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GATE Aptitude - Coggle Diagram
GATE Aptitude
Quantitative Aptitude
Statistics
Simple Average
Sum of Terms / No. of Terms
Arithmetic Progression/Series
Arithmetic Mean = (First Term + Last Term) / 2
Common Difference
Sum = ((a + an) / 2) = (n / 2) * (2a + (n-1)d)
(a + c) / 2 = b
a and c are both or both even
Probability
P(A' OR B') = 1 - P(A AND B)
P(A - B) = P(A AND B')
P(A' AND B') = 1 - P(A OR B)
Terms
Eperiments
Sample Space
Set of all outcomes of an expreiment
Event
A collection of outcomes or a subset of sample space
Types
Equally likely events
Exhaustive events
Mutually exclusive events
Probability
Required Outcome / Total Outcome
Complementary Probability
1 - P(A)
Union Probability
Chance of occurrence of either event
Intersection Probability
P(A*B) = P(A)*P(B|A)
P(A*B) = P(B)*P(A|B)
Tricks
a^n + b^n is divisible by (a + b) for odd n
Selecting numbers in AP
For a, b, c to be in AP, b = (a + c)/2
So a, c should both be even or both odd
So selecting becomes choosing pair of odd or even numbers from a range of numbers
Number of ways to form n from 1 and 2
Select all 1s
1 choice
Select one 2
n-1C1
Select two 2s
n-2C2
n/2Cn/2
Sum all the values
Coin Toss
For coin toss type experiment the probability of each outcome is 1/2
No need to count total and then divide, just multiply probabilities
Each event has count 1, and total event count is 2^n
This boils down to 1/2 for each event
Sum is divisible by 3
(x+y) is divisible by 3 when
x and y both divisible by 3
x has a remainder 1 and y has a remainder 2 when divided by 3
a^m + b^m is divisible by k
Check the cycle of remainder
Find combinations where the sum is compatible with the division
Symmetry
Symmetry can be used to find the probability of an event
If the chances of something happening is same for a n-variables, and this encompasses the entire sample space, then
Probability is 1/N
Example
Example 1
Example 2
Problem Types
A,B,C,D probabilities are given
X/A,X/B,X/C,X,D probabilities are given
X happens
Find probability of A
Tips
Remember cases of with and without replacement
Remember cases of duplicates in permutation and combination
Remainder
Fermat's little theorem
a^p mod p = a mod p
Eulers Totient Method
Euler Number
The Euler number of a number x means the number of natural numbers which are less than x and are co-prime to x
Z = P1^n1*P2^n2..
E(Z) = Z*[1-1/P1]*[1-1/P2]...
When y^E(z) is divided by z, the remainder will always be 1 Where, E(z) is Euler number of z and y and z are co-prime to each other
When y^(E (z).k) is divided by z, where k is an integer, remainder will always be 1 That is if the power is any multiple of the Euler number of the divisor, even in that case the remainder will be 1. z and y and z are co-prime to each other
Number inside bracket with exponent can be converted to mod
Conversion of number to mod works because other digits is MSB are not important
Can be converted to negative remainder when it is closed to the max value which will make it smaller
Geometry
Triangle
Triangle Inequality
If a, b, c are the length of the 3 sides of a triangle
a + b > c
b + c > a
c + a > b
Centres
Circumcenter
The circumcenter of a triangle is defined as the point where the perpendicular bisectors of the sides of that particular triangle intersect
Center of the circle passing through the vertices of the triangle
Circumradius
Equilateral Triangle
a/sqrt(3)
abc/4* Area
Right Angled Triangle
c/2
Centroid
The centroid of a triangle is obtained by the intersection of its medians
Incenter
The incenter is the point where all of the angle bisectors meet in the triangle, like in the video. It is not necessarily the center of the triangle.
