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THE COMPLETE HANDBOOK ON

QUANTITATIVE

APTITUDE

FOR CAT AND ALL OTHER MBA

ENTRANCE EXAMS

BROUGHT TO YOU BY

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MODULE 1: ARITHMETIC

ALL CONCEPTS AND

FORMULAE

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Percentage Profit Loss

Discount & Interest

Percentages: To convert a fraction into a

percentage, we need to multiply with 100. For

example, the fraction

, or

simply

.

Some fractions and their conversions are shown

below:

Fraction % Fraction % Fraction %

1 100 1/8 12.5 1/15 6.66

1/2 50 1/9 11.11 1/16 6.25

1/3 33.33 1/10 10 1/17 5.88

1/4 25 1/11 9.09 1/18 5.56

1/5 20 1/12 8.33 1/19 5.26

1/6 16.67 1/13 7.69 1/20 5

1/7 14.28 1/14 7.14

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Percentage change: Percentage change is

calculated as the change between the initial and

final values calculated as a percentage of the initial

value.

Given that the percentage change be n%, we have:

. Rearranging the terms, we get:

Percent

change

Fraction

multiplied

Percent

change

Fraction

multiplied

100%

increase

100%

decrease

50%

increase

50%

decrease

33.3%

increase

33.3%

decrease

25%

increase

25%

decrease

20%

increase

20%

decrease

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Percent change in product of 2 values: The net

percentage change in the product of two variables

when one changes by and the other changes by

is also given by:

Percent change in sum/difference of 2 values: Let

there be two commodities whose values are and

have percentage changes of and , the net

percentage change in the sum/difference of the

two is given by:

Here and have to be taken with their sign,

positive for increase and negative for decrease.

Profit loss and discount: involves the same

concepts as in percentages. Some of the important

points are:

1. The price at which an article is bought is

called Cost Price (C.P.)

2. The price at which the article is sold is called

Selling Price (S.P.)

3. Profit = S.P. – C.P.

4. Profit % and Loss % are calculated on C.P.

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5. Marked price (M.P.) is the price at which an

article is sold without any discount.

6. Discount % is calculated on M.P.

7. When there is profit / loss:

When there is discount:

8. S.P. and C.P. in terms of number of articles:

If S.P. of n1 articles = C.P. of n2 articles, % loss

or gain =

If we buy articles and get articles free, it

means that articles are sold at the

cost price of articles.

9. If two items are sold, each at the same price,

one at a profit of % and the other at a loss

of %, there is an overall loss given by:

%.

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Simple and Compound Interest

The table gives an example of how simple interest

(SI) and compound interest (CI) operate. The

principal at the beginning of 1st year, of Rs.10000

and a rate of 10% p.a. are considered. The details

for 5 years are shown:

There are certain observations regarding the

difference between the compound and simple

interests and also difference between the interests

in consecutive years. They are given below:

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Ratio Proportion & Variation

1. Ratio remains unchanged if each term is

multiplied or divided by a constant:

2. Thus, for a ratio, say: , we have:

, where is a constant

3. If

, then

4. If

, then

5.

6. If and are positive:

7. If is directly proportional to

where is a constant

8. If is inversely proportional to

where is a constant

9. Joint variation: If when is constant and

, when is constant, then these can be

combined as shown:

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Average Mixture & Alligation

The alligation method:

When w1 quantity of cheaper ingredient A of cost

C1 and w2 quantity of dearer ingredient B of cost C2

are mixed to get an average cost of mixture Cm:

The rule can also be represented as follows:

To find the mean value of a mixture:

When x1 and x2 quantities of A and B of cost C1 and

cost C2 respectively are mixed, cost of mixture:

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Time Speed & Distance

(i) If the ratio of speeds of A and B is a : b, then

the ratio of the times taken by them to cover

the same distance is b : a.

(ii) If the ratio of speeds of A and B is a : b, then

the ratio of the distances travelled by them in

the same time is a : b.

(iii) If the ratio of time of travel of A and B is a : b,

then the ratio of the distances travelled by

them with the same speed is a : b.

Average Speed:

If a man covers certain distances at different

speeds, the average speed is given by:

(i) If a man covers a certain distance at miles/h

and an equal distance at miles/h, the

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average speed during the whole journey is

miles/h.

