if the sum of its digit is divisible by 9 e.g: The number 15606 is divisible by 9 as the sum of the digits 1 + 5 + 6 + 0 + 6 = 18 is divisible by 9. Divisibility by 10 A number is divisible by 10, if it ends in zero e.g: The last digit of 4470 is zero, therefore 4470 is divisible by 10. Divisibility by 11 A number is divisible by 11 if the difference of the Sum of the digits at odd places and sum of the digits at the even places is either zero or divisible by 11.

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e.g: In the number 9823, the sum of the digits at odd places is 9 + 2 = 11 and the sum of the digits at even places is 8 + 3= 11. The difference between it is 11 - 11 = 0 the given number is divisible by 11. Divisibility by 12 A number is divisible by 12 if it is divisible by 3 and 4 e.g: The number 1644 is divisible by 12 as it is divisible by 3 and 4 Divisibility by 18 An even number satisfying the divisibility test by 9 is divisible by 18. e.g: The number 80388 is divisible by 18 as it satisfies the divisibility test of 9. Divisibility by 25 A number is divisible by 25 if the number formed by the last two digits is divisible by 25 or the last two digits are zero e .g: The number 7975 is divisible by 25 as the last two digits are divisible by 25. Divisibility by 88 A number is divisible by 88 if it divisible by 11 and 8 e.g: The number 10824 is divisible by 88 as it is divisible by both 11 and 8 Divisibility by 125 A number is divisible by 125 if the number formed by last three digits is divisible by 125 or the last three digits are zero e .g: ' 43453375' is divisible by 125 as the last three digits '375' are divisible by 125.

Common factors

A common factor of two or more numbers is a number which divides each of them exactly e.g: 3 is a common factor of 6 and 15. Highest Common factor Highest common factor of two or more numbers is the greatest number that divides each of them exactly. e.g: 3,4,6,12 are the factors of 12 & 36 among them the greatest is 12 and hence the H. C. F of 12 and 36.is 12 H. C. F is also called as Greatest common divisor or Greatest Common measure. Symbolically written as G. C. D or G. C. M

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Method of finding H.C.F Method of division • For two numbers: Step1: Greater number is divided by the smaller number Step2: Divisor of (1) is divided by its remainder Step3: Divisor of (2) is divided by its remainder. This could be continued until the remainder is '0' Then H. C. F is the divisor of the last step e.g: Find the H. C. F of 96 and 348 Solution:

Here the divisor of the last step is 12 H. C. F of 96 and 348 is 12 Method of finding H. C. F of 3 or more numbers : Step 1: Take any two numbers as your wish and find their H.C.F. Step 2: In the given numbers take the third number and find the H. C. F of the number taken and the H. C. F of previous pair. Step 3: In the given numbers take the fourth number and find H. C. F of the fourth number and the previous H.C.F Continue the same process till the end The final H.C.F is concluded to be the H.C.F of all the given numbers e.g Find the H.C,F of 128,246,100

Now 6 is H.C.F of 120,246, is 6 then take 100 with respect to 6

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H.C.F of decimals Let us observe the example Find the H.C.F of 3.2, 4.12, 1.3, 7 Solution : First write the numbers eliminating the decimals with the appropriate multiplication by either '10' or 100 or 1000 etc Here multiplying all the numbers with 100. the numbers are 320,412,130,700 Now we will find the H.C.F of the above numbers. Here the H.C.F of above numbers is 2 Did you remember we multiplied all the numbers by 100 to eliminate the influence of decimals. Here we divide the answer by 100 to get H.C.F of the given numbers The H.C.F is 2 /100 =0.02 L.C.M (Least common multiple) Least common multiple of two or more given numbers is the 'least or lowest number' which is divisible by each of them exactly, in the sense without a non zero remainder Method of finding L.C.M Step 1: Write the given numbers in a line separated by a comma Step 2: Divide any two of the given numbers exactly with a prime number then the quotients and the undivided numbers are written in a line below the first Step 3: Repeat the same process till all the numbers in the line are prime to each other. We mean without any common factor Conclusion : The product of all divisors and the numbers in the last line is the required L.C.M of the numbers Example : Find the L.C.M of 14, 18, 24, 30 214, 18, 24, 30

37, 9, 12, 15

7, 3, 4, 5L.C.M of 14, 18, 24 and 30 is 2 × 3 × 3 × 7 × 4 × 5 = 2520 L.C.M of decimals

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Let us observe the example Find the L.C.M of 1.6, 0.28, 3.2, 4.9 Solution : First write the numbers eliminating the decimals with the appropriate multiplication by either '10' or 100 or 1000 etc Here multiplying all the numbers with 100. the numbers are 160, 28, 320, 490 Now we will find the L.C.M of the above numbers. 2160, 28, 320, 490

280, 14, 160, 245

240, 7, 80, 245

220, 7, 40, 245

210, 7, 20, 245

55, 7, 10, 24571, 7, 2, 245 1, 1, 2, 35

The L.C.M of 160,28,320 and 490 is 2 . 2. 2 . 2 . 2 . 5 . 7 .1 . 1 . 2 . 35 = 78400. Did you remember we multiplied all the numbers by 100 to eliminate the influence of decimals. Here we divide the answer by 100 to get L.C.M of the given numbers The L.C.M is 78400 /100 = 784

key points on L.C.M and H.C.F

Finding L.C.M and H.C.F of Fractions

Example1 : Find the L.C.M of 2/5, 81/100 and 125/302 Solution : First find the "L.C.M of the numerators" As there is no common number (prime) which divides any two of the numbers. Hence the product itself is L.C.M

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L.C.M = 2 × 81 × 125 = 20250 Now find the H.C.F of denominators H.C.F of 5 and 100 is ' 5 ' H.C.F of 5 and 302 is ' 1 ' H.C.F of 5,100 and 302 is ' 1 ' L.C.M of the given fractions = 20250/1 = 20250.

Example 2 : Find the H.C.F of 4/9, 10/21 and 20/63

Solution :H.C.F of numerators 4,10 and 20 is 2L.C.M of denominators 9,21 and 63 is 63H.C.F of the given fractions=2/63

H.C.F of a number of fractions is always a fraction but L.C.M may be a fraction or an integer.

The product of any two numbers is equal to product of "L.C.M" and "H.C.F" of the numbers. e.g Let the two numbers be 32 & 450 Their product is 14400 The L.C.M of 32 & 450 is 7200 The H.C.F of 32 & 450 is 2 Now you can check the above statement

To find the greatest number that will exactly divide x, y and z Required number = H.C.F of x, y and z

To find the greatest number that will divide x, y and z leaving remainders a, b and c respectively .Required number = H.C.F of (x-a), (y-b) and (z-c)

To find the least number which is exactly divisible by x, y and z .Required number = L.C.M of x, y and z

To find the least number which when divided by x, y and z leaves the remainders a, b and c respectively . It is always observed that (x - a) =(y - b) = (z - c) = K(say) . Required number =(L.C.M of x, y and z) - K

To find the least number which when divided by x, y and z leaves the same remainder 'r' in each case. Required number = (L.C.M of x, y and z) + r

To find the greatest number that will divide x, y and z leaving the same remainder in each case,

1. When the value of remainder r is given: Required number= H.C.F of (x-r), (y-r) and (z-r)

2. When the value of remainder is not given: Required number=H.C.F of | (x-y) | , | (y-z) | and | (z-x) | .

SIMPLIFICATION

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In simplification we are supposed to follow the order which is essentially demanded by mathematics and given by a common note of remembrance as VBODMAS

Where

V indicates Vinculum B indicates 'Bracket' O indicates 'Of' D indicates 'Division M indicates 'Multiplication' A indicates 'Addition' S indicates 'Subtraction' Use of Algebraic identities : A student can use algebraic identities given below in the simplification The Algebraic identities are 1.(a + b)2 = a2 + 2ab + b2

2.(a - b)2 = a2 - 2ab + b2 3.(a + b)2 + (a - b)2 = 2(a2 + b2 ) 4.(a + b)2 - (a - b)2 = 4ab 5.a2 - b2 = (a + b) (a - b) 6.(a + b)3 = a3 + 3a2b +3ab2 + b3 = a3 + b3 + 3ab(a + b) 7.(a - b)3 = a3 - 3a2 b + 3ab2 - b3 = a3 - b3 - 3ab(a - b) 8. a3 + b3 = (a + b) (a2 - ab + b2 ) 9. a3 - b3 = (a - b) (a2 + ab + b2 ) 10 .a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - bc - ca )11.a4 - b4 = (a2 + b2 ) (a + b) (a - b)

The number of divisors of a composite number:

If N is a composite number in the form Where a, b, c are primes, then the number of divisors of N,

represented by 'N' is given by e.g Let the number be 600

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Now 600= 23 x 31 x 52 Therefore the number of divisors of 600 is ( 3+1 ) ( 1+ 1) ( 2+ 1) = 24

The number of divisors of 600 excluding 1 & its self is 24 - 2 = 22 The sum of divisors of a composite number:

If N is a composite number in the form of

Where a, b, c are primes, then the sum of the divisors , is given by

e.g Let the number be 600 600 = 23 x 31 x 52

Now =

=

Sum of natural numbers from 1 to n =

e.g Sum of natural numbers from 1 to 40 =

Number of odd numbers from 1 to n is

Number of even numbers from 1 to n is

Sum of even numbers from 1 to n is k ( k +1) : where k indicates number of even numbers from 1 to n e.g Sum of even number from 1 to 80 is 40 (40+1) = 1640 Here from 1 to 80 there exists 40 even numbers

Sum of odd numbers from 1 to n is ( Number of odd numbers from 1 to n ) 2

e.g Sum of odd numbers from 1 to 60 is ( 30) 2 = 900 Here from 1 to 60 there exists 30 odd natural numbers

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Sum of squares of first n natural numbers is . e.g Sum of squares of first 20 natural numbers is

Sum of the squares of first n even natural numbers is

e.g Sum of squares of first 10 even natural numbers is

Sum of squares first n odd natural numbers is e.g

Sum of squares first 30 odd numbers is

Sum of cubes of first n natural numbers is .e.g Sum

cubes of first 9 natural numbers is

Any number N can be represented in the decimal system of

number as

.e.g

A sum of any5 consecutive whole numbers will always be divisible by 5e.g Here 25 i.e (3 + 4 +5 + 6 +7 ) is divisible by 5

The difference between 2 numbers will be divisible 9 e.g Let the 2 numbers be 95& 59 Here 59 is just the reversal of 95 Now 95 -59 = 36 You can observe that 36 is divisible by 9

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Products :

n! = eg. 6! =

The product of any 'r' consecutive integers ( numbers) is divisible by r!

If m and n are two integers then ( m+ n) ! is divisible by m! n!

In the kind of numbers , finding the units digit Observe the working Here the unit digit of 252 is 2 and the index is 54

we know that , . hence unit digit is

repeated after each 4 indices the units digit of is same

as the unit digit in which is also same as the unit digit in

which is nothing but 4

Complete Remainder

A remainder obtained by dividing a given number by the method of successive division is called complete remainder. Ex : A certain number when successively divided by 2, 3 and 5 leave remainders 1,2 and 4 respectively. What is the complete remainder or

remainder when the same number be divided by 30 ?. Here this 30 is the product of 2, 3 and 5

When there are two divisors and two remainders the

complete remainder is given by

When there are three divisors and three remainders

the complete remainder is given by

When there are four divisors and four remainders

the complete remainder is given by In any case if there are no remainders consider them as zeros Two numbers when divided by a certain number gives

remainders r 1 and r 2 . When their sum is divided by the same divisor, the remainder is r 3 . The divisor is given by r 1 + r 2 - r

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3 . Ex Two numbers when divided by a certain divisor give remainders 437 and 298 respectively. When their sum is divided by the same divisor, the remainder is 236. Then the divisor is given by 437 + 298 - 236 = 499

Fractions and ordering fractions

A number of the type which represent x number of parts of y number of equal parts of a thing is called a fraction , remember y is never be zero

Fraction Such a fraction is known as common fraction

A fraction whose denominator is 10 or 100 or 1000 etc. is called a decimal fraction .

Fractions whose denominators are same, are called like fractions,

Ex are like fractions.

Fractions whose denominators are different, are called unlike

fractions, Ex are unlike fractions.

When two fractions have the same denominator, the greater fraction is that which has the grater numerator.

When two fractions have the same numerator, the greater fraction is that which has the smaller denominator. In the above two arguments if the identity is not possible convert one fraction according to your convenience

Let two given fractions be To insert a fraction lying

between the following is an identity which is always true , using this identity we can insert any number of fractions between two fractions

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Ex

Ex Arrange in ascending order Sol L.C.M of 5, 16,7, 104 is 7280

then

After making all the denominators same and having the comparison between numerators the fractions in ascending

order are

Ex Insert three fractions between

� �

Surds and Indices

* (a is called the base) is called a surd if x is a fraction and is called an index if x is an integer. ax x a y = a x+y

ax /a y = a x-y

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We can use the above identities in arithmetic

Ex : Find the expression with rational denominator.

Sol : Given expression =

Finding the square root

Division method : Let us have an example 64009

square root of 64516 is 254 Let us observe the above working rule in words

first consider the given number then from right consider, two digits pairs leaving left most digit as it is

For the left most part guess the approximate square root, in the example it is 2, then start the division

Between 6 and 4 the reminder is 2 adjacent to it bring down 45 now the number is 245

As in the example for this 245 the divisor is 45 that its " 4 is double of the previous quotient 2" and �5 is a guess" such that for 245, 45 must produce very nearer multiple of 45 with 5

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In the last step in above example 20 is the reminder, then bring down 16 than it becomes 2016 for this the divisor is 504 where �50 is double of the previous quotient 25" and �3 is the guess" such that for 2016, 504 must produce very nearer multiple of 504 with 5

Here the complete quotient is the square root of the given number i.e254 is the square root here

After having the argument once again you can observe the example Square root of 119716 is 346

Key points on finding square root

1. A number ending with 2,3,7, 8 cannot be a perfect square the last digit of a perfect square must be any one among 0,1,4,5,6,9 2. A number ending with odd number of zeros can never be a perfect square 3. The difference between squares of two consecutive numbers is always an odd number. 4. A square root of a decimal fraction first eliminate the decimal point by dividing and multiplying with even powers of 10 then find the square root of both numerator and denominator separately and then you can conclude the square root Ex Find the square root of 1190. 25

DECIMAL FRACTIONS

The method of representing one part of the system to whole system is termed as fraction. They are of 2 types.

1) Vulgar or common fractions eg : etc

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2) Fraction whose denominators are powers of 10 is known as decimal fractions

eg : etc A fraction is also called a rational number A fraction in its lowest terms always possess numerator and

denominator which are prime to each other . if numerator and denominator are large numbers & if their

common factor cannot be easily guessed then we find its H.C.F A fraction is unity if its numerator and denominators are equal. A fraction with numerator lesser than denominator is termed as

proper fraction. eg: etc A fraction with numerator equal to or greater than denominator

is termed as improper fraction eg: etc A mixed fraction contains a whole number and a fraction eg:

In adding mixed fractions the integral and fractional parts should be collected separately.

The easier method of comparing two or more fractions is to express each fraction into decimal fraction and then compare.

Adding zero to the extreme right of a decimal does not alter its value.

The decimal number in which a set of digits is repeated again and again is known as recurring decimal.

A decimal fraction in which all the digits after the decimal point

are repeated is called pure recurring decimal. eg: etc. A decimal fraction in which at least one digit after the decimal

point is non recurring is termed as mixed recurring decimal.