Properties
The incenter is equidistant from the sides of the triangle
It is the center of the incircle
Inradius
r * s = Area
Equilateral Triangle = a / 2sqrt(3)
Right Angled Triangle
(a + b -c)/2
Orthocenter
The orthocenter of a triangle is the point where the perpendicular drawn from the vertices to the opposite sides of the triangle intersect each other
For an acute angle triangle, the orthocenter lies inside the triangle
For the obtuse angle triangle, the orthocenter lies outside the triangle
For a right triangle, the orthocenter lies on the vertex of the right angle
Line Segments
Median
A line segment joining a vertex of a triangle with the mid-point of the opposite side
Each median of a triangle divides the triangle into two smaller triangles which have equal area
The 3 medians divide the triangle into 6 smaller triangles of equal area
Angle Bisector
A line segment joining a vertex of a triangle with the opposite side such that the angle at the vertex is split into two equal parts
Altitude
A line segment joining a vertex of a triangle with the opposite side such that the segment is perpendicular to the opposite side
Product of any altitude and length of it corresponding side is a constant for a triangle
This can be used to find the possible values of the sides
Then use triangle inequality to put constraints on the altitudes
Cevian
Special Triangles
Isosceles Triangle
An isosceles triangle is a triangle which has any two of its sides equal to each other
The angles opposite these equal sides are equal
The unequal side is called the base of the triangle
Equilateral Triangle
All three sides are equal
All angles are 60 degrees
Area
sqrt(3)*a^2/4
Scalene Triangle
No two sides are equal
No two angles are equal
Angle
Acute Angled Triangle
An acute angle triangle (or acute-angled triangle) is a triangle in which all the interior angles are acute angles
Obtuse Angled Triangle
A triangle whose any one of the angles is an obtuse angle or more than 90 degrees, then it is called obtuse-angled triangle
Area
base*height/2
sqrt(s(s-a)(s-b)(s-c))
s = (a+b+c)/2
a*b*sin(θ)/2
Base Proportionality theorem
If same height area is proportional to base
If same base area is proportional to height
Pythagoras Theorem
For Right Angled Triangle
AC^2 = AB^2 + BC^2
Obtuse Angled Triangle
AC^2 > AB^2 + BC^2
Acute Angled Triangle
AC^2 < AB^2 + BC^2
Sine Rule
In any triangle, the ratio of a side length to the sine of its opposite angle is the same for all three sides
Basic Proportionality Theorem
Midpoint Theorem
The line segment in a triangle joining the midpoint of two sides of the triangle is said to be parallel to its third side and is also half of the length of the third side
Appolonius's Theorem
The sum of the squares of any two sides of any triangle equals twice the square on half the third side, together with twice the square on the median bisecting the third side
Internal Angle Bisector Theorem
AB/AC = BD/DC
Stewart's Theorem
b^2m+c^n = a(d^2 + mn)
Ladder Theorem
Mass Point Geometry
m1/m2 = l2/l1
Fulcrum concept
Technique 1
Check the ratios if they follow any known values for which can determine the degrees of the angles
Based on the angles determine other segments
Apply mass point geometry
Calculate all unknowns which can be directly calculated
Technique 2
If a line is divided into multiple segments and we draw a line to end points on the segment then the ratio of the segments is equal to the area of the triangles formed
Technique 3
Represent area of the triangle in various forms
Area value can be substituted in another equation where there are unknowns
Circle
Triangle with edge as diameter
Other point on Circle
Right Angled Triangle
Other point outside Circle
Obtuse Angled Triangle
Other point inside Circle
Acute Angled Triangle
Angles made by the chord on the same arc are same
Secant
Line passing through a circle
Chord
Line segment with end points on circle
Inscribed Angle Theorem
The inscribed angle theorem states that an angle θ inscribed in a circle is half of the central angle 2θ that subtends the same arc on the circle
The angle does not change as its vertex is moved to different positions on the circle
Alternate Segment Theorem
The angle measure between a chord of a circle and a tangent through any of the endpoints of the chord is equal to the measure of angle in the alternate segment.