(ii) If a man covers a certain distance at miles/h

and another distance at miles/h, such that

the time of travel is same, the average speed

during the whole journey is

miles/h.

Relative Speed: Relative speed refers to the speed

of a moving body with respect to another moving

body:

1. If 2 bodies move in the same direction with

speeds V1 & V2 (V1 > V2), then their relative

speed is given as V1 – V2 => to cover a gap D

between them, the time the faster body takes

to catch up with the slower one is

– .

2. If 2 bodies are moving in opposite directions

with speeds V1 & V2, then their relative speed

is given as V1 + V2 => To cover a gap D

between them, the time taken is

.

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Effective or Resultant Speed: Let the speed of an

object without the effect of any medium be M

miles/h, and the speed of the medium be R

miles/h:

(i) If an object travels in a medium against the

flow of the medium: Upstream motion. Speed

Upstream = (M – R).

(ii) If an object travels in a medium with the flow of

the medium: Downstream motion. Speed of

object Downstream = (M + R).

Time taken to meet in to and fro motion: Let two

objects P and Q start moving from two positions A

and B in opposite directions towards each other

and meet after time Tm. After meeting they

continue their motion and reach their respective

destinations T1 and T2 time after the meeting, then,

we have:

Circular races:

When the two people, PA and PB, having respective

speeds of A and B (A > B) are running around a

circular track (of length L) starting at the same

point and at the same time, we have:

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Clocks:

Minute-hand covers 360° in 60 minutes; hence its speed is 6° per minute

Hour-hand covers 30° in 60 minutes; hence its speed is 0.5° per minute

Relative speed (same direction) = 5.5° / minute.

Speed ratio = 6 : 0.5 = 12 : 1, hence there are (12 – 1) = 11 distinct meeting positions; in a day there are 22 meetings

In a period of 12 hours, the hands coincide 11 times; hence time gap between two successive coincidences is 12/11 hours = 65 5/11 minutes

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Time & Work

1. If N1 people with efficiency E1 do a work W1 in

D1 days and N2 people with efficiency E2 do a

work W2 in D2 days, then:

2. If 2 men take and days to do the same

work, their efficiencies are in the ratio

3. If A and B take P days more and Q days more

than the time taken by A and B together to

complete a work, then time taken by P and Q

together is give by the relation:

4. When money is divided among people for their work, the total money earned should be shared by people in the ratio of the work done by each of them.

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MODULE 2: NUMBERS

ALL CONCEPTS AND

FORMULAE

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Number Properties

Introduction

Factors, Multiples, HCF and LCM

If the HCF of two numbers is H, the numbers

can be assumed as and such that

and are co-prime.

While calculating HCF or LCM of fractions, the

fractions should be in the most reducible form.

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The HCF of a set of numbers has to be a factor

of the LCM of the set of numbers OR in other

words, the LCM of a set of numbers must be a

multiple of the HCF of the set of numbers.

For 2 numbers and having HCF as and

LCM as :

If a series of numbers is of the type ,

then consecutive numbers of this series differ

by . Conversely any series of numbers having

difference of (basically an AP with common

difference N) can be represented as ,

where is an integer

If , when divided by , , … etc. leave the

same remainder , then the number can be

expressed as:

If , when divided by , , … etc. leave the

remainders , … etc. such that

, the number can be expressed

as:

If , when divided by , , … etc. leave the

remainders , … etc. such that

, etc. then first determine the smallest

number satisfying the above conditions. The

number can be expressed as:

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Tests of Divisibility:

: A number is divisible by if the number

formed by its last ‘ ’ digits is divisible by .

3 or 9: A number is divisible by 3 or 9 when

the sum of its digits is divisible by 3 or 9

respectively.

: A number is divisible by if the number

formed by its last ‘ ’ digits is divisible by .

11: A number is divisible by 11, if the

difference between the sum of the digits in

the odd places and in the even places of the

number is divisible by 11.

7: For 7, we have a series: {-2, -3, -1, 2, 3, 1}.

We need to find the sum of the product of

consecutive digits of the number with the

numbers in the series from the right to left.

The number is divisible by 7 if the above sum

is divisible by 7.

13: The rule is similar to that of 7, only that

the series is {4, 3, -1, -4, -3, 1}.

