GATE Aptitude
Quantitative Aptitude
Verbal Aptitude
Spatial Aptitude
Logical Reasoning
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
Syllogism
Syllabus
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
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Probability
P(A' OR B') = 1 - P(A AND B)
P(A - B) = P(A AND B')
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Statement
Conclusion
Types
Positive/Yes
Negative/No
Neutral/Can't Say
Do not use restatement, statement restated in conclusion should not be used
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Possibility
Some Not
Clear
No
Some
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
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All A's are B
Some A's are not B
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Some A's are B
No A's are B
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Some A's are B
No B's are A
Examples
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Some copy are pen
All Pen are copy
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All book are pen
All pen are copy
Some book are copy
A and B order can be changed in some/some not case
Answer in these cases will be either/or
Reverse
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All A's are B
Some B's are not A
Diagrams
Basic/Main Diagram
Possibility Diagram
Possibility allows the basic condition along with other conditions not stated but are possibilities
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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
Negative
Some A's are B's
Some B's are not A's
If some As are not B's, all B's can be A's
Direction Sense
Left-Right
Default direction of movement is north
Draw direction lines as reference
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
The circumcenter of a triangle is defined as the point where the perpendicular bisectors of the sides of that particular triangle intersect
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
Line Segments
Median
Angle Bisector
Altitude
A line segment joining a vertex of a triangle with the mid-point of the opposite side
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
A line segment joining a vertex of a triangle with the opposite side such that the segment is perpendicular to the opposite side
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
Scalene Triangle
No two sides are equal
No two angles are equal
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
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
Area
sqrt(3)*a^2/4
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
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a*b*sin(θ)/2
sqrt(s(s-a)(s-b)(s-c))
s = (a+b+c)/2
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Pythagoras Theorem
For Right Angled Triangle
Obtuse Angled Triangle
AC^2 = AB^2 + BC^2
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
Center of the circle passing through the vertices of the triangle
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Circumradius
Equilateral Triangle
abc/4* Area
a/sqrt(3)
Right Angled Triangle
c/2
Inradius
r * s = Area
Equilateral Triangle = a / 2sqrt(3)
Right Angled Triangle
(a + b -c)/2
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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
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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
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
b^2m+c^n = a(d^2 + mn)
Mass Point Geometry
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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
Other point outside Circle
Other point inside Circle
Right Angled Triangle
Obtuse Angled Triangle
Acute Angled Triangle
Angles made by the chord on the same arc are same
Secant
Chord
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
Line passing through a circle
Line segment with end points on circle
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.
Three cases
Meet outside and both are secants
Meet inside and both are secants
Meet outside and one is secant and another tangent
Base Proportionality theorem
If same height area is proportional to base
If same base area is proportional to height
Congruency
Types
SSS
SAS
ASA
P(A' AND B') = 1 - P(A OR B)
Terms
Eperiments
Sample Space
Event
Set of all outcomes of an expreiment
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
Select one 2
1 choice
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
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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
Problem Types
a^m + b^m is divisible by k
Check the cycle of remainder
Find combinations where the sum is compatible with the division
Tips
Remember cases of with and without replacement
Remember cases of duplicates in permutation and combination
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A,B,C,D probabilities are given
X/A,X/B,X/C,X,D probabilities are given
X happens
Find probability of A
a^p mod p = a mod p
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
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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
Grammar
Tense
Types
Past
Present
Future
Types
Prefect
Prefect Continuous
Types
Simple/Indefinite
Continuous/Progressive
Prefect
Prefect Continuous
Types
Simple/Indefinite
Continuous/Progressive
Prefect
Prefect Continuous
I is treated as a plural in present tense
Singular
Sentences
Types
Declarative
Imperative
Negative
Exclamatory
A declarative sentence simply makes a statement or expresses an opinion
An imperative sentence gives a command or makes a request
An interrogative sentence asks a question
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
Third Person
First
The person being talked to
The person being talked about
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
Articles
Articles are words that define a noun as specific or unspecific
Types
Definite
Indefinite
The
a
an
Negative
Sub + does not + Verb + obj
Interrogative
Does + Sub + Verb + obj?
Interrogative + -Negative
Does + Sub + Not + Verb + obj?
