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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)+ 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 =