Any composite number: For any composite

number, check divisibility with its factors such

that the factors are co-prime to each other

and the LCM of those factors is that number.

For example, for 120, check for 3, 5 and 8.

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Last Digit of a number: To find the last digit of a

number raised to any exponent, we need to

consider only the last digit of the original number

in the base. Thus, the last digit of is the same

as the last digit of .

Last digits of numbers follow a pattern. For

example, last digits of 2n: 21 = 2, 22 = 4, 23 = 8, 24 =

16, 25 = 32, 26 = 64, 27 = 128, 28 = 256, etc.

Note that they end in 2, 4, 8 and 6 and this pattern

gets repeated every fourth power. The patterns are

shown below:

Number Pattern

1 1

2 2, 4, 8, 6

3 3, 9, 7, 1

4 4, 6

5 5

6 6

7 7, 9, 3, 1

8 8, 4, 2, 6

9 9, 1

0 0

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Highest power of a number in the factorial of

another number:

To find the highest power of x that divides N!,

keep dividing N successively by x and add the

integer part of all the quotients (Successive

division means dividing the quotient of the

earlier division). While this is the process of

getting the answer, do understand the concept

behind find the highest power as it may be

used in other application. The interpretation of

the highest power of x that divides a factorial is

that if from 1 × 2 × 3 × 4 × 5 × 6 × …… × N, if all

the powers of x is separated, the sum of all

those powers; for e.g. from 19!, if we separate

all powers of 3 we have:

1 × 2 × (3) × 4 × 5 × (3×2) × 7 × 8 × (3×3) × 10 ×

11 × (3×4) × 13 × 14 × (3×5) × 16 × 17 × (3×3×2)

× 19 = 38 × N (N ≠ multiple of 3)

Thus, the required highest power:

; here, refers to

the integer part of the number

The same logic as above can be used for any

product and not necessarily a factorial. Thus if

the questions is what is the highest power of 3

that can divide the product of squares of all

odd numbers from 1 to 20, one should identify

that the powers of 3 would appear only in the

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squares of 3, 9 and 15. Thus the largest power

of 3 dividing the given product is 2 + 4 + 2 = 8.

When the number of zeroes at the end of a

product of series of numbers is asked, think of

the highest power of 2 and 5 in the product.

If any number is expressed as 10n × m, where m

is not a multiple of 10, then n is the number of

zeroes at the end of the given number. n is also

the highest power of 10 that divides the given

number.

The above property is not just limited to 10. If a

number N can be expressed as 7n × m, where m

is not a multiple of 7, then in this case also n is

the largest power of 7 that can divide the given

number.

The above rules can be used effectively to

factorize a factorial. Thus if one needs to find

the number of ways in which 15! can be written

as a product of 2 numbers.

We can factorize the given number, i.e. write it

in the form of 2a × 3b × 5c × 7d × 11e × …

Thus we need to find the highest power of

prime numbers that can divide the given

number. Thus in our case 15! can be written as

211 × 36 × 53 × 71 × 111 × 131.

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Factors of a Natural Number: Any number can be

expressed as a product of its prime factors raised

to suitable exponents.

Let , where are distinct

prime factors of and are the exponents of

those primes.

1. Let F be the number of factors of the number

N. We have:

2. The number of ways in which N can be

expressed as the product of two factors,

including N and 1 is given by:

, where is odd ( is a perfect square)

, where is even ( is a non-perfect square)

3. The number of ways in which can be

expressed as a product of 2 co-prime factors is:

, here is the number of different prime

factors of .

4. Sum of all the factors of :

5. Product of all the factors of :

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Remainders: There are multiple ways of calculating

remainders. The rules are given below:

1. If the remainder when is divided by is ,

then the remainder when is divided by is

the same as the remainder when is divided

by , where is some positive integer.

For example: To find the remainder when

is divided by 11:

We know that when divided by 11 leaves a

remainder 10, which can be thought of as a

remainder of .

Thus:

is

the final remainder.

2. Use of binomial theorem (for positive integer

index) to find remainders: when

divided by ‘ ’ leaves a remainder equal to the

remainder when is divided by ‘ ’.

For example: To find the remainder when

is divided by 12:

Thus:

is the final

remainder (since 1728 = i.e. a multiple of

12).

