CAT Preparation

Sunday 18 March 2012

How to solve Analogies

Analogy asks to find the relationship between the pairs of words. The relationship between the words in the question asked is similar to the options, you have to find out the correct option.

Types of Analogies

Antonyms: Words that have opposite meaning
Synonyms: Words that have same meaning
Descriptive: One word describes the other
Part to Whole: One word is part or piece of the other
Item to Category: One word belongs to the category of the other
Degree: One word is more intense than other

Three-Step Bridge Method for solving analogies

Step 1: Build a strong bridge (relationship) sentence relating the words in the question pair. The bridge should be as short and clear as possible.
Step 2: Use this bridge with each answer choice, inserting them in place of the words in the question pair.
Step 3: The answer will be the sentence which is the most logical. If after completing steps 1 and 2, you still have not found the answer pair that works, it may be necessary to adjust the bridge sentence.

Example 1: Bad : Terrible

hard : difficult
hot : warm
funny : hilarious
right : wrong
young : new

The abhorrent individual was spurned by his fellow citizens because of his aberrant behavior.
With her speaking skills, she has the ability to fill the auditorium to its capacity.
The minister adjured his wayward congregation to abjure the sins of the flesh.
I would accept your excuse, except the part about losing the watch.
The number of students who wanted access to the computer labs was in excess of two hundred.
The government would often adopt policies that required people to adapt to a harsh regime.
The trouble with many adolescents is that they never seem to grow out of adolescence.
I need your advice. Please advise me on this.
She was confused, displaying ambivalent feelings about the ambiguous situation they'd gotten into.
Sometimes it seems more shocking to be amoral than to be immoral .
After we have the jeweler appraise the diamond, we will apprise you of its value.
When they got the assent of the weather bureau, they allowed the enormous balloon to begin its ascent.
I am averse to traveling in such adverse weather conditions.
We need a lot of money. She will allot funds according to need.
Are you all ready already, or do we have to wait for you?

My favorite show, Seinfeld, is currently doing re-runs; the new episodes will begin presently.
I kept a weekly diary during those years that I worked on the dairy farm.
If you wish to seem demure, you will have to demur less vociferously.
Any cool dessert would taste great out here in the sandy desert.
The prisoner tried to devise a clever device to help him escape.
She thought her dog would die after it drank that bowl of blue dye.
He went from a dilemma to a quandary.
The conductor seemed discomfited on the podium by the rude, discomforting behavior of the visiting pianist.
They kept their love affair discreet by living discrete lives.
You will want a disinterested judge. An uninterested judge, however, is a liability.
When asked to disassemble his old jalopy, he agreed, seeming to dissemble.
What effect does this have on you? How does it affect you?
It was part of the government's economic strategy to direct the military to purchase the most economical material available.
We should elect a president before he or she selects members of the cabinet.
How did the politicians plan to elicit these obviously illicit campaign funds without getting caught?

The insidious nature of her argument suggests an invidious comparison.
There were, for instance, several instances in which the latch failed and the door floor open, just at the the most dangerous instant.
In the intense heat, the team of scientists did an intensive study of the extensive crop damage.
The scientists were intensely focused on the problem. They studied it intently for months.
She made a laudatory speech concerning the students' laudable accomplishments.
As he led his soldiers into battle, his feet seemed made of lead.
The lightning striking all around them, the sailors proceeded in their task of lightening the cargo.
I am loath to associate with people who loathe me.
My shoes are so loose that I'm going to lose them.
I hope the bank can arrange a loan for me. If not, I hope my sister can lend me some money.
A luxuriant tropical garden was planted on the grounds of the most luxurious hotel in town.
It has been raining way too much and for too many days.
John and Mary thought that studying the martial arts, like judo, would improve their marital relationship.
He hardly deserves a medal, nor did he show true mettle when he tried to meddle in our affairs.
The moral of this story is that the morale of a military unit is extremely important.

Grammar - Introduction

Grammar is the set of structural rules that govern the composition of sentences, phrases, and words in any given language.

The grammar is approached in two different ways: Descriptive and Prescriptive. The descriptive way tries to look at the grammar of any spoken language or dialect as it actually exists and it judges whether a sentence is grammatical or not based on the rules of the speech rather than on a set of rules. The perspective grammar prescribes rules for the proper usage of language.

