MATHEMATICS:
MATHEMATICS Arithmetic Progression ( A. P. ):
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)+ 1Pascal’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 1Permutation:
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 outcomeProperties 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/AVerbal Problems:
Verbal Problems Work Problem
Rate of working x Time working
= Completion of the work
Rate x Time = 1Clock Problem:
Clock ProblemClock Problem:
Clock ProblemVariation 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 proportionalityRate 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 =