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3. When is divided by

, the remainder obtained is .

Thus, if the remainder is zero, is a factor

of .

4. Three important rules are:

Base System: The different digits used in different

base systems are:

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Base conversion:

1. Conversion from any base to base 10:

The number is converted to base 10

by finding the value of the number; i.e.

2. Conversion from base 10 to any base:

A number written in base 10 can be converted

to any base ' ' by first dividing the number by

' ', and then successively dividing the quotients

by ' '. The remainders, written in reverse order,

give the equivalent number in base ' '.

Example: The number 25 (base 10) in base 4 is:

Writing the remainders in reverse order, 25 in

base 10 is equal to the number 121 in base 4.

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MODULE 3: ALGEBRA

ALL CONCEPTS AND

FORMULAE

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Linear & Quadratic Equations

Linear Equations: Linear equations are

equations in which the highest index of the

variables is 1. Consider the equation .

Here, and are the variables. Indexes of both

and are 1.

Simultaneous Linear Equations (in two variables):

The two equations represent two straight lines and

the solution represents their point of intersection.

Eg. ……………… Equation (1)

……………… Equation (2)

Here, , , etc. are constants and , are the

variables.

Different cases for simultaneous equations:

(i)

: This implies that both the

equations are identical equations and hence,

there are infinite solutions.

(ii)

: This implies that both the

equations are equations of parallel lines and

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hence, there exists no possible solution since

parallel lines never intersect.

(iii)

: This refers to the case where a unique

solution can be obtained from the 2 equations.

a.

: Since the ratio of coefficients

of is the same as the ratio of the constant

terms, the value of y in the equation is zero.

b.

: Since the ratio of coefficients

of is the same as the ratio of the constant

terms, the value of x in the equation is zero.

Quadratic Equations: An equation is said to

be quadratic when the variables contain the

highest exponent of 2, thus is a

quadratic expression of .

Roots of a Quadratic Equation: After suitable

reduction, every quadratic equation can be

written in the form: and the

solution of the equation is:

and

where and are the roots of the

equation.

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Discriminant: The value given by – is

called the discriminant and depending upon its

value we can determine the nature of the roots of

the quadratic equations.

(i) : Roots are real

(ii) Roots are real and equal

(iii) : Roots are real and unequal

(iv) : Roots are imaginary or complex

(v) is a perfect square: Roots are real & rational

(vi) is not a perfect square: Roots are real &

irrational (roots are and , where

and are rational)

Sum and Product of the Roots: If the roots of the

quadratic equation are and ,

then we have:

and

Suppose we have to form the equation whose

roots are and . So and

Thus, we have:

– (sum of roots) + product of roots = 0

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Maxima and Minima: Any quadratic expression

can be expressed in the form

where is the value of

where the expression takes a minimum / maximum

value and is the corresponding minimum or

maximum value of the expression. Thus, are

the coordinates of the vertex of the parabola.

Real roots; value of is

negative,

Minimum

value

Real roots; value of is

positive,

Maximum

value

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Inequalities

It deals with cases where variables or numbers are

less than or more than other variables or numbers.

The following symbols are used:

>: More than <: Less than

: Less than or equal to : More than or equal to

1. Polynomial inequalities: For example:

First we plot the roots and decide the sign of

the expression between any two of the three

roots. Since the sign of the expression will

alternate between consecutive roots, we can

decide the region. The above expression at x =

0 gives a value 8 > 0. Thus, we have the

following diagram:

The shaded region in the diagram depicts the

given inequality. Thus: – 2 < < 1 OR > 4.

2. Modulus: Modulus or absolute value is a

function that returns the magnitude ignoring

– 2 1 4

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the sign. Thus, |2| = |– 2| = 2. Hence, we have:

Another way of interpreting modulus is

distance from a point on the number line. For

example, | – 2| = 6 implies that the distance

of the point from the point 2 on the number

line is 6 units. Hence, we attain the points 8 and

– 4 on the number line (shown below).

Thus, from the above concept, we have:

a.

or

b.

Important points:

does not exist

Minimum value of is ‘0’ (Note: Minimum value of is also ‘0’)

– 4 2 8

6 6

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3. An inequality (with respect to zero) involving a

ratio of two quantities is equivalent to the

same inequality involving the product of the

same quantities:

a.

b.