Average

The Average of any number of quantities is sum of their quantities by the number of quantities (n). ⇒ Average

If there are two types of items say A and B , A has m number of sub items and B has n number of sum items then the average of A and B is (Am+Bn)/(m+n)

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If a vehicle travels from one place to another at a speed of a kmph but returns at the speed of b kmph then its average speed

during the whole journey is kmph. Out of three numbers, first number is x times of the second

number and y times of the third number. If the average of all the

three numbers is z then the first number is Let the average age of men and women in a town be x years and

the average age of women be y years and the average age of

men be z years. Then the number of men in that town is if N indicates the total number of men and women of the town.

The average age of N persons is x years. If one new person joins them. Then the average age is increased by y years. Then the age of new comer is x + (1 + N) y years.

The average age of N persons is x years. If M persons joins them, the average age is increased by y years then the average

age of newcomers is y years The average age of N persons is x years. If M persons joins

them, the average age is decreased by y years then the average

age of new comers is y years The average age of N persons is x years. If M persons left, then

the average age is increased by y years, then the average age

ofoutgoing persons is y years. The average age of N persons is x years. If M persons left, then

the average age is decreased by y years. Then the average age

of outgoing persons is y years In a group of N persons whose average age is increased by y

years when a person of x years is replaced by a new man. Then the age of new comer is x + Ny years.

The average temperature of Sunday, Monday, Tuesday and Wednesday was Xo C. The average temperature for Monday, Tuesday, Wednesday and Thursday was Yo C. If the temperature on Thursday is ao C then the temperature on Sunday (bo C) can be given as bo C = No of days (X -Y) + a Here No of days = 4.

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Percentage

For converting a percent into a fraction, divide it by 100. For converting a fraction into a percent, multiply it by 100. For converting a percent into a decimal. Shift the decimal point

two places to the left. For converting a given quantity x as a percentage of another

given quantity y , write There is no unit for percentage.

Tricks to solve the problems

Of the given two numbers if the first is x % more than the

second, then the second will be less than the first. Of the given two numbers if the first is x % less than the second,

then the second will be more than the first. If two numbers are respectively x % and y % more than the third

number, then the first number will be of the second.

If two numbers are respectively x % and y % less than a third

number, then the first number will be of the second.

If a number or quantity is increased by x % then in order to

restore its original value it must be decreased by If a number or quantity is decreased by x % then in order in

restore its original value it must to be increased by . If a number is successively increased by x % and y % then a

single equivalent increase in that number will be

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If two successive discounts of x % and y % are allowed on a particular amount, then a single discount that is equivalent to

the two successive discounts will be If a number is successively increased by x %, y % and z %, then

a single equivalent increase in that number will be

If three successive discounts of x %, y % and z % are allowed on an amount, then a single discount that is equivalent to the three

successive discounts will be If a number is increased by x % and thereafter reduced by x %,

then the number will be reduced by If a number is reduced by x % and thereafter increased by x %

then the number will be reduced by If due to an increase by x % in the selling price of certain

commodity the Sale or consumption of the commodity decreased by y %, then gross receipts on account of sale of that commodity

will be increased or decreased by and

will be decreased by If the price of a commodity increases by x %, by how much

percent should a family reduce its consumption so as not to increase the expenditure on commodity ; Percentage of

reduction In an election between two candidates a candidate gets R% of

the votes cast and loses by x votes to be elected. How many

votes were casted at the election. Number of votes cast A man gives x % of his money to his son, y % of the remaining to

his daughter and z % of the remaining to his wife. If he is left now with Rs. R, what is his initial amount. Total amount in the

beginning

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If m boys and n girls appeared in an examination and x % of the boys and y % of the girls passed in it, then the total percentage

of students who passed is If in a examination, in which the minimum pass percentage is x

%, a candidate secures y marks and fails by z marks, then the

total number of marks in this examination will be . If in an examination x % and y % candidates respectively fail in

two different subjects while z% candidates fail in both the subjects, then the percentage of candidates who pass in both

the subjects will be

Time & Work

TIME AND WORK

Let us form a relationship between the three variables i.eThe amount of work (W)The number of persons doing it (P) And the time period of doing work (T).

If 'P 1 ' persons can do 'W 1 ' part of work a day for 'T 1 ' days and 'P 2 ' persons can do 'W 2 'part of work in 'T 2 ' days, then we have a very general formula P 1 T 1 W 1 = P 2 T 2 W 2 .

From our common sense, we can also derive the following relations

If the number of persons engaged to do the work remains same, then More work requires more time and vice-versa. i.e., for constant P, W and T are directly proportional W α T

If the amount of work (W) to be done is kept unchanged, then more persons required less time and vice-versa. i.e., when W is kept constant P and T are inversely proportional

P α If time to finish the work remains unchanged. More work

requires more persons and vice-versa. i.e., for constant T, P and W are directly proportional P α W We can combine the above three relations between P, T and W

then is Constant or

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If A can do a piece of work in n days then work done by A in 1

day is

If A's 1 day work is , then A can finish the whole work in n days.

If A can do a piece of work in x days and B can do the same work in y days, then both of them working together would take

i.e days to complete the same work. If A, B and C can complete a piece of work in x , y and z days

respectively, then they together would take days to complete the same work.

If A and B working together can complete a piece of work in x days and A alone can complete the same work in y days, then B

alone would complete the same piece of work in days If A, B and C can complete a piece of work in x , y and z days

respectively, the portions of work done by them would be in the

ratio of If A can do a work in x days, while B can do the same work in y

days. They began to work. But A left n days before its completion. Then the time, when whole work in finished

days. 'A' persons can complete a work in x days.

If 'B' more persons join them after n days of the day they started working, then the no. of days they will complete the remaining

work A takes n times the time in completing a work as taken by B and

C together to complete the same work. If all of them can complete the same work in t days, then time taken by A to do

the work alone = days. A contractor undertook a work in T days. He employed 'A' men

to carryout the job. But after T 1 days be found that only x part

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of work had been done. Then the no. of more workers should be

employed to finish the work in time is Where, T 2 = No. of remaining days, i.e., T - T 1 y = Remaining work, i.e., (1 - x ) A 1 = No. of required workers to do the remaining work.

If a men or b women can do a piece of work in n days, then c

men and d women can do the work in days If A is k times efficient than B and is therefore able to finish a

work in n days less than B, then

1. A and B, working together, can finish the work in days

2. A, working alone, can finish the work in days

3. B, working alone, can finish the work in If A working alone takes a days more than A and B working

together. B alone takes b days more that A and B working together, then the number of days taken by A and B, working

together, to finish the job is given by

If A can complete part of work in n days, then part of the

work will be done in days A servant was engaged on the condition that he will get Rs. a

per day for work but he will be fined Rs. b per day on being absent. After n days if he get Rs. c in all,

Then no. of days of his absence

Surds

If is a surd where a is not a perfect square, then is called

a quadratic surd. Ex: ,

If is a surd, then is called a cubic surd. Ex:

If is surd, then is called a biqudratic surd. Ex:

1


If a is nonzero rational number and is a surd, then ,

are called mixed surds. Ex:

If a = 1; A surd, expressed in the form is called an entire

surd or a pure surd Ex:

A surd, expressed in the form where b is the least positive rational number, is called the simplest form of the surd. Ex:

. A surd consisting of a single term is called a monomial surd, i.e.,

a simple surd is also called as a monomial surd. Ex:

The sum or difference of a rational number and one or more surds is called a compound surd if it contains at least two terms.

Ex: A compound surd consisting of two terms is called a binomial

surd. Ex: A compound surd of three terms is called a trinomial surd. Ex:

Two simple surds are said to be similar surds or like surds if the

quotient of the two surds is a rational number. Ex: Two simple surds are said to be dissimilar surds or unlike surds

if the surds are not similar. Ex: The product of two dissimilar quadratic surds is a surd. Ex:

, a surd. The quotient of two dissimilar quadratic surds is a surd. Ex:

If are two dissimilar surds, then there exists no surd of

the form such that

If are two surds then there exists no nonzero rational

number c such that

. Where a, c are rational numbers

are surds

1


where where

are positive rational numbers

and is a surd

where a, b ,x, y are positive rational numbers such that x + y = a, 4xy = b

If then x + y + z = a and

Where a, b, c, d, x, y, z are positive

rational numbers such that x + y + z = a,

Method of finding Cube root of a surd

let

Then

...............................(1)

We apply

( )( ) = ........(2) Solving (1) & (2) we get the values of x & y and hence the result

Ex: Find

Sol: Let

Now

∴

1


(1)

(2)

Ex: Find the cube root of

Let

Solving (1) & (2)

Rationalising Factors

The binomial surds of the form are called conjugate

surds, each surd is called the conjugate of the other. Ex: is

the conjugate of The product of two conjugate surds is a rational number. Ex:

a rational number. If the product of two surds is a rational number then each surd

is called a rationalising factor (R.F) of the other

1


is a rationalising factor of

is a rationalising factor of Rationalising factor of a surd is not unique.

A surd may have infinite number of rationalising factors.

are rationalising factors of

Rationalising factor of is

A rationalising factor of is

Progressions

1. Sequence 2. Series 3. Progressions4. Arithmetic Progression (A.P) 5. Geometric progression (G.P) 6. Harmonic Progression (H.P) 7. Arithmetico Geometric Progression 8. Sum of Natural Numbers 9. Exponential series 10. Logarithmic series 11. Method of 'converting Decimals into Rational Numbers'

Sequence

A sequence is a function of natural numbers. Here co-domain is the set of real numbers. If range is a subset of real numbers (complex numbers), then the sequence is called a real sequence ( or complex sequence)

You can avail the meanings of domain, co - domain and range in the chapter ' functions'

Example2,4,6,8,10....; 5,3,1,-1,-3....; 1,3,9,27....;

: A Sequence is said to be finite or infinite sequence according as, it has finite or infinite number of terms

Series

http://catmaterial.com/lessons/progressions.html#practice:1:30:1

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By adding or subtracting the terms of a sequence we get a series :Thus 2 + 4+ 6+ 8 +10.....; 5 +3 + 1- 1-3 ...; 1 - 3 +9 -27 + ....; are series

A Series is said to be finite or infinite series according as, it has finite or infinite number terms

If is a sequence, then the expression

is a series

In this chapter generally t indicates a term , indicates n th term

Example :

Write down a sequence whose n th term is

Sol . Let Let n = 1 , 2, 3, 4.... Successively, we get

Hence we obtain the sequence

Progressions

It is not necessary that the terms of a sequence, always follow a certain pattern, or they are described by some explicit formula for the n th term. Those sequences whose terms follow a certain pattern are called progressions

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The progressions are of types 1. Arithmetic progression 2. Geometric progression 3. Harmonic progression 4. Arithmetico - Geometric progression

Arithmetic Progression (A.P)

A sequence is called an arithmetic progression if the difference of a term and the previous term is always a constant i.e

d for all The constant difference, which is denoted by d is called the common difference

The common difference of an A.P can be zero, positive or

negative If where

d is common difference of A.P. and is first term.

n th term is

last term

Sum of n terms in A.P

Properties :

If are the terms of an A.P., then

are also the terms of an A.P. with same common difference.

If are the terms of an A.P. with the common

difference ' d ', then are also terms of an A.P. with common difference ' kd '. ( where k is a non-zero constant)

In an A.P. the sum of terms equidistant from the beginning and the end is always same l,

i.e.,

1


While doing the problems we adopt any three terms in A. P as

. Similarly we adopt any four terms as

respectively

If are in A.P. then a set of terms taken at regular intervals from this A.P. are also in A.P.

If and are two A.P.s Then are also in A.P

If and are two A.P.s then .... And

are not in A.P

If are in A.P, then

If n th term of any sequence is a linear expression then the sequence is A.P

If the sum of n terms of any sequence is a quadratic expression

in n i.e then the sequence is A. P

If a,b,c are in arithmetic progression, then b is arithmetic mean

of a and c , i.e., .

Single A.M of n positive numbers is

Between two given quantities we can insert n quantities

from an A.P. In this insertion

we say that are n arithmetic means between a and b .Let'd' be common difference of this A.P. Then

, Similarly, we

can find .

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Geometric Progression (G.P.)

are in G.P. If where r is a constant and is called common ratio of the given G.P, which cannot be zero.

ExampleThe sequence 4, 12, 36,108,324,... is a G.P, because

Which is a constant

n th term of G.P . t n =

Sum of n terms

Sum of infinite terms

Properties of G.P.

if are in G.P., then are also in G.P. with same common ratio.

If are in G.P., then are also in G.P.

If are in G.P then are also in G. P with

the common ratio

If are the terms of a G.P., then are also the

terms of a G.P. with the common ratio In a G.P. the product of terms equidistant from beginning and

end is always same i.e.,

1


If and are two G.P. s then and

are also in G. P

If and .. are two G.P. s then are not in G.P

If are terms of an A.P. with common difference d ,

then are in G.P. with common ratio

If are in G.P., then the terms chosen at regular intervals also from a G.P

If a,b,c are in geometric progressions then b is geometric mean

of a and c, i.e., b =

If are in G. P then

Three terms in G. P can taken be taken as . Five numbers

in G. P. can be taken as etc

If are in G. P then are in A.P.

Single G. M of the n positive numbers is

Between two given quantities we can insert n quantities

Let be n geometric means between a and b Then

is a geometric progression. Let ' r ' be common ratio of this G.P., then

similarly we can

write

Harmonic Progression (H.P)

If are in A.P., then are in H.P.

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n th term of this H.P

n th term of this H.P from the end No term of H.P. can be zero There is no general formula for finding out the sum of n terms of

H.P If a and b are two non-zero numbers and H is harmonic mean of

a and b then from H.P. and then

Single H. M of n positive numbers is H where

Insertion of n Harmonic means between two given

numbers

Let a , b two given numbers. If n numbers are inserted

between a and b such that the sequence b is a

HP, then are called n harmonic means between a and

b are in AP

Let D be the common difference of this A.P. Here term

Then Substituting the

value of D, we can obtain the values of

If a progression is given then the way to decide the kind of the progression Let a, b, c are any three successive terms of a progression

Then

1


If then a , b, c are in A. P

If then a , b, c are in G.P

If then a , b, c are in H.P

Arithmetic mean Geometric mean Harmonic mean (property holds if the given numbers are positive).

If and H are arithmetic, geometric and harmonic means of two numbers a and b respectively, then G2 = AH.

