(MAINS & ADVANCE)
Er. L.K.Sharma
Er. L.K.Sharma an engineering graduate from NIT, Jaipur (Rajasthan), {Gold medalist, University of Rajasthan} is a well known name among the engineering aspirants for the last 15 years. He has been honored with BHAMASHAH AWARD two times for the academic excellence in the state of Rajasthan. He is popular among the student community for possessing the excellent ability to communicate the mathematical concepts in analytical and graphical way. He has worked with many premiere IITJEE coaching institutes of Delhi, Haryana, Jaipur and Kota, {presently associated with THE GUIDANCE, Kalu Sarai, New Delhi as senior mathematics faculty and Head of Mathematics department with IGNEOUS, Sonipat (Haryana)}. He has worked with Delhi Public School, RK Puram, New Delhi for five years as a senior mathematics {IITJEE} faculty. Er. L.K.Sharma has been proved a great supportive mentor for the last 15 years and the most successful IITJEE aspirants consider him an ideal mathematician for Olympiad/KVPY/ISI preparation. He is also involved in the field of online teaching to engineering aspirants and is associated with www.100marks.in and iProf Learning Sols India (P) Ltd for last 5 years , as a senior member of www.wiziq.com (an online teaching and learning portal), delivered many online lectures on different topics of mathematics at IITJEE {mains & advance} level . “Objective Mathematics for IITJEE” authored by Er. L.K.Sharma has also proved a great help for engineering aspirants and its ebook format can be downloaded from http://mathematicsgyan.weebly.com.
Contents 1. Basics of Mathematics
1
2. Quadratic Equations
14
3. Complex Numbers
20
4. Binomial Theorem
29
5. Permutation and Combination
32
6. Probability
36
7. Matrices
42
8. Determinants
55
9. Sequences and Series
61
10. Functions
67
11. Limits
76
12. Continuity and Differentiability
80
13. Differentiation
86
14. Tangent and Normal
91
15. Rolle's Theorem and Mean Value Theorem
93
16. Monotonocity
95
17. Maxima and Minima
97
18. Indefinite Integral
101
19. Definite Integral
109
20. Area Bounded by Curves
114
21. Differential Equations
118
22. Basics of 2DGeometry
123
23. Straight Lines
125
24. Pair of Straight Lines
129
25. Circles
132
26. Parabola
138
27. Ellipse
142
28. Hyperbola
146
29. Vectors
152
30. 3Dimensional Geometry
161
31. Trigonometric Ratios and Identities
170
32. Trigonometric Equations and Inequations
175
33. Solution of Triangle
177
34. Inverse Trigonometric Functions
180
1. BASICS of MATHEMATICS {1} Number System : (i) Natural Numbers : The counting numbers 1,2,3,4,..... are called Natural Numbers. The set of natural numbers is denoted by N. Thus N = {1,2,3,4, .....}. (ii) Whole Numbers : Natural numbers including zero are called whole numbers. The set of whole numbers, is denoted by W. Thus W = {0,1,2, ...............} (iii) Integers : The numbers ... 3, 2, 1, 0, 1,2,3,....... are called integers and the set is denoted by I or Z. Thus I (or Z) = {... 3, 2, 1, 0, 1, 2, 3...} (a)
Set of positive integers denoted by I+ and consists of {1,2,3,....} called as set of Natural numbers.
(b)
Set of negative integers, denoted by I and consists of {......,
(c)
Set of nonnegative integers {0,1,2 ......}, called as set of Whole numbers.
(d)
Set of nonpositive integers {..........,
3, 2, 1}
3, 2, 1,0]
(iv) Even Integers : Integers which are divisible by 2 are called even integers. e.g. 0, 2, 4..... (v) Odd Integers : Integers, which are not divisible by 2 are called as odd integers. e.g. 1, 3, 5, 7...... (vi) Prime Number : Let 'p' be a natural number, 'p' is said to be prime if it has exactly two distinct factors, namely 1 and itself. so, Natural number which are divisible by 1 and itself only are prime numbers (except 1). e.g. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ... (vii) Composite Number : Let 'a' be a natural number, 'a' is said to be composite if, it has at least three distinct factors.
Note: (i)
'1' is neither prime nor composite.
(ii)
'2' is the only even prime number.
(iii)
Number which are not prime are composite numbers (except 1).
(iv)
'4' is the smallest composite number.
(viii) Coprime Number : Two natural numbers (not necessarily prime) are coprime, if there H.C.F (Highest common factor) is one. e.g. (1,2), (1,3), (3,4), (3, 10), (3,8), (5,6), (7,8) etc. Mathematics Concept Note IITJEE/ISI/CMI
Page 1
These numbers are also called as relatively prime numbers.
Note: (a) (b)
Two prime number(s) are always coprime but converse need not be true. Consecutive numbers are always coprime numbers.
(ix) Twin Prime Numbers : If the difference between two prime numbers is two, then the numbers are twin prime numbers. e.g. {3,5}, {5,7}, (11, 13}, {17, 19}, {29, 31} (x) Rational Numbers : All the numbers that can be represented in the form p/q, where p and q are integers and q 0, are called rational numbers and their set is denoted by Q. p Thus Q = { q : p,q I and q 0}. It may be noted that every integer is a rational number
since it can be written as p/1. It may be noted that all recurring decimals are rational numbers . (xi) Irrational Numbers : There are real numbers which can not be expressed in p/q form. These numbers are called irrational numbers and their set is denoted by Qc. (i.e. complementary set of Q) e.g. 2 , 1 + 3 , e, etc. Irrational numbers can not be expressed as recurring decimals.
Note: e 2.71 is called Napier's constant and
3.14.
(xii) Real Numbers : The complete set of rational and irrational number is the set of real numbers and is denoted by R. Thus R = Q Qc. The real numbers can be represented as a position of a point on the real number line. The real number line is the number line where in the position of a point relative to the origin (i.e. 0) represents a unique real number and vice versa. Positive side
Negative side –3
–2
–1
0
1 2 2
3
Real line
All the numbers defined so far follow the order property i.e. if there are two distinct numbers a and b then either a < b or a > b.
Note: (a)
Integers are rational numbers, but converse need not be true.
(b)
Negative of an irrational number is an irrational number.
(c)
Sum of a rational number and an irrational number is always an irrational number
(d) (e) (f)
ma r a h e.g. 2 + 3 S . 82 K . 6 L 7 The product of a non zero rational number & an irrational 58be an Er. 8number 027will5always 1 0 irrational number. 9 801 If a Q and b Q, then ab = rational number, only if a = 0. 839 sum, difference, product and quotient of two irrational numbers need not be a irrational number or we can say, result may be a rational number also.
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
Page 2
(xiii) Complex Number : A number of the form a + ib is called complex number, where a,b R and i = 1 , Complex number is usually denoted by Z and the set of complex number is represented by C. Note : It may be noted that N W I Q R C.
{2} Divisibility Test : (i)
A number will be divisible by 2 iff the digit at the unit place of the number is divisible by 2.
(ii)
A number will be divisible by 3 iff the sum of its digits of the number is divisible by 3.
(iii)
A number will be divisible by 4 iff last two digit of the number together are divisible by 4.
(iv)
A number will be divisible by 5 iff the digit of the number at the unit place is either 0 or 5.
(v)
A number will be divisible by 6 iff the digits at the unit place of the number is divisible by 2 & sum of all digits of the number is divisible by 3.
(vi)
A number will be divisible by 8 iff the last 3 digits of the number all together are divisible by 8.
(vii)
A number will be divisible by 9 iff sum of all it's digits is divisible by 9.
(viii)
A number will be divisible by 10 iff it's last digit is 0.
(ix)
A number will be divisible by 11, iff the difference between the sum of the digits at even places and sum of the digits at odd places is 0 or multiple of 11. e.g. 1298, 1221, 123321, 12344321, 1234554321, 123456654321
{3} (i)
Remainder Theorem :
Let p(x) be any polynomial of degree greater than or equal to one and 'a' be any real number. If p(x) is divided by (x a), then the remainder is equal to p(a). (ii)
Factor Theorem :
Let p(x) be a polynomial of degree greater than of equal to 1 and 'a' be a real number such that p(a) = 0, then (x a) is a factor of p(x). Conversely, if (x a) is a factor of p(x), then p(a) = 0. (iii)
Some Important formulae :
(1)
(a + b)2 = a2 + 2ab + b2
= (a b)2 + 4ab
(2)
(a b)2 = a2
= (a + b)2
(3)
a2 b2 = (a + b) (a b)
(4)
(a + b)3 = a3 + b3 + 3ab (a + b)
(5)
(a b)3 = a3 b3
(6) (7) (8)
2ab + b2
4ab
ma r a h a + b = (a + b) 3ab(a + b) = (a + b) (a + b ab) S . 82 K . 6 L 7 a b = (a b) 3ab (a b) = (a b)(a + b + ab)r. E 81027 5058 9 801 (a + b + c) = a + b c + 2ab + 2bc + 2ca 39 1 1 1 8 = a + b + c + 2abc 3
3
3ab(a b)
3
3
3
2
3
2
2
2
2
2
2
Mathematics Concept Note IITJEE/ISI/CMI
2
2
2
2
a
b
c
Page 3
1 [(a b)2 + (b c)2 + (c a)2 ] 2
(9)
a2 + b2 + c2 ab bc ca =
(10)
a3 + b3 + c3 3abc = (a + b + c) (a2 + b2 + c2 ab bc ca) =
1 (a + b + c) [(a b)2 + (b c)2 + (c a)2 ] 2
(11)
a4 b4 = (a + b)(a b) (a2 + b2)
(12)
a4 + a2 + 1 = (a2 + 1)2
a2 = (1 + a + a2) (1 a + a2)
{4} Definition of indices : If 'a' is any non zero real or imaginary number and 'm' is the positive integer, then am = a.a.a....a (m times). Here a is called the base and m is the index, power or exponent. (I) Law of indices : (1)
a0 = 1,
(2)
am =
(3)
am + n = am an, where m and n are rational numbers
(4)
am n =
(5)
(am)n = amn
(6)
ap/q =
(a 0) 1
am
q
,
am an
(a 0)
, where m and n are rational numbers, a 0
ap
{5} Ratio & proportion : (i)
Ratio :
1. If A and B be two quantities of the same kind, then their ratio is A : B; which may be denoted by the fraction
A (This may be an integer or fraction ) B
2. A ratio may represented in a number of ways e.g.
a ma na = = = .....where m, n, .... b mb nb
are nonzero numbers. 3. To compare two or more ratio, reduced them to common denominator. 4. Ratio between two ratios may be represented as the ratio of two integers e.g.
a c : : b d
ma r a Sh ace82 a .cKe. Ratios are compounded by multiplying them together i.e. L. . ..... = 6 ... . b d f 277 bdf 8 r E 810 505 9 ratio If a : b is any ratio then its duplicate ratio is a : b ; triplicate 0is1a : b .....etc. 8 9 83 If a : b is any ratio, then its subduplicate ratio is a : b ; subtriplicate ratio is
a /b ad c / d = bc or ad : bc. duplicate, triplicate ratio.
5. 6.
7. a1/3 : b1/3 etc.
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
2
2
1/2
3
3
1/2
Page 4
(ii)
Proportion :
When two ratios are equal , then the four quantities compositing them are said to be proportional. If
a c = , then it is written as a : b = c : d or a : b :: c : d b d
1.
'a' and 'd' are known as extremes and 'b and c' are known as means.
2.
An important property of proportion : Product of extremes = product of means.
3.
If
a : b = c : d, then b : a = d : c(Invertando)
4.
If
a : b = c : d, then a : c = b : d(Alternando)
5.
If
a : b = c : d, then a b c d = (Componendo) b d
6.
If
a : b = c : d, then a b c d = = (Dividendo) b d
7.
If
a : b = c : d, then a b c d = (Componendo and Dividendo) ab cd
{6} Cross Multiplication : If two equations containing three unknown are a1x + b1y + c1z = 0 ............(i) a2x + b2y + c2z = 0 ............(ii) Then by the rule of cross multiplication x y z b1c2 b2c1 = c1a2 c2a1 = a1b2 a2b1
............(ii)
In order to write down the denominators of x, y and z in (3) apply the following rule, "write down the coefficients of x, y and z in order beginning with the coefficients of y and repeat them as in the diagram" b1 c1 a1 b1 b2 c2 a2 b2 Multiply the coefficients across in the way indicated by the arrows; remembering that informing the products any one obtained by descending is positive and any one obtained by ascending is negative.
{7} Intervals : Intervals are basically subsets of R and are commonly used in solving inequalities or in finding domains. If there are two numbers a,b R such that a < b, we can define four types of intervals as follows : Symbols Used (i) Open interval : (a,b) = {x : a < x < b} i.e. end points are not included ( ) or ] [
ma r a h S . 82 K . 6 L 7 r. also 1included 27 058 Eare 0 (ii) Closed interval : [a,b] = {x : a x b} i.e. end points 98 8015 This is possible only when both a and b are finite. 839 (iii) Openclosed interval :(a,b] = {x : a < x b} ( ] or
(iv)Closedopen interval :[a,b) = x : a x < b} Mathematics Concept Note IITJEE/ISI/CMI
[] ]]
[ ) or [ [ Page 5
The infinite intervals are defined as follows : (i)
(a, ) = {x : x > a}
(ii)
[a, ) = {x : x a}
(iii)
(
(iv)
( ,b] = {x : x b}
(v)
( , ) = {x : x R}
,b) = {x : x
Note: (i)
For some particular values of x, we use symbol { } e.g. If x = 1, 2 we can write it as x {1,2}
(ii)
If their is no value of x, then we say x (null set)
Various Types of Functions : (i)
Polynomial Function : If a function f is defined by f(x) = a0 xn + a1 xn 1 + a2xn 2 + ......... + an 1 x + an where n is a non negative integer and a0, a1, a2, ........, an are real numbers and a0 0, then f is called a polynomial function of degree n.
There are two polynomial functions, satisfying the relation; f(x).f(1/x) = f(x)+f(1/x), which are f (x) = 1 xn
(ii)
Constant function : A function f : A B is said to be a constant function, if every element of A has the same f image in B. Thus f : A B; f(x) = c , x A, c B is a constant function.
(iii)
Identity function : The function f : A A define dby, f(x) = x x A is called the identity function on A and is denoted by IA. It is easy to observe that identity function is a bijection.