Intersecting Secants Theorem
Three cases
Meet outside and both are secants
Meet inside and both are secants
Meet outside and one is secant and another tangent
Congruency
Types
SSS
SAS
ASA
Digits
Number of digits in X^n
Express number in log
Multiply with n
It will decide the value of the number of bits
Set Theory
Set problems where number of elements only in sets are given
Total number of elements = N(A) + N(B) + n(C) + N(A∩B) + N(B∩C) + N(C∩A) - 2*n(A∩B∩C)
Vocab
Umpteen
Fulminate
Vituperative
Vengeance
Retribution
Retaliation
Logical Reasoning
Syllogism
Statement
Conclusion
Types
Positive/Yes
Negative/No
Neutral/Can't Say
Do not use restatement, statement restated in conclusion should not be used
Possibility
Some Not
Some
Clear
No
All
All A's are B's
Some A's are B's
Some B's are A's
Reverse is not true
Some A's are B's
Some B's are A's
Some A's are not B's
x
No A's are B's
No B's are A's
Some A's are not B's
Some B's are not A's
Complementary Pairs
Types
All A's are B
Some A's are not B
A and B order can be changed in some/some not case
All A's are B
Some B's are not A
Some A's are B
No A's are B
Some A's are B
No B's are A
Negative
Some A's are B's
Some B's are not A's
Answer in these cases will be either/or
Examples
Some copy are pen
All Pen are copy
All book are pen
All pen are copy
Some book are copy
If explicit relation between two sets is not specified then we can assume full overlap as a possibility
Some people are buyers
No buyer is market
Some market are not malls
All market being people is a possibility
If some As are not B's, all B's can be A's
Table
Reverse
Diagrams
Basic/Main Diagram
Possibility Diagram
Possibility allows the basic condition along with other conditions not stated but are possibilities
Direction Sense
Left-Right
Default direction of movement is north
Draw direction lines as reference
Syllabus
Video 1
Video 2
Verbal Aptitude
Grammar
Tense
Types
Past
Types
Continuous/Progressive
Normal
Sub + is + V ing + obj
Negative
Sub + is not + V ing + obj
Interrogative
Interrogative Negative
Prefect
Present Perfect for continuing situation
We use
for
to talk about a period of time: five minutes, two weeks, six years
We use
since
to talk about a point in past time: 9 o'clock, 1st January, Monday
Normal
Negative
Interrogative
Interrogative Negative
Just finished
Prefect Continuous
Simple/Indefinite
I is treated as a plural in present tense
Singular
Negative
1 more item...
Interrogative
1 more item...
Interrogative + -Negative
1 more item...
Sub + Verb + [s/es] + obj
Plural
Usage
Habitual Action
Universal Truth
Newspaper Headings
Commentary
Historical event
Present
Types
Simple/Indefinite
Continuous/Progressive
Prefect
Prefect Continuous
Future
Types
Simple/Indefinite
Continuous/Progressive
Prefect
Prefect Continuous
Sentences
Types
Declarative
A declarative sentence simply makes a statement or expresses an opinion
Imperative
An imperative sentence gives a command or makes a request
Negative
An interrogative sentence asks a question
Exclamatory
An exclamatory sentence is a sentence that expresses great emotion such as excitement, surprise, happiness and anger, and ends with an exclamation point
Person
Types
Second Person
The person being talked to
Third Person
The person being talked about
First
The person talking
Grammatican persons are accomplished by pronouns
Pronoun
Pronouns are often used to take the place of a noun, to avoid repeating the noun
When a pronoun replaces a noun, the noun is called the antecedent
Types
Personal
I, We, You, They
Reflexive
He hurt
himself
Emphatic
She
herself
opened the door
Indefinite
Someone, anyone, nobody etc.
Relative
This is the place
where
we met
Demonstrative
This, That, These, Those
Interrogative
What, who, which, whose, whom
Distributive
Each, either, neither, none
Possessive
his, hers, mine, yours
Articles
Articles are words that define a noun as specific or unspecific
Types
Definite
The
Indefinite
a
an
Clause
Subject + Verb combination
Types
Independent
Dependent
Types
Noun
Adjective
Adverb
No complete thought
Adverb
A word that modifies the meaning of a verb, an adjective or another adverb
Types
Time
I have spoken to him
already
I hurt my knee
yesterday
He comes here
daily
I have heard this
before
We shall
now
begin to work
Frequency
I have told you
twice
I have not seen him
once
He
often
makes mistakes
Sam called
again
He
frequently
comes late
Place
Stand here
He looked up
Walk backward
Go there
He went away
Manner
He reads clearly
This story is well written
He fought bravely
The girl works hard
You should not do so
Adverbs of degree or quantity
You are partly right
I am fully prepared
He is rather busy
He was too careless
These mangoes are almost ripe
Adverbs of affirmation or negation
He certainly went
Surely you are mistaken
I do not know him
Teachers should never agree to the illogical demands
Things turned out to be exactly the same as expected
Adverbs of reason
There was no network hence I switched off my phone
He therefore left school
I started running so that I didn't miss the train
He was left because he was late
Why is it so hot inside the bus?
Prepositions
A word governing, and usually preceding, a noun or pronoun and expressing a relation to another word or element in the clause
Types
Time
On, in, at, by, since, for, from, before/after, beside, besides
On
I will see you
on
7th June
I will see you
on
Monday
Indicates days, dates or part of a particular day
In
He started his business in 1999
He is born in the 21st century
I will see you
in
spring season
I got this surprise
in
January
Indefiinite periods of time
At 9'0 clock
In a minute, in an hour, in a day, in a week, in a year
Indicates long indefinite periods of time
At
Indicating precise/specific periods of time
Time
9'o clock
Holidays
At christmas, At weekend
Specific periods of time
At night, at lunch, at sunset, at present
in/at
We can use in or at with names of cities, towns, villages
Use in with a place or an area
We stayed in Mumbai
He lives in Church street
Use at when pointing to a specific part on area
Plane stopped at Mumbai Airport
He lives at House number- 14, Church Street
by
not later than(at or before)/travelling
He had promised to be back home
by
4'o clock
We travelled
by
train
We travelled by car
We travelled
in
a car
We travelled
in
his car
From (....to....)