If A, G are given then a, b are the roots of the equation

If A, G, H are the arithmetic, Geometric , Harmonic means between three given numbers a, b, c then the numbers a, b, c

are the roots of the equation

Arithmetico Geometric Progression

A Progression is said to be an arithmetico geometric Progression if its terms are formed by multiplying the corresponding terms of an A. P and a G. P

Example :

Here 1 , 3, 5,7,9, ... are in A. P and are in G. P

Sum of n terms of an Arithmetico - Geometric Series

Let --------1 (Multiplying both sides of ( 1) by common ratio r we get)

-------------------2 (Subtracting (2) from (1) we get )

1


=

-----------3

Sum of Natural Numbers

If nth term of a series is of the from then sum of n

terms is

Example :- Find

Sol :-

Exponential Series

( e is an irrational number) Coefficient

of ;

1


Logarithmic Series

Method of converting Decimals into Rational Numbers

Example : Express the recurring decimal 0.125125125.... As a rational number Sol Let X = 0.125 125125.......(1) 1000 X = 125.125125.........(2) Subtracting (1) and (2) we get 999 X = 125

X =

Example : Find the value of 0.32595959 Sol. Let X = 0.3259595959 Then 100 X = 32.595959 ----------2 and 10000 X = 3259.5959--------3 Subtracting eq 2 from eq 3 we get 9900 X = 3227

X =

Quadratic Equations

1. Definition2. Nature of the roots 3. Formation of a Quadratic equation4. Transformation of equations5. The minimum and maximum values of f(x)=ax 2 +bx+c

1


6. Location of roots and its necessary conditions7. Key points on roots 8. Theory of equations

t

Definition

An equation of the form ax 2 +bx+c=0 where a, b, c belongs to the real numbers and a ≠ 0 is a quadratic equation. Do you guess why 'a' should not be equal to zero If a = 0 then the equation becomes a linear equation. If ax 2 + bx + c = 0 is a quadratic equation given then the quantity b 2 -4ac is known as Discriminant . And is denoted by 'D'. The roots of a quadratic equation ax 2 +bx + c = 0 are

Argument: The roots are of the form , if D is not a perfect square. If 'D' is a perfect square then both the roots are rational numbers

Nature of the roots

If D>0, the roots are real and distinct.

If D=0, the roots are real and equal

If D<0, the roots are complex with nonzero imaginary

If a, b, c are rational and if 'D' is a perfect square then the roots are rational

If a=1, b, c belongs to 'I' and the roots are rational numbers then the roots must be integers.

In the quadratic equation ax 2 + bx + c =0, if a=b=c=0 then it has infinitely many roots because it is an identity in x. Let us have an example for this

Example : The number of values of a for which ( a 2 - 3a + 2) x 2 + (a 2 -5a+6)x+a 2 -4=0 is an identity in x is 1) 0 2)2 3) 1 4) 3 Sol :- It is an identity in x if a 2 -3a+2=0, a 2 -5a+6=0, a 2 -4=0

1


Solving the above a=1,2 and a=2,3 and a=2,-0 ∴equation is an identity if a=2 which is common in all the three Hence '2' is correct answer

Formation of a Quadratic equation

If the roots are α and β then the quadratic equation is x 2 -( α + β )x + αβ =0 If any two quadratic equations a1 x 2 + b1 x+ c1 =0 and a2 x 2 + b2 x+c2 =0 have a root α in common then the condition follows is (b1c2 - b 2 c1 )( a1 b2- a2 b 1 )=( c1a 2 - c 2 a 1 ) 2 and the common root is

Transformation of equations

If are the roots of the quadratic equation then,

Sum of roots

Product of roots

If f(x) = 0 is a quadratic equation with the roots α and β (say) then

The equation whose roots are reciprocals i.e 1/α , 1/ β , is =0

The equation whose roots are K times the given roots i.e Kα , Kβ

, is = 0 The equation whose roots are squares of the given roots i.e α

2 ,β2 , is =0, here you need to remember that rational index must be eliminated

The equation whose roots are α +k, β +k is If we replace x by x 2 , the resulting equation is of fourth degree

and its roots are

An expression in is called a symmetric function of if the

function is not affected by interchanging and .

1


Above relation holds for any quadratic equation weather the coefficients are real or non real complex

With the help of above relations many other symmetric

functions of and can be expressed in term of the coefficients a, b, and c

Let us have a discussion about the region which x belongs to in the quadratic inequalities ax 2 +bx+c ≥ 0 (or) ax 2 +bx+c ≤ 0

Step 1: Solve ax 2 +bx+c =0 Say α , β are the roots of the equation. Step 2: On the real number line mark α and β

Step3: If a, (ax 2 +bx+c) are of same sign then x occupy the region on both the sides of α and β as shown in diagram

If a, (ax 2 +bx+c) are of different signs x occupy the inner region between α , β

Step 4:

In the conclusions If the inequality contains equality sign then the boundaries must be closed

In the conclusions If the inequality does not contain equality sign then the boundaries are open

Example : Find The domain of f(x) = x 2 + x - 2 ≤ 0

1)x ∈ [-2,1]

2) x ∈ (-2, 1)

3)

4)

Sol :-Step 1: Solving x 2 +x-2=0, the roots are (-2,1) Step 2:

1


Step 3: f(x) is negative , x 2 coefficient is positives as it is 1 which are of opposite signs x belong to the region in between -2, and 1

Step 4 : The expression also has equality sign , hence the boundaries must be closed ∴x ∴ [-2, 1] Hence the correct answer is '1'

The minimum value of f(x) = ax 2 +bx+c where a > 0 is -D/4a we can obtain this value at f(-b/2a) or we can apply the value of 'D' as we know D=b 2 -4ac

The maximum value of f(x)=ax 2 +bx+c where a < 0 is -D/4a we can obtain this value at f(-b/2a) or we can apply the value of 'D' as we know D=b 2 -4ac

Location of roots and its necessary conditions

When a real number k lies between the roots α and β i.e α<k<β Conditions to be checked i) D >0 ii) a f(K)<0

When both the roots α and β lie in the interval (k1 k2 ) Conditions to be checkedi) D ≥ 0 i.e roots are real ii) a f(k 1 ) >0, a f (k 2 ) <0 iii) k 1 < (-b/2a) < k 2 When a real number k is less than or greater than the roots α and β . Conditions to be checked i) D ≥ 0 ii) a f(k) >0

The quadratic function ax 2 +2hxy+by 2 +2gx+2fy+c is resolvable into linear rational factors if and only if D= abc+2fgh-af 2 -bg 2 -ch 2 =0

1


Key points

A complex number a is called solution or root of a quadratic equation ax 2 +bx+c=0 if a α2 + bα + c=0

If a quadratic equation in x cannot have more than two roots and if the two roots are equal to a then a is called double root.

α , β are the roots of ax 2 +bx+c=0 then• α + β = -b/a • α β = c/a • ax 2 +bx+c = a(x-α) (x- β ).

Some symmetric functions of roots are

i)

ii)

iii)

iv)

v)

vii) If a+b+c=0 then the roots are 1 and c/a If a-b+c=0 then the roots are -1 and -c/a the roots are negative then a, b, c will have the same sign If the roots are positive then a ,c will have the same sign

different from the sign of b If a 1 x 2 + b 1 x+ c 1 =0 and a 2 x 2 + b 2 x+ c 2 =0 have the

same roots then

= = o If α, β are the roots of ax 2 +bx+c=0 and if α β =1

then the roots are reciprocal to each other. If one root of quadratic equation is irrational then the

other root is its conjugate one root of quadratic equation is a complex number then

the other root is its conjugate. Condition for the roots of ax 2 +bx+c=0 to be in the ratio

m : n is (m + n) 2 ac = m n b 2

Conditions for one root of ax 2 +bx+c=0 is the square of the other is b 3 +a 2 c+ac 2 =3abc

Theory of equations

1


Polynomial function If a0 , a1 , a2 ,...... an , are real and n is a positive integer, then f(x)= a 0 +a 1 x+a 2 x 2 +......+a n x n is called a polynomial function in x. Polynomial equationIf a0 , a1 , a2 ,...... an , are real and n is a positive integer then f(x) = a 0 +a 1 x+a 2 x 2 ....... a n x n =0 is called polynomial equation in 'x' with real coefficients If a n ≠ 0, then f(x)=0 is an equation of degree n Degree of the polynomialThe highest power of 'x' for which the coefficient is nonzero in a polynomial function, is called the degree of the function. Constant polynomialIf the degree of the polynomial function is zero, then that polynomials are called as constant polynomials Zero polynomialIf the coefficients of a polynomial are all zeros, then that polynomial is called zero polynomial zero polynomial has no degree the domain of a zero polynomial is R. Linear, Quadratic cubic, Biquadratic equation Polynomial equations of degrees 1,2,3 and 4 are called as linear, quadratic , cubic and biqudratic equations respectively

The equation ax+b=0, a ≠ 0 is of degree one and is called a linear equation

The equation ax 2 +bx+c, a ≠ 0 is of degree 2 and is called quadratic equation

The equation ax 3 +bx 2 +cx+d=0, a ≠ 0 is of degree 3 and is called cubic equation

The equation ax 4 +bx 3 +cx 2 +dx+c=0, a ≠ 0 is f degree 4 and is called a biquadrate equation.

Root of an equationThe value of x which satisfies f(x) =0 is called root of the equation f(x) =0 If f(a)=0, then x = a is a root of equation f(x)=0 Also (x-a) is a factor of the polynomial f(x). Remainder theorem: If a polynomial f(x) is divided by (x-a) then the remainder is f(a)

1


Relation between the roots and coefficientsSuppose a 1 , a 2 ,..... a n are the roots of the equation a 0 x n + a 1 x

n-1 + a 2 x n-2 �. a n =0, a 0 =0, say P 1 = (a 1 /a 0 ), P 2 =(a 2 /a 0 ),............... P n =(a n /a 0 ) then we can write

s 1 = sum of all roots Σα1 = - P1 s 2 = sum of products of the roots taken two at a time

Σα1α2 = P2

s 3 = Σα1α2α3 = - P3 s 4 = Σα1α2α3α4 = P4

.............................................................................

.............................................................................. s n = α1α2α3...................... α n = (-1) n P n

Inequalities

1. Number line 2. Intervals3. Absolute value of a real number4. Basic inequalities with respect to real numbers 5. Arithmetic & Geometric means inequality6. Weighted A.M., G.M. Inequality7. Weierstras inequality 8. Cauchy - Schwartz inequality9. Number line rule

Number line

A line on which we can plot all the real numbers, usually which is parallel to x- axis or x- axis it self is called the number line .

Intervals

Closed interval : If a real number is such that , then lies in closed interval a and b and written as [ ,b] .(Here end points are included) e.g.

Open interval : If a real number is such that < x< b, then lies open interval and b and written as ( ,b) or ] ,b [ Here end points are not included. e.g; 1< <3 (1,3) or x ]1,3 [

Half open, half closed interval : If is such that a < b (Left end point is included but right end point excluded). Written as [ ,b) or [a,b[. e.g. 1< 5 (1,5]

Absolute value of a real number

1


The absolute value of a real number 'x' is denoted by and defined by

=

is also defined as

If x is positive = x

If x is negative =

a

=

(Triangle inequality)

(Triangle inequality

iff ab 0

iff, ab 0

Basic inequalities with respect to real numbers

We shall learn elementary proprieties of inequalities ,which will be used in the subsequent discussion.

If a, b, c are real numbers such that a > b and b > c, then a > c.

More generally, if are real

numbers such that . If a, b are real numbers, then a > b ⇒ a + c > b + c, For all c ∈

R

If a, b are two real numbers, then a>b ⇒ ac>bc and > for all

c>0, c ∈ R Also, a>b ⇒ ac<bc and < for all c<0, c ∈ R

If a>b>0, Then .

If then

1


If are positive real numbers such that

then If a, b are positive real numbers such that a<b and if n is any

positive rational number, then

If 0<a<1 and n is any positive rational number, then

0< <1

If a>1 and n is any positive rational number, then

>1

0< <1 If 0<a<1 and m, n are positive rational numbers, then

m>n ⇒ >

m<n ⇒ <

If > >0 and >1 then

If >0 and 0< <1 then

Arithmetic & Geometric means inequality

If are n distinct positive real numbers, then

i.e. A.M.>G.M.

If = ......= , then A.M. = G.M. Thus, equality occurs when all quantities are equal.

Weighted A.M., G.M. Inequality

The mean of some terms which are the products of a common ratio or

of different multiplicants, say is called weighted mean. and so correspondingly weighted arithmetic mean and geometric mean.

If are n positive real numbers and are n positive rational numbers, then

1


i.e. weighted A.M.> Weighted G.M.

Two Important inequalities If are n positive distinct real numbers, then

, If m<0 or m>1

, if 0<m<1. i.e. the arithmetic mean of the m th power of n positive quantities is greater than m th power of their arithmetic mean except when m is positive proper fraction known as m th power mean.

If and are rational numbers and m is a rational number, then

if m<0 or m>1 and

if 0<m<1

If are n distinct positive real numbers and p, q, r are natural numbers, then

Weierstras inequality

if are n positive real numbers, then for n 2

are positive numbers less than unity. Then,

Cauchy ' Schwartz inequality

If and are 2n real numbers, then

With the

equality holding if and only if

1


If are n positive variables such that

(Constant), then the product is greatest

when and the greatest value is

If are positive variables such that (Constant), then

the sum is least when and the

least value of the sum is n

If are variables and are positive real

numbers such that (Constant ), Then

is greatest , when

Number line rule

Let us have an example .

Ex: solve f (x) = Sol : Here critical points are w,r,t

On the real no line plot all the above values

1


Observing the above diagram try to understand the below argument carefully.

The right most is +ve

At the critical point 9 as the exponent of its factor is even (100) The function continues its nature. i.e. +ve But till the critical point 3.

At the critical point 3 as the exponent of its factor is odd (7) The function changes its nature i.e. +ve becomes -ve. till the critical point 0.

At the critical point 0 as the exponent of its factor is odd (13) The function changes its nature i.e. -ve becomes +ve.

At the critical point -1/5 as the exponent is even. The function continues its nature.

At the critical point -1 as the exponent is even. The function continues its nature. At the critical point -4 as the exponent is odd thefunction changes its nature.

Now w r t to the question we need f(x) 0

As in the diagram we defined the interval but here you can observe that we took closed intervals because the question is w r t to . If it is strictly greater than we should take only the open intervals.

Now again there is a key note to observe that some factors are in Denominators, as they don't be equal to 0 we should delete thecorresponding values from our established answer.

Finally the answer is

SET

Contents

1. Set2. Union & Intersection of sets3. Important Identities on Sets4. Relations

1


Set

A set is a "well defined collection of objects"Ex A = a set of vowels in English alphabet = { a, e, i, o, u}

There are two methods to define a set Roster form (Tabulation method): In this method a set is described by listing elements, separated by commas, within the braces. Ex A = set of odd natural number less than 9 = {1,3,5,7}

Set builder form (Rule method) : In this method a set is described by properties satisfied by the elements Ex A = set of even natural numbers less than 8 = { x : x : odd natural number and x < 8 }

Types of sets

Null set : A set having no elements is called a null set or empty

set and is denoted by or { } Unit set or Singleton : A set having one and only one element is

called singleton or unit set. Subset : If every element of a set A is also an element of a set B,

then A is called subset of B, and written as

Ex The sets are the subsets of the set Proper subset : A set A is said to be proper subset of a set B if

every. element of A is an element of B and has at least one element in B which is not an element of A and is denoted by A B Ex If A = {1, 2, 4,5} and B = {5,1,2,4,3}

Power set :The set of all the subsets of a given set A is called power set and is denoted by P(A)

Ex If A = =

Super set : In the statement , B is called the super set of A Equal sets : Two sets A and B are said to be equal if every

element of A is an element of B, and every element of B is an element of A. If A and B are equal, we write A = B

i.e., Ex sets {1,2,3} and {2, 3,1} are equal sets.

1


Finite and Infinite set : A set in which the elements are countable is called a finite set, otherwise it is called an infinite set Ex set of natural number less than 400 (finite set) set of all integers (infinite set)

Universal set : If all the sets under consideration are likely to be subsets of a set then the set is called the universal set and is denoted by U or S.

A as a set of vowels which is subset of

then U is a universal set.

Number of elements in a finite set is called cardinal number or order of a finite set.

The total number of subsets of a finite set containing n elements is 2n .