(iv)
Algebraic Function : y is an algebraic function of x, if it is a function that satisfies an algebraic equation of the form, P0(x) yn + P1(x) yn 1 + ........ + Pn 1 (x) y + Pn(x) = 0 where n is a positive integer and P0(x), P1(x) ........ are polynomials in x. e.g. y = x is an algebraic function, since it satisfies the equation y2 x2 = 0.
All polynomial functions are algebraic but not the converse.
A function that is not algebraic is called Transcendental Function .
(v)
Rational Function : g(x) A rational function is a function of the form, y = f(x) = h(x) , where g(x) & h(x) are
polynomial functions. (vi)
Irrational Function : An irrational function is a function y = f(x) in which the operations of additions, substraction, multiplication, division and raising to a fractional power are used
ma r a h S . x x 82 K For example y = is an irrational function . 6 L 7 2x x Er. 81027 5058 (a) The equation f(x) = g(x) is equivalent to the following system 9 8 01 9 f(x) = g (x) & g(x) 0 83 3
1/3
2
(b)
The inequation
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
f(x) < g(x) is equivalent to the following system Page 6
f(x) < g2(x) & f(x) 0 & g(x) > 0 (c)
The inequation
f(x) > g(x) is equivalent to the following system
g(x) < 0 & f(x) 0 (vii)
or g(x) 0
&
f(x) > g2(x)
Exponential Function : A function f(x) = ax = ex ln a (a > 0, a 1, x R) is called an exponential function. Graph of exponential function can be as follows : Case  I For a > 1
Case  II For 0 < a < 1
f(x)
f(x)
(0,1)
(0,1) x
0
x
0
(viii) Logarithmic Function : f(x) = logax is called logarithmic function where a > 0 and a 1 and x > 0. Its graph can be as follows Case  I For a > 1
Case  II For 0 < a < 1
f(x)
0
f(x)
(0,1)
x
0
(1,0) x
LOGARITHM OF A NUMBER : The logarithm of the number N to the base 'a' is the exponent indicating the power to which the base 'a' must be raised to obtain the number N. This number is designated as loga N.
Hence :
logaN = x ax = N , a > 0 , a 1 & N > 0
If a = 10 , then we write log b rather than log10 b . If a = e , we write ln b rather than loge b . The existence and uniqueness of the number loga N follows from the properties of an experimental functions .
ma r a h S . log N 82 K . identity : =N , a>0 , a1 & N>0 a 6 L 7 Er.. 81027 5058 This is known as the FUNDAMENTAL LOGARITHMIC IDENTITY 9 801 NOTE :loga1 = 0 (a > 0 , a 1) 839 loga a = 1 (a > 0 , a 1) and
From the definition of the logarithm of the number N to the base 'a' , we have an a
log1/a a =  1 Mathematics Concept Note IITJEE/ISI/CMI
(a > 0 , a 1) Page 7
THE PRINCIPAL PROPERTIES OF LOGARITHMS : Let M & N are arbitrary posiitive numbers , a > 0 , a 1 , b > 0 , b 1 and is any real number then ; (i) loga (M . N) = loga M + loga N (ii) loga (M/N) = loga M loga N (iii)
loga M = . loga M
logb M =
(iv)
NOTE : logba . logab = 1 logba = 1/logab.
log a M log a b
logba . logcb . logac = 1 x
ln a e = ax
logy x . logz y . loga z = logax.
PROPERTIES OF MONOTONOCITY OF LOGARITHM : (i) For a > 1 the inequality 0 < x < y & loga x < loga y are equivalent. (ii)
For 0 < a < 1 the inequality 0 < x < y & loga x > loga y are equivalent.
(iii)
If a > 1 then loga x < p
0 < x < ap
(iv)
If a > 1 then logax > p
x > ap
(v)
If 0 < a < 1 then loga x < p
x > ap
(vi)
If 0 < a < 1 then logax > p
0 < x < ap
If the number & the base are on one side of the unity , then the logarithm is positive ; If the number & the base are on different sides of unity, then the logarithm is negative. The base of the logarithm ‘a’ must not equal unity otherwise numbers not equal to unity will not have a logarithm & any number will be the logarithm of unity. n
For a non negative number 'a' & n 2 , n N
(ix)
Absolute Value Function /Modulus Function :
a = a1/n.
x if x 0 The symbol of modulus function is f(x) =x and is defined as : y = x = x if x 0
y y
=
–x
y
0
=
x
x y = x
Properties of Modulus : For any a, b R (i)
a 0
(iii)
a a, a
(v) (vii)
(ii)
a
a a b = b
a b a b
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
(iv) (vi)
ma r a h S . 82 ab = a b K . 6 L 7 Er. 81027 5058 a + b a + 9 b 01 8 9 83 a =  a
Page 8
(x)
Signum Function : A function f (x) = sgn (x) is defined as follows :
Y
1 for x 0 0 for x 0 f(x) = sgn (x) = 1 for x 0
y = 1 if x > 0
O
(x); x 0 (x) It is also written as sgn x = 0; x 0
(xi)
x
y = –1 if x < 0 y = sgn x
f(x); f(x) 0 f(x) sgn f(x) = 0; f(x)0 Greatest Integer Function or Step Up Function : The function y = f(x) = [x] is called the greatest integer function where [x] equals to the greatest integer less than or equal to x. For example : for 1 x<0 ; [x] = 1 ; for 0 x<1 ; [x] = 0 ; [x] = 2 and so on. for 1 x<2 ; [x] = 1 ; for 2 x<3 Properties of greatest integer function :
(a)
x 1 < [x] x
(b)
[x m] = [x] m iff m is an integer..
(c)
[x] + [y] [x + y] [x] + [y] + 1
(d)
[x] + [ x] =
(xii)
Fractional Part Function : It is defined as, y = {x} = x [x]. e.g. the fractional part of the number 2.1 is 2.1 2 = 0.1 and the fractional part of 3.7 is 0.3.
0 ;
if x is an integer
1 ;
otherwise
The period of this function is 1 and graph of this function is as shown. y 1
y=1 x
–2
–1
0
1
2
y = (x)
Properties of fractional part function (a)
{x m} = {x} iff m is an integer
(b)
{x} + { x} =
Mathematics Concept Note IITJEE/ISI/CMI
0 ; if x is an integer 1; otherwise
3
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Page 9
Graphs of Trigonometric functions : (a)
y = sin x
y [ 1,1]
x R;
y 1
0
x
–1
(b)
y = cos x
y [ 1,1]
x R;
y 1 0
x
–1
(c)
x R (2n+1) /2; n I;
y = tan x
y R
y
0
(d)
x
x R n ; n I; y R
y = cot x
y
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
0
a m r a x h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Page 10
(e)
x R n ; n I; y ( , 1,] [1, )
y = cosec x
y 1
0
x
–1
(f)
x R (2n+1) /2; n I;
y = sec x
y ( , 1,] [1, )
y 1
0
x
–1
Trigonometric Functions of sum or Difference of Two Angles : (a)
sin (A B) = sinA cosB cosA sinB
(b)
cos (A B) = cosA cosB sinA sinB
(c)
sin2A sin2B = cos2B cos2A = sin(A+B). sin(A B)
(d)
cos2A sin2B = cos2B sin2A = cos(A+B). cos(A B)
(e)
tan A tanB tan (A B) = 1 tan A tanB
(f)
cot A cot B 1 cot (A B) = cot B cot A
(g)
tan (A+B+C) =
tan A tanB tanC tan A tanB tanC 1 tanA tanB tanB tanC tanC tanA
Factorisation of the Sum or Difference of Two sines or cosines : (a)
sinC + sin D = 2sin
C D C D cos 2 2
(b)
sin C sinD = 2cos
C D C D sin 2 2
a C D Cm D r a sin h 22 2 S . K . 68 L 7 7 . 58 : Er of 02 &5Cosines Transformation of Products into Sum or Differences Sines 1 0 8 9 8 01 (a) 2sinA cosB = sin (A+B) + sin(A B) (b) 2cosA sinB = sin(A+B) sin(A B) 9 83 (c) 2cosA cosB = cos(A+B) + cos(A B) (d) 2sinA sinB = cos(A B) cos(A+B) (c)
cosC + cosD = 2cos
Mathematics Concept Note IITJEE/ISI/CMI
C D C D cos 2 2
(d)
cosC cosD=
2sin
Page 11
Multiple and Submultiple Angles : (a)
sin 2A = 2 sinA cosA ; sin = 2sin
(b)
cos2A = cos2A
(c)
2
cos
2
sin2A = 2cos2A 1 = 1 2 sin2A; 2cos2 2 =1 + cos ,2sin2 2 = 1 cos .
2tan 2 2 tanA tan 2A = ; tan = 2 2 1 tan 1 tan A
2
2 tanA 1tan2 A ; cos 2A = 2 1 tan A 1 tan2 A
(d)
sin 2A =
(f)
cos 3A = 4 cos3A 3 cosA
(e)
sin3A = 3 sinA 4sin3A
(g)
tan 3A =
3tanA tan3 A 13tan2 A
Important Trigonometric Ratios : (a)
sin n = 0 ;
(b)
sin 15o or sin
cos n = ( 1)n ; tan n = 0, where n I
3 1 5 = = cos 75o or cos 12 12 2 2
cos 15o or cos
tan 15o =
(c)
3 1 5 = = sin 75o or sin 12 12 2 2
3 1 = 2 3 = cot 75o ; tan 75o = 3 1
sin 10 or sin 18o =
5 1 & cos 36o or cos 5 = 4
3 1 = 2 3 or cot 15o 3 1 5 1 4
Conditional Identities : If A + B + C = then : (i)
sin 2A + sin 2B + sin 2C = 4 sin A sinB sinC
(ii)
sinA + sinB + sinC = 4 cos
(iii)
cos 2A + cos 2B + cos 2C =
(iv)
cos A + cos B + cos C = 1 + 4 sin
(v)
ma r a h A B B C C A S . tan + tan + tan tan + tan tan =1 82 K . 2 2 2 2 2 2 6 L 7 r. 1027 058 E A B C A B C 98 8015 cot + cot + cot = cot . cot . cot 2 2 2 2 2 2 39 8 cot A cot B + cot B cot C + cot C cot A =1
(vi)
(vii) (viii)
A B C cos cos 2 2 2
1 4 cosA cos B cos C A B C . sin . sin 2 2 2
tanA + tan B + tan C = tan A tan B tanC
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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(ix)
A+B+C=
then tan A tan B + tan B tan C + tan C tan A = 1 2
Range of Trigonometric Expression : E = a sin + b cos E= =
b a2 b2 sin( + ) , where tan = a a2 b2 cos(
a
), where tan = b
Hence for any real value of ,
a2 b2 E
a2 b2
Sine and Cosine Series : sinn 2 n1 sin + sin( + ) + sin( +2 ) + ..... + sin n1 = sin 2 sin 2
cos + cos( + ) + cos( +2 ) + .... + cos n1
n sin 2 n1 = cos 2 sin 2
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Mathematics Concept Note IITJEE/ISI/CMI
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2. QUADRATIC EQUATION If a0 0, polynomial equation of 'n' degree is represented by a0 x n a1 x n 1 a2 x n 2 .......an 0 , where n N . If n = 1, equation is termed as 'linear' and if n=2, equation is termed as quadratic equation
Note: (i) Values of 'x' which satisfy the polynomial equation is termed as its roots or zeros. (ii) If general, polynomial equation of 'n' degree is having n roots, but if it is having more than n roots, then it represents an identity. for example:
( x a)( x b) ( x b)( x c ) ( x c)( x a) 1 is quadratic equation but (c a)(c b) (a b)(a c) (b c)(b a)
have more than two roots (viz: x = a,b,c, ....... etc.) (iii) If ax2 + bx + c = 0 is an identity, then a=b=c=0. (iv) An identity is satisfied for all real values of the variable. for example: sin2x + cos2x = 1 x R.
2.
Relation Between Roots and Coefficients : The solutions of quadratic equation ax2 + bx + c = 0, (a 0) is given by
(i)
2 x = b b 4ac , where b2 4ac=D is called discriminant of quadratic equation. 2a
If , are the roots of quadratic equation ax2 + bx + c = 0, a 0, then:
(ii)
ax2 bx c a(x )(x )
+ =
b a
=
c a

D
 = a
A quadratic equation whose roots are and is (x )(x ) = 0
(iii)
x2 {sum of roots}x + {product of roots} = 0
Note: If , , are the roots of ax3 + bx2 + cx + d = 0, then b c d , S2 and S3 = ; (where Sk represents sum of a a a roots taking k roots at a time)
S1 =
If , , , are the roots of the equation ax4 + bx3 + cx2 + dx + e = 0, then
= –b/a , ( )( ) c / a , ( ) ( ) d / a , e / a.
ma r a h S . then, f(x) a (x )(x )......(x ) 82 K . 6 L 7 7 58 a x a x ........ a x a a (x )(x )........(x 1 ) 02 Er. 8 0 5 Comparing the coefficients of like powers of x on both sides,9 we get 801 39 a coefficient of x 8 S ......
If 1 , 2 , 3 ,........n are roots of the equation f(x) anxn an1xn1 ...... a1x a0 ....(1) n
n
1
2
n
n1
n
n1
1
0
n
1
n
n 1
n
1
2
n 1
i
1
2
i 1
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
3
n
an
coefficient of xn
Page 14
S2
i
12 ...... (1)2.
j
i j
S3
i
j
k
an2 coefficient of xn2 an coefficient of xn
12 3 2 3 4 ...... (1)3.
i jk
: : :
an3 coefficient of xn3 (1)3 an coefficient of xn
: : :
Sn a1a2a3 ......................................an (1)n
3.
: : : a0 constant term (1)n . an coeficient of xn
Nature of Roots : Consider the quadratic equation ax2 + bx + c = 0 having , as its roots, D=b2 4ac Nature of Roots
D=0
D 0
(Roots are equal, α = β =–b/2a)
(Roots are unequal)
a, b, c R and D > 0 Roots are real and unequal
a,b,c R and D < 0 Roots are imaginary
a, b, c Q and D is a perfect square Roots are rational
a,b, c Q and D is not a perfect square Roots are irrational
a = 1, b, c I and D is a perfect square Roots are integral.
Note:
ma r a (i) If the coefficients of the equation ax + bx + c = 0 are all real and h + i is its root, S . 82 K . 6 then iβ is also a root. i.e. imaginary roots occur in conjugate pairs. L 7 Er. 81027 5058 (ii) If the coefficients in the equation are all rational and α 9 β is one of1its roots, then 80 9 3 α β is also a root where α , β Q and β is not a perfect 8square. 2
(iii) If a quadratic expression f(x) = ax2 + bx + c is a perfect square of a linear expression then D=b2–4ac = 0. Mathematics Concept Note IITJEE/ISI/CMI
Page 15
4.