For MBA Admission Tests, we need to focus on prescriptive grammar.

Parts of Speech

All words in English language are divided into nine categories. These categories are called the Parts of Speech.

Nouns: are the names of people, places, animals, things, and abstract ideas.
Verbs: are words that indicate what is being done in a sentence, and tell us about the activities of its subject and about states and conditions.
Pronouns: are words that are used instead of nouns and have exactly the same functions as nouns.
Adjectives: are words that go with or qualify or modify nouns and sometimes pronouns. They are often used to describe the thing that the noun refers to.
Adverbs: are words that typically modify verbs. They tell you how, when, where, why or how often an action takes place. They can also modify adjectives, other adverbs and even whole clauses.
Determiners: are small words used before nouns to tell you which one, or how many, or whose, and so on. They include articles (a, an, the), demonstratives (this, that, these, those), possessives (my, his, our, ...) and quantifiers (all, some, no, ...).
Prepositions: typically come in front of noun phrases and pronouns and tell you how the noun /pronoun is related to the rest of the sentence in terms of place, time, reason and so on.
Conjunctions: are joining words that typically link clauses in a sentence. They can also link nouns, pronouns, verbs, etc.
Interjection: are words used to express some sudden feeling or emotion and they are not grammatically related to the other words in a sentence. For example: Hello! Alas! Hurrah! Ah! Oh! Well! etc. NEXT

Grammar - Terms

Clause: group of related words containing a subject and verb
Gerund: a verb form acting as a noun
Adjective: word that modifies a noun
Participle: verb form acting as an adjective
Adverb: word that modifies a verb, adjective or adverb
Phrase: group of related words not containing a subject and verb
Infinitive: to plus the root of the verb
Preposition: locates something in time or place
Predicate: completer of a sentence
Interjection: command, protest, or exclamation. NEXT

Noun

There is no fixed definition of Noun. Simply, a noun is a person, place, thing or idea.
Examples
Person: man, woman, teacher, doctor, John
Place: home, office, town, country, India
Thing: table, car, banana, money, music, love, dog, monkey

Countable Nouns

Countable nouns are things that can be counted. For example: Book. You can count the number of books. You may have one, two, three or ten books.

Countable nouns can be singular or plural.
Countable nouns use indefinite article a/an.
Some and Any are used with countable nouns.
A few and Many are also used with countable noun.

Uncountable Nouns

Uncountable nouns are things which cannot be counter. For example: Milk. You can count bottles of milk or litres of milk, but cannot count milk itself.

Uncountable nouns are treated as singular.
Indefinite article a/an is not used with uncountable nouns.
Some and Any are used with uncountable nouns.
A little and Much are used with uncountable nouns.

Common and Proper Nouns

A common noun is any non-specific person, place or thing.
A proper noun is any specific person, place or thing: such as names, places, companies.
NEXT

Pronouns

Pronouns are words that take the place of a noun. Pronouns are used instead of a noun. If we didn't have pronouns, we would have to repeat a lot of nouns!

Types of Pronouns

Personal Pronouns

I, me, mine
you, your, yours
she, her, hers,
it, its
we,us, our, ours
they, them, their,
theirs
myself
yourself

Interrogative Pronouns

who
whom
what
which
whose

Demonstrative Pronouns

this
that
these
those

Indefinite Pronouns

anybody
each
either
none
someone, one

Reflexive Pronouns

myself, yourself
himself, herself
ourselves, itself

Verb

Verb is a word that expresses action or otherwise helps to make a statement. Every sentence must have a verb.
Kinds of verbs

Action verbs which express physical or mental action. Example: He rode the horse to victory.
Linking verbs which make a statement by connecting the subject with a word that describes it or explains it. Example: He has been sick. NEXT

Adjective

Adjective is a word that modifies or describes a noun or pronoun.

Adjectives are invariable: They do not change their form depending on the gender or number of the noun.
Where a number of adjectives are used together, the order depends on the function of the adjective. The usual order is: Value/opinion, Size, Age/Temperature, Shape, Colour, Origin, Material. However, this is not strictly followed.