4. Arithmetic Mean Geometric Mean: For two

(or more) positive numbers and :

5. Exponents: for all

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Logarithms & Indices

Logarithms and their properties:

is real if , ,

and

where is any base

(if ) or (if )

Indices and their properties:

and

provided

(if is odd) or (if is even)

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Functions & Graphs

Functions: When y is expressed in terms of such

that for any , a unique value of is obtained, is

called a function of .

Functions of are usually denoted by symbols of

the form , , etc.

Domain and Range: Let , Then the set is

known as the domain of ‘ ’ and the set is known

as the co-domain or range of ‘ ’ and is denoted by

.

For example, Let = {-1, 0, 1} and . Then

, , .

Here domain is {-1, 0, 1}, and range is {0, 1}.

Inverse of a function: If and

and the above occurs for all possible values of

and , we say that is the inverse of

and vice-versa. To generate the inverse of a

function, we need to follow the following steps:

1. Express in terms of

2. Interchange and

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Say, , let us determine the

inverse function :

Thus, the inverse function

Even, Odd and Inverse – Graphical significance:

Even function: If , it is even.

Even functions are symmetric about Y-axis.

Odd function: If , it is odd.

Odd functions are symmetric about Origin.

A function and its inverse is symmetric about

the line .

Composite functions: , ,

, , etc. are composite functions.

Let and :

,

.

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Periodic function: A function is periodic if

there exists a number so that

for all . Here, is the period of the function.

For example, is a periodic function

with period since .

Piece-wise functions: Functions which have

different expressions over different values of are

piece-wise functions.

An example (modulus function ) is shown below:

Max – Min functions:

implies that if

or if .

implies that if

or if .

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Graphs of functions:

Look at the graph of and :

The graph of will also be similar to

the above.

The graph of will also be

similar except that the co-ordinates will change.

The center of the diamond (square) shape rather

than being will just be and accordingly

the other co-ordinates will also change but the size

of the diamond will remain the same i.e. a square

of side .

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Sequences

Sequence: A series of numbers a1, a2 … an formed

by some definite rule is a sequence or progression.

Arithmetic Progression: A succession of numbers is

said to be in arithmetic progression (A.P.) if the

difference between any two consecutive terms is

always constant (called the common difference):

, , , , …

(1st term , common difference )

Geometric Progression: A progression, in which

every term bears a constant ratio with its

preceding term, is called a geometrical progression

(G.P.). The constant ratio is the common ratio.

, , , , …

(1st term , common ratio )

Some important results:

1. Sum of first ‘ ’ natural numbers:

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2. Sum of first ‘ ’ odd numbers:

3. Sum of the first ‘ ’ even numbers:

4. Sum of the squares of first ‘ ’ natural numbers:

5. Sum of the cubes of first ‘ ’ natural numbers:

6. Arithmetic-co-geometric series: An example of

how to proceed is shown below:

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MODULE 4: GEOMETRY

ALL CONCEPTS AND

FORMULAE

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Geometry

Triangles: A triangle is a figure bounded by three

lines (AB, BC and CA) in a plane. A, B, C are called

the vertices of the triangle.

Properties of triangles:

1. Sum of any 2 sides in a triangle is greater than

the third side.

2. The greatest side in a triangle has the

greatest angle opposite to it.

3. Sum of the interior angles is and sum of

its exterior angles is .

4. Any exterior angle in a triangle is equal to

sum of the interior opposite angles.

5. If two sides of a triangle are equal it is called

isosceles. The angles opposite to the equal

sides are also equal.

6. If all the three sides are equal it is called

equilateral. Each angle is equal to .

7. A triangle with no two sides equal is called

scalene.

A

B C

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8. If one angle of a triangle is it is called

right-angled.

If ABC is right-angled at A, then:

AB2 + AC2 = BC2 (Pythagoras’ theorem)

9. If one angle is more than it is called

obtuse-angled.

10. If all the angles are less than it is called

acute-angled.

11. In a triangle ABC, if a, b & c are the 3 sides,

where c is the greatest side, then,

(i) If c2 < a2 + b2, is acute.

(ii) If c2 = a2 + b2, is right angle.

(iii) If c2 > a2 + b2, is obtuse.