Common Roots : Consider two quadratic equations, a1x2 + b1x + c1=0 and a2x2 + b2x + c2=0. (i) If two quadratic equations have both roots common, then the equation are identical and their coefficient are in proportion. i.e. a1 b1 c 1 a2 b2 c2
(ii) If only one root is common, then the common root ' ' will be :
c1a2 c2a1 b c b2c1 1 2 a1b2 a2b1 c1a2 c2a1
Hence the condition for one common root is (a1b2  a2b1 )(b1c2  b2 c1 ) = (c1a2  c2 a1 )2
Note: (i) If f(x) = 0 and g(x) = 0 are two polynomial equation having some common root(s) then these common root(s) is/are also the root(s) of h(x)=af(x)+bg(x)=0. (ii) To obtain the common root, make coefficients of x2 in both the equations same and subtract one equation from the other to obain a linear equation in x and then solve it for x to get the common root.
5.
Factorisation of Quadratic Expressions : If a quadratic expression f(x,y) = ax2 + 2hxy + by2 + 2gx + 2fy + c may be resolved into two linear factors, then a h g Δ =abc + 2fgh–af2–bg2–ch2 = 0 OR
6.
h b f g f
=0
c
Graph of Quadratic Expression : y = f(x) = ax2 + bx + c; a 0
2 y D b 4a = a x 2a
Note: (i) f(x) = ax2+bx+c represents a parabola.
ma r a h S 2 shape (iii) If a>0 then the shape of the parabola is concave upwards and if.a < 0 then 8the K . 6 L 7 of the parabola is concave downwards. r. 1027 058 (iv) the parabola intersect the yaxis at point (0,c). E 1the5real roots 98xaxis8are 0 (v) the xcoordinate of point of intersection of parabola with of the quadratic equation f(x) = 0. 839
b D , (ii) the coordinate of vertex are 2a 4a
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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7.
Range of Quadratic Expression f(x) = ax2 + bx + c. (1) Absolute Range :
D D , If a > 0 f(x) and if a < 0 f(x) , 4a 4a Hence maximum and minimum values of the expression f(x) is and it occurs at x =
D in respective cases 4a
b (at vertex). 2a
(2) Range in restricted domain :
8.
(a) If
b [x1, x2], then f(x) min f(x1 ), f(x2 ) , max f(x1 ), f(x2 ) 2a
(b) If
D D b , max f(x1 ), f(x2 ), [x1, x2], then f(x) min f(x1 ), f(x2 ), 4a 4a 2a
Sign of quadratic Expressions : The value of expression f(x) = ax2 + bx + c at x = x0 is equal to ycoordinate of a point on parabola y = ax2 + bx + c whose xcoordinate is x0. Hence if the point lies above the xaxis for some x = x0 then f(x0) > 0 as illustrated in following graphs:
Note: (i) ax2 bx c 0 x R a 0 and D 0. (ii) ax2 bx c 0 x R a 0 and D 0. (iii) ax2 bx c 0 x R a 0 and D 0. (iv) ax2 bx c 0 x R a 0 and D 0.
Mathematics Concept Note IITJEE/ISI/CMI
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Page 17
10.
Location of Roots : Let f(x) = ax2 + bx + c, where a b c R.
(x0,f(x0))
(A)
x0
(x0,f(x0))
x
(B)
x0
x
(C)
x0
x
(x0,f(x 0))
(i) Conditions for both the roots of f(x) = 0 to be greater than a specified number 'x0' are b2–4ac 0 ; af(x0) > 0 and (–b/2a) > x0. (refer figure no. (A)) (ii) Conditions for both the roots of f(x) = 0 to be smaller than a specified number 'x0' are b2–4ac 0 ; af(x0) > 0 and (–b/2a) < x0. (refer figure no. (B)) (iii) Conditions for both roots of f(x) = 0 to lie on either side of the number 'x0' (in other words the number 'x0' lies between the roots of f(x) = 0) is b2 –4ac > 0 and af(x0) < 0. (refer figure no. (C))
(x1,f(x1)) (D)
x1
(x1,f(x1))
(x2,f(x2))
(E)
x
x2
x1
x2
(x2,f(x2))
x
(iv) Condition that both roots of f(x) = 0 to be confined between the numbers x1 and x2, (x1 < x2) are b2 4ac 0 ; af(x1) > 0 ; af(x2) > 0 and x1<( b/2a)<x2. (refer figure no. (D)) (v) Conditions for exactly one root of f(x) = 0 to lie in the interval (x1, x2) i.e. x1 < < x2 is f(x1).f(x2) < 0. (refer figure no. (E))
Important Results (i) If α is a root of the equation f(x) = 0, then the polynomial f(x) is divisible by (x α ) or (x α ) is a factor of f(x) (ii) An equation of odd degree will have odd number of real roots and an equation of even degree will have even numbers of real roots.
ma r a .Sh root (iv) The quadratic equation f(x) = ax + bx+c = 0, a 0 has as.aK repeated 8if2and only 6 L 7 if f( ) = 0 and f'( ) = 0. In this case f(x) = a(x – r 27 058 E ) . 81=0–b/2a. 5 (v) If polynomial equation f(x) = 0 has a root of multiplicity9 r (where r0 >1 1), then f(x) can 8 be written as f(x) = (x– ) g(x), where g( ) 0, also, f'(x) 3=90 has as a root with 8 multiplicity r–1. (iii) Every equation f(x) = 0 of odd degree has at least one real root of a sign opposite to that of its last constant term. (If coefficient of highest degree term is positive). 2
2
r
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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(vi) If polynomial equation f(x) = 0 has n distinct real roots, then f(x)= a0 (x 1 )(x 2 )...(x n ) , where 1 , 2 , ... n are n distinct real roots and f '(x) f(x)
n
1
(x
k 1
k
)
(vii) If there be any two real numbers 'a' and 'b' such that f(a) and f(b) are of opposite signs, then f(x) = 0 must have odd number of real roots (also at least one real root) between 'a' and 'b' (viii) If there be any two real numbers 'a' and 'b' such that f(a) and f(b) are of same signs, then f(x) = 0 must have even number of real roots between 'a' and 'b' (ix) If polynomial equation f(x) = 0 has n real roots, then f'(x) = 0 has atleast (n – 1) real roots. (x) Descartes rule of signs for the roots of a polynomial The maximum number of positive real roots of a polynomial equation f(x) = 0 is the number of changes of the signs of coefficients of f(x) from positive to negative or negative to positive. The maximum number of negative real roots of the polynomial equation f(x) = 0 is the number of changes from positive to negative or negative to positive in the signs of coefficients of f(–x) = 0. (xi) To form an equation whose roots are reciprocals of the roots in equation
a0 xn a1xn1 ... an1x an 0 , x is replaced by 1/x and then both sides are multiplied by xn.
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Mathematics Concept Note IITJEE/ISI/CMI
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3. COMPLEX NUMBER (1)
Complex number system : A number of the form a + ib where a , b R . and i 1 is called a complex number. If z = a + ib , then the real part of z is denoted by Re(z) and the imaginary part of z is denoted by Im(z). A complex number z is said to by purely real if Im(z) = 0 and is said to be purely imaginary if Re(z) = 0.
Note:
Set of all complex numbers is denoted by C , where C a ib : b R i
1
Zero is the only number which is purely real as well as purely imaginary.. i
1 is the imaginary unit and is termed as 'iota'.
If n I , then (i)4n 1, (i)4n 1 i , (i)4n 2 1 and (i)4n 3 i Sum of four consecutive powers of 'i' is always zero ( i.e. (i)n (i)n 1 (i)n 2 (i)n 3 0 ) , where n I .
The property of multiplication ( a)( b) ab is valid only is atleast one of a or b is nonnegative.
(2)
Algebraic Operations : 1. Addition :
(a + bi) + (c + di) = (a + c) + (b + d)i
2. Subtraction : (a + bi) – (c + di) = (a – c) + (b – d)i 3. Multiplication : (a + bi)(c + di) = ac + adi + bci + bdi2 = (ac – bd) + (ad + bc)i 4. Division :
a bi a bi c di ac bd (bc ad)i ac bd bc ad i = . 2 2 2 c2 d2 c di c di c di c d2 c d
Note: (i) In real numbers if a2 + b2 = 0 then a = b = 0 but in complex numbers, z12 + z22 = 0 does not imply z1 = z2 = 0. (ii) Inequalities in complex numbers are not defined (Law of order is not applicable for complex numbers).
(3)
Equality In Complex Number : Two complex numbers z1 = a1 + ib1 and z2 = a2 + ib2 are equal if and only if their corresponding real and imaginary parts are equal respectively
ma r a h S . 82 K Conjugate of Complex Number : . 6 L 7 r. 1027 058 E Conjugate of a complex number z = a + ib is denoted by z and a–ib. 1(a5,asb) zon=the 98and is8defined 0 Geometrically a complex number z = a + ib is represented by ordered pair 9 axis represents the complex plane (or Argand plane) and the mirror image of z about 83real
(4)
z1 = z2
Re(z1) = Re(z2) and Im (z1) = Im (z2).
conjugate of z.
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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Properties of Conjugate of a Complex Number : z1 z2 z1 z2
(z) z
z z 2 Re(z)
z z 2i Im(z)
z z z is purely real
z z 0 z is purely imaginary.
zz [Re(z)]2 [Im(z)]2  z 2
z1 z2 z1 z2
z1 z2 z1 z2
z1z2 z1 z2
z z 1 1 if z2 0 z2 z2
(z)n zn
If f(z) a0 a1z a2z2 ... anzn where a0 , a1 ,....an and z are complex number , then f(z) a0 a1 (z) a2 (z)2 ... an (z)n 2z 3z3 For example : 4z 1
(5)
2z 3(z)3 4z 1
Modulus of a Complex Number : If z = a + ib , then modulus of complex number z is denoted by z and defined as  z 
a2 b2 . Geometrically z is the distance of z from origin on the complex plane.
Properties of modulus :  z  0 z 0  z  Re(z)  z  zz  z 2 Mathematics Concept Note IITJEE/ISI/CMI
ma r a h S . 82 K . 6 L 7 Er. z1 027 5058  z   z   z  8 9 801  z  Im(z)  z 3 8 9  z1 z2   z1  z2  Page 21
z1 z1 , if z2 0 z2 z2
 z1 z2 2  z1 2  z2 2 z1z2 z1z2
 z1 z2 2  z1 2  z2 2 z1z2 z1z2  z1 2  z2 2 2 Re(z1z2 )  z1 z2 2  z1 z2 2 2( z1 2  z2 2 )
 z1 2  z2 2 2 Re(z1z2 ) If z1 , z2 0, then  z1 z2 2  z1 2  z2 2
z1 is purely imaginary. z2
Geometrically z1 – z2 represents the distance between complex points z1 and z2 on argand plane. Triangle Inequality : If z1 and z2 are two complex numbers , then  z1 z2   z1   z2  .
 z1 z2   z1   z2  The sign of equality holds iff z1 , z2 and origin are collinear and z1 , z2 lies on same side of origin.
(6)
Argument of a Complex Number : If nonzero complex number z is represented on the argand plane by complex point 'P' , then argument ( or amplitude ) of z is the angle which OP makes with positive direction of real axis. 1 Let complex number z = a + ib and tan
b , then principal argument of z is given as a
arg(z) , if z lies in I quadrant. arg(z) ( ) , if z lies in II quadrant. arg(z) ( ) , if z lies in III quadrant. arg(z) , if z lies in IV quadrant.
Note: Argument of a complex number is not unique , according to definition if is a value of argument , then 2n n I is also the argument of complex number..
If is argument of complex number and ( , ] , then the argument is termed as principal argument. Unless otherwise mentioned , arg(z) implies the principal argument. Properties of Argument of Complex Number :
ma r a h S . 82 K . 6 arg(z ) = n arg(z) + 2k for some integer k. L 7 r. 1027 058 E arg(z) = 0 z is real, for any complex number z 0 98 15 0 8 39 z 0 arg(z) = π /2 z is purely imaginary, for any complex8 number
arg(z1z2) = arg(z1) + arg(z2) + 2k for some integer k. arg(z1/z2) = arg(z1)–arg(z2) + 2k for some integer k.
n
arg( z ) = –arg(z) Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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(7)
Representation Of A Complex Number : (a) Cartesian Form (Geometric Representation) : Complex number z = x + i y is represented by a point on the complex plane (Argand plane/ Gaussian plane) by the ordered pair (x , y). (b) Trigonometric/Polar Representation : z r(cos i sin ) , where  z  r and arg z , represents the polar form of complex number
Note: cos i sin is also written as cis or ei . cos x
eix eix eix eix and sin x are known as Euler ' s Identities 2 2
(c) Euler's Representation : z rei , where  z  r and arg z , represents the Euler's form of complex number
(d) Vectorial Representation : Every complex number can be considered as if it is the position vector of a point. If
the point P represents the complex number z then , OP= z and OP=z.
(8)
Demoivre's Theorem : If n is any rational number , then (cos i sin )n cos n i sinn. If z (cos 1 i sin 1 )(cos 2 i sin 2 )(cos 3 i sin 3 )....(cos n i sin n ) then z cos(1 2 3 .... n ) i sin(1 2 .... n ) , where 1 , 2 , 3 ..... n R
If z r(cos i sin ) and n is a positive integer , then 2k+ 2k z1 / n r1 / n cos i sin , where k = 0 , 1 , 2 , 3 .... (n–1) n n
ma r a 2k p 2k h p ,2 S If p , q z and q 0 , then (cos i sin ) cos i sin . K q L. q 68 7 7 . r 2 E 810 5058 where k = 0 , 1 , 2 , 3 .... (q–1). 9 801 Note: 9 83complex This theorem is not valid when n is not a rational number or the number is not in p /q
the form of cos i sin . Mathematics Concept Note IITJEE/ISI/CMI
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(9)
Cube Root Of Unity : Let x (1)1 / 3 x3 1 0 (x 1)(x2 x 1) 0. x 1 ,
1 i 3 1 i 3 , 2 2
1 i 3 1 i 3 , then 2 2 2
If
Cube Roots of unity is given by : 1 , , 2 .