Union & Intersection of sets

Is set of elements which belong either to A or B or

both A and B

Ex A =

B =

Intersection of sets

(A intersection B) means set of elements which belong to both sets A and B. thus

1


is also written as AB

i.e.,

Ex

then

Disjoint Sets : Two non-empty set A and B are said to be disjoint if

.

Complement Set or Ac (A complement) means set of elements which are not in set A (i.e., U - A )

Difference of sets A - B means the set of elements belong to A but do not belong to B.

this can also be written as Symmetric Difference of Two sets

Let A and B be two non-empty sets, the set is called symmetric difference of two sets and is denoted by A Δ B. ∴ AΔB = (A - B) ∪ (B - A).............................(1)

1


A - B = Symbolically,

Similarly elements belong to only B set = B -A or

Ex

Important Identities on Sets

n (A) is number of elements in set A then

Relations

1


Let A and B be any two sets. Then any subset of is a

relation R from A to B i.e.,

Ex If

R= being a subset of is a relation A to B. If set A consist m elements and B consist n elements then consist mn elements. 2mn different relations from A to B.

If A is a non-empty set, then a relation R such that is called a relation on set A. Here, is universal relation on set A.

and so it is a relation on A This relation is called the universal relation on A.

If every element of set A is related to itself only then it is called

Identity relation. e.g., If Let A be any set, then a relation R on A is said to be Transitive

relation iff

Ex Let A = If a relation is reflexive, symmetric and transitive then it is

called Equivalence relation.

Some important Relations The intersection of two equivalence relations on a set is an

equivalence relation on the set. The union of two equivalence relations on a set is not

necessarily an equivalence relation on the set. The inverse of an equivalence relation is an equivalence

relation.

Cartesian Product of Sets Let A and B be any two non-empty sets. The set of all ordered pairs (a,b). Such that and is called the Cartesian product of the sets A and B and is denoted by

If set A contains m elements and set B contains n elements, then the number of elements in is mn. Some Important Results on Cartesian product of Sets

1


Functions

Contents

1. Functions2. Intervals 3. Even and odd functions4. Greatest integer function 5. Absolute value function (modulus function) 6. Signum function7. Periodic Function8. Rules for finding period of a periodic function: 9. Explicit Function 10. Implicit function 11. Invertible functions12. Key Notes

Functions

Let x and y be two non empty sets. A function f : x into y (or from x to y) written as f : x y is a rule or a correspondence which connects every member x of X to exactly one member y of Y

Here X is domain of f, and Y is called co-domain of f. The set of f(x) = {f(x), x ∈ X} consisting of all images f(x), is called the range of f i.e Range is always a subset of the co domain, Range is set of out puts.

Y = f(x) is not a function if there is an element in the set of x for which we don't have its image (corresponding element in set y).

Algebraic operations on functions (combination of functions) If f and g are two functions with domains D1 and D2 respectively, then

( f + g ) ( x ) = f(x) +g(x) with domain D1 D2

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( f - g ) ( x ) = f(x) -g(x) with domain D1 D2

( f g ) ( x ) = f(x).g(x) with domain D1 D2

( f/g )(x) = f(x)/g(x) provided g(x) ≠ 0 with domain D1 D2

One - one and Many one functions :

If f(x1 ) = f(x2 ) x1= x2

Then function f(x) is one-one i.e. if distinct elements in domain of f have distinct images in co domain, then f is said to be one -one and if, there exists atleast two distinct elements in domain having same image, then it is many-one function.

One-one functions are also called as injective functions

Onto and Into functions If each element in co-domain have atleast one pre-image in the set of domain, then the function is onto and if there is at least a single element in the set of co-domain which does not have its pre image, then it is into and if range is proper subset of co-domain, then the function is onto. Onto functions are also called subjective functions

Function which are one-one and onto are called bijective Functions

Intervals

Closed interval If a real number x is such that a ≤ x ≤ b, then x lies in closed interval a and b, and is written as x ∈ [a,b]. Ex If x ∈ [2,3] 2 ≤ x ≤ 3 [here endpoints are included ] Open interval If a real number x is such that a < x < b, then x lies in the open interval a and b, and written as x ∈ (a,b) or x ∈ ]a,b[ Ex x ∈ (2,3) 2 < x < 3 [ Here endpoints are not included] Half open, half closed interval If x is such that a ≤ x < b Ex x ∈ [2,3) or [2,3[ 2 ≤ x < 3 [Here left end point is included but right end point is not included]

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Even and odd functions

f(x) is an even function if f(-x) = f(x) for every x ∈ domain Ex f(x) = cos x f(x)= x2 Their graphs are symmetrical about y axis If f(x) = cos x, f(-x) = cos(-x) = cos x f(-x) = f(x) (even function) f(x) is an odd function if f(-x) = - f(x) for every x ∈ Domain

Ex : Their graphs are symmetric about origin If f(x) = sin x f(-x)= - sin x f(-x) = - f(x)

Greatest integer function

[ x ] Denotes the greatest integer function which means greatest integer less than or equal to x, i.e [ x ] ≤ x. Ex [-2] = - 2 , [ -3.14 ] = - 4 , [2.71] = 2 In general [ x ] = n if n ≤ x < n+1 (where n ∈ integer and x ∈ R)

F(x) = [x] , then Domain of this function is R. Range of this function is set of integers Greatest integer function is also called step function. It is many-one function It is not a periodic function Properties of Greatest integer function [x+n]=[x]+n, where x ∈ R and n ∈ integer If x = [x]+{x}, then {x} denotes fractional part of x. {x} [01)for any real number x -1 < | x | ≤ , x ∈ R [-x] = -x if x ∈ integer = -[x] -1 if x ∉ integer [x+y] = [x]+[y] when 0 ≤ | x | + | y | < 1 = [x]+[y]+1 when 1 ≤ | x | + | y | <2

Absolute value function (modulus function)

1


y = | x | =

Also = | x |

Signum function

Domain is R ; Range R ∈ {-1,0,1}

Periodic Function

A function f : x → y is said to be a periodic function if there exist a positive real number T such that f ( x + T ) = f( x ) ∀ x ∈ X. the least value of T is called the fundamental period of a function. In general, the fundamental period or principal period is called the period of a function. Graphically speaking graph gets repeated after each period of the function Rules for finding period of a periodic function:

If f(x) is a periodic function with fundamental period T, then 1/f(x) will also be a periodic function with same fundamental period T

If f(x) is a periodic function with period T, then kf(x) is also a periodic function with same fundamental period T.

If f(x) and g(x) are two functions with fundamental periods T 1 and T 2 respectively then f(x) +g(x) is a periodic function with fundamental period of L.C.M of T 1 and T 2 , provided f(x) and g(x) cannot be interchanged by adding a positive number in x which is less then L.C.M of T 1 and T 2 , in that case period becomes that number and also L.C.M of T 1 and T 2 should exist otherwise this is not a periodic function. Ex f(x) = sin (x/3) + cos (2x/3) is a periodic function with

1


L.C.M of 6 π and 3 π , so period of f(x) is 6π . Ex f(x) = | sin x | + | cos x | , here period of | sin x | and | cos x | is π so L.C.M is π . But there exist a number π /2 (less than π ), which on adding in x can interchange | sin x | into | cos x | and | cos x | into | sin x | by adding into xHence period of f(x) is π/2

If is a periodic function with period T, Then is also

period function with period .

Explicit Function

y is said to be explicit function of independent variable x, if we can write y = f(x) i.e if y can be expressed directly in terms of the independent variable x Eg y = x2 , y = 2x3 + 3x2 + x + 9

Implicit functions

y is said to be implicit function of the independent variable x, if it cannot be expressed directly in terms of x.

Ex

Invertible functions

All one-one and onto (bijective) functions are invertible functions. Their graphs are symmetrical about the line y = x Method of finding inverse of a function Suppose y is written as function of x. First find x in terms of y And write it as f-1(y) Now replace y by x Ex y = sin x D [- π /2, π /2] and co-domain is [-1,1] Let x = sin-1 y = f-1 (y) then Sin-1 x = f-1 (x) Domain and range of Inverse Trigonometric Functions Function Domain Range Sin-1 x [-1, 1]

1


Cos-1 x [-1, 1]

Tan-1 x

Cot-1 x

Sec-1 x (- ∞ -1] [1, ∞ )

Cosec-1 x (- ∞ -1] [1, ∞ )

�Key Notes

If a polynomial satisfies the property f(x) +f(1/x) = f(x) f(1/x) then f(x) = 1 ± xn

Domain of polynomial functions is

Range of odd degree polynomial functions is Even functions are many one functions

• Any function can be written as sum of an even function and

an odd function Every constant functions is an even function and f(x) = 0 is an

even function as well as an odd function. f(x) = x function is an identity function Constant functions are periodic functions with no fundamental

period If f(x) is inverse function of g(x) then f(g(x)) = x or g (f(x))=x i.e.

same as f-1 f(x) = x log x2n =2n log | x | , where n ∈ N x + (1/x) ≥ 2 (if x ∈ R + ) and x + (1/x) ≤ -2 (if x ∈ R-1 ) Two functions y = f(x) and y = g(x) can be combined (added,

subtracted, multiplied, or divided) only in common domains L.C.M of a rational number and an irrational number does not

exist. If number of elements in A = m and number of elements in B =

n, then Total number of functions from A → B is nm

Total of injective function

Number of sujective functions Number of bijective functions = n! Note that a function can be bijective only if m = n

1


Chain Rule

If p : q :: r : s . The method of finding the 4 th term of a proportion when three

are given is known as rule of three as above . Three or more quantities are said to be in compound proportion

if one quantity depends on the other remaining quantities. If p,q,r,s are four quantities & if p : q :: r : s then

1) Componendo

2) Dividendo

3) Componendo & Dividendo

4) Invertendo

5) Alternendo Direct proportion is indicated by arrows in the same direction ,

Inverse proportion is indicated by arrows in opposite direction.

Ratio and Proportions

Ratio ratio is the relation between two numbers which is expressed by a fraction, If the terms of a ratio be multiplied or divided by the same quantity the value of the ratio remains unaltered.

Two quantities which are being compared are called its terms Then the first term is called antecedent and second term is called consequent The Ratio of two quantities is always an abstract number (without any units). Proportion

The equality of two ratio is called proportion consider the two ratios 1 st ratio 2 nd ratio a : b x : y

1


The proportion may be written as a : b : : x : y

The numbers a, b, x and y are called the terms, a and y are called extremes (end terms), and b and x are called the means (middle terms)

Key notes

If a and b are two quantities, then 1) Duplicate ratio of a : b = a2 : b2

2) Sub-duplicate ratio of a : b = 3) Triplicate ratio of a : b = a3 : b3

4) Sub-triplicate ratio

5) Inverse or reciprocal ratio of

6) Third proportional to 'a' and If a : b = x : y and b : c = p : q , then

1)

2) The Ratio in which two kinds of substances must be mixed

together one at Rs. x per kg and another at Rs. y per kg, so that

the mixture may cost Rs. n per kg The ratio is Let the incomes of two persons be in the ratio of a : b and their

expenditure be in the ratio of x : y .and If the saving of each

person is Rs.n then income of each is Rs. and Rs.

respectively. In a mixture the ratio of milk and water is a : b . in this mixture

another n liter of water is added, then the ratio of milk and water in the resulting mixture becomes a : m .

1) Then the quantity of milk in the original mixture is

2) Then the quantity of water in the original mixture is

1


In a mixture of n liter, the ratio of milk and water is x : y . If another m litres of water is added to the mixture, the ratio of

milk and water in the resulting mixture If four numbers a, b, c and d are given then

1) should be added to each of these numbers so that the resulting numbers may be proportional

2) should be subtracted from each of these numbers so that the resulting Numbers may be proportional

Speed

If a man walks a distance 6 km in each hour, we say that his speed is

6 km per hour, i.e., 6 Thus the speed of a body is the rate at which it is moving

Speed = Distance = Time × Speed

Time =

m/sec

km/hr. ¨ If the speed of a body is changed in the ratio m : n , then ratio

of the time taken changes in the ratio n : m .

Average Speed

When certain distance is covered by a body in parts at different

speeds, then the average speed

Caution average speed should not be calculated as average of

1


different speeds, i.e., Ave. speed ≠ There are two different cases when average speed is required.

Case I When time remains constant and speed varies : If a man travels at the rate of x km/h for t hours and again at the rate of y km/h for another t hours, then for the whole journey, his average speed is given by

Average speed

Case II When the distance covered remains same and the speeds vary : When a man covers a certain distance with a speed of x km/h and another equal distance at the rate of y km/h. then for the whole

journey, the average speed is given by Average speed km/h.

Velocity The speed of a moving body is called as its velocity if the direction of motion is also taken into consideration

Velocity

Relative speed

a) Bodies moving in same direction When two bodies move in the same direction, then the

difference of their speeds is called the relative speed of one with respect to the other.

When two bodies move in the same direction, the distance between them increases (or decreases) at the rate of difference of their speeds.

b) Bodies moving in opposite direction

The distance between two bodies moving towards each other will get reduced at the rate of their relative speed (i.e., sum of

their speeds). The

1


Relative speed of one body with respect to other body is sum of their speeds.

Increase or decrease in distance between them is the product of their relative speed and time.

Key notes to solve problems

When a moving body covers a certain distance at x km/h and another same distance at the speed of y km/h, then average

speed of moving body during its entire journey will be km/h

A man covers a certain distance at x km/h by car and the same distance at y km/h by bicycle. If the time taken by him for the whole journey by t hours, then Total distance covered by him

km. A boy walks from his house at x km/h and reaches the school 't 1

' minutes late. If he walks at y km/h he reaches ' t 2 ' minutes earlier. Then, distance between the school and the house.

km

If a man walks with of his usual speed he takes t hours more to cover a certain distance.

Then the time to cover the same distance when he walks with

his usual speed, hours. If two persons A and B start at the same time in opposite

directions from the points and after passing each other they complete thejourneys in 'x ' and ' y ' hrs. respectively, then

If the speed is of the original speed, then the change in time taken to cover the same distance is given by Change in time =

original time

Key notes to solve problems on Trains

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The time taken by a train in passing a signal post or a telegraph

pole or a man standing near a railway line The time taken by a train passing a railway bridge or a

platform or a tunnel or a train at rest where, x = length of the train y = length of the bridge or platform or standing train or tunnel

Time taken by faster train to pass the slower train in the same

direction where, x = length of the first train y = length of the second train u = speed of the first train v = speed of the second train and u > v Time taken by the trains in passing each other while moving in

opposite direction

Time taken by the train to cross a man where, both are moving in the same direction and x = length of the train u = speed of the train and v = speed of the man. Time taken by the train to across a man running in the opposite

direction If two trains start at the same time from two points A and B

towards each other and after crossing, they take a and b hours in reaching B and A respectively. Then.

A train starts from a place at u km/h and another fast train starts from the same place after t hours at v km/h in the same direction. Find at what distance from the starting place both the trains will meet and also find the time of their meeting.

