CAT Preparation

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

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 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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Saturday 25 February 2012