Properties of cube roots of unity : 3n 1 ; 3n 1 ; 3n 2 2 n I
1 0
3 1
2 and 2
2
0 , if k is not multiple of 3 1 k 2k 1 , if k is multiple of 3
1 1 and 2
In polar form the cube roots of unity are : 2 2 4 4 + i sin , cos + i sin 3 3 3 3 The cube roots of unity , when represented on complex plane , lie on vertices of an equilateral triangle inscribed in a unit circle having centre at origin , one vertex being on positive real axis. cos 0 + i sin 0 , cos
A complex number a + ib , for which a : b = 1 :
3 or
3 : 1 , can always be
expressed in terms of i , , 2 .
Note: If x , y , z R and is nonreal cube root of unity , then x2 x 1 (x )(x 2 )
x2 x 1 (x )(x 2 )
x2 xy y2 (x y)(x y2 )
x2 xy y2 (x y)(x y2 )
x2 y2 (x iy)(x iy) x3 y3 z3 3xyz (x y z)(x y 2 z)(x 2 y z)
(10) nth Roots of Unity : The nth roots of unity are given by the solution set of the equation xn = 1 = cos 0 + i sin 0 = cos 2 k + i sin 2 k
ma r a 2k 2k h S x cos i sin , where k 0 , 1 , 2 ,...., (n 1). . 82 K n n . 6 L 7 Er. 81027 5058 Properties of n roots of unity 9 801 2 2 3be9expressed in the form i sin e , then n roots of unity can (i) Let cos 8 n n x = [cos 2 k + i sin 2 k ]1/n
th
i(2 / n)
th
of a G.P. with common ratio . (i.e. 1, , 2 ,..... n 1. ) Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
Page 24
(ii) The sum of all n roots of unity is zero i.e., 1 2 ..... n 1 0 (iii) Product of all n roots of unity is (–1)n–1. (iv) Sum of pth power of nth roots of unity 0 , when p is not multiple of n 1 p 2p .... (n 1)p n , when p is a multiple of n
(v) The n, nth roots of unity if represented on a complex plane locate their positions at the vertices of a regular plane polygon of n sides inscribed in a unit circle having centre at origin, one vertex on vertex on positive real axis.
(11) Rotation Theorem : (i) If P(z1) and Q(z2) are two complex numbers such that z1 = z2 , then :
(ii) If P(z1), Q(z2) and R(z3) are three complex numbers and PQR =
θ , then:
(12) Logarithm Of A Complex Number : Let z = x + iy , then z  z (e)i
i
logeZ logeZ logee
logeZ
2 2 1 loge(x y ) i arg(z) 2
ma r a h (13) Standard Loci in the Argand Plane : S . 82 K . 6 L 7 . 27 05of8z is a Erarg(z) If z is a variable point in the argand plane such that =0 , then locus 1 15 98 straight line (excluding origin) through the origin inclined at an angle8 with xaxis. 0 39 8 If z is a variable point and z is a fixed point in the argand plane such that 1
arg(z – z1) = , then locus of z is a straight line passing through the point representing z1 Mathematics Concept Note IITJEE/ISI/CMI
Page 25
and inclined at an angle with xaxis. Note that line point z1 is excluded from the locus. If z is a variable point and z1 , z2 are two fixed points in the argand plane , then (i) z – z1 = z – z2 Locus of z is the perpendicular bisector of the line segment joining z1 and z2 (ii)  z z1   z z2  constant ( z1 z2 ) Locus of z is an ellipse (iii) z – z1 + z – z2 = z1 – z2 Locus of z is the line segment joining z1 and z2 (iv) z – z1 – z – z2 = z1 – z2 Locus of z is a straight line joining z1 and z2 but z does not lie between z1 and z2. (v) z – z1 – z – z2 = constant ( z1 – z2) Locus of z is hyperbola. (vi) z – z12 + z – z22 = z1 – z22 Locus of z is a circle with z1 and z2 extremities of diameter (vii) z – z1 = k z – z2 k 1 z z1 (viii) arg z z2
Locus of z is a circle
Locus of z is a segment of circle.
z z1 (ix) arg / 2 Locus of z is a circle with z1 and z2 as the vertices of diameter.. z z2 z z1 (x) arg 0 or Locus z is a straight line passing through z1 and z2. z z2
(xi) The equation of the line joining complex numbers z 1 and z 2 is given by z z1 z z1 or z2 z1 z2 z1
z z1 z2
z 1 z1 1 0 z2 1
(13) Geometrical Properties : (i) Distance formula : If P(z1) and Q(z2) are two complex points on complex plane then distance between P and Q is given by : PQ = z1 – z2 (ii) Section formula : If P(z1) and Q(z2) are two complex points on complex plane and R(z) divides the line segment PQ in ratio m : n , then : z
mz2 nz1 mn
(for internal division)
z
mz2 nz1 mn
(for external division)
Note: If a, b, c are three real numbers such that az1 + bz2 + cz3 = 0 , where a + b + c = 0 and a, b, c are not all simultaneously zero, then the complex numbers z1, z2 and z3 are collinear.
ma r a h (iii) Triangular Properties : S . 8z2, z K . 6 If the vertices A, B and C of a triangle is represented by the complex numbers L 7 z respectively and a, b, c are the length of sides , then Er:. 81027 5058 9 801 z z z Centroid (Z ) of the ABC = 3 839 1
2
and
3
1
2
3
G
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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Orthocentre (ZO) of the ABC =
z1 tan A z2 tanB z3 tan C tan A tanB tan C
Incentre (ZI) of the ABC =
az1 bz2 cz3 abc
Circumcentre of the ABC =
z1 sin2A z2 sin 2B z3 sin 2C sin2A sin2B sin 2C
Note: Triangle ABC with vertices A(z1) , B(z2) and C(z3) is equilateral if and only if 1 1 1 0 z1 z2 z2 z3 z3 z1 z12 z22 z23 z1z2 z2 z3 z3z1
(iv) Equation of a Straight Line :
An equation of a straight line joining the two points A(z1) and B(z2) is
z z1
z 1 z1 1 0
z2
z2 1
An equation of the straight line joining the point A(z1) and B(z2) is z = tz1 + (1 – t)z2 where t is is a real parameter. The general equation of a straight line is az az b 0 where a is a nonzero complex number and b is a real number (v) Complex Slope of a Line : If A(z1) and B(z2) are two points is the complex plane , then complex slope of AB is defined to be
z1 z2 two lines with complex slopes 1 and 2 are z1 z2
parallel , if 1 2 perpendicular , if 1 2 0 the complex slope of the line az az b 0 is given by (a / a) (vi) Length of Perpendicular from a Point to a Line : Len gt h
of
perpen di cu lar
of
poi n t
where a C {0} , b R , is given by : p
A()
from
t he
l i ne
az az b 0 ,
 a a b  2  a
(vii) Equation of Circle : An equation of the circle with centre at z0 and radius r is r – z0 = r or z z0 rei , 0 2 (parametric from)
ma r a General equatio of a circle is h S . 82 K . (1) 6 zz az az b 0 L 7 . 27 058 Erthat where a is a complex number and b is a real number such aa10 b0 98 8015 Centre of (1) is –a and its radius is aa b . 839 Diameter Form of a Circle
or zz z0 z0z z0 z0 r3 0
An equation of the circle one of whose diameter is the segment joining A(z1) and B(z2) is Mathematics Concept Note IITJEE/ISI/CMI
Page 27
(z z1 )(z z2 ) (z z1 )(z z2 ) 0
An equation of the circle passing through two points A(z 1) and B(z 2 ) is
z (z z1 )(z z2 ) (z z1 )(z z2 ) k z1 z2
z 1 z1 1 0 where k is a parameter z2 1
Equation of a circle passing through three noncollinear points. Let three noncollinear points be A(z1) , B(z2) and C(z3). Let P(z) be any point on the circle. Then either ACB APB [when angles are in the same segment] ACB APB [when angles are in the opposite segment]
or
z z2 z z2 arg 3 arg 0 z z1 z3 z1 z z2 z z1 arg 3 arg z3 z1 z z2 z z2 z z1 arg 3 0 z3 z1 z z2 z z1 z3 z2 arg z z2 z3 z1
z [using arg 1 arg(z1 ) arg(z2 ) and arg(z1z2 ) arg(z1 ) arg(z2 ) ] z2 (z z1 )(z3 z2 ) In any case , we get (z z )(z z ) is purely real. 2 3 1
(z z1 )(z3 z2 ) (z z1 )(z3 z2 ) (z z2 )(z3 z1 ) (z z2 )(z3 z1 )
Condition for four points to be concyclic. Four points z1 , z2 , z3 are z4 will lie on the same (z4 z1 )(z3 z2 ) circle if and only if (z z )(z z ) is purely real. 4 2 3 1
(z1 z3 )(z2 z4 ) is purely real. (z1 z4 )(z2 z3 )
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
Page 28
4. BINOMIAL THEOREM An algebraic expression which contains two distinct terms is called a binomial expression. 3 3 For example : (2x y) , x , x 1 etc. y x
General form of binomial expression is (a+b) and the expansion of (a+b)n, n N is called the binomial theorem. For example : (a + b)1 = a + b (a + b)2 = a2 + 2ab + b2 (a + b)3 = a3 + 3a2b + 3ab2 + b3
Note: Coefficients in the binomial expansion is termed as binomial coefficients and these binomial coefficients have a fixed pattern which can be seen through pascal triangle
(2)
Statement of Binomial theorem : If a, b C and n N, then : (a + b)n = nC0 anb0 + nC1 an–1 b1 + nC2 an–2 b2 + ... + nCr an–r br + ... + nCn a0 bn n
or
(a b)n
n
Cr an rbr
r 0
General term in binomial expansion (a + b)n is given by Tr + 1, where Tr + 1 = n Cr an rbr Now, putting a = 1 and b = x in the binomial theorem (1 + x)n = nC0 + nC1 x + nC2 x2 + ... + nCr xr + ... + nCn xn n
(1 x)n
n
Cr xr
r 0
General term in binomial expansion (1 + x)n is given by Tr + 1, where Tr + 1 = n Cr xr
(3)
ma r a h S . 82 K . 6 L 7 27 058 (i) The number of terms in the expansion is n + 1.Er. 0 1 98 8015 (ii) The sum of the indices of 'a' and 'b' in each term is n. 39 equidistant from 8terms (iii) The binomial coefficients (i.e. C , C ........ C ) of the Properties of Binomial Theorem :
n
n
0
n
1
n
the beginning and the end are equal, i.e. C0 = nCn , nC1 = nCn–1 etc. { nCr = nCn–r} (iv) Middle term (s) in expansion of (a + b)n , n N : Mathematics Concept Note IITJEE/ISI/CMI
n
Page 29
If n is even, then number of terms are odd and in this case only one middle term n 2 exists which is 2
th
therm if n is odd, then number of terms are even and in this case
n 2 two middle terms exist which are 2
7.
th
n 3 and 2
th
terms .
Properties of Binomial coefficients : (1 + x)n = C0 + C1x + C2x2 + ......... + Crxr + ........ + Cnxn (1)
The sum of the binomial coefficients in the expansion of (1 + x)n is 2n Putting x = 1 in (1) n C0 + nC1 + nC2 + ....... + nCn = 2n ......(2) n
n
or
r 0
(2)
.....(1)
C = 2n r
Again putting x = 1 in (1), we get C0 nC1 + nC2 nC3 + ............... + ( 1)n nCn = 0
n
......(3)
n
(1)r nCr = 0
or
r 0
(3)
The sum of the binomial coefficients at odd position is equal to the sum of the binomial coefficients at even position and each is equal to 2n 1. from (2) and (3) n C0 + nC2 + nC4 + ............... = 2n 1 n C1 + nC3 + nC5 + .............. = 2n 1
(4)
Sum of two consecutive binomial coefficients n Cr + nCr 1 = n+1Cr L.H.S. = nCr + nCr 1
n!
=
(n r)!r!
n! 1 1 (n r)!(r 1)! r n r 1
=
n! (n 1) (n r)!(r 1)! r(n r 1) (n 1)! = (n r 1)!r!
n+1
Cr
= R.H.S.
Ratio of two consecutive binomial coefficients n
n
(6)
n! (n r 1)!(r 1)!
=
=
(5)
+
C
C
r
= r1
n
Cr =
n r
n
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
nr 1 r
1Cr
1
=
n(n 1) r(r 1)
n
2Cr
2
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 1 n (r 1)) 9n(n 1)(n802).......( 9 = .......... = 83 r(r 1)(r 2).........2.1 Page 30
8.
Binomial Theorem For Negative Integer Or Fractional Indices If n R then, (1 + x)n = 1 + nx +
.............. +
n(n 1) 2 n(n 1)(n 2) 3 x + x + ............. 2! 3!
n(n 1)(n 2).......(n r 1) r x + ............ . r!
Remarks (i)
The above expansion is valid for any rational number other than a whole number if x < 1.
(ii)
When the index is a negative integer or a fraction the number of terms in the expansion of (1 + x)n is infinite, and the symbol nCr cannot be used to denote the coefficient of the general term.
(iii)
The first terms must be unity in the expansion, when index 'n' is a negative integer or fraction
(iv)
The general term in the expansion of (1 + x)n is Tr+1 =
n(n 1)(n 2).......(n r 1) r x r!
(v)
When 'n' is any rational number other than whole number then approximate value of (1 + x)n is 1 + nx(x2 and higher powers of x can be neglected)
(vi)
Expansions to be remembered (x< 1) (a) (1 + x) 1 = 1 x + x2 x3 + ....... + ( 1)r xr + ....... (b) (1 x) 1 = 1 + x + x2 + x5 + .............. + xr + ....... (c) (1 + x) 2 = 1 2x + 3x2 4x3 + .....( 1)r (r + 1) xr + ....... (d) (1 x) 2 = 1 + 2x + 3x2 + 4x3 + ..... + (r + 1)xr + .......
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Mathematics Concept Note IITJEE/ISI/CMI
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5. PERMUTATION AND COMBINATION Permutations are arrangements and combinations are selections. In this chapter we discuss the methods of counting of arrangements and selections. The basic results and formulas are as follows :
(1)
Fundamental Principle of Counting : (i) Principle of Multiplication : If an event can occur in 'm' different ways, following which another event can occur in 'n' different ways, then total number of different ways of simultaneous occurrence of both the events in a definite order is m + n. (ii) Principle of Addition : If an event can occur in 'm' different ways, and another event can occur in 'n' different ways, then exactly one of the events can happen in m + n ways.