The distance between two places A and B is x km. A train starts from A to B at u km/h. One another train after t hours starts from B to A at v km/h. At what distance from A will both the

1


train meet and also find the time of their meeting

Two trains starts simultaneously from the stations A and B

towards each other at the rates of u and v km/h respectively. When they meet it is found that the second train had traveled x km more than the first. Then the distance between the two stations

(i.e., between A and B)

PIPES & CISTERNS

1. Pipes & Cisterns

PIPES & CISTERNS

If a pipe can fill a tank in x hours, then the part filled in 1 hour

= If a pipe can empty a tank in y hour, then the part of the full

tank emptied in 1 hour = If two pipes can fill a tank in x and y hours respectively and both

the pipes are opened simultaneously then, time taken to fill the

tank hours. A tap fills a cistern in x hours and the other can empty the

cistern in y hours. If both the taps are opened simultaneously,

then, time taken to fill the tank hours If two taps A and B together can fill a tank in x hours and the

tap A only can fill the tank in y hours, The time taken by B alone

to fill the tank is hours. Two pipes can fill a cistern in x and y hours respectively. How

much,,after opening both the pipes, second pipe should be

1


turned off so that the cistern be filled in z hours ?

Required time hours. Three taps A, B and C can fill a tank in x,y and z hours

respectively. If all the three taps are opened simultaneously,

then time taken to fill the tank hours. Two pipes A and B can fill a cistern in x hours and y hours

respectively. A third pipe C can empty the cistern in z hours. If all the three pipes are opened at the same time, then

time taken to completely fill the cistern hours.

Two taps can fill a cistern in n min. and min. respectively. If these taps are opened alternately for a minute, Then the time taken to fill the cistern

Mixture & Allegation

Allegation Mixture is a method to solve some sort of problems where the ratio and averages are involved. Now let us observe the example. A trader has two kinds of sugars with their selling prices Rs. 8 and Rs. 12 But as he is not comfortable with the demand of sales he want to mix both and sell it at the price Rs. 9.5. Now the problem is finding in what ratio the two kinds must be mixed ? Remember all are selling prices. Here we mean same kind of data. Now consider the diagram.

1


Now take the difference of 8 and 9.5 i.e 1.5 put it in the blank in the same line similarly take the difference of 12 and 9.5 i.e 2.5. Put it in the blank in the same line. Now the completed diagram as follows:

Now the ratio of sugars is (1 st Kind) : (2 nd Kind) = (2.5) : (1.5) ie the ratio = 5 : 3 Be carefull while taking the data which must be in the same units.

Replacements

1)

2) Amount of 'B' left = M - Amount of 'A' left

3)

1


Now you follow and have a keen observation on conceptual examples.

CLOCKS , CALENDARS & AGES

1. Clocks Concept2. Calendars Concept3. Ages Concept

Clocks Concept

The dial of the clock is circular in shape and was divided into 60 equal minute spaces

60 minute spaces traces an angle of 3600 ∴ 1minute space traverses an angle of 60

In 1 hour, Minute hand traverses 60 minute space or 3600 ,Hour hand traverses 5 minute space or 300

The hands of the clock are perpendicular in 15 minute spaces

apart

The hands of the clock are in straight line and apposite to each other in 30 minute spaces apart.

The hands of the clock are in straight line when they coincide or opposite to each other.

The hands of the clock are perpendicular to each other for 22 times in 12 hours and for 44 times in a day.

The hands of the clock are opposite to each other for 11 times in 12 hours and 22 times in a day.

The hands of the clock coincides with each other for 11 times in 12 hours and 22 times per day

The hands of the clock are 44 times in a straight line per day

1


The minute hand gain 55 minutes over hour hand per hour.

Hence x minute space to be gained by minute hand over hour

hand can be calculated as Ex : At what time between 2'O clock and 3'O clock the hands of

the clock are opposite to each other.

1. 34( ) past 2'Oclock 2. 43( ) past 2'Oclock

3. 56( ) past 2'Oclock 4. 64( ) past 2'Oclock Sol At 2'O clock the minute hand will be at 12 as shown below

The minutes hand to coincide with the hour hand it should trace at first 10 minute spaces And then the hands of the clocks to be opposite to each other minute hand should trace 30 minute spaces i.e. totally it should gain 10+30=40 minute spaces to be opposite to that of hour hand We know that, Minute hand gains 55 minute spaces over hour hand in 1 hour ∴ Minute hand gain 40 minute spaces over hour hand in 40 × (60/55) = 43(7/11) Hence the hand of the clock will minutes be opposite to each at

43( ) past 2'Oclock ∴ Correct option is �2'

When clock is too fast, too slowIf a clock or watch indicates 6 hr 10 min when the correct time is 6, it is said that the clock is 10 min too fast If it indicates 6. 40 when the correct time is 7, it is said to be 20 min too slow.

1


Now let us have an example based on this concept Ex My watch, which gains uniformly, is 2 min, & show at noon on Sunday, and is 4 min 48 seconds fast at 2 p.m on the following Sunday when was it correct ? Sol: From Sunday noon to the following Sunday at 2 p.m there are 7 days 2 hours or 170 hours.

The watch gains min in 170 hrs.

∴ the watch gains 2 min in hrs i.e., 50 hours Now 50 hours = 2 days 2 hrs. ∴ 2 days 2 hours from Sunday noon = 2 p.m on Tuesday.

Calendars Concept

The time in which the earth travels round the sun is a solar year

and is equal to 365 days 5 hrs. 48 minutes and seconds Year is 365.2422 days approximately. The common year consists of 365 days. The difference between a common year and a solar year is

therefore 0.2422 of a day and we consider it by adding a whole day to every fourth year.

Consequently in every 4 th year there are 366 days. The years which have the extra day are called leap years. The

day is inserted at the end of February, The difference between 4 common years and 4 solar years is 0.969 of a day.

If therefore, we add a whole day to every 4 th year, we add too much by 0.0312 of a day. To take account of this, we omit the extra day three times every 400 years,

The thing is to ensure that each season may fall at the same time of the year in all years. In course of time, without these corrections, we should have winter in July and summer in January also .

With the very small variation, the present divisions of the year are those given in B.C 46 by Julius Caesar . The omission of the extra day three times in 400 years is called the Gregorian Correction. This correction was adopted at once in 1582 in Roman Catholic Countries. but not in England until, 1752. The Gregorian mode of reckoning is called the New Style, the

1


former, the Old Style. The New Style has not yet been adopted in Russia , so that they are now 13 days behind us as an Example What we call Oct. 26 th they call 13 th Oct . They have Christmas day on 7 th of January and we have on 25 th December every year.

In an ordinary year there are 365 days i.e., 52 weeks + 1 day Therefore an ordinary years contains 1 odd day. A leap year contains 2 odd days.

100 year = 76 ordinary years + 24 leap years. = 76 odd days + 48 odd days = 124 odd days = 17 weeks + 5 days.(in the consideration of weeks) ∴ 100 years contain 5 odd days. 200 years contains 3 odd days. 300 years contain 1 odd days

Since there are 5 odd days in 100 years, there will be 20 days in 400 years. But every 4 th century is a leap year. ∴ 400 years contain 21 days. Here 400 years contain no odd days.

As First January 1 A.D was Monday. we must count days from Sunday i.e Sunday for 0 odd days , Monday for 1 odd day , Tuesday for 2 odd days and so on.

Last day of a century cannot be either Tuesday. Thursday or Saturday.

The first day of a century must either be Monday. Tuesday, Thursday or Saturday.

Now let us observe the Examples Ex How many times does the 29 th days of the month occur in 400 consecutive years 1) 97 times 2) 4400 times 3) 4497 times 4) none Sol: In 400 consecutive years there are 97 leap years. Hence in 400 consecutive years, February has the 29 th day 97 times, and the remaining 11 months have the 29 th day 400 x 11 or 4400 times. ∴ 29 th day of the month occurs (4400 + 97) or 4497 times Ex Given that on 10 th November 1981 is Tuesday, what was the day on 10 th November 1581 1) Monday 2) Thursday 3) Sunday 4) Tuesday

1


Sol: After every 400 years, the same day comes. Thus if 10 th November1981 was Tuesday, before 400 years i.e on 10 th November 1581 , it has to be Tuesday.

Ages Concept

In such problems . There may be three types Age some years ago Present age Age some years after Numerically giving two of these situations you will be asked to find the third. The relation between the ages of two or more persons may also be given. Now let us have an Example and then you can have a keen observation on Conceptual Examples Ex The sum of the ages of a mother and daughter is 72 years. After four years the product of their ages was 20 times the age of the mother at that time. The ratio of present ages of the mother and daughter are 1)5:3 2)4:7 3)7:2 4)none Sol: Let the present ages of the mother and daughter be x and (72 - x). ⇒ (x + 4) (72 - x + 4) = 20(x + 4) ⇒ x 2 - 52x - 224 = 0 i.e x = 56. ∴ mother's age = 56 years and daughter's age = 16 years. Ratio=7:2 Hence correct option is 3

LOGARITHMS & LOGARITHMIC SERIES

1. logarithm2. Logarithmic Inequalities3. Exponential series4. Logarithmic series

Logarithm

1


If y = ax (where a > 0 and a ≠ 1) then x = logay As a x > 0, log is defined only for positive real numbers Loga 0 or log of negative number is not real number. Ex :- If log3 5 = x ↔ 5 = 3x Some key points on logarithms:

ak = x ⇔ loga x = k ; a>0,a≠0,x>0

loge(ab) = loge(a) + loge(b)

log a 1 = 0 log a a = 1

a log

b a = b

logba→-α if b > 1 and a→0+ logba→α if b > 1 and a→α logba→α if 0 < b < 1 and a→0+ logba→-α if 0 < b < 1 and a→α

Logarithmic Inequalities

o log y x > z → x > Yz , when y > 1 o log y x < z → 0< x < yz , when y > 1 o log y x > z → 0 < x < yz , when 0 < y < 1 o log y x < z → x > yz , when 0 < y < 1

o

o If the characteristic of log 10 n is k, then the number of digits in n is k+1

o If the characteristic of log 10 n is k i.e -k, then the number of zeros between decimal point and significant (non zero figure) is k-1

1


Exponential series

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

Logarithmic series

1


On subtracting (ii) from (i), we get

INTERESTS & DISCOUNTS

Contents

1. Introduction2. Simple interest 3. Key notes on Simple interest4. Compound interest5. Key notes on Compound interest6. True Discount7. Bankers Discount

�

Introduction

Interest is the money paid for the use of money borrowed, i.e., extra money paid for using other's money is called interest.

Here the sum borrowed is called the principal. The total sum of interest and principal is called the Amount.

Simple interest

For a certain period , If the interest on a certain sum borrowed is reckoned uniformly, then it is called simple interest, denoted by S.I. Here

1


A = Amount P = Principal I = Interest T = Time (in year) R = Rate percent per annum

The following are the relations

...............(1)

.................(2)

..................(3)

..............(4)

.............(5) ...................(6)

Key notes on Simple interest

If some Principal of money at simple interest amounts to Rs. A 1 in T 1 years and Rs A 2 in T 2 years, then the sum and rate of interests are

Principal

Rate A sum of money becomes n times itself in T years at simple

interest , then the rate of interest is Rate If a sum of money becomes n times in T years at S.I then it will

be m times of itself in ..... years

Required time years

If S.I on a sum of money is th of the principal and the time T is

equal to the rate percent R then Rate = Time

1


A certain sum is at S.I at a certain rate for T years. And if it had been put at R 1 % higher rate, then it would fetch Rs. x more

Then the Principal The annual payment that will discharge a debt of Rs. P due in T

years at the rate of interest R% per annum is Annual payment

Let the rate of interest for first t 1 years is r 1 % per annum, for

the next t 2 years is r 2 % per annum and for the period beyond that is r 3 % Suppose all together the simple interest for t 3 years is Rs.

I .Then Principal The simple interest on a certain sum of money at r 1 % per

annum for t 1 years is Rs. m .The interest on the same sum for t 2 years at r 2 % per annum is n

Then the sum

Compound interest

If interest as it becomes due and is not paid to the lender but is added on to the principal, then the money is said to be lent at compound interest And the total sum owed after a given time is called the amount at compound interest for that time. In a certain period, the difference between the amount and the original principal is called the compound interest (C.I).

Here Principal = P Time = n years Rate = R% p.a Then

when interest is compounded annually the amount is when interest is compounded half yearly the amount is

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When interest is compounded quarterly the amount is

When interest is compounded k times a year the amount is

Key notes on Compound interest

The amount of Rs. P for 3 years when the rate of interest for first year is R 1 %, for second year is R 2 % and third year is R 3 % when compounded annually. Amount

If a certain sum of money becomes x times in t years put at compound interest compounded annually , then it will be y times

in ..... years Required time years. If a Principal at compound interest amounts Rs. x in t 1 years

and Rs. y in t 2 years , then the rate of interest per annum is .

Rate% where, n= t 2 - t 1

Rate (where n = t 2 - t 1 ) If a loan of Rs. P at r% compound interest p.a is to be repaid in

two equal yearly instalments, then the value of each instalment is.

If a loan of Rs. P at r% compound interest p.a is to be repaid in

three equal yearly instalments, then the value of each instalment is

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The difference between compound interest and simple interest

on Rs. P for 2 years at R% p.a Difference The difference between compound interest and simple interest

on Rs. P for 3 years at R% p . a Difference

True Discount

Suppose X owes Y Rs. R at the end of one year and if Y demands it now. Naturally X will pay Y some amount less in consideration of immediate payment. This reduction is called True- Discount . Interest is calculated on the P.W., whereas True Discount is calculated on the Sum Due. Formulae: When Interest is compounded

Present Worth

True Discount

When Interest is simple T is the time and R is the rate per cent.

Sum Due (S.D) = P.W + T.D

P.W = S.D - T.D at simple interest

Bankers Discount

Banker Discount (B.D) It is the simple interest on the sum due for the unexpired period ( time ). Banker Gain (B.G) It is the difference between the simple interest

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on the sum due and that on the present worth of the sum due. B.G is the simple interest on the T.D.

Formulae

B.D

B.D B.G = B.D - TD

Sum due

Rate

Time Money paid by the banker = S.D - B.D Three days of grace are always added while calculating time in

questions on Bankers Discount

Let us observe the Example Ex A bill is drawn for Rs. 6565 on the 2 nd April at 5 months credit. And it is discounted on 24 th June at 5% per annum then Find: • Bankers Discount, • Money received by the holder of the bill, and • The Bankers Gain. Sol: S.D = Rs. 6565 Date of drawing = 2 nd April (for 5 months) Date of maturing = 5 th Sep (including 3 days grace) Date of discounting = 24 th June

No. of days from 24 th June to 5 th September = 73 years

Bankers discount

= Rs. 65.65 P Amount received by the holder of the bill = S.D - B.D = Rs. 6565 - Rs. 65.65 P = Rs. 6499.35 P

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True discount on Rs. 6565 B.G = B.D - T.D = 65 P

Profit & Loss

1. Profit and Loss2. Key Notes

Profit and Loss

Cost PriceCost price (C.P) is the price at which a particular article is bought. Profit and loss both are marked on cost price.

Selling Price (S.P)Selling price is the price at which a particular article is sold. OverheadsThe expenses incurred on transportation, maintenance, Packaging, advertisement and the like are considered as Overhead. These overheads and the profit when added to the cost price determine the selling price.