Examples

He is a lonely man (Quality)
This clock is German (Origin)
My coat is very old (Age)
This is a very long film (Size)
He wore a red shirt (Colour)
A square envelope (Shape)
Grammar is boring (Judgement)

Subject Verb Agreement

Basic Principle: Singular subjects need singular verbs; plural subjects need plural verbs. My brother is a nutritionist. My sisters are mathematicians.

The indefinite pronouns anyone, everyone, someone, no one, nobody are always singular and, therefore, require singular verbs.
Phrases such as together with, as well as, and along with are not the same as and. The phrase introduced by as well as or along with will modify the earlier word (mayor in this case), but it does not compound the subjects (as the word and would do).
The pronouns neither and either are singular and require singular verbs even though they seem to be referring, in a sense, to two things.
The conjunction or does not conjoin (as and does): when nor or or is used the subject closer to the verb determines the number of the verb. Whether the subject comes before or after the verb doesn't matter; the proximity determines the number.
The words there and here are never subjects.
Sometimes modifiers will get betwen a subject and its verb, but these modifiers must not confuse the agreement between the subject and its verb.
Words such as glasses, pants, pliers, and scissors are regarded as plural (and require plural verbs) unless they're preceded the phrase pair of (in which case the word pair becomes the subject).
Fractional expressions such as half of, a part of, a percentage of, a majority of are sometimes singular and sometimes plural, depending on the meaning. (The same is true, of course, when all, any, more, most and some act as subjects.)
If your sentence compounds a positive and a negative subject and one is plural, the other singular, the verb should agree with the positive subject.

Saturday 25 February 2012

Probability

Probability is the likelihood or chance of an event occurring.

Some Concepts

When we toss a coin, then either a Head (H) or a Tail (T) appears.
A dice is a solid cube ,having 6 faces,marked 1, 2, 3, 4, 5, 6 respectively. When we throw a die, the outcome is the number that appears on its upper face.
A pack of cards has 52 cards. It has 13 cards of each suit, namely spades, clubs, hearts and diamonds. Cards of spades and clubs are balck cards. Cards of hearts and diamonds are red cards. There are four honours of each suit. These are Aces, Kings, Queens and Jacks. These are called Face cards.

The probability of a certain event occurring can be represented by P(A). The probability of a different event occurring can be written as P(B). Therefore, for two events A and B,

$\displaystyle P(A) + P(B) - P(A\cap B) = P(A\cup B)$

Mutually Exclusive Events

Events A and B are mutually exclusive if they have no events in common. If two events are mutually exclusive,

$\displaystyle P(A) + P(B) = P(A\cup B)$

Independent Events

Two events are independent if (and only if)

$\displaystyle P(A\cap B) = P(A)P(B)$

Conditional Probability

Conditional probability is the probability of an event occurring, given that another event has occurred.

$\displaystyle P(A|B)$ means the probability of A occurring, given that B has occurred.

For two events A and B,

$\displaystyle P(A\cap B) = P(A|B)P(B)$

$\displaystyle P(A\cap B) = P(B|A)P(A)$

Permutation & Combination

Combination & Permutation deals with arrangement of thing. If the order doesn't matter, then it is called Combination. If the order does matter, then it is a Permutation.

In other words, Permutation is an ordered Combination.

Permutation

${^nP_r} = {n! \over {(n-r)!}$

There are basically two types of permutation:

When repetition is allowed
No repetition

1. Permutations with Repetition
To choose r things from n when repetition is allowed, the permutations are:

n × n × ... (r times) = n^r

(Because there are n possibilities for the first choice, then there are n possibilites for the second choice, and so on.)

Combinations

Number of ways objects can be selected from a group.
${^nC_r} = {{^nP_r} \over r!}$ NEXT

Logarithms

$x = b^y$
$\therefore \log x = \log b^y$
$\log x = y \log b$
${{\log x} \over {\log b}} = y$
$\log_b x = y$
Therefore,
$\text{ if }x = b^y,\text{ then }y = \log_b (x)$