(2)
The Factorial : Let n be a positive integer. Then the continued product of first n natural numbers is called factorial n, be denoted by n! or n . Also , we define 0! = 1. when n is negative or a fraction, n ! is not defined. Thus , n ! = n (n–1) (n–2).......3.2.1. Also , 1! 1 (0!) 0! 1.
(3)
Arrangement or Permutation : The number of permutations of n different things, taking r at a time without repetition is given by nPr n!
n
Pr = n(n–1) (n–2) .... (n–r+1) =
(n r)!
Note: (i) The number of permutations of n different things, taking r at a time with repetition is given by nr (ii) The number of arrangements that can be formed using n objects out ow which p are identical (and of one kind) q are identical (and of another kind), r are identical (and of another kind) and the rest are distinct is
(4)
n! . p!q!r !
Conditional Permutations :
ma r a h S . 82 K . 6 C r! always occur L 7 r. 1which 27p particular 58things 0 (ii) Number of permutation of n dissimilar things takenE r at a time 0 5 8 9 801 C r! never occur 839 (i) Number of permutation of n dissimilar things taken r at a time which p particular things n p
r p
n p
r p
(iii) The total number of permutation of n dissimilar things taken not more than r at a time,
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
Page 32
n(nr 1) . n1 (iv) Number of permuutations of n different things, taken all at a time, when m specified when each thing may be repeated any number of times, is
things always come together is m! (n m 1)! (v) Number of permutation of n different things, taken all at a time when m specified things never come together is n! m! (n m 1)!
(5)
Circular Permutation: The number of circular permutations of n different things taken all at a time is given by (n 1)! , provided clockwise and anticlockwise circular permutations are considered to be differnt
Note: (i) The number of circular permutations of n different things taken all at a time , clockwise and anticlockwise circular permutations are considered to be same , is given by
(6)
(n 1)! . 2
Selection or Combinations : The number of combinations of n different things taken r at a time is given by nCr , where n
Cr =
n n! Pr = where r n ; n N are r W.. r!(n  r)! r!
Note: (i) nCr is a natural number. (iii) nCr = nCn–r
(ii) nC0 = nCn = 1, nC1 = n (iv) nCr + nCr–1 = n+1Cr
(v) nCx = nCy x = y or x + y = n (vi) If n is even then the greatest value of nCr is nCn/2 (vii) If n is odd then the greatest value of nCr is nC(n+1)/2 or nC(n–1)/2 n
n n 1 n (viii) Cr r . Cr 1
(x)
(7)
n
Cr
(ix)
r 1 n 1 . Cr 1 n1
n
Cr
Cr 1
nr 1 r
(xi) nC0 + nC1 + nC2 + .... + nCn = 2n
Selection of one or more objects (a) Number of ways in which atleast one object be selected out of 'n' distinct objects is n C1 + nC2 + nC3 + ........ + nCn = 2n–1 (b) Number of ways in which atleast one object may be selected out of 'p' alike objects of one type 'q' alike objects of second type and 'r' alike of third type is (p + 1) (q + 1) (r + 1)–1
ma r a h'n' objects (c) Number of ways in which atleast one object may be selected . from S 2 where 8 K . 'p' alike of one type 'q' alike of second type and 'r' alike of third type and rest 6 L 7 n (p + q + r) are different , is Er. 81027 5058 (p + 1) (q + 1) (r + 1) 2 –1 9 801 39 8 Formation of Groups : n–(p + q + r)
(8)
Number of ways in which ( m + n +p) different things can be divided into three different Mathematics Concept Note IITJEE/ISI/CMI
Page 33
groups contiaining m, n & p things respectively is
(m n p)! , m!n!p!
If m = n = p and the groups have identical qualitative characteristic then the number of groups =
(3n)! . n!n!n!3!
However , if 3n things are to be divided equally among three people then the number of (3n)!
ways =
8.
(n!)3
.
Multinomial Theorem : Coefficient of xr in expansion of (1 x) n = n + r 1Cr (n N) Number of ways in which it is possible to make a selection from m + n + p = N things, where p are alike of one kind , m alike of second kind & n alike of third kind taken r at a time is given by coefficient of xr in the expansion of (1 + x + x2 + ....... + xp) (1 + x + x2 + ..... + xm) (1 + x + x2 + ..... + xn). (i)
For example the number of ways in which a selection of four letters can be made from the letters of the word PROPORTION is given by coefficient of x4 in (1 + x + x2 + x3) ( 1 + x + x2) (1 + x + x2) (1 + x) (1 + x).
(ii)
Method of fictious partition : Number of ways in which n identical things may be distributed among p persons if each person may receive one, one or more things is; n+p 1Cn
Let N = pa qb rc ... where p, q, r... are distinct primes & a, b, c... are natural numbers then :
9.
10. 11.
(a)
The total numbers of divisors of N including 1 & N is = (a +1)(b + 1)(c + 1).......
(b)
The sum of these divisors is = (p0 + p1 + p2 + ... + pa) (q0 + q1 + q2 + ....... + qb ) (r0 + r1 + r2 + ... + rc)....
(c)
Number of ways in which N can be resolved as a product of two factors is 1/2 (a+1)(b+1)(c+1).... if N is not a perfect square 1/2 [(a+1)(b+1)(c+1)...+1] if N is a perfect square
(d)
Number of ways in which a composite number N can be resolved into two factors which are relatively prime ( or coprime) to each other is equal to 2n 1 where n is the number of different prime factors in N.
ma r a h S . 82 K . 6 L 7 Dearrangement : Er. 81027 5058 9 envelopes Number of ways in which 'n' letters can be put in 'n' corresponding 01 such that no 8 9 letter goes to correct envelope is 83
Let there be 'n' types of objects, with each type containing atleast r objects. Then the number of ways of arranging r objects in a row is nr.
1 n! 1 1!
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
1
2!
1
3!
1
4!
.......... ....... ( 1)
n
1 . n!
Page 34
Exponent of Prime p in n! If p is a prime number and n is a positive integer , then Ep(n) denotes the exponent of the prime p in the positive integer n. n n n n Ep (n!) 2 3 .... s where S is the largest naturla number. Such that p p p p ps n ps 1.
Some Important Results for Geometical problems (i) Number of total different straight lines formed by joining the n points on a plane of which m(< n) are collinear is nC2 – mC2 + 1. (ii) Number of total triangle formed by joining the n points on a plane of which m (< n) are collinear is nC3 – mC3. (iii) Number of diagonals in a polygon of n sides is nC2 – n. (iv) If m parallel lines in a plane are intersected by a family of other n parallel line. Then total mn(m 1)(n 1) 4 (v) Given n points on the circumference of a circle, then (a) Number of straight lines = nC2 (b) Number of traingles = nC3 n (c) Number of quadrilaterals = C4. (vi) If n straight lines are drawn in the plane such that no two lines are parallel and no three lines are concurrent. then the number of part into which these lines divide the plane is
number of parallelograms so formed is
m
C2 nC2 i.e.
1 n. n
(vii) Number of rectangle of any size in a square of n n is
r
3
and number of squares of any
r 1 n
size is
r
2
r 1
(viii) In a rectangle of n p(n p) number of rectangle of any size is
np (n 1)(p 1) and 4
n
number of squares of any size is
(n 1 r)(p 1 r). r 1
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Mathematics Concept Note IITJEE/ISI/CMI
Page 35
6. PROBABILITY Theory of probability measures the degree of certainty or uncertainty of an event, if an event is represented by 'E', then its probability is given by notation P(E), where 0 P(E) 1.
(1)
Basic terminology: (i) Random Experiment : It is a process which results in an outcome which is one of the various possible outcomes that are known to happen, for example : throwing of a dice is a random experiment as it results in one of the outcome from {1, 2, 3, 4, 5, 6}, similarly taking a card from a pack of 52 cards is also a random experiment. (ii) Sample Space : It is the set of all possible outcomes of a random experiment for example : {H, T} is the sample space associated with tossing of a coin. In set notation it can be interpreted as the universal set. (iii) Event : It is subset of sample space. for example : getting a head in tossing a coin or getting a prime number is throwing a die. In general if a sample space consists 'n' elements, then a maximum of 2n events can be associated with it.
Note: Each element of the sample space is termed as the sample point or an event point. The complement of an event 'A' with respect to a sample space S is the set of all
elements of 'S' which are not in A . It is usually denoted by A' , A or AC.
AA S
P(A) P(A) 1
Complement of event
(iv) Simple Event or Elementary Event : If an event covers only one point of sample space, then it is called a simple event, for example : getting a head followed by a tail in throwing of a coin 2 times is a simple event. (v) Compound Event or mixed event (Composite Event) : when two or more than two events occur simultaneously then event is said to be a compound event and it contains more than one element of sample space. for example : when a dice is thrown, the event of occurrence of an odd number is mixed event. (vi) Equally likely Events : If events have same chance of occurrence, then they are said to be equally likely events. For example : In a single toss of a fair coin, the events {H} and {T} are equally likely.. In a single throw of an unbiased dice the events {1}, {2} , {3}, {4}, {5} and {6} are equally likely. In tossing a biased coin the events {H} and {T} are not equally likely..
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 9 (vii) Mutually Exclusive / Disjoint / Incompatible Events8 : 3
Two events are said to be mutually exclusive if occurrence of one of them rejects the possibility of occurrence of the other (i.e. both cannot occur simultaneously). Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
Page 36
Mathematically, A B
P(A B) 0
For example: When a coin is tossed the event of occurrence of a head and the event of occurrence of a tail are mutually exclusive events. When a dice is thrown sample space S = (1, 2, 3, 4, 5, 6), Let A = the event of occurrence of a number greater than 4 = {5, 6} B = the event of occurrence of an odd number = {1, 3, 5} C = the event of occurrence of an even number = {2, 4, 6} Now B and C are mutually exclusive events but A and B are not mutually exclusive because they can occur together (when the number 5 turns up). (viii) Exhaustive System of Events : If each outcome of an experiment is associated with at least one of the events E1, E2, E3, .... En, then collectively the events are said to be exhaustive. Mathematically, E1 E2 E3........ En = S.. For example. In random experiment of rolling an unbiased dice, S = {1, 2, 3, 4, 5, 6} Let E1 = the event of occurence of prime number = {2, 3, 5} E2 = the event of occurence of an even number = {2, 4, 6} E3 = the event of occurence of number less than 3 = {1, 2} Now, E1 E2 E3 S and hence events E1, E2 and E3 are mutually exhaustive events.
(2)
Classical Definition of Probability : If a random experiment results in a total of (m + n) outcomes which are equally likely and mutually exclusive and exhaustive and if 'm' outcomes are favourable to an event 'E' while 'n' are unfavourable, then the probability of occurrence of the event 'E', denoted by P(E), is given by : P(E) =
number of favourable outcomes m = total number of outcomes mn
P(E)
n(E) n(S)
Note: (i) In above definition, odds in favour of 'E' are m : n, while odds against 'E' are n : m . (ii) P(E) or P(E ') or P(Ec ) denotes the probability of nonoccurrence of E.
ma r a h S . 82 K . 6 L 7 Notation of an event in set theory: Er. 81027 5058 9 801 In dealing with problems of probability, it is important to convert the 39verbal description of an event into its equivalent set theoretic notation. Following8 table illustrates the verbal P(E) 1 P(E)
(3)
n mn
description and its corresponding notation in set theory. Mathematics Concept Note IITJEE/ISI/CMI
Page 37
Verbal description of Event
Notation is Set theory.
Both A and B occurs.
AB
Atleast one of A or B occurs
AB
A occurs but not B.
AB
Neither A nor B occurs.
AB
Exactly one of A and B occurs
(A B) (A B)
All three events occur simultaneously
A BC
Atleast one of the events occur.
A BC
Only A occurs and B and C don't occur.
A BC
Both A and B occurs but C don't occur.
A BC
Exactly two of the event A, B, C occur.
(A B C) (A B C) (A B C)
Atleast two of the events occur.
(A B) (B C) (A C)
exactly one of the events occur.
(A B C) (A B C) (A B C)
None of the events A, B, C occur.
A B C A B C.
Note: (i) De Morgan's Law : If A and B are two subsets of a universal set, then (A B)c A c Bc and (A B)c Ac Bc.
(ii) Distributive Law :
A (B C) (A B) (A C) A (B C) (A B) (A C). (iii) Venn diagram for events A , B , C :
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
Page 38
(4)
Addition theorem of probability : If 'A' and 'B' are any two events associated with a random experiment, then (i) P(A B) = P(A) + P(B) P(A B) or P(A B) P(A) P(B) P(AB). (ii) P(A B) P(B) P(A B) (iii) P(A B) P(A) P(A B)
(iv) P (A B) (A B) P(A) P(B) 2P(A B) P(A B) P(A B) . If A , B , C are three events associated with a random experiment, then (v) P(A B C) P(A) P(B) P(c) P(A B) P(B C) P(A C) P(A B C) (vi) P(at least two of A, B c occur) = P(B C) + P(C A) + P(A B)–2P(A B C) (vii) P(exactly two of A, B, c occur) = P(B C) + P(C A) + P(A B)–3P(A B C) (viii) P(exactly one of A, B , C occur) = P(A) + P(B) + P(C)–2P(B C)–2P(C A)–2P(A B) + 3P(A B C)
(5)
Independent and dependent events If two events are such that occurence or nonoccurence of one does not effect the chances of occurence or nonoccurence of the other event, then the events are said to be independent. Mathematically : if P(A B) P(A).P(B) then A and B are independent. For example When a coin is tossed twice, the event of occurrence of head in the first throw and the event of occurrence of head in the second throw are independent events. When two cards are drawn out of a full pack of 52 playing cards with replacement (the first card drawn is put back in the pack and the second card is drawn), then the event of occurrence of a king in the first draw and the event of occurrence of a king in the second draw are independent events because the probability of drawing a king in the second draw 4 whether a king is drawn in the first draw or not. But if the two cards are drawn 52 without replacement then the two events are not independent.
is
Note: (i) If A and B are independent , then A' and B' are independent , A and B' are independent and A' and B are independent. (ii) Three events A,B and C are independent if and only if all the three events are pairwise independent as well as mutually independent. (iii) Complementation Rule : If A and B are independent events, then P(A B) 1 P(A B) 1 P(A B) 1 P(A).P(B)
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839
Similarly, if A, B and C are three independent events, then P(A B C) 1 P(A).P(B).P(C).