12. The perimeter of a triangle is the sum of its

three sides.

13. Area of a triangle

14. Area of a triangle =

where are the sides and is the semi-

perimeter

15. The medians of a triangle refer to the side

bisectors drawn from the opposite vertex. All

three medians always intersect at a single

point called the centroid.

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The centroid divides each median in the ratio

(2 units from the vertex and 1 unit from

the side).

16. The altitudes of a triangle refer to the

perpendicular drawn from a vertex to the

opposite side. All three altitudes always

intersect at the ortho-centre.

The ortho-center of a right-angled triangle is

the vertex containing the right angle.

17. The in-centre is the point of intersection of

the three lines which bisect the angles at the

vertex of the triangle. A circle can be drawn

with its centre as the in-centre which touches

the three sides of the triangle. The radius of

the circle is called the in-radius .

The Circum-center of a right-angled triangle is

the midpoint of the hypotenuse.

Area of the triangle =

18. The circum-centre refers to the point of

intersection of the perpendicular side

bisectors of the three sides of the triangle. A

circle can be drawn with its centre as the

circum-centre which passes through the three

vertices of the triangle. The radius of the

circle is called the circum-radius .

Area of the triangle =

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Similar Triangles: Similar triangles are similar in

shape, do not have same size but are proportional

i.e. the ratio of their sides is constant. Two or more

triangles can be called similar if:

i) There are two angles of the same measure in all

the triangles (since sum of angles is , if

two angles are equal, the third angle must also

be equal)

ii) The ratio of two corresponding sides of both

triangles are equal and the angle included

between the above two sides of the two

triangles are equal.

If two triangles are similar:

a) Ratio of their corresponding sides is the same

and also equal to the ratio of any

corresponding length measure of the triangles;

example: ratio of their heights or medians etc.

b) Ratio of their areas = square of the ratio of their

corresponding sides.

c) Ratio of their perimeters is equal to the ratio of

their corresponding sides.

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Polygons: For a polygon with sides:

a) Sum of all interior angles =

b) Sum of all exterior angles =

c) Number of diagonals =

d) Each interior angle in a regular polygon =

e) Each exterior angle in a regular polygon =

f) Area of a regular polygon =

(Perimeter) (Altitude from centre to a side)

Circles: Some properties of circles are mentioned

below:

1. O = Centre of the circle, OA = Radius, AB =

Tangent, XY = Chord

O

Radius

A

B

X Y Z

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2. Tangent is perpendicular to the radius i.e. AB

OA.

3. Perpendicular from the centre bisects the

chord i.e. OZ bisects XY.

4. Similarly, if Z is the mid-point of XY, then OZ is

perpendicular to XY.

5. Angle subtended by an arc at the center is

twice the angle subtended by the same arc at

the circumference, i.e. QOR = 2 x QPR.

6. The angle between a tangent (QP) and a

chord (RP) at the point of contact (P) is equal

to the angle in the alternate segment i.e.

QPR = PSR.

7. A quadrilateral with all vertices on a circle is

cyclic, and sum of its opposite angles is .

8. Two tangents can be drawn from any external

point on a circle, and the tangents are equal

in length.

P Q

R

S

Q R

O

P

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Quadrilaterals: Properties of various quadrilaterals

are mentioned below:

Parallelogram:

Opposite sides are parallel and equal

Diagonals bisect each other

Area = Base Height

Rectangle:

Special parallelogram with all angles

Diagonals are equal

Area = Length Width

Rhombus:

Special parallelogram with all sides equal

Diagonals are perpendicular to each other

Area =

Product of the diagonal lengths

Square:

Special parallelogram with all sides equal and all angles

Diagonals are equal and also perpendicular to each other

Area =

Product of the diagonal lengths =

Square of the side

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Some special points:

1. Among all quadrilaterals with the same area,

the square has the least perimeter.

2. Among all quadrilaterals with the same

perimeter, the square is has the greatest area.

3. Among all 2-dimensional figures with the same

perimeter, the circle has the maximum area.

4. Among all 2-dimensional figures with the same

area, the circle has the least perimeter.

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Mensuration & Solids

Right Prism: A prism is a shape that has the same

uniform cross-section at any point of its height. A

right prism is a prism where the lateral edges are

perpendicular to the base (examples: cube, cuboid,

cylinder).