Logarithmic Identities

$\log_b(xy) = \log_b(x) + \log_b(y)$
$\log_b\!\left(\begin{matrix}\frac{x}{y}\end{matrix}\right) = \log_b(x) - \log_b(y)$
$\log_b(x^d) = d \log_b(x)$
$\log_b\!\left(\!\sqrt[y]{x}\right) = \begin{matrix}\frac{\log_b(x)}{y}\end{matrix}$
$x^{\log_b(y)} = y^{\log_b(x)}$
$c\log_b(x)+d\log_b(y) = \log_b(x^c y^d)$
$\log_b(1) = 0$
$\log_b(b) = 1$
$b^{\log_b(x)} = x$
$\log_b(b^x) = x$
$\log_a b = {\log_c b \over \log_c a}$ NEXT

Indices and Surds

Laws of Indices

$a^m \times a^n = a^{m+n}$
$a^m \div a^n = a^{m-n}$
$(a^m)^n = a^{mn}$
$a^{1 \over m} = \sqrt[m]{a}$
$a^{-m} = \frac{1}{a^m}$
$a^{\frac{m}{n}} = \sqrt[n]a^m$
$a^0 = 1$

$a^1 = 1$ NEXT

Binomial Theorem

Binomial Expression: An algebraic expression consisting of two terms with a positive or negative sign between them. Example: (x+y)
The expansion of binomial expression raised to power n is called Binomial Theorem.
${(x + y)^n = x^n + ^nC_1x^{n-1}y + ^nC_2x^{n-2}y^2 + \dots + y^n}$
$^nC_1, ^nC_2, \dots , ^nC_n$ are Binomial Coefficients.
Points to Note:

There are total of (n+1) terms in the expansion.
In each term, sum of the indices of x and y is equal to n.

Functions

A function is a rule which indicates an operation to perform.

Graph Transformations

y = f(x) + a is the same as the graph y = f(x), shifted upwards by a units.
y = f(x - a) shifts the graph a units to the right.
y = f(ax) is a stretch with scale factor 1/a parallel to the x-axis.
y = a.f(x) is a stretch with scale factor a parallel to the y-axis. NEXT

Progressions

A Progression is a sequence of numbers which have some kinf of relation. This relation determines what kind of a progression is. Generally, there are two types of progressions:

Arithmetic Progression (AP)
Geometric Progression (GP)

Any progression (AP or GP) can be generally expressed as
$a_1 + a_2 + a_3 +\dots+ a_{n-1} + a_n$
Total Terms: $n$
First Term: $a_1$
Last Term: $a_n$

Arithmetic Progression

In AP, the relation amoung sequence of numbers is that the difference between any two successive numbers is same.
Example: 3, 5, 7, 9, 11, 13, ... is an AP with difference 2. This difference is called common difference.
$a_n = a_1 + (n - 1)d$
$S_n=\frac{n}{2}( a_1 + a_n)=\frac{n}{2}[ 2a_1 + (n-1)d]$

Geometric Progression

In GP, each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio.

Example: 2, 6, 18, 54, ...

$a + ar + ar^2 + ar^3 + ar^4 + \cdots$

$a_n = a\,r^{n-1}$ NEXT

Quadratic Equations

$ax^2+bx+c=0$
Linear Equation can have degree of atmost 1 and has only one solution. Quadratic Equation can have degree of atmost 2 and has two solutions.
General form of quadratic equation: ax² + bx + c = 0, where a, b and c are constants. Note that maximum degree of x is 2.

Solving Quadratic Equations

Unlike linear equations, any quadratic equation always has two solutions called roots of quadratic equation. After solving quadratic equation, you will get two values of x. To solve quadratic equation, you can use directly quadratic formula.

$x={-b\pm\sqrt{b^2-4ac} \over 2a}$

Discriminant

In the above quadratic formula, the expression underneath the square root sign is called the discriminant of the quadratic equation. Discriminant is used to find the nature of roots.

$\Delta = b^2 - 4ac$

Case 1: $\Delta > 0$

Real and distinct roots

Case 2: $\Delta = 0$

Real and one distict root (two same roots)

Case 3: $\Delta < 0$

Roots are imaginary and occur as complex conjugates of each other

Sum and Product of Roots

Let $\alpha$ and $\beta$ be the roots of quadratic equation $x^2+px+q=0$
$x^2+px+q=(x-\alpha)(x-\beta)$
$x^2+px+q=x^2-(\alpha+\beta)x+\alpha \beta$
$p=-(\alpha+\beta)$
$q=\alpha \beta$
$\text{Sum of the roots} = -p$

$\text{Product of roots} = q$ NEXT

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