(6)
Conditional Probability
Let A and B be any two events, B , then P(A/B) denotes the conditional probability of occurrence of event A when B has already occurred. Mathematics Concept Note IITJEE/ISI/CMI
Page 39
For example : When a die is thrown, sample space S = {1, 2, 3, 4, 5, 6} Let A = the event of occurrence of a number greater than 4. = {5, 6} B = the event of occurrence of an odd number. = {1, 3, 5} Then P(A/B) = probability of occurrence of a number greater than 4, when an odd number has occurred
Note: A A A A (i) If B , then P P 1 and P P 1. B B B B
(ii) If A , and B is dependent on A, then B B P(B) P(A).P P(A).P . A A
(7)
Multiplication Theorem of Probability : If A and B are any two events, then A A P(A B) P(B). P , where B and P denotes the probability of occurrence of B B event A when B has already occurred. If A and B are independent events, then probability of occurrence of event A is not affected by occurrence or nor occurrence of event B, therefore A P P(A) B
(8)
Total Probability Theorem If an event A can occur with one of the n mutually exclusive and exhaustive events B1, B2,..., Bn and the probabilities P(A / B1 ),P(A / B2 ),........,P(A / Bn ) are known , then n
P(A)
P(B ).P(A / B ). i
i
i 1
(9)
Bayes' Theorem (Inverse Probability) : Let B1 , B2 , B3 , ........... , Bn is a set of n mutually exclusive and exhaustive events and event A can occur with any of the event B1 , B2 , ...... , Bn . If event 'A' has occured, then Bi probability that event 'A' had occured with event Bi is given by P A
A P(Bi ).P B Bi P i n A A P(Bi ).P Bi i 1
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
, where
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Page 40
Particular Case : Let A , B , C be three mutually exclusive and exhaustive events and event 'E' can occur with any of the events A , B and C as shown in diagram: Now, if event 'E' had occurred, then y A P E x y z E P(A).P A A P E P(A)P E P(B)P E P(c)P E A B C
(10) Binomial Probability Distribution : Let a random experiment is conducted for n trials and in each trial an event 'E' is defined for which the probability of its occurence is 'p' and probability of nonoccurence is 'q', where p+q = 1. Now if in n trials, event 'E' occurs for r times, then the probability of occurence of event 'E' for r times is given by: P(r) nCr (p)r (q)n r .
Note: (i) (q p)n nC0 qn nC1(p)(q)n 1 n C2 (p)2 (q)n 2 ........ nCn (p)n
p(0) p(1) p(2) ........p(n) 1
(ii) For Binomial Probability distribution, it mean and variance is given by np and npq respectively
(10) Geometrical Applications : If the number of sample points in sample space is infinite, then classical definition of probability can't be applied. For uncountable uniform sample space, probability P is given by:
P=
favourable dimension total dimension
For example: If a point is taken at random on a given straight line segment AB, the chance that it falls on a particular segment PQ of the line segment is PQ/AB. If a point is taken at random on the area S which includes an area , the chance that the point falls on is /S.
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7. MATRICES
Any rectangular arrangement of numbers (real or complex) is called a matrix. If a matrix has 'm' rows and 'n' columns then the order of matrix is denoted by m n. If A = [aij ]mn , where aij denotes the element of ith row and jth column , then it represents the following matrix :
a13 a11 a12 a21 a22 a23 A ai1 ai2 ai3 am1 am2 am3
(1)
aij
a2j
aij
amj
a1n a2n ain amn
Basic Definitions : (i) Row matrix : A matrix having only one row is called as row matrix. General form of row matrix is A = [a11 a12 a13 ...... a1n] (ii) Column matrix : A matrix having only one column is called as column matrix. a11 a21 A General form of column matrix is am1
(iii) Singleton matrix : If in a matrix there is only one element then it is called singleton m a t r i x . A [aij]mn is a singleton matrix if m = n = 1. For example : [2] , [3] , [a] , [–3] are singleton matrices. (iv) Square matrix : A matrix in which number of rows and columns are equal is called a square matrix. General form of a square matrix is A = [aij]n. a12 a11 a a 22 A 21 an1 an2
a1n a2n ann
ma r a h S Note: . 82 K . 6 L 7 r. 1027 058 E (i) In a square matrix , the elements of the form a is termed as diagonal where 15 98 elements, 0 8 i = 1 , 2 , 3 , ...... , n and line joining these elements is called the principal diagonal or 839 leading diagonal or main diagonal. ii
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(ii) Summation of the diagonal elements of a square matrix is termed as its trace. n
If A [aij ]nn , then tr(A)
a
ii
i 1
tr(A) a11 a22 a33 ......... ann
(iii) If A [aij ]nn , B [bij ]nn and be a scalar , then : tr(A) tr(A)
tr(A B) tr(A) tr(B)
tr(AB) tr(BA)
tr(A) tr(A ') or tr(A T )
tr(In ) n
tr(O) 0
tr(AB) tr A.tr B
(v) Rectangular Matrix : A matrix in which number of rows in not equal to number of columns is termed as rectangular matrix. A [aij]mn is rectangular if m n . If m > n, matrix is termed as vertical matrix and if m < n , matrix is termed as horizontal matrix. (vi) Zero matrix : A = [aij]m n is called a zero matrix if aij = 0 i and j. For example :
0 0 O2 0 0
;
O32
0 0 0 0 0 0
(vii) Diagonal matrix : A square matrix [aij]n is said to be a diagonal matrix if aij = 0 for i j. (i.e. all the elements of the square matrix other than diagonal elements are zero). Diagonal matrix of order n is denoted as diag (a11, a22, ...... , ann). (viii) Scalar matrix : Scalar matrix is a diagonal matrix in which all the diagonal elements are same. A = [aij]n is a scalar matrix if aij = 0 for i j and aij = k for i = j. (ix) Unit matrix (Identity matrix) : Unit matrix is a diagonal matrix in which all the diagonal elements are unity. Unit matrix of order 'n' is denoted by In (or I). A = [aij]n is a unit matrix if aij = 0 for i j and aii = 1 for i = j For example :
1 0 I2 = , I3 = 0 1
1 0 0 0 1 0 . 0 0 1
(x) Triangular Matrix : A square matrix [aij]n is said to be triangular matrix if each element above or below the principal diagonal is zero. It is of two types :
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 Lower triangular matrix : 39 8 A = [a ] is said to be lower triangular matrix if a = 0 for i < j. (i.e. all the elements
Upper triangular matrix : A = [aij]n is said to be upper triangular if aij = 0 for i > j (i.e. all the elements below the diagonal elements are zero).
ij n
ij
above the diagonal elements are zero.) Mathematics Concept Note IITJEE/ISI/CMI
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Note:
Minimum numbers of zero in a triangular matrix is given by
n(n 1) , where n is order of 2
triangular matrix. Diagonal matrix is both upper and lower triangular. (xi) Singular and Nonsingular matrix : Any square matrix A is said to be nonsingular if  A  0 , and a square matrix A is said to be singular if A = 0. det (A) or A represents determinant of square matrix A. (xii) Comparable matrices : Two matrices A and B are said to be comparable if they have the same order. (xiii) Equality of matrices :Two matrices A and B are said to be equal if they are compa rable and all the corresponding elements are equal. Let A = [aij]m n and B = [bij]p q , then A = B m = p, n = q and aij = bij i and j. (2)
Scalar Multiplication of matrix : let be a scalar (real or complex number ) and A = [aij]m n be a matrix , then A is defined as [ aij ]mn Note: If A and B are matrices of the same order and , are any two scalars , then
(3)
(i) (A B) A B
(ii) ( )A A A
(iii) (A) ()A (A)
(iv) (A) (A) (A)
Addition of matrices : Let A and B be two matrices of same order (i.e. comparable matrices) , then A + B is defined as: A + B = [aij]m n + [bij]m n = [cij]m n , where cij = aij + bij i and j. Note: If A , B and C are matrices of same order , then A + B = B + A (Commutative law) (A + B) + C = A + (B + C) (Associative law) A + O = O + A = A , where O is zero matrix which is additive identity of the matrix. A + (–A) = 0 = (–A) + A , where (–A) is obtained by changing the sign of every element of A, which is additive inverse of the matrix.
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839
A B A C B C (Cancellation law) B A C A Mathematics for IITJEE Er. L.K.Sharma
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(4)
Multiplication of Matrices : Two matrices A and B are conformable for the product AB if the number of columns in A (permultiplier) is same as the number of rows in B (post multiplier). Thus , if A [aij]mn and B [bij ]np are two matrices of order m n and n p respectively , n
then their product AB is of order m p and is defined as AB [cij ]mp , where cij airbrj r 1
1
2
1
0 2
14
02
22
3 2 4
For example : 1 322 2 1 123 1 6 0 3 2 323 7 3 1 23
Note: (i) If A , B and C are three matrices such that their product is defined , then (Generally not commutative) AB BA (AB)C = A(BC) (Associative Law) IA = A = AI (I is identity matrix for matrix multiplication)
A(B+C) = AB + AC
If AB = AC B = C If AB = 0
(Distributive law) (Cancellation law is not always applicable) (It does not imply that A = 0 or B = 0 , product of zero matrix may be a zero matrix.)
two non
(ii) If A and B are two matrices of the same order , then (A + B)2 = A2 + B2 + AB + BA (A – B)(A + B) = A2 – B2 + AB – BA A(–B) = (–A)B = –(AB)
(A – B)2 = A2 + B2 – AB – BA (A + B)(A – B) = A2 – B2 – AB + BA
(iii) The positive integral powers of a matrix A are defined only when A is a square matrix A2 = A.A , A3 = A.A.A = A2A. For any positive integers m , n AmAn = Am+n I n = I , Im = I
(Am)n = Amn = (An)m
(5)
A0 = In where A is a square matrix of order n.
Transpose of a Matrix : Let A = [aij]m n , then the transpose of A is denoted by A' (or AT) and is defined as AT = [bij]n m , where bij = aji i and j. a1
For example : Transpose of matrix b
Note: Let A and B be two matrices then (AT)T = A Mathematics Concept Note IITJEE/ISI/CMI
1
a2 b2
a3 b3 2 3
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839
a1 is a2 a3
b1 b2 b3
32
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(A B)T A T B T ,
(kA)T = kAT , k be any scalar (real or complex) (AB)T = BT AT , A and B being conformable for the product AB (A1A2A3 .... An–1 An)T = AnT An–1T .... A3T A2T A1T (Reversal law) (An)T = (AT)n IT = I AT = A (A–1)T = (AT)–1
(6)
A and B being comparable.
Symmetric and Skewsymmetric Matrix : A square matrix A = [aij] is called symmetric matrix if aij = aji i , j (i.e. AT = A) a h g For example: A = h b f is a symmetric matrix. g f c A square matrix A = [aij] is called skewsymmetric matrix if aij = – aji i, j (i.e. AT = –A) 0
x
y
For example: B x 0 z is a skew symmetric matrix. y z
0
Note: Every unit matrix , scalar matrix and square zero matrix are symmetric matrices.
Maximum number of different elements in a symmetric matrix of order 'n' is
n(n 1) . 2
All principal diagonal elements of a skewsymmetric matrix are always zero because for any diagonal element. aii aii aii 0 Trace of a skew symmetric matrix is always 0. If A is skew symmetric of odd order , then det(A) = 0.
(7)
Properties of Symmetric and skewsymmetric Matrices: (i) If A is a square matrix , then A + AT , AAT , AT A are symmetric matrices , while A – AT is skewsymmetric matrix. (ii) If A is a symmetric matrix , then –A , KA , AT , An , A–1 , BT AB are also symmetric matrices , where n N, K R and B is a square matrix of order as that of A. (iii) If A is a skewsymmetric matrix, then A2n is a symmetric matrix for n N A2n+1 is a skewsymmetric matrix for n N kA is skewsymmetric matrix, where K R BTAB is skewsymmetric matrix , where B is a square matrix of order as that of A. (iv) If A and B are two symmetric matrices , then A B , AB BA are also symmetric matrix Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Page 46
AB–BA is a skewsymmetric matrix AB is symmetric matrix when AB = BA.
(v) If A and B are two skewsymmetric matrices , then A B , AB BA are skewsymmetric matrices AB + BA is a symmetric matrix. (vi) If A a skewsymmetric matrix and C is a column matrix, then CT AC is a zero matrix. (vii) Every square matrix A can uniquely be expressed as sum of symmetric and
1
1
T T skewsymmetric matrix i.e. A 2 (A A ) 2 (A A )
(8)
Submatrix, Minors, Cofactors : Submatrix : In a given matrix A , the matrix obtained by deleting some rows or columns (or both) of A is called as submatrix of A. a b c
For example :
d s , then x y z w
If A p q r
a b a b c a b c p q , p q r , p q r are all submatrices of A. x y z x y
Minors and Cofactors : If A = [aij]n is a square matrix , then minor of element aij, denoted by Mij , is defined as the determinant of the submatrix obtained by deleting ith row and jth column of A. Cofactor of element aij , denoted by Cij , is defined as Cij (1)i j.Mij a b For example: If A = c d , then M11 = d = C11 M12 = c , C12 = –c M21 = b , C21 = –b M22 = a = C22
If
a b c A d e f , then g h i
M11 = ei–hf , M32 = af–dc and C22 = ai–cg , C31 = bf–ec , C23
Mathematics Concept Note IITJEE/ISI/CMI
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 = bg–ah , etc. 839 Page 47
(9)
Determinant of A matrix : If A = [aij]n is a square matrix of order n > 1 , then determinant of A is defined as the summation of products of elements of any one row (or any one column) with the corresponding cofactors. For example:
a11 a12 A a21 a22 a31 a32
a13 a23 a33
A = a11C11 + a12 C12 + a13 C13 . a11
a22
a23
a32
a33
a12
a21 a23 a31 a33
a13
a21 a22
a31 a33 , or
A = a12 C12 + a22 C22 + a32 C32 . a12
a21 a23 a31 a33
Properties of determinant
a22
a11
a13
a31 a33
a32
a11
a13
a21 a23
:
If A = [aij]n , then the summation of the products of elements of any row with corresponding cofactors of any other row is zero. (Similarly , the summation of the products of elements of any column with corresponding cofactors of any other column is zero).
For example: a11 C21 + a12 C22 + a13 C23 = 0 , and a12 C11 + a22 C21 + a32 C31 = 0.
A = AT for any square matrix A.