1. Volume = Area of base x Height

2. Lateral surface area = Perimeter of base x

Height

3. Total surface area = Lateral surface area +

Base area(s)

Right Pyramid: A pyramid is a shape whose area of

cross-section decreases at a uniform rate till it

converges to a point, called the vertex. A pyramid

whose base is a regular polygon, the centre of

which coincides with the foot of the perpendicular

dropped from the vertex on base is a right pyramid

(examples: cone, triangular-base pyramid, square-

base pyramid).

1. Volume =

x Area of base x Height

2. Lateral surface area =

x Perimeter of base x

Slant height

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3. Total surface area = Lateral surface area +

Base area

4. Slant height is the perpendicular from the

vertex to the midpoint of a side of the base.

In a cone, the slant height is the line joining

the vertex to any point on the base circle.

Some special solids:

Cuboid:

Volume = , where these are the length, width and height respectively

Surface area =

A cube is a cuboid with

Cylinder:

Volume = , where is the radius and is the height

Surface area =

Cone:

Volume =

, where is the radius

and is the height

Surface area = , where is

the slant height

Sphere:

Volume =

, where is the radius

Surface area =

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Solids inside other solids:

1. The largest possible sphere circumscribed by

a cube of edge , has radius

2. The largest possible cube inscribed in a

sphere of radius , has edge

3. The largest possible sphere inscribed in a

cylinder of radius and height , will have the

radius given by the smaller value between

and

4. The largest possible cube inscribed in a

cylinder of radius and height , will have the

edge given by the smaller value between

and

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Co-ordinate Geometry

Rectangular axes: The figure shows the XY

Cartesian plane. The line XOX is called the X axis

and YOY the Y axis. A point P(x, y) in the plane

denotes the x value or abscissa of P and the y value

or the ordinate of P.

The plane is divided into four equal parts by the

two axes, called the Quadrants (I, II, III, and IV).

Straight lines:

i) The X-axis divides the line joining the points and in the ratio and the Y-axis divides the line joining the same two points in the ratio .

ii) Two lines are parallel if their slopes are equal

iii) Two lines are perpendicular if the product of their slopes is .

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iv) The equation of a straight line parallel to the X-

axis is , where is a constant, and that of a straight line parallel to the Y-axis is , where is a constant.

v) The slope of a line parallel to X-axis is zero and that of a line parallel to Y-axis is undefined.

vi) Equation of a straight line is:

Here is the slope and is the Y-intercept

vii) The equation of a line passing through 2 points

and is:

Circles: If P be a point on a circle with centre

C and radius , the equation of the circle is:

The equation is:

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MODULE 5: COMBINATORICS

ALL CONCEPTS AND

FORMULAE

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Set Theory

1. Union: The Union of two sets A and B, (i.e. A

B), is a set that contains all the elements

contained in A or B.

A = {2, 3}, B = {1, 3, 5}

Then A B = {1, 2, 3, 5}

2. Intersection: Intersection of two sets A and B,

(i.e. A B), that contains the elements

common to both A and B.

A = {2, 3}, B = {1, 3, 5}

Then A B = {3}

3. Difference of Two Sets: Difference of two sets A

and B, (i.e. A – B), is a set of elements present

in A but not in B.

A = {2, 4}, B = {2, 6, 5}

Then: A – B = {4} (as shown in the diagram

beside).

Similarly: B – A = {6, 5}

A

B

A

B

A B

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4. Complement: Complement of a set A, (i.e.

or ), is a set that contains the elements

outside A.

If = {1, 2} and ξ = {1, 2, 7, 8}

Then ’ = {7, 8}

Relation for two-set and three-set Venn diagrams:

For 2 sets and :

For 3 sets , and :

A

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Permutation & Combination

Basic Principle of Counting:

If an event P can be performed in ‘ ’ ways and

another event Q can be performed in ‘ ’ ways and

both operations are mutually exclusive, i.e. there is

nothing common to P and Q, then P OR Q can be

performed in ways.

If an event P can be performed in ‘ ’ ways and for

each of these ‘x’ ways, there are ‘ ’ different ways

of performing another independent event Q, then

P AND Q can be performed in ways.