Use to show the time when something starts
The shop is open from Monday to Friday
The museum is open
from
9am
to
6pm
since
Used before a noun denoting a particular time and is preceded by a verb in perfect tense
I have been eating nothing
since
yesterday
She has been ill
since
Monday
for
Period of time in number
I have been waiting
for
two hours
I was in France for two months
before/after
She met them day before yesterday
It was a long time back before they were married
I will do this painting after sometime
beside/besides
She always sits beside her sister
He is famous for singing, beside other talents
Place
on top of
When two objects are touching or when it is an unusual place to put something
the keys are on top of the refregirator
above
When two objects are not touching
The pictures are above the couch
on
A small thing on the surface of a big thing
A pen is on the table
under
Under is preferred when something is covered by what is over it
The keys are under the table
below
Below is preferred when one thing is not directly under another
A climber stopped several hundred meters below the top of mountain
behind
At or towards the back of somebody/something, and often hidden by it or them
Stay close behind me
The sun disappeared behind the clouds
in front of
Something before/in font of somebody
Mohan did his homework in front of me
near
a short distance away
His house is very near
Direction
into, in through, to, towards, from, off
into
They divided the cake into 4 pieces
She came into the room
in
She is in the garden
We live in an apartment
Conjunctions
Word or set of words that connects two words, phrases or clauses in a sentence
Phrase
Group of words and does not have a subject and a verb
Group of words that has a subject and a verb
Types
Coordinating Conjunctions
The connected phrases/clauses are of equal importance
for, and, nor, but, or, yet, so
Punctuation Rule
Join two words without a comma
Join more than two word with comma to join words except the last one
Independent Clause + (,) + Coordinating conjunction + Independent Clause
We can go to the market, or we can study at home
Subordinating Conjunction
Joins a clause to another on which it depends for its full meaning
Types
Time
After
As soon as
As long as
Before
Until/till
Since
When
Whenever
While
Concession
Although
Though
Even though
Comparison
Whether
Whereas
As much as
than
Rather than
Condition
if
provided that
in case
only if
unless
assuming that
Reason
as
in order that
since
because
so that
that
Place
where
wherever
Correlative Conjunction
Pairs of joining words that we frequently use to connect two ideas in a sentence
both...and
whether...or
either...or
neither...nor
not only...but also
Adjective
Adjective
Words which describe the nouns or pronouns
Types
Quality
He is a clever boy
Quantity
Some, little, all, whole, double, few, half, any, etc. (used with uncountable)
I don't have much time
I have little faith in God
Number
On/First, 2/Second, each, all, several some etc. (used with countable)
Demonstrative
This, that, these, those, latter, Former, Such...
Difference with demonstrative pronoun
Demonstrative Adjective modifies the noun and is always followed by the noun
This car is mine
Demonstrative Pronoun takes the place of noun phrase
This is my car
Interrogative
What manner of man is he?
Difference with interrogative pronoun
Interrogative Adjective asks a question and describes a noun
Which color looks better?
Interrogative pronoun asks a question, but stands alone
Which should I buy you on your birthday?
Distributive
Each, Every, Neither, Either, Any, One, Both etc.
Difference with Distributive Pronoun
Each man was given a pen
Distributive adjective modifies a noun or pronoun. There is always a noun next to the distributive adjective
Distributive pronoun used as a subject or object. There is never a noun next to the distributive pronoun
Each of us will get a pen
Proper
Originate from proper noun
Italian, Russian, Indian etc.
Possessive
My, Your, Its, Theirs, His/Her etc.
Difference between possessive pronoun
Possessive adjective is used to show ownership and comes before a noun in a sentence
My book is on the table
Possessive pronoun does show ownership but it does not come before a noun. It can also be used to replace a noun
This phone is yours
Spatial Aptitude
Bisector of two vectors
Syllabus
Verbal Aptitude
Basic English grammar
tenses
articles
adjectives
prepositions
conjunctions
verb-noun agreement
parts of speech
Basic vocabulary
words
idioms
phrases in context
Reading and comprehension
Narrative sequencing