AB = AB = BA
If λ be a scalar , then λ A is obtained by multiplying any one row (or any one column) of A by λ
 λA  = λn  A  , where A aij n
If A is a skew symmetric matrix of odd order then A = 0
If A = diag (a1 , a2 , ...... an) then A = a1a2...an
1 A
ma r a h S . 82 K . 6 L 7 . 27 058 ErA If two rows are identical (or two columns are identical) then =8 0. 10 9 8015 39 If any two rows (or columns) of a determinant be interchanged8 , the determinant is unaltered in  A n  An  , n N and  A 1 
numerical value but is changed in sign only. Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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If each element of any row (or column) can be expressed as a sum of two terms , then the determinant can be expressed as the sum of two determinants.
The value of a determinant is not altered by adding to the elements of any row (or column) the same multiples of the corresponding elements of any other row (or column).
(10) Cofactor matrix and adjoint matrix : If A = [aij]n is a square matrix , then matrix obtained by replacing each element of A by corresponding cofactor is called as cofactor matrix of A, denoted as cofactor A. The transpose of cofactor matrix of A is called as adjoint of A, denoted by adj A. If A = [aij]n , then cofactor A = [Cij]n where Cij is the cofactor of aij adj(A) = [Cji]n a11 a12 If A a21 a22 a31 a32
a13 C11 C12 a23 , then adj A C21 C22 C31 C32 a33
i and j.
T
C13 C11 C21 C31 C23 C12 C22 C32 ; C13 C23 C33 C33
Where Cij denotes the cofactor of aij in A. Note: If A and B are square matrices of order n and In is unit matrix, then A(adj A) = A In = (adj A)A adj A = An–1 adj(adj A) = An–2 A
adj(adj A) =  A (n 1)2
adj(AT) = (adj A)T
adj(AB) = (adj B)(adj A)
adj(Am) = (adj A)m , m N
adj(kA) kn 1 (adj A), k R
adj(In) = In
adj(O) = O
A is symmetric adj A is also symmetric.
A is diagonal adj A is also diagonal
A is triangular adj A is also triangular..
A is singular  adj A  0
(11) Inverse of a matrix (reciprocal matrix) : A nonsingular square matrix of order n is invertible if there exists a square matrix B of the same order such that AB = In = BA. In such a case, we say that the inverse of A is B and we write A–1 = B
ma r a h S . 1 82 K The inverse of A is given by A . .adj A 6 L 7 A Er. 81027 5058 The necessary and sufficient condition for the existence of the inverse matrix 1 A 9 of a square 0 8 is that  A  0 839
AA 1 A 1 A I. 1
Note: If A and B are invertible matrices of the same order , then
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(A–1)–1 = A
(AT)–1 = (A–1)T
(AB)–1 = B–1A–1
(Ak )1 (A 1 )k , k N
adj(A–1) = (adj A)–1
 A 1 
A = diag (a1 a2 ... an) A–1 = diag (a1–1 a2–1 ... an–1)
A is symmetric A–1 is also symmetric.
A is diagonal,  A  0 A 1 is also diagonal.
A is scalar matrix A–1 is also scalar matrix.
A is triangular ,  A  0 A 1 is also triangular..
Every invertible matrix possesses a unique inverse.
If A is a nonsingular matrix , then AB = AC B = C and BA = CA B = C.
In general AB = O does not imply A = O or B = O. But if A is non singular and AB = O, then B=O. Similarly B is non singular and AB = O A = O. Therefore , AB = O either both are singular or one of them is O.
1  A 1 A
(12) Important Definitions on Matrices : (i) Orthogonal matrix : A square matrix A is said to be an orthogonal matrix if , AAT = I = ATA. cos sin cos
For example : A sin
(ii) Involutory matrix : A square matrix A is said to be involutory if A2 = I, I being the identity matrix. For example: A =
1 0 0 1
is an involutory matrix.
(iii) Idempotent matrix : A square matrix A is said to be idempotent if A2 = A. 1/2 1/2
For example: 1/2 1/2 is an idempotent matrix.
ma r a (iv) Nilpotent matrix : h S . 82 K . 6 A square matrix is said to be nilpotent of index p, if p is the least positive integer such that L 7 Er. 81027 5058 A = O. 9 801 0 1 For example : 0 0 is nilpotent matrix of index 2. 839 p
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(v) Periodic matrix : A matrix A will be called a periodic matrix if Ak+1 = A , where k is a positive integer. If however k is the least positive integer for which Ak+1 = A , then k is said to be the period of A. (vi) Differentiation of a matrix: f(x) g(x)
dA
f '(x) g '(x)
If A then is the differentiation of matrix A. dx h'(x) l '(x) h(x) l(x)
2
dA 2x cos x sin x then dx 2 0 2 2x
For example: If A x
(vii) Conjugate of a matrix : If matrix A [aij]mn , then the conjugate of matrix A is given by A , where A [ aij ]mn i 2 3i 2 3i i For example : If A ; then A 3 i 3 i 4 4
(viii) Hermitian and skewHermitian matrix: A square matrix A = [aij] is said to be Hermitian matrix if aij aji i, j A A a
For example: b ic
b ic d
A square matrix, A = [aij] is said to be a skewHermitian if aij aji i, j A A 0
For example: 2 i
2 i 0
(ix) Unitary matrix : A square matrix A is said to be unitary if AA I , where A is the transpose of complex conjugate of A. (13) System of Linear Equations and Matrices Consider the system a11 x1 + a12x2 + ......... + a1nxn = b1 a21 x1 + a22x2 + ......... + a2nxn = b2 ................................................... am1 x1 + am2x2 + ......... + amnxn = bn
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Let A =
a11 a12 ....... a21 a22 ....... .... .... ....... am1 am2 .......
a1n a2n .... , amn
X=
x1 x2 .... x n
and B =
b1 b2 ... ... b n
Then the above system can be expressed in the matrix form as AX = B. The system is said to be consistent if it has atleast one solution. (i) System of linear equations and matrix inverse : If the above system consist of n equations in n unknowns, then we have AX = B where A is a square matrix. If A is nonsingular, solution is given by X = A 1B . If A is singular, (adj A) B = 0 and all the columns of A are not proportional, then the system has infinite many solution. If A is singular and (adj A) B 0, then the system has no solution (we say it is inconsistent). (ii)
Homogeneous system and matrix inverse : If the above system is homogeneous, n equations in n unknowns, then in the matrix for m it is AX = O. ( in this case b1 = b2 = ..... bn = 0) , where A is a square matrix. If A is nonsingular, the system has only the trivial solution (zero solution) X = 0. If A is singular, then the system has infinitely many solutions (including the trivial solution) and hence it has non trivial solutions. (iii)
Rank of a matrix : Let A = [aij]m n . A natural number is said to be the rank of A if A has a nonsingular submatrix of order and it has no nonsingular submatrix of order more than . Rank of zero matrix is regarded to be zero.
e.g.
3  1 2 5 A = 0 0 2 0 0 0 5 0
3 2 we have 0 2 as a non singular submatrix.
The square matrices of order 3 are 3  1 2 3  1 5 3 2 5  1 2 5 0 0 2 0 0 0 0 2 0 0 2 0 0 0 5 . 0 0 0 0 5 0 0 5 0
ma r a h S . 82 K . 6 L 7 (iv) Elementary row transformation of matrix : Er. 81027 5058 The following operations on a matrix are called as elementary 9 row transformations. 01 8 9 (a) Interchanging two rows. 83 and all these are singular. Hence rank of A is 2.
(b) (b)
Multiplications of all the elements of row by a nonzero scalar. Addition of constant multiple of a row to another row.
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Note : Similar to above we have elementary column transformations also. Remark : 1. 2.
Elementary transformation on a matrix does not affect its rank. Two matrices A & B are said to be equivalent if one is obtained from other using elementary transformations. We write A B. (v)
Echelon form of a matrix : A matrix is said to be in Echelon form if it satisfy the following : (a) The first nonzero element in each row is 1 & all the other elements in the corresponding column (i.e. the column where 1 appears) are zeroes. (b) The number of zeroes before the first non zero element in any non zero row is less than the number of such zeroes in succeeding non zero rows.
Result : Rank of a matrix in Echelon form is the number of non zero rows (i.e. number of rows with atleast one non zero element). Remark : 1.
To find the rank of a given matrix we may reduce it to Echelon form using elementary row transformations and then count the number of non zero rows. (vi) System of linear equations & rank of matrix : Let the system be AX = B where A is an m n matrix, X is the ncolumn vector & B is the mcolumn vector. Let [AB] denote the augmented matrix (i.e. matrix obtained by accepting elements of B as (n + 1)th column & first n columns are that of A). (A) denote rank of A and ([AB]) denote rank of the augmented matrix. Clearly (A) ([AB]) (a)
If (A) < ([AB]) then the system has no solution (i.e. system is inconsistent). (b) If (A) = ([AB]) = number of unknowns, then the system has unique solution. (and hence is consistent) (c) If (A) = ([AB]) < number of unknowns, then the systems has infinitely many solutions (and so is consistent).
ma r a h S . (vii)Homogeneous system & rank of matrix : 82 K . 6 L 7 let the homogenous system be AX = 0, m equations in 'n' In7 this case 8 r.unknowns. 2 E 0 1 B = 0 and so (A) = ([AB]). 8 01505 9 Hence if (A) = n, then the system has only the trivial solution. If98 (A) < n, then 3 8 the system has infinitely many solutions. Mathematics Concept Note IITJEE/ISI/CMI
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CayleyHemilton Theorem : Every matrix satisfies its characteristic equation e.g. let A be a square matrix then A–xI=0 is characteristics equation of A. If x3 – 4x2 – 5x – 7 = 0 is the characteristic equation for A,then A3 – 4A2 + 5A – 7I = 0 Roots of characteristic equation for A are called Eigen values of A or characteristic roots of A or latent roots of A. If is characteristic root of A, then 1 is characteristic root of A–1.
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8. DETERMINANTS Consider the equations a1x + b1y = 0 and a2x + b2y = 0
a a y y 1 and 2 x b1 x b2
a1 a 2 b1 b2
a1b2 a2b1 0
a1 b1 Now the obtained eliminant a1b2 – a2b1 is expressed as b b , which is called the 2 2
determinant of order two.
(1)
a1 b1 a1b2 a2b1 b2 b2
Minors and Cofactors : If A = [aij]n is a square matrix , then minor of element aij, denoted by Mij , is defined as the determinant of the submatrix obtained by deleting ith row and jth column of A. Cofactor of element aij , denoted by Cij , is defined as Cij (1)i j.Mij For example:
a b A = c d , then
If M11 M12 M21 M22
= d = C11 = c, C12 = –c = b, C21 = –b = a = C22 a b c A d e f , then g h i
If
M11 = ei–hf , M32 = af–dc and C22 = ai–cg , C31 = bf–ec , C23 = bg–ah , etc.
(2)
Determinant of A matrix : If A = [aij]n be a square matrix of order n > 1 , then determinant of A is defined as the summation of products of elements of any one row (or any one column) with the corresponding cofactors.
For example:
a11 a12 A a21 a22 a31 a32
a13 a23 a33
A = a11C11 + a12 C12 + a13 C13 . a11
a22
a23
a32
a33
Mathematics Concept Note IITJEE/ISI/CMI
a12
a21 a23 a31 a33
a13
a21 a31
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 a 39 8 , or a 22
33
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A = a12 C12 + a22 C22 + a32 C32 a21 a23
a12
a31 a33
a22
a11
a13
a31 a33
a32
a11
a13
a21 a23
Note: If A = [aij]n , then the summation of the products of elements of any row with corresponding cofactors of any other row is zero. (Similarly , the summation of the products of elements of any column with corresponding cofactors of any other column is zero). For example:
a11 C21 + a12 C22 + a13 C23 = 0 , and a12 C11 + a22 C21 + a32 C31 = 0.
(3)
Properties of Determinants : (i) The value of a determinant remains unaltered, if the rows and columns are interchanged. a1 b1 i.e. D a2 b2 a3 b3
c1 c2
a1
c3
a1
a3  A   A T 
b1 b2 b3 c1
c2
c3
(ii) If any two rows (or columns) of a determinant be interchanged, the value of determinant is changed in sign only. For example: a1
b1
c1
If D a2 b2 a3 b3
a2 b2
c2
c2 , R1 R2 D ' a1 b1
c1 , then D' D
c3
c3
a3 b3
(iii) If a determinant has all the elements zero in any row or column then its value is zero. For example: 0
0
D a2 b2 a3 b3
0 c2 0. c3
(iv)If a determinant has any two rows(or columns) identical, then its value is zero. a1
b1
For example: D a1 b1 a3 b3
c1 c1 0.
(R1 R 2 )
c3
(v) If all the elements of any row (or column) is multiplied by the same number, then the determinant is multiplied by that number. For example: a1
b1
c1
D a2 b2
c2
a3 b3
c3
Ka1 Kb1 Kc1 and D '
a2
b2
c2
a3
b3
c3
, then D ' KD
ma r a h S Note: . 82 K . 6 L 7 7 one column) r. 10or2any 58 of If λ be a scalar , then λ A is obtained by multiplying E any one row( 0 5 8 9 801 A by λ . 39 8  λA  = λ  A  , where A a n
ij n
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
Page 56
(vi) If each element of any row (or column) can be expressed as a sum of two terms then the determinant can be expressed as the sum of two determinants. For example: a1 x b1 x c1 x a2
b2
c2
a3
b3
c3
a1
b1
c1
x
y
z
a2 b2
c2 a2 b2
c2
a3 b3
c3
c3
a3 b3
(vii) The value of a determinant is not altered by adding to the elements of any row (or column) a constant multiple of the corresponding elements of any other row ( or column). For example: a1
b1
c1
If D a2 b2 a3 b3
c2 , R1 R1 mR 2 and R3 R 3 nR1 c3
a1 ma2 b1 mb2 D'
(4)
c1 mc2
a2
b2
c2
a3 na1
b3 nb1
c3 nc1
, then D ' D
Summation of Determinant : f(r) g(r) h(r)
Let (r) a1 b1
a2
a3
b2
b3
where a1, a2, a3, b1, b2, b3 are constants (independent of r) , then n n
(r)
n
f(r)
r 1
r 1
n
g(r)
r 1
h(r) r 1
a1
a2
a3
b1
b2
b3
Note: If more than one row or one column are function of r , then first the determinant is simplified and then summation is calculated.