Plural
Sub + Verb + [s/es] + obj
Usage
Habitual Action
Universal Truth
Newspaper Headings
Commentary
Historical event
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
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Just finished
Normal
Negative
Interrogative Negative
Sub + is + V ing + obj
Sub + is not + V ing + obj
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
Total number of elements = N(A) + N(B) + n(C) + N(A∩B) + N(B∩C) + N(C∩A) - 2*n(A∩B∩C)
Types
Personal
Reflexive
Emphatic
Indefinite
Relative
Demonstrative
Interrogative
Distributive
Possessive
She herself opened the door
He hurt himself
Someone, anyone, nobody etc.
This is the place where we met
I, We, You, They
This, That, These, Those
What, who, which, whose, whom
Each, either, neither, none
his, hers, mine, yours
Clause
Subject + Verb combination
Types
No complete thought
Independent
Dependent
Types
Noun
Adjective
Adverb
Adverb
A word that modifies the meaning of a verb, an adjective or another adverb
Types
Time
Frequency
Place
Manner
Adverbs of degree or quantity
Adverbs of affirmation or negation
Adverbs of reason
I have spoken to him already
I have told you twice
I hurt my knee yesterday
He comes here daily
I have heard this before
We shall now begin to work
I have not seen him once
He often makes mistakes
Sam called again
He frequently comes late
Stand here
He looked up
Walk backward
Go there
He went away
He reads clearly
This story is well written
He fought bravely
The girl works hard
You should not do so
You are partly right
I am fully prepared
He is rather busy
He was too careless
These mangoes are almost ripe
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
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
Place
Direction
On, in, at, by, since, for, from, before/after, beside, besides
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On
I will see you on 7th June
In
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He started his business in 1999
He is born in the 21st century
Indefiinite periods of time
At 9'0 clock
I will see you on Monday
Indicates days, dates or part of a particular day
I will see you in spring season
Indicates long indefinite periods of time
I got this surprise in January
In a minute, in an hour, in a day, in a week, in a year
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
Use at when pointing to a specific part on area
We stayed in Mumbai
He lives in Church street
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
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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
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She always sits beside her sister
He is famous for singing, beside other talents
on top of
above
When two objects are not touching
on
A small thing on the surface of a big thing
A pen is on the table
The pictures are above the couch
When two objects are touching or when it is an unusual place to put something
the keys are on top of the refregirator
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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
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
Subordinating Conjunction
Correlative Conjunction
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
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Independent Clause + (,) + Coordinating conjunction + Independent Clause
We can go to the market, or we can study at home
Joins a clause to another on which it depends for its full meaning
Types
Time
Concession
Comparison
Condition
Reason
Place
After
As soon as
As long as
Before
Until/till
Since
When
Whenever
While
Although
Though
Even though
Whether
Whereas
As much as
than
Rather than
if
provided that
in case
only if
unless
assuming that
as
in order that
since
because
so that
that
where
wherever
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
Quantity
Number
Demonstrative
Interrogative
Distributive
Proper
Possessive
He is a clever boy
Some, little, all, whole, double, few, half, any, etc. (used with uncountable)
I don't have much time
I have little faith in God
On/First, 2/Second, each, all, several some etc. (used with countable)
This, that, these, those, latter, Former, Such...
Difference with demonstrative pronoun
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Demonstrative Adjective modifies the noun and is always followed by the noun
This car is mine
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Demonstrative Pronoun takes the place of noun phrase
This is my car
What manner of man is he?
Difference with interrogative pronoun
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Interrogative Adjective asks a question and describes a noun
Which color looks better?
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Interrogative pronoun asks a question, but stands alone
Which should I buy you on your birthday?
Each, Every, Neither, Either, Any, One, Both etc.
Difference with Distributive Pronoun
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Each man was given a pen
Distributive adjective modifies a noun or pronoun. There is always a noun next to the distributive adjective
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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
Originate from proper noun
Italian, Russian, Indian etc.
My, Your, Its, Theirs, His/Her etc.
Difference between possessive pronoun
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Possessive adjective is used to show ownership and comes before a noun in a sentence
My book is on the table
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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
Vocab
Umpteen
Fulminate
Vituperative
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Vengeance
Retribution
Retaliation
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