Arithmetic Progression ( A. P. ) - a sequence of numbers if the differences between consecutive terms are the same.
7 , 10 , 13 , 16 , 19 …
a1 a2 a3 a4 a5 … an
a1 = the first term
an = the nth term
d = common difference
= a2 – a1 = a3 – a2
nth term of A.P.
an = a1 + ( n-1 ) d
Sum of terms in A.P.
Sn = ( a1 + an )
or Sn = ( 2a1 + (n – 1)d )

Geometric Progression (G.P.):

- a sequence of numbers if the ratios of consecutive terms are the same.
2 , 6 , 18 , 54 , 162 …
a1 a2 a3 a4 a5 … an
a1 = the first term
an = the nth term
r = common ratio
= a2/a1 = a3/a2 nth term of G.P.
an = a1 rn-1 Geometric Progression (G.P.)

Slide4:

Sum of terms in G.P. : Sn = a1 (1-rn-1 ),
r < 1
Sum of Infinite Geometric Progression
S= a1 / 1-r

Harmonic Progression (H.P.):

Harmonic Progression (H.P.) -a sequence of numbers in which their reciprocals forms an arithmetic progression.

Binomial Theorem:

Binomial Theorem Expansion of (a + b)n
Properties:
The number of terms in the expansion of (a + b)n is n + 1.
The first term is an , the last term is bn.
The exponent of “a” descends linearly from n to 0.
The exponent of “b” ascends linearly from 0 to n.
The sum of the exponents of a and b in any of the terms is equal to n.
The coefficient of the second term and the second to the last term is n.

Slide7:

rth term of (a + b)n
rth term = n!
_________ an-r+1 br-1
(n-r+1)! (r-1)!
if middle term : r = (n/2)+ 1

Pascal’s Triangle:

Pascal’s Triangle - used to determine coefficients of the terms in a binomial expansion.
(a + b)0 1
(a + b)1 1 1
(a + b)2 1 2 1
(a + b)3 1 3 3 1
(a + b)4 1 4 6 4 1
(a + b)5 1 5 10 10 5 1
(a + b)6 1 6 15 20 15 6 1

Permutation:

Permutation
-is an ordering of the elements such that one element is first, one is second, one is third, and so on.
Permutations of n elements
P = n!

Slide10:

Permutations of n elements taken r at a time
nPr = n!
_____
(n-r)!

Distinguishable Permutations:

Distinguishable Permutations Suppose a set of n objects has n1 of one kind of object, n2 of a second kind, n3 of a third kind, and so on, with n = n1 + n2 + n3 + …+ nk.
P = n!
___________
n1!n2!n3!...nk!

Slide12:

Cyclical Permutation (Permutation of n things in a circle)
P = (n – 1)!

Combination:

Combination - a method of selecting subsets of a larger set in which order is not important.
Combinations of n elements taken r at a time
nCr = n!
_______
(n-r)!r!

Probability of an Event:

Probability of an Event If an event E has n(E) equally likely outcomes and its sample space S has n(S) equally likely outcomes, then the probability of event E is
P(E) = favorable / probable outcome

Properties of Exponents:

Properties of Exponents am an = am + n
am/ an = am – n
a-n = 1/ an
a0 = 1 , a is not = 0
(ab)m = am bn
(am)n = am n
(a/b)m = am /bm
|a²| =|a²| = a2

Properties of Logarithms:

Properties of Logarithms Base Logarithm
log (uv) = log u + log v
log u/v = log u – log v
log un = n log u
loga a = 1
logu v = logv/logu
loga m = n then an = m
log m = log n then m = n

Natural Logarithm:

Natural Logarithm ln (uv) = ln u + ln v
Ln (u/v) = ln u – ln v
ln un = n ln u
ln u = loge u , e = 2.718

Quadratic Equation:

Quadratic Equation If Ax2 + Bx + C = 0
x =
where B2 – 4AC is called the discriminant
if B2 = 4AC , the roots are equal
if B2 > 4AC , the roots are real, unequal
if B2 < 4AC , the roots are imaginary

Properties of Roots:

Properties of Roots
Sum of roots : x1 + x2 = -B/A
Product of roots : x1 x2 = C/A

Verbal Problems:

Verbal Problems Work Problem
Rate of working x Time working
= Completion of the work
Rate x Time = 1

Clock Problem:

Clock Problem

Clock Problem:

Clock Problem

Variation Problem:

Variation Problem x is directly proportional to y
x Q y ? x = ky
x is inversely proportional to y
x Q ? x = k(1/y)
k = constant of proportionality

Rate Problem & Age Problem:

Rate Problem & Age Problem Rate Problem
- motion of body with uniform velocity.
Distance = Rate x Time
Age Problem
Past Present Future
was is will be
ago now
10 8
A – 10 A A + 8

Complex Numbers:

Complex Numbers For real numbers a and b, the number
a + bi
is a complex number, bi is an imaginary number. i2 = -1
Operations of complex numbers
Addition
(a + bi) + (c + di) = (a + c) + (b + d)i
Subtraction
(a + bi) - (c + di) = (a - c) + (b - d)i
Multiplication
(a + bi)(c + di)
Division
(a + bi)/(c + di)

PLANE GEOMETRY & MENSURATION:

PLANE GEOMETRY & MENSURATION b a c Right triangle – is a triangle having one right angle.

Isosceles triangle:

Isosceles triangle Area, A = or A =

You do not have the permission to view this presentation. In order to view it, please
contact the author of the presentation.

Send to Blogs and Networks

Processing ....

Premium member

Use HTTPs

HTTPS (Hypertext Transfer Protocol Secure) is a protocol used by Web servers to transfer and display Web content securely. Most web browsers block content or generate a “mixed content” warning when users access web pages via HTTPS that contain embedded content loaded via HTTP. To prevent users from facing this, Use HTTPS option.

By: suchitraswain20102re (86 month(s) ago)

wowo...amzng.