Profit or Gain

Whenever a person sells an article at price greater than the cost price he is said to have profit or gain.Profit or Gain = S.P - C.P

If S.P is less than the C.P. there is loss.Loss = C.P - S.P Loss or gain are always reckoned on the cost price only Gain for

Rs. 100 is Gain percent

Gain % Loss for Rs. 100 is Gain percent Loss %

in case of profit

in case or loss

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S.P. in case of gain or profit

S.P. in case of loss If the C.P. of m goods = S.P. of n goods. Then

Gain %

Loss %

Key Notes

When an article is sold at a profit of x %. If it would be sold for Rs. n less, there would be a loss of y %, then the cost price of

the article C.P. A man sells an article at a gain of x %. If it would have been sold

for Rs. n more, there would have a profit of y %, then C. P =

A person brought two articles for Rs n. On selling one article at x % profit and other at y% profit, he get the same selling price of each,then

C.P. of first article

C.P. of second article When m articles are bought for Rs. n and n articles are sold for

Rs. m and m > n, then Profit % If A sells an article to B at a profit of r 1 %, B sells it to C at a

profit of r 2 % and C sells it to D at a profit of r 3 %, then, cost

price of D. If A sells an article to B at a loss of r 1 %, B sells it to C at a loss

of r 2 % and C sells it to D at a loss of r 3 %, then, Cost Price of

D = Cost price of A dealer purchases a certain number of articles at x articles for

a rupee and the same number at y articles for a rupee. He mixes

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them together and sells at z articles for a rupee.

Then his gain or loss % according to positive or negative sign.

If 'P 1 ' is rate gain w r t selling price S 1 and 'P 2' is rate gain w r t selling price S 2

Then, C.P. = difference between selling prices. If 'P 1 ' is rate gain w r t selling price S 1 and 'P 2' is rate loss w r t selling price S 2

C.P. difference between selling prices.

When a man sells two things at the same price each and in this process his loss on first thing is x % and gain on second thing is x %, then in such a type question, there is always a loss. and

Loss When a man buys two things on equal price each and in those

things one is sold on the profit of x % and another is sold on the loss of x %, then there is no loss or no gain per cent.

A sells an article at a profit of r 1 % to B . and B again sells it to C at a profit of r 2 %.. If C pays Rs. P to B, then, C. P. of the

article for A dishonest seller prefers to sell goods at his cost price but uses

a false weight of x grams for each kilogram. Then his gain per cent,

When a shopkeeper on selling an article for Rs. n, gains as much per cent as the cost price of it, then C.P. of the article

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If there is loss in place of profit, then C.P. of the article =

Partnerships

Kinds of partnershipKey notes

Partnership

When two or more than two persons agree to invest money to run a business jointly, this association or deal is called partnership and those who invest money are called partners. The total investment is called the capital. Kind of partners There are two kinds of partners. Working or active partner When a partner devotes his time for the business in addition to invest his money, he is called a working partner. With mutual agreement, the active partners get some fixed percentage of profit as working allowance. Sleeping or non � active partner A partner who simply invests money, but doest not attend to the business is called a sleeping partner. Kinds Of Partnership

Simple partnership If the capitals of several partners are invested for the same period. It is called a simple partnership. Compound or complex partnership : If the capitals of the partners are invested for different intervals of time, the partnership is called compound or complex.

Key notes

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If the capitals of two partners are invested for the same duration of period and let A 1 and A 2 be their investments and total profit is Rs. P, then share of the partners in the profits are

Rs. If the capitals of two partners be Rs. A 1 and A 2 for the periods

t 1 and t 2 respectively and the profit be Rs. P, then shares of

the partners in the profit are Rs.

Permutations & Combinations

Contents

1. Factorial

2. The symbols & or C(n, r)

3. Relation between and

4. Properties of or C(n,r) 5. Fundamental Principal of multiplication 6. Fundamental Principle of Addition 7. Permutation 8. Circular Permutation9. Grouping

Factorial

The continued product of first 'n' natural numbers is called the 'n factorial' and is denoted by n! i.e n! = 1 × 2 × 3 × 4 × ......... × (n-1) × n Ex : 4! = 1 × 2 × 3 × 4 5! = 1 × 2 × 3 × 4 × 5

Note : 0! = 1 Factorials of proper fractions (or) negative integers are not

defined Note : The product of r consecutive positive integers is divisible

by r!(kn)! Is divisible by (n!) kThe product of 2n consecutive negative integers is divisible by (2n)!

Useful symbols

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In this section we can have two symbols v i z, nPr or P(n,r) and nCr (or) C(n,r)

: If 'n' is a natural number and 'r' is a positive

integer satisfying 0 ≤ r ≤ n, then the natural number is

denoted by the symbol or P(n,r)

Ex : = 20

or c (n,r) : If n is a natural number and r is a positive integer

satisfying 0 ≤ r ≤ n, then the natural number Is denoted

by the symbol or c(n,r) thus

Ex : = C (9,3) = = 84

Relation between and

Properties of

= nCn-r for 0 ≤ r ≤ n If x and y are non negative integers such that x + y = n then nCx

= nCy Let n and r be non negative integers such that r ≤ n then nCr +

nCr-1 = n+1Cr

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Let n and r be non-negative integers such that 1 ≤ r ≤ n then nCr

= . n-1Cr-1 If 1 ≤ r ≤ n, then n. n-1Cr-1 = (n - r+1) nCr-1 nC x = nCy ⇒ x = y or x+y =n

If n is even, then the greatest value of nCr (o ≤ r ≤ n) is

If n is odd, the greatest value of nCr (o ≤ r ≤ n) is or

Fundamental Principal of multiplication

In an experiment If there are two operations such that operation 1 can be performed in m ways and operation 2 can be performed in n ways then the experiment could be completed in mn ways. Here we can extend the same principle to any number of operations. Ex: A letter lock consists of 5 rings each marked with 10 different letters. What is the maximum number of unsuccessful attempts to open the lock. Sol:E ach ring is marked with 10 different letters. Hence each ring has 10 positions. Thus, the total number of attempts that can be made to open the lock is 10 x 10 × 10 × 10 × 10 = 105 . Out of these, there must be one attempt in which the lock will open. ∴ Total number of unsuccessful attempts = 105 -1

Fundamental Principle of Addition

If there are two operations such that they can be performed independently in m and n ways respectively, then either of the two

operations can be performed in ways Permutation : Each of the different arrangements which can

be made by taking some or all of a given number of distinct objects is called a permutation

Combinations : Each of the different selections made by taking some or all of a given number of distinct objects, irrespective of their arrangements or order in which they are placed is called a combination.

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The number of all permutations of n distinct items or objects taken 'r' at a time is n(n-1)(n-2)...........(n-(r-1)) = nPr

The number of all permutations of n distinct objects taken all at a time is n!

The number of ways of selecting r items or objects from a group

of n distinct items or objects is The number of ways of selecting zero or more things from n

different things, is . The number of ways of selecting at least r things from n

different thins is . The number of ways of selecting at most r things from n

different thins is . Ex: Sanjeev has 7 friends. In how many ways can he invite one or more of them to dinner. Sol : The required number of ways

=

= . The number of ways selecting one or more things from

things of which p are alike of one kind, q are alike of second kind, r alike of third kind while s are different , is

The sum of the digits in the units place of all numbers formed

from the digits taken all at a time and without

repetition, is The sum of all the numbers that can be formed by using the

digits without repetition, is

Let N = where are different primes

and are natural numbers, then the total number of

divisors of N including 1 and N, is And sum of these divisors , is

..................................(

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The number of ways in which N can be resolved as a product of

two factors, is if N is not perfect

square and if N is a perfect square. The number of all permutations of n different objects taken r at

a time When a particular object is to be always included in each arrangement is n-1Cr-1 × r!

When a particular object is never taken in each arrangement is n-1Cr × r! When two specified objects always occur together is n-

2Cr-2 × (r-1)! × 2!

Permutations of n things, taken all at a time, of which p are alike of one kind, q alike of second kind such that p + q = n is

Permutations of n things of which P1 are alike of one kind, P2 are alike of second kind, P3 are alike of third kind ...................... Pr are alike of r th kind such that

Circular Permutation

The number of circular permutations of 'n' distinct objects is (n-1)!If anticlockwise and clockwise order of arrangements are not distinct like arrangements of flowers in a garland etc, then the number of circular permutations of n distinct items is ½ {(n-1)!}

Grouping

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Number of ways in which (m + n + p) different objects can be divided into three groups of sizes m, n and p objects respectively

The total number of ways of dividing 'n' identical objects into 'r' groups, if blank group are allowed is m+r-1Cr-1

The total number of ways of dividing 'n' identical objects into r groups, if blank groups are not allowed is n-1Cr-1

Derangement If n arranged different objects are to be rearranged such that none of the object is at its original position then number of ways is

Probability

Contents

1. key notes on probability 2. Addition theorem on probability-->Multiplication theorem of

probability3. Probability related to sets4. Bayes' theorem 5. Calendar Problems

key notes on probability

An experiment which can be repeated any number of times under identical conditions and which is associated with a set of known results, is called a random experiment or trail

The result of any single repetition of a random experiment is called an elementary event or simple event.

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The list of all elementary events in a trail is called list of exhaustive events.

Elementary events are said to be equally likely if they have the same chance of happening.

The favorable cases to a particular event of an experiment are called successes and the remaining cases are called failures with respect to that event.

If there are n exhaustive equally likely elementary events in a

trail and m of them are favorable to an event A. then is called the probability of A. it is denoted by P(A)

If A is an event in a sample space S , then .

Two events A , B in a sample S are said to be complementary if

. Let A be an event in sample space S . An event B in S is said to

be complement of A . If A , B are complementary in S. The

complement B of A is denoted . The complement of an event A in a sample space S is unique. If

is the complement of A of A the= A

If A is an event in a sample space S, then Let A,B be two events in a sample space S. If

The set of all possible outcomes (results) in a trial is called sample space for the trail. It is denoted by S. The elements of S are called sample points.

Let S be a sample space of a random experiment. Every subset of S is called an event.

Let S be a sample space. The event Æ is called impossible event and the event S is called certain event in S.

Let S be as sample space. An event E in S is said to be a simple event if E contains only one sample point.

Two events A, B in a sample space S are said to be disjoint or mutually exclusive if

The events A 1 ,A 2 ,.....A n in a sample space S are said to be mutually exclusive or pair wise disjoint if every pair of the events A 1 ,A 2 ,.....A n are disjoint

Two events A, B in a sample space S are said to be exhaustive if .

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The events A1, A2...A n in a sample space S are said to be

exhaustive . If . If A 1 , A 2 ,.....A n are n mutually exclusive events in a sample

space S then

If A is an event in a sample space S , then the ratio is

called the odds in favor to A and is called the odds against to A.

Addition theorem on probability :

If A , B are two events in a sample space S. Then

If A, B, C are three events in a sample space S, then

If A, B are two events in a sample space then the event of happening B after the event A happening is called conditional

event. It is denoted by If A, B are two events in a sample space S such that P(A) ≠ 0.

Then

Multiplication theorem of probability:

Let A, B be two events in a sample space S such that P(A) ≠ 0, P(B) ≠ 0. Then

Two events A, B in a sample space S are said to be independent

if Two events A, B in a sample space S are independent if

. If A,B are two independent events in a sample space S, then

1. are independent

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2. are independent

3. are independent. If p, q are the probabilities of success , failure of a game in

which A , B play. If A starts the game then Probability of A ...s

win =

Probability of B's win= if A starts the game If n letters are put at random in the n addressed envelopes, the

probability that all the letters are in right envelopes = at least

one letter may be in wrongly addressed envelope = 1 all the letters may be in wrong envelopes =

Probability related to sets:

If A and B are two finite sets and if a mapping is selected at random from the set of all mappings from A into B then the probability that the mapping is a

one one function is

many one function is

Constant function is

One one onto function is

Bayes' theorem :

If A 1 , A 2 ......,A n are mutually exclusive and exhaustive events in a sample space S such that

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is any event with

Calendar Problems

Now observe the essential example carefully.

Discuss the Probabilities under mentioned • Exactly 52 Sunday in a leap year. • Exactly 53 Fridays in a leap year. • Exactly 53 Thursdays in a non leap year. Sol : In a leap year as you know that there will be 366 days in which every week day will appear 52 times cyclically (because if we divide 366 with 7 we will get quotient as 52 and the remainder as 2. it means that finally there will be 2 days left) Those two days may be Sunday, Monday, or Monday, Tuesday Tuesday, Wednesday Wednesday, Thursday Thursday, Friday Friday, Saturday Saturday, Sunday Now discussing about the first bit, exactly 52 Sundays in the sense we should not find any matching which is of Sunday. Here the number of favorable cases are 5. The total number of cases are 7

Hence the probability is . Now discussing about the second bit, exactly 53 Fridays in the sense we should get the matching which gives us Friday. Here the number of favorable cases are 2 Total number of cases are 7

Hence the probability is Now discussing about the Third bit, as a non leap year has 365 days, apart

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from the quotient 52 the remainder is 1. it means that that one day may be any one among Sunday Monday Tuesday Wednesday Thursday Friday Saturday Here as we need 53 Thursdays The number of favorable cases is 1 Total number of cases is 7

Hence the probability is Similarly we can have the probability of exactly 52 Sundays in a non leap

year is .

Basic Geometry

Contents

1. Angle 2. Types of angles 3. Parallel and non parallel lines4. Triangle 5. Quadrilaterals6. Polygons7. circles

Angle

When two non-parallel and coplanar lines intersect , at the point of intersection the measure of rotational displacement is called an angle

1


Types of angles

If ' ' is an angle such that

Parallel and non parallel lines

Two lines are said to be parallel lines if they are coplanar and non intersecting

The point of intersection of parallel lines is at infinite places which is not real

The angle between two parallel lines is either 0 0 or 180 0 or 360 0

Two lines are said to be inclined lines ( non parallel ). If they are coplanar and intersect at a real point

The point of intersection of inclined lines is real

Transversal if a line intersects two parallel lines it is called transversal Suppose l

1 , l 2 are two parallel lines and 'l' is a transversal, then we will have eight angles as shown in figure

1


Vertical angles : The angles pair wise are called Pairs of vertical angles. The corresponding pairs of vertical angles are always equal.

Corresponding angles : The angles

pair wise are called corresponding angles. The corresponding pairs of corresponding angles are equal.

Alternate angles: The angles are called pairs of alternate angles. The corresponding pairs of alternate angles are equal.

Complementary angles: Two angles whose sum is 90 o , are called Complementary angles.

Supplementary angles : Two angles whose sum is 180 o are called supplementary angles.

TRINAGLE

Sum of the three angles is always 180 o , i.e. The exterior angle is equal to the sum of the two other interior

opposite angles. i.e. Sum of the lengths of any two sides is greater than the third

side, i.e. AB + BC > AC AB + AC > BC BC + CA > AB:

Area of the Δ ABC = The side opposite to greatest angle is greatest and opposite to

smallest angle is smallest and vice versa. The line joining mid points of two sides of a Δle is always parallel

to the third side and it is half of the third side.

1


i.e. &

Basic proportionality theorem : If a line is drawn parallel to one side of a triangle. then it will divide other two sides in the same ratio.

If The converse of this theorem is also true

i.e. If Pythagoras theorem : In a Δle if the sum of the squares of any

two sides is equal to the square of third side then it is a right angled triangle and right angled at longest side.

The converse of the above theorem is also true.

Key points on Triangles In a triangle ABC if AB 2 > BC 2 + CA 2 then it is obtuse angled Δle

where ∠C is obtuse. In a Δle ABC if AB 2 + BC 2 > AC 2 and AB 2 + AC 2 > BC 2 then the

Δle is acute angled Δle.

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In

If If 'n' is even natural number then the D le with the sides

form a right angled D le.

If 'n' is odd then the triangle with sides n, form a right angled D le.

Median of Triangle : A line joining the mid point of a side to the opposite vertex is called Median of a triangle.

Properties of median :

In a ΔABC, if 'AD' is the median then it divides ΔABC into two equal parts (i.e.)ΔADB =ΔADC.