Combination: Combination refers to selection. If

there are distinct objects from which objects

need to be selected, it is referred to as

combination of objects from (where ). It

is written as

Some important points:

1.

2.

3.

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4. is maximum in value if

( even) or

or

–

(n odd)

Selection of some or all of different things: Of

the things, one can select either 1 or 2 or 3 or …

things. This is calculated as:

Selection of some or all of identical things: If

there are identical things, then selection of any

number of them can be done in only 1 way as there

cannot be a choice between identical objects. Thus,

the number of ways in which we can select either

none or all of identical things is

The number of ways in which we can select some

or all of things, of which are alike of one

kind, are alike of another kind is therefore given

by . However, this includes the case

in which none of the things are chosen and hence

needs to be removed.

Thus, the final answer is –

Distribution of distinct items equally in identical

groups: Let there be ‘ ’ distinct objects that need

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to be distributed equally in ‘ ’ identical groups. This

is possible only if is divisible by . Let

, i.e.

each group has items. Thus, the number of ways

of achieving the distribution:

Distribution of distinct items equally in distinct

groups: Let there be ‘ ’ distinct objects that need

to be distributed in ‘ ’ distinct groups. the number

of ways of achieving the distribution:

Distribution of identical items in distinct groups:

The number of ways of distributing ‘ ’ distinct

objects in ‘ ’ groups (not necessarily equally) is

given by:

The above also gives us the number of non-

negative integer solutions of the sum of distinct

terms which is equal to .

Distribution of distinct items in distinct groups:

The number of ways of distributing ‘ ’ distinct

objects in ‘ ’ groups (not necessarily equal

distribution) is given by:

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Permutation: Permutation is the number of

arrangements of things which can be made by

taking some or all of different things (where

). It can be written in the form of formula as:

Permutations of items where some of them are

identical: If there are things of which are

identical things of one kind, are identical things

of another kind, are identical things of another

kind and so on , and all of them

are arranged, the number of ways is:

Permutations of different items in a circle: Since

a circle has no starting point, we need to first

assign one of the items in any of the available

positions to denote the starting point. The rest

items can be arranged in ways.

Thus, the number of arrangements in a circle (of

people sitting around a circular table) is .

However, for cases where clockwise and anti-

clockwise arrangements become identical (of

arranging different flowers in a garland), the

answer becomes

ways.

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Probability

Random Experiment: A random experiment is an

experiment where all the possible outcomes of the

experiment are known in advance and the exact

outcome is unpredictable.

For example: while tossing a coin, the possible

outcome is either a head or a tail. However, we

cannot predict what the outcome will be.

Sample Space: This constitutes the set of all

outcomes than can possibly occur in an

experiment. However, it must be maintained that

no two of the above outcomes can occur

simultaneously.

For example, the sample space for throwing a dice

is {1, 2, 3, 4, 5, 6}.

Event: This constitutes a set of outcomes which are

a part of the sample space.

In the above example of throwing a dice, a possible

event could be the occurrence of prime numbers.

This event is denoted by {2, 3, 5}. Thus, we can see

that the event is a subset of the sample space.

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Mutually Exclusive Events: If two or more events

have nothing in common, then they are said to be

mutually exclusive events.

For example, if we choose a card from a deck of 52

cards, getting a diamond and a spade are mutually

exclusive events since both cannot happen

simultaneously. On the contrary, getting a diamond

and a queen are not mutually exclusive since a card

can be a queen of diamonds.

Exhaustive Events: A set of events is exhaustive if

there is no scope of any other situation coming up.

For example, on asking a question, a man’s reply

can either be a true statement or a false

statement. There is no other possible scenario.

Independent Events: Two events are said to be

independent if one event does not affect the other

event.

For example, for a number of students taking a

test, the performance of one student does not

affect the performance of other students. Hence,

these are independent events.

Classical Definition of Probability: Probability is

the measure of chances of occurrence of an event.

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If a random experiment, the probability of the

occurrence of an event , usually denoted by

is given by:

Some important points:

if the events

and are independent

Expected value: Expected value of an event is the

sum of the product of probabilities of every

outcome with the value (monetary) associated with

the outcome.

Expected value

Where etc. are the probabilities and ,

etc. are the corresponding values associated with

the respective probabilities.

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