Centroid : The point of concurrence of the medians of a triangle is called centroid and is denoted by 'G'

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If 'G' is the centroid of ΔABC, then AD is one of its median then 'G' divides AD in the ratio 2 : 1.

If 'G' is the centroid ; then AB 2 + BC 2 + CA 2 = 3 (AG 2 + BG 2 + CG 2 )

Δ ABG =ΔBCG =ΔACG = ΔABC In a Δ ABC if D, E, F are the mid points of the sides BC, CA and

AB respectively.

then Δ AEF = ΔBDF = ΔCDE =Δ DEF = Δ ABC and

Angle bisector

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Internal angle bisector: A line bisecting an internal angle of a Δle is called internal angular bisector.

In a Δ ABC, if AD is the (internal) angular bisector of A, then it divides the side BC in the ratio of remaining sides. i.e. BD : DC = AB : AC

Incentre : The point of concurrence of internal angular bisectors of a Δle is called Incentre and is denoted by 'I'

Incentre of Δle is equidistance from three sides. If 'I' is the Incentre of a Δle ABC, and AD is angular bisector then

External angule bisector : A line bisecting an external angle of a triangle is called external angular bisector.

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Ex-centre: A point of concurrence of two external angle bisectors and one internal angle bisector is called Excentre.

Opposite the corresponding three vertices there exists three

Excentres I 1 , I 2 ,I 3

The Incentre of a Δle is the orthocentre of the Excentral Δle which is of 3 Excentres.

Altitude A line passing through a vertex and perpendicular to its opposite side is an altitude of a triangle.

Orthocentre : The point of concurrence of altitudes of a triangle is called orthocentre and is denoted by 'O'.

Perpendicular bisector : A line perpendicular to a side bisecting the side is called perpendicular bisector of a triangle

Circumcentre : The point of concurrence of perpendicular bisectors of a triangle is called circumcentre and is denoted by 'S'.

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Circumcentre of D le is equidistant from all the 3 vertices.

Key Points In a rightangle triangle Orthocentre is the rightangled vertex and Circumcentre is the midpoint of hypotenuse.

In a equilateral triangle centroid. Orthocentre, Circumcentre, Incentre are all coincident.

In a triangle O, G, S are collinear and G divides OS in 2 : 1 ratio. If the lengths of 2 medians of a D le are equal then it is an

isosceles D le. If the lengths of 3 medians of D le are equal then it is an equilateral Δle.

Similar triangles Two triangles ABC, DEF are said to be similar if their corresponding angles are equal (or) the ratio of their corresponding sides are equal

(i.e) Δ ABC ~ ΔDEF ↔ Note : Two congruent triangles are similar but the converse need not be true.

A.A.A. Similarity: If the corresponding angles of two triangles are equal then they are similar.

S.S.S. Similarity: If the ratio of the corresponding sides of two triangles are equal then they are similar.

S.A.S Similarity : If the ratio of two corresponding sides of two triangles are equal and their included angles are equal then they are similar.

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Similarity in Right angled triangle

If ABC is a triangle such that then ΔACD ~ΔCBD ~ ΔABC

There are three similar right angled triangles in the above figure. From the above figure. CD2 = AD. DB AC2 = AD. AB BC2 = BD. AB

The ratio of the areas of two similar triangles is equal to ratio the squares of their corresponding sides.

In two similar triangles, the ratio of the altitudes is same as the ratio of their corresponding sides.

Congruent triangles

If two triangles have the corresponding sides and angles are equal then they are called congruent triangles i.e. the triangles having with same size and shape are called congruent triangles. i.e. triangles ABC ΔBEF ↔

AB = DE; BC = EF; CA = FD and

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Note : The areas of two congruent triangles are also equal.

In a triangle ABC, the ratio of the sides is equal to the ratio of the their corresponding sine angles. i.e. a : b : c = sin A : sin B : sin C

In a right angled Isosceles triangle, the hypotenuse is times of its equal side, Here the angle are 90o , 45o , 45o ,

In an Equilateral triangle with side 'a' and altitude 'h', we have h

= and

The area of a Δ ABC is

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Two triangles on the same base and in between the same parallel lines have equal area.

Two triangles with equal areas, on the same base should lie between the same parallel lines (or) their heights are equal.

If in a triangle ABC, a circle is inscribed by touching the sides at P.Q. R respectively then

AP + BQ + CR = PB + QC + RA = (AB + BC + CA)

QUADRILATERALS

Quadrilateral : A closed polygonal region with four sides is a quadrilateral.

The sum of the four angles of a quadrilateral ABCD is 360o .

Area of quadrilateral ABCD = Here AC, BD are called diagonals of ABCD.

Types of Quadrilaterals .

You will find the entire measuring phenomena in Mensuration chapter

Cyclic quadrilateral : A quadrilateral inscribed in a circle is called a cyclic quadrilateral. In a cyclic quadrilateral ABCD, A + C = 180o = B + D

1


Trapezium : A quadrilateral in which a pair of opposite sides

are parallel is called Trapezium Area =

Parallelogram : A quadrilateral in which opposite sides are parallel and equal is called a parallelogram.

Rectangle : A parallelogram in which each angle at the vertex is a right angle.

Rhombus : A parallelogram in which all sides are equal.

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Square : A parallelogram which is both rectangle and rhombus.

Properties.

Quadrilateral

Lengths of the diagonals

Perpendicularity of diagonals

Area

a)Parallelogram

Not equal

No Base × height

b) Rectangle

Equal No

Length × breadth

c) RhombusNot equal

Yes

d) Square Equal Yes (side)2

A cyclic trapezium is a parallelogram. A cyclic parallelogram is a rectangle.

In a rectangle ABCD, P is a point then PA2 + PC2 = PB2 + PD2

The figure formed by joining the mid-points of the sides of quadrilateral is given by

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Quadrilateral ⇒ Quadrilateral

Trapezium ⇒ Parallelogram

Parallelogram ⇒ Parallelogram

Rectangle ⇒ Rhombus

Rhombus ⇒ Rectangle

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Square ⇒ Square

In a Rhombus ABCD, AC2 + BD2 = 4AB2

In a square ABCD, diagonal is times of its side and its

Two parallelograms on the same base and in between two parallel lines have the same areas.

(i.e.) If AB | | CF then ABCD = ABEF

A parallelogram and a triangle are on the same base and in

between two parallel lines then the area of the triangle is half of the area of the parallelogram.

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If AB | | CD then D ABE =

If a circle is inscribed in quadrilateral ABCD, Then AB + CD = BC + AD

POLYGONS

Polygon : A closed diagram with a finite number of sides is a polygon. If a polygon has all equal sides then it is called a regular polygon.

The number of diagonals of a regular polygon with 'n' sides is

Name of the regular polygon with a given number of sides and its diagonals :

Name of the polygon Number of sides Number of diagonals Quadrilateral 4 2 Pentagon 5 5 Hexagon 6 9 Heptagon 7 14 Octagon 8 20 Nonagon 9 27 Decagon 10 35

In a regular pentagon, the number of sides and the number of diagonals are equal.

Interior and Exterior angles of a Regular polygon of 'n' sides :

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Exterior angle : The angle subtended by a side of the regular polygon at the vertex out side

Exterior angle : Sum of all exterior angles = 2 p

Interior angle : The interior angle of the regular polygon is

The sum of all the interior angles of the regular polygon = (2n ' 4) right angles

Area of a regular polygon :

The area of a regular polygon of 'n' sides with side 'a' = n

The area of a regular hexagon with side a = 6

The area of a regular polygon of 'n' sides escribed on a circle of

radius 'r' = n

1


The area of a regular polygon of 'n' sides inscribed in a circle of

radius 'R' = n

CIRCLES

? Centre & Radius of Circles: In a plane The locus of all points which are at a constant distance r from a fixed point. 'C' is called the circle and C is centre and 'r' is called radius.

Properties :

Chord : A line which cuts a circle at two distinct points is called a chord. if a chord AB is drawn from P then it is called a secant of P.

1


A perpendicular from the centre to a chord bisects the chord.

The length of the chord where 'r' is radius of the circle and'd' is perpendicular distance from centre to the chord.

Two chords AB, CD of a circle intersect at E, then AE. EB = CE. ED

Two equal chords of circle are at equal distances from its centre.

Tangent : A line which touches a circle at a single point 'P' is called the tangent to the circle at 'P'. Here 'P' is called point of contact of the tangent line.

The tangent and radius of the circle are perpendicular to each

other at the point of contact.

1


From an external point, two tangents can be drawn to a circle.

Here the lengths of the two tangents are equal. Here PA = PB

The length of the tangent from 'P' to a circle of radius 'r' and

centre 'C' such that CP = d is

If PAB is a secant and PT is a tangent to a circle then PA . PB = PT 2

Position of two circles and the number of their common tangent : Suppose r 1 , r 2 are the radii of two circles whose distance between their centres is 'd' then

If , The circles touch each other internally and the number of common tangents possible are 1 i.e. it is direct common tangent

1


If , The circles touch each other externally and the number of common tangents possible are 3 i.e. two direct common tangents and one transverse common tangent

If , The circles don't touch as shown in figure and Hence no common tangents are possible

If , The circles don't touch as shown in figure Here four common tangents are possible i.e. Two direct common tangents and two transverse common tangents

If , The circles intersect each other as shown in figure Here two common tangents are possible which are direct common tangents

1


The length of direct common tangents is

The length of transverse common tangent is

Two circles having same centers with different radii are called concentric circles

The angles on the same segment of a circle are equal.

The angle subtended by an arc of a circle at the centre is double the angle subtended by the same are at any point on the circumference of the circle.

Alternate segment theorem: If a line touches a circle and from the point of contact if a chord is drawn to the circle then the angles between the tangent and the chord are equal to the angles in the alternate segment.

1


If AB is a segment (chord) of a circle and 'C' is a point on the circle such that

If ∠ ACB is acute then C lies on the major segment of the are AB.

If ∠ACB is obtuse then C lies on the minor segment of the are AB.

Mensuration

Contents

1. Quadrilaterals2. Path ways3. Regular polygons4. Circle5. Solids6. Cube & Cuboids7. Prism8. Cylinder9. Pyramid10. Cone11. Sphere

TRIANGLES

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a, b and c are three side of the triangle; h is the altitude and AC is the base

If then

Perimeter 2s = a + b +c

Area = base height = any side length of perpendicular

dropped on it =

Right angled triangle

b,c are the the lengths of the two sides of right angle as shown in figure

1


( Pythagoras theorem )

Perimeter 2s = a+ b +c

Area = product of two sides of right angle

Equilateral triangle

Perimeter = 3a

Isosceles triangle

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b is the length of two equal sides a is the base, AD is the perpendicular dropped on base such that it divides the base equally

Here, AB = AC = b and CB = a corresponding two angles , are equal

QUADRILATERALS

Quadrilateral

AC is the diagonal = d

DF and BE are two perpendiculars drawn on the diagonal (AC) p 1 , and p 2 are the lengths of that perpendiculars

1


Perimeter = sum of all four sides.

Parallelogram

b is the base

h is the perpendicular distance between the base and its opposite side

opposite sides are equal and parallel to each other.

a,b are two sides, d is diagonal , p is the perpendicular dropped on that diagonal as shown in diagram

; ;

Opposite angles are equal ;

Diagonals bisect each other but not equal

Area

1


AB║DC

If θ is the angle between a and b

Diagonals bisect each other

If then

where

Square

a = length of side

d = diagonal

All sides are equal and opposite sides are parallel to each other

AB = BC = CD = DA = a

1


All the angles are equal to 90 o ,

Diagonals are Equal and bisect each other at 90 o

BD = AC and

Perimeter = 4a =

(Where p is the perimeter of the square)

Rectangle

l = length

b = breadth

d = diagonal

Opposite sides equal and parallel to each other

Here,

All angles are equal to 90 o

Both diagonals are equal and bisect each other

O is the bisecting point of diagonals

1


if area = where p and A are known and l and b are unknown.

The two values of x will give l and b

Rhombus

a = each side

d 1 = diagonal

d 2 = another diagonal

h = height

All sides are equal Opposite sides are parallel to each other

Hence ; ;

Opposite angles are equal ;

Diagonals not equal but bisect each other at 90 o ; ,

; here A stands for Area

: Here A stands for Area

1


Trapezium

a and b are two parallel sides, h is the height

;

As shown in diagram

=

Pathway along a rectangle

Case I

If Pathway is outside a given rectangle

Let the length and breadth of the rectangle be l, b and, Width of path way be W, then

Area of pathway = 2W (shaded portion)

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Case II

If path way is inside a given rectangle

Area of pathway = 2W (shaded portion)

Parallel path (Shaded portion As shown in figure )

In a rectangle, if there are two paths drawn parallel to the length and breadth respectively as shown in figure, where w is the width of each parallel path.

Then, area of two parallel paths (shaded portion)

Regular polygons

Let us imagine a regular polygon of n sides as shown in the diagram at

Just for a example sake take n = 8

1


Here is exterior angle of a regular polygon of n sides, =

is interior angle of a regular polygon of n sides,

Let a be the length of each side then Area of the regular polygon of n

Sides is

Area of a circle inscribed inside a regular polygon of n sides is

where a is length of each side

1


Area of circumscribing circle of a regular polygon of n sides is

Circle

Let us observe the diagram

O is the centre of the circle

r is the radius of the circle

diameter of the circle is d = 2r

Circumference (or perimeter) of the circle is : Here

Area of the circle

If C is circumference and A is area, then ;

Sector

Sector of a circle is a part of the area of a circle between two radii.

1


Length of the part of the arc xy = where is angle enclosed by to radii and r is radius as shown in figure

Area of a sector (where q is the angle between two radii)

Segment: "sector minus the triangle formed by the two radii�" is called the segment of the circle.

Area of the segment = Area of the sector -Area D OAB

=

Perimeter of segment = length of the are + length of segment AB

Circular pathway

Case I: A circular path way is constructed out side a given circle pf, radius r the width of the path way is

1


Circle is of radius = r, there is pathway, outside the circle of width = W

Area of circular pathway

Case II: When, the pathway is inside a given circle

Area of circular pathway =

SOLIDS

Any figure enclosed by one or more surfaces is called a solid figure.

When plane surfaces are forming a solid, they are called it Faces and the solid is called a polyhedron.

The lines which bind the faces of a solid figure (or solid) are called its Edges.

The volume of a solid figure is the amount of space enclosed by its binding surfaces.

1


The Area of the Whole surface is equal to the sum of the areas of its binding surfaces.

Cuboid: It is a figure bounded by six rectangular faces which are

perpendicular each other

the opposite faces of a cuboid are equal rectangles lying

in parallel planes.

l is the length

b is the breadth

h is the height

Total number of faces = 6

Rectangular side face = 4

Top and bottom Rectangular faces = 2

Curved surface area or lateral surface area = 2( bh + lh)

Total surface area = 2( bh + lh + lb )

Volume = where A1,A2 and A 3 are areas of base, side and end faces respectively

Diagonal of cuboid =

1


If the external length, breadth and height of a wooden box without lid be l, b, h respectively and let the thickness of the wood be x, then the

volume of wood used in making the box =

If the box is with lid, the volume of wood used is

If the length, breadth and height of a cuboid are increased, by a%, b% and c% respectively, then its volume increased by

Cube : It is solid figure bound by six equal dimensional faces which are perpendicular to each other

Curved surface area or lateral surface area = 4a2

Total surface area = 6 a 2

Volume = a 3

If the total surface area of a cube be S, then its Volume =

Three cubes of iron whose edges are a, b, and c respectively are melted and formed into a single cube, then the edge of new cube

Diagonal of the cube = ( edge ) =

1


Volume of the cube =

Total surface area of the cube = Let any two cubes, then

Ratio of volumes = ( ratio of sides ) 3

Ratio of surface areas = ( Ratio of sides )2

( Ratio of surface areas) 3 = ( Ratio of volumes ) 2

If the edge of a cube is increased by a%, then whole surface of the

cube increased by

If sides of a cube is increased , by a%, then its volume increased by

Prism

It is a solid whose sides are parallelograms and whose both ends lie on parallel planes. The end on which a prism may be supposed to stand is called the base and the perpendicular distance between both the ends of a prism is called the height of a prism.

A prism is called a Right Prism when its edges formed by side faces adjacent to one another are perpendicular to its ends.

Otherwise it is said to be an Oblique Prism. When the ends of a prism are parallelograms, the prism is called a Parallelepiped.

Right Prism

1


Surfaces are Rectangular

Base Polygon (may be triangle rectangle, etc.)

Curved surface area or lateral surface area = (base perimeter) × (height)

Volume = Base area × height

Cylinder

The limit of a prism whose number of sides is infinite while the breadth of each side is indefinitely diminished, is called a cylinder

Volume of prism = Base area × Height

Right Circular Cylinder

r is the radius of base

h is the height

Curved surface area or lateral surface area = base perimeter × (height) = 2 π rh

1


Total surface area = 2 π r (h + r)

Volume = π r 2 h

Let any two cylinders whose radii are equal then

Ratio of volumes = Ratio of heights

Ratio of volumes = Ratio of curved surface areas = Ratio of heights

Let any cylinders whose heights are equal then

Ratio of volumes = ( Ratio of radii) 2

Ratio of volumes = ( Ratio of curved surface areas) 2

Ratio of curved surface areas = Ratio of radii

Let any two cylinders whose volumes are equal then

Ratio of radii =

Ratio of curved surface areas = Inverse ratio of radii

Ratio of curved surface areas =

Let any two cylinders Whose curved surface areas are equal then

Ratio of radii = Inverse ratio of heights

Ratio of volumes = Inverse ratio of heights

Ratio of volumes = Ratio of radii

If height of a cylinder is changed a % keeping radius same, then its volume changes by a%

If radius is changed by a% keeping height same then the volume

changes by

1


radius changes by a % and height changes by b%, then volume of the

cylinder changes by

Pyramid

It is a solid whose sides are triangles, having a common vertex and whose base is a plane rectilinear figure.

The perpendicular drawn from the vertex of a pyramid to its base is called the height of the pyramid

The straight line joining the vertex to the middle point of the base is called the axis of the pyramid and if this axis is perpendicular to the base, then the pyramid will be a Right Pyramid.

Pyramid

the slant height = l

the altitude = h

Surface is of Triangles

Base is a Polygon

Curved surface area or lateral surface area

(Where P is the base perimeter l is the slant height )

Volume = ( base area ) (altitude)

Cone

1


It may be defined as the he limit of a pyramid whose number of sides of the base in indefinitely increased.

Cone (Right Circular)

r is the radius of base

h is the altitude

l is the slant height

Base is Circle

Curved surface area or lateral surface area = πr l where

Total surface area = π r ( + r )

Volume = ( base area altitude ) =

If a cone of height h is melted and a solid sphere of radius r is formed

then the radius of the base of the cone =

If a cone is inscribed into a cylinder, then the relation in the volumes=

Let any two cones whose volume are equal then Ratio of radii =

1


Let any two cones whose radii are equal then Ratio of volumes = Ratio of heights

Let any two cones when heights are same then Ratio of volumes = ( ratio of radii) 2

Let any two cones Whose curved surface areas are equal then Ratio of radii = Inverse ratio of slant heights

If the radius of the base of the cone is changed by a% and height

remains same, then the volume changed by

If the height of a cone is changed by a % and radius remains same, then its volume changed by a%

If the radius and height of a cone are changed by a % and b%

respectively, then its volume changed by

Frustum of a Cone

Curved surface area or lateral surface area = π (R + r )

Total surface area = π (R 2 + r 2 + R l + r )

Volume =

Sphere

1


Total surface area = 4 π r2 where r is radius of the sphere Volume =

If the surface area of a sphere is 'A' , then its volume =

If three solid spheres of radii and are melted and a big solid

sphere is made, then the radius of this sphere is

If by melting n small equal spheres, a big sphere is made, then the radius of the big sphere is

If a largest possible sphere is circumscribed by a cube of edge 'a' cm,

then the radius of the sphere =

If a largest possible cube is inscribed in a sphere of radius 'a' cm, then

the edge of the cube =

If the largest possible sphere is inscribed in a cylinder of radius'r' cm,

and height'h' cm, then radius of the sphere =

If a largest possible sphere is inscribed in a cone of radius'r' cm and slant height equal to the diameter of the base, then radius of the

sphere =

1


If a sphere of radius R is melted to form smaller spheres each of radius r, then The Number of smaller sphere =

=

the radius or diameter of a sphere is changed by a%, then its volume

changed by

the radius of a sphere is changed by a %, then its surface area

changed by

Hemi- Sphere ( Half part of the sphere )

Let r is radius of the sphere as shown in diagram

1. Volume =

2. Curved surface area =

3. Total surface area =

If a largest possible cube is inscribed in a hemisphere of radius'r' cm,

then the edge of the cube =

Costing

Example: Flooring cost = Floor Area cost per unit area

Example: Fencing cost = perimeter of the plot cost per unit length

1


Coordinate Geometry

Contents

1. Cartesian Plane2. Distance Between two points3. Area of a Polygon4. Section Formula5. Centroid ,Orthocentre Incentre , Ex-centre and Circumcentre6. Slope of the a straight line7. Equations of a straight line8. Locus9. Equations of the bisectors of the lines10. Concurrent lines11. Pair of straight lines (Just basics )12. Key points >

Cartesian Plane

The plane in which x-axis and y-axis, two mutually perpendicular lines intersect at origin o is called x-y plane or Cartesian plane.

These lines divide the plane in to 4 quadrants. Any point in this plane is represented by p ( x , y )

Here = distance of the point from y-axis ( Abscissa of the point )

= distance of the point from x-axis. ( ordinate of the point)

The adjacent figure shows the four Quadrants In the Cartesian plane

1


If the point P ( x, y ) lies in first quadrant then

If the point P ( x, y ) lies in second quadrant then

If the point P ( x, y ) lies in Third quadrant then

If the point P ( x, y ) lies in fourth quadrant then If the point P ( x, y ) lies on X axis then y = 0 , x

If the point P ( x, y ) lies on Y axis then

Distance Between two points

The distance between two points A(x1, y1 )and B(x2 ,y2 ) is AB =

In order to prove that a given figure is a square, prove that the four sides are equal and the diagonals are also equal.

In order to prove that a given figure is a rhombus, prove that all the four sides are equal.

In order to prove that a given figure is a rectangle, prove that opposite sides are equal and the diagonals are also equal.

In order to prove that a given figure is a parallelogram, prove that the opposite sides are equal.

In order to prove that a given figure is a rhombus but not a square, prove that that its all sides are equal but the diagonals are not equal.

In the question if it is given that it is a parallelogram then to decide whether it as square, rectangle, rhombus, parallelogram follow the working rule square AB = AD AC = BD rhombus AB = AD AC ≠BD rectangle AB ≠AD AC = BD parallelogram AB ≠AD AC ≠BD

Area of a Polygon

The area of the polygon whose vertices are (x1 ,y1 ),(x2, y2 ),

(x3 ,y3 )............(xn ,yn ) is

1


Area of Triangle with vertices A(x 1 ,y 1 ),B(x 2, y 2 ) and C(x 3 , y 3 )

or

Condition for Colinearity of Three Point If three points A(x 1 ,y 1 ),B(x 2, y 2 ) and c(x 3 , y 3 ) are collinear, then Area of

(or)

(or) Write the equation of straight line using any two points and check whether third point satisfies it or not.

Section Formula

If P(x, y) divides the line joining A , B in the ratio m : n

then (internally) (externally)

If lies in the line joining P divides

in the ratio

x -axis divides the line segment joining in the ratio - y 1 : y 2

y -axis divides the line segment joining in the ratio - x 1 : x 2 Ex Find the ratio in which x -axis & y- axis divides the line

segment joining .

Sol Let A

1


divides in the ratio -3 : 5 or 3 : 5 externally .

divides in the ratio - (-2) : 3 = 2 : 3

If are three consecutive vertices of a

parallelogram, then the fourth vertex is

If D are the midpoints of sides of a D le as shown in figure

then A =

then B =

then C = If a point P divides the line segment joining the points A, B in

the ratio then the point Q which divides A , B in the ratio is called harmonic conjugate of P with respect to A and B

Let the given line ax + by + c = 0 divide the line segment joining A(x1 ,y1 ) and B(x2 ,y2 ) in the ratio m : n, then

If A and B are on same side of the given line, then is negative

If A and B are on opposite sides of the given line, then in positive

Centroid ,Orthocentre Incentre , Ex-centre and Circumcentre

Centroid of a Δ The point of intersection of the medians is called centroid of triangle This point divides each medians in the ratio 2 : 1. Its coordinates are

triangle

Incentre This is point of intersection of internal angle bisectors of the triangle

its co-ordinates are where the data is as

1


shown in figure

Excentres This is the point of intersection of one internal angle bisector and two external angle bisectors of the angles of a triangle.

Excentre 1 = which is opposite to the vertex A

Excentre 2 = which is opposite the vertex B

Excentre 3 = which is opposite the vertex C

Orthocentre of a triangle This point is the point of intersection of the altitudes (i.e. the lines through the vertices and perpendicular to opposite sides). Its coordinates are

where A, B, C are the angles at the vertices A,B,C

Circum-centre of a Δ : The point of intersection of right bisectors of the sides of a triangle. (i.e. the line through the mid point of a side and perpendicular to it) is called circum-centre of the triangle. Its coordinates are

1


Where the data is involved as shown in figure

The circum centre of a right angled triangle is the mid point of hypotenuse and half the hypotenuse is circum radius

The circle opposite to the vertex A and with centre I 1 and touching all the lines through the sides is called the escribed circle opposite A

similarly there will be two more escribed circles opposite to the vertices B and C.

The circle which touches all the three sides of the triangle and with incentre as it centre is called incircle

The circle which passes through all the three vertices is called the circumcircle its centre is circumcentre

In a Δle the maximum number of circles which touches all the three lines through the sides is four

In any Δle O, N, G, S, are collinear where O is orthocentre, N is nine point circle centre, G centroid , S circumcentre Let us observe the trick now

N divides O and G in the ratio 3 : 1 internally G divides O and S in the ratio 4 : 2 = 2 : 1 internally S divides O and G in the ratio 6 : 2 = 3 : 1 externally G divides N and S in the ratio 1 : 2 internally N divides O and S in the ratio 1 : 1 internally

Slope of the a straight line

Slope of a line is 'tan' of angle made by the line form the positive direction of in anti-clockwise direction and the slope is denoted by ,

i.e. Here

1


If line passes through the points and B then

Equations of a straight line

Equation of the straight line in slop, Y intercept form is y = m x + c

The equation of a straight line passing through two points(x 1 y

1 ),( x 2 y 2 ) is Equation of a straight line whose X intercept and Y intercept are

a, b respectively is The general equation of a straight line is ax + by + c = 0

The area of the triangle formed by the line with the

coordinate axes is The perpendicular form or normal form of the equation of the

straight line is Where P is length of perpendicular, and a is angle between the perpendicular and X-axis as shown in diagram

The image in the line ax + by + c = 0 is given by:

1


Angle between the lines

If the lines are parallel then

then

If are the vertices of a triangle, then the area is

If are the sides of a triangle, then the

area of the triangle is given by Where C1 ,C2 ,C3 are the co-factors of c1 ,c2 ,c3 in the determinant.

Equation of the straight line passing through and making

an angle ' q ' with x-axis

Any point on this line is There are two points on the line at a distance 'r' from Q (x 1 ,y 1

). The coordinates of the other points are obtained by replacing 'r' by -r or q by + q

It's distance from the given point

Here 'q' is the angle made by the straight line with the +ve x-axis.

1


Locus

Locus of a point is a path traveled by the point under some given Mathematical conditions. Ex If the sum of distances of a point from two perpendicular lines in a plane is 1, then it's locus is 1) a circle 2) a pair of straight line 3) a straight line 4) a square Sol

Let point be then

Hence the correct option is 4

Equations of the bisectors of the lines

Equations of angle bisectors of where c1 and c2 are of same sign

If a1 a2 +b1 b2 < 0 then Acute angle bisector

obtuse angle bisector is given by

If a1 a2 +b1 b2 > 0, then

Acute angle bisector is

Obtuse angle bisector is The bisector of the angle in which origin lies is

If a1 a2 +b1 b2 < 0

1


Equation of the bisector of the angle in which the point

lies, is given by

If are of same sign

Concurrent lines

................................................1

For different values of it represents straight lines, passing through

the point of intersection of and

Pair of straight lines

Pair of straight line are represented by a second degree equation

will represent two straight lines if

and

1


If represents a pair of lines and

then, the point of intersection of the lines is

ax2 + 2hxy + by2 = 0 always represents two straight lines passing through origin

Combined eq�. of straight lines, passing through origin and

with given slopes is : Splitting ax2 + 2hxy + by2 = 0 into to linear equations

If h2 - ab < 0, the straight lines are imaginary and intersect at a

real point . Because satisfies both the equations.

Angle is

be coincident, if h2 - ab = 0

Equation to the straight lines bisecting the angle between the

lines given by

is

Key points

If are two vertices of a triangle whose centroid is

then the third vertex is

If are two vertices of an equilateral triangle, then

its third vertex is

If are extremities of the hypotenuse of a right

angled isosceles triangle, then its third vertex is

1


If P is the orthocenter of are the

orthocenters of

If G is the centroid of then The area of the parallelogram formed by

is

The area of the rhombus formed by The condition that the slopes of the lines represented by

in the ratio p : q is

If is the centroid of the triangle whose sides

are and

The length of the intercept on the

cut by the pair of lines

is . The length of the intercept on the cut by the pair of lines

is The lines represented by

form a

Parallelogram

rectangle

rhombus

square

Basic Trigonometry

1


Contents

1. Basic Trigonometric Ratios2. The nature of Trigonometric ratios 3. Basic Trigonometric Identities4. Combinations of Trigonometric Ratios5. Maximum & Minimum values 6. Sum of the sin and cosine series when the angles are in A.P.

Basic Trigonometric Ratios

tan A and cot A may take any real value.

The nature of Trigonometric ratios

1


All ratios sin q , cos q , tan q , cot q , sec q and cosec q are positive in 1 st quadrant.

sin q ( or cosec q ) is positive in IInd quadrant, rest are negative.

tan q ( or cot q ) is positive in IIIrd quadrant, rest are negative cos q ( or sec q ) is positive in IVth quadrant, rest are negative.

Basic Trigonometric Identities

or

Ex :

1+

The values of trigonometric ratios at different angles.

0o

30o

45o

60o

90o

sin q

0

1

cos q

1

0

1


tan q

0

1

∞

cosec q

∞

2

1

sec q

1

2

∞

cot q

∞

1

0

Compound Angles

Ex :

1


quantitative recipe !!! by veera karthik

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