(5)
Integration of a Determinant : f(x) g(x) h(x)
Let (x) a1 b1
a2
a3
b2
b3
where a1, a2, a3, b1, b2, b3 are constants (independent of x) , then
ma r a h S . (x)dx a a a 82 K . 6 L 7 b b b Er. 81027 5058 9 801 Note: 39 If more than one row or one column are function of x , then first8 the determinant is simplified b
b
a
b
f(x)dx
a
b
g(x)dx
a
h(x)dx a
1
2
3
1
2
3
and then it is integrated. Mathematics Concept Note IITJEE/ISI/CMI
Page 57
(6)
Differentiation of Determinant : f1 (x)
f2 (x)
f3(x)
Let (x) g1(x) g2 (x) g3 (x)
, then
h1(x) h2 (x) h3(x) f1 '(x) f2'(x) f3'(x) '(x) g1(x)
f1(x)
f1(x)
f2 (x)
f3(x)
g2 (x) g3 (x) g1'(x) g2'(x) g3'(x) g1(x)
g2 (x)
g3 (x)
h1 (x) h2 (x) h3 (x)
(7)
h1(x)
f2 (x) h2 (x)
f3(x) h3 (x)
h1'(x) h2'(x) h3'(x)
Multiplication Of Two Determinants : a1 b1 m1 a b1 2 1 1 1 a2 b2 2 m2 a2 1 b2 2
a1 b1
c1
a2 b2
c2
a3 b3
c3
1
m1
a1m1 b1m2 a2m1 b2m2
a11 b12 c1 3
n1
a1m1 b1m2 c1m3
2 m2 n2 a2 1 b2 2 c2 3 3 m3 n3
a1n1 b1n2 c1n3
a2m1 b2m2 c2m3 a2n1 b2n2 c2n3
a3 1 b3 2 c3 3
a3m1 b3m2 c3m3
a3n1 b3n2 c3n3
Multiplication expressed in the calculation is row column which also applicable in matrix multiplication , but in case of multiplication of determinants , row column , row row , column row and column column all are applicalble.
Note: Power cofactor formula : If = aij is a determinant of order n , then the value of the determinant Cij = n 1 , where Cij is cofactor of aij.
(8)
Standard Results :
a 0 0 a d e a 0 0 0 b 0 0 b f d b 0 abc 0 0 c
e
f
c
a h g h b f abc 2fgh af 2 bg2 ch2. g f
0 0 c
c
a b c b c a (a3 b3 c3 3abc) (a b c)(a2 b2 c2 ab bc ac) c a b
1 a
1 b
1 c
2
2
2
a
b
c
1
1
1
2
2
a
b
a3 b3
(a b)(b c)(c a)
1 (a b c) (a b)2 (b c)2 (c a)2 2
1 a a3
c2 (a b)(b c)(c a)(ab bc ca)
ma r a h a)(a2 b c) .S c)(c (a . K b)(b 68 L 7 7 . b E cr 02 5058 1 8 9 801 839 1 b
1 c
3
3
c3
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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(8)
Cramer's Rule : System of Linear equations (I) Consider the system of nonhomogenous equations : a1x + b1y + c1z = d1 a2x + b2y + c2z = d2 a3x + b3y + c3z = d3 a1
b1
c1
d1 b1
c1
a1
Where D a2 b2 c2 , D1 d2 b2 c2 , D2 a2 a3 b3 c3
d3 b3 c3
d1
c1
a1
b1
d1
d2 c2 and D3 a2 b2 d2
a3 d3 c3
,then
a3 b3 d3
D D1 D , y 2 , z 3 D D D Nonhomogenous system of equations may or may not be consistent ,following cases explains the consistency of equations :
according to Cramer's rule : x
If D 0 , then the system of equations are consistent and have unique solution.
If D = D1 = D2 = D3 = 0 , then the system of equations have either infinite solutions or no solution.
If D = 0 and atleast one of D1, D2, D3 is not zero then the equations are inconsistent and have no solution.
(II) Consider the system of homogenous equations : a1x + b1y + c1z = 0 a2x + b2y + c2z = 0 a3x + b3y + c3z = 0 In homogenous system , D1 = D2 = D3 = 0 , hence : If D 0 , then the system of equations are consistent and have unique solution (i.e. Trivial solution)
If D = 0 , then the system of equations consitent and have infinite solutions. (i.e. Nontrivial solution)
Note: Homogenous system of equations are always consistent , either having Trivial solution or nontrivial solution (III) System of equations , a1x + b1y = c1 and a2x + b2y = c2 represents two straight lines in 2dimonseional plane , hence :
If
a1 b 1 : System of equations are consistent and have uniqe solution a2 b2
(i.e. Intersecting lines)
a1 b1 c1 If a b c : Wywtem of equation are consistent and have infinite solution 2 2 2
(i.e. coincident lines)
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839
a1 b c 1 1 : System of equation in inconsistent and have no solution If a2 b2 c2
(i.e. parellel lines).
Note:
Three equation in two variables : If x and y are not zero, then condition for a1x + b1y + c1 = 0; a2x + b2y + c2 = 0 and Mathematics Concept Note IITJEE/ISI/CMI
Page 59
a3x + b3y + c3 = 0 to be consistent in x and y is a1
b1
c1
a2 b2
c2 0.
a3 b3
c3
(10) Application of Determinants : (i) Area of a triangle whose vertices are (x1 , y1) , (x2 , y2) and (x3 , y3) is :
=
1 2
x1
y1 1
x2
y2 1
x3
y3 1
If = 0 , then the three points are collinear.. (ii) Equation of a straight line passing through (x1, y1) and (x2, y2) is : x1
y1 1
x2
y2 1 = 0 y3 1
x3
(two point form of line)
(iii) If lines a1x + b1y + c1 = 0 , a2x + b2y + c2 = 0 and a3x + b3y + c3 = 0 are nonparallel and noncoincident , then lines are concurrent , if a1
b1
c1
a2 b2
c2 0
a3 b3
c3
(iv) ax2 + 2 hxy + by2 + 2gx + 2fy + c = 0 represents a pair of straight lines if : a h g h b f g f
c
= 0 abc + 2 fgh af2 bg2 ch2 = 0
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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9. SEQUENCE AND SERIES (1)
Basic Definitions : Sequence : A sequence is a function whose domain is the set of natural numbers.Since the domain for every sequence is the set N of natural numbers, therefore a sequence is represented by its range. If f : N R , then f(n) = Tn , n N is called a sequence and is denoted by {Tn} {f(1), f(2), f(3), ...........} = {T1, T2, T3, ...................}
Note: (i) A sequence whose range is a subset of R is called a real sequence. (ii) Finite sequences : A sequence is said to be finite if it has finite number of terms. (iii) Infinite sequences : A sequence is said to be infinite if it has infinite number of terms. (iv) It is not necessary that the terms of a sequence always follow a certain pattern or they are described by some explicit formula for the nth term. Sequences whose terms follow certain patterns are called progressions. (v) All progressions are sequences, but all sequences are not progressions. For example: Set of prime numbers is a sequence but not a progression Series : By adding or substracting the terms of a sequence, we get an expression which is called a series. If a1, a2, a3,........... an is a sequence, then the expression a1 + a2 + a3 + ....... + an is a series. For example : (i) 1 + 2 + 3 + 4 + ............ + n (ii) 2 + 4 + 8 + 16 + ........... nth term of sequence: If summation of n terms of a sequence is given by Sn , then its nth term is given by : Tn = Sn – Sn–1 .
(2)
Arithmetic progression (A.P.) : A.P. is a sequence whose terms increase or decrease by a fixed number. This fixed number is called the common difference. If a is the first term and d is the common difference , then A.P. can be written as : a , a + d, a + 2d ,.......... a + (n–1) d , ......... nth term of an A.P. : Let a be the first term and d be the common difference of an A.P., then Tn = a + (n–1)d and Tn – Tn–1 = Constant. (i.e. , common difference) Sum of first n terms of an A.P. : If a is first term and d is common difference then
ma r a h S . 82 K where is the last term . 6 L 7 Er. 81027 5058 p term of an A.P. from the end : 9 801 Let 'a' be the first term and 'd' be the common difference of an A.P. 9 3 having n terms. Then p term from the end is (n – p + 1) term from the beginning. 8
Sn =
n (2a + (n 1)d) 2
=
n (a + ) 2
th
th
th
pth term from the end = T(n–p+1) = a + (n – p)d Mathematics Concept Note IITJEE/ISI/CMI
Page 61
(3)
Properties of A.P. : (i) If a1 , a2 , a3..... are in A.P. whose common difference is d, then for fixed nonzero number K R.
a1 K , a2 K , a3 K ,.... will be in A.P. , whose common difference will be d.
Ka1 , Ka2 , Ka3 ..... will be in A.P. with common difference = Kd.
a1 a2 a3 , , ..... will be in A.P. with common difference = d/k. K K K
1 1 1 , , ,........ will be in H.P.. a1 a2 a3 If x R {1}, then xa1 , x a2 , xa3 ,...... will be in G.P..
(ii) The sum of terms of an A.P. which are equidistant from the beginning and the end is constant and is equal to sum of first and last term. i.e. a1 + an = a2 + an–1 = a3 + an–2 = .... (iii) Any term (except the first term) of an A.P. is equal to half of the sum of terms equidistant from the term i.e. an
1 (an k an k ),k n. 2
(iv) If number of terms of any A.P. is odd, then sum of the terms is equal to product of middle term and number of terms. (v) If number of terms of any A.P. is even then A.M. of middle two terms is A.M. of first and last term. (vi)If the number of terms of an A.P. is odd then its middle term is A.M. of first and last term. (vii) If a1, a2 , ........ an and b1 , ........ bn are the two A.P.'s. Then a1 b1 , a2 b2 ,......an bn are also A.P.'s with common difference d1 d2 , where d1 and d2 are the common difference of the given A.P.'s . (viii) Three numbers a , b , c are in A.P. iff 2b = a + c. (ix) If the terms of an A.P. are chosen at regular intervals, then they form an A.P. (x) Three numbers in A.P. can be taken as a–d, a, a + d; four numbers in A.P. can be taken as a–3d, a–d, a + d, a + 3d; five numbers in A.P. are a–2d, a–d, a , a + d, a + 2d and six terms in A.P. are a–5d, a–3d, a–d, a+d, a+3d, a+5d etc.
(4)
Arithmetic Mean (A.M.) : If three terms are in A.P. then the middle term is called the A.M. between the other two, so if a, b, c are in A.P. , b is A.M. of a and c. ac 2 nArithmetic Means Between Two Numbers : b
ma r a h S . 82 K . 6 L 7 r. a) 1027 058 En(b =a+ 15 n 9 18 0 8 839
If a, b are any two given numbers and a, A1, A2....., An, b are in A.P. then A1, A2, ... An are the n A.M.'s between a and b. A1 = a +
ba 2(b a) , A2 = a + ,...., An n1 n 1
Note:
Sum of n A.M.'s inserted between a and b is equal to n times the single A.M. between Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
Page 62
n
a and b i.e.
A
r
nA where A is the single A.M. between a and b.
r 1
(5)
Geometric Progression (G.P.) : G.P. is a sequence of numbers whose first term is non zero and each of the succeeding terms is equal to the preceding terms multiplied by a constant. Thus in a G.P. the ratio of nth term and (n–1)th term is constant. This constant factor is called the common ratio of the series a, ar, ar2, ar3, ar4,....... is a G.P. with a as the first term and r as common ratio. nth term of G.P. : Tn = a(r)n–1
a(r n 1) , r 1 Sum of the first n terms : Sn = r 1 na , r 1 Sum of an infinite G.P. , when r < 1. S
a . 1r
pth term from the end of a finite G.P. : If G.P. consists of 'n' terms, p th term from the end = (n – p + 1) th term from the beginning = arn–p. n 1
1 Also, the pth term from the end of a G.P. with last term and common ratio r is r
(5)
Properties of G.P. : (i) If each term of a G.P. be multiplied or divided or raised to power by the some nonzero quantity, the resulting sequence is also a G.P. (ii) The reciprocal of the terms of a given G.P. form a G.P. with common ratio as reciprocal of the common ratio of the original G.P. (iii) In a finite G.P. , the product of terms equidistant from the beginning and the end is always the same and is equal to the product of the first and last term. i.e. , if a1 , a2 , a3 , ........ an be in G.P. Then a1 an = a2 an–1 = a3 an–2 = a4 an–3 = ...... = ar . an–r+ 1 (iv)If the terms of a given G.P. are chosen at regular intervals, then the new sequence so formed also forms a G.P. (v) Three nonzero numbers a , b , c are in G.P. iff b2 = ac.
ma r a h S . 82 K . 6 L 7 (vii) If first term of a G.P. of n terms is a and last term is r,.then the product 8 of E 81027 5of0all5terms the G.P. is (a) . 9 801 83S9denotes the sum of the (viii) If there be n quantities in G.P. whose common ratio is r and
(vi) Every term (except first term) of a G.P. is the square root of terms equidistant from it. i.e. Tr
Tr p .Tr p ; [r p]
n/2
m
first m terms , then the sum of their product taken two by two is Mathematics Concept Note IITJEE/ISI/CMI
r Sn Sn 1 . r 1 Page 63
a a, ar, in general we take e r
(ix) Any three consecutive terms of a G.P. can be taken as a r
,
k
a r
k 1
,
a r
k 2
,....a, ar, ar2, .........ark in case we have to take 2k + 1 terms in a G.P..
(x) Any four consecutive terms of a G.P. can be taken as a r
,
2k 1
a r
2k 3
a r
3
,
a , ar, ar3, in general we take e r
a ,...... , ar, ....... ar2k 1 in case we have to take 2k terms in a G.P.. r
(xi) If a1, a2, a3, ... and b1, b2, b3......., are two G.P.'s with common ratio r1 and r2 respectively then the sequence a1b1, a2b2, a3b3,.. is also a G.P. with common ratio r1r2. (xii) If a1, a2, a3,....... are in G.P. where each ai > 0, then log a1, log a2, log a3.............. are in A.P. and its converse is also true.
(6)
Geometric Means (G.M.) : If a, b, c are in G.P., b is the G.M. between a and c. b2 = ac nGeometric Means Between a, b: If a, b are two given numbers and a, G1, G2, ....., Gn, b are in G.P.. Then G1, G2, G3,...., Gn are n G.M.s between a and b. G1 = a(b/a)1/n+1, G2 = a(b/a)2/n+1 ,........., Gn = a(b/a)n/n +1
Note: The product of n G.M.s between a and b is equal to the nth power of the single G.M. between a and b n
G
r
(G)n , where G is the single G.M. between a and b.
r 1
(7)
Harmonic Progression : The sequence a1 , a2 , .... , an , ..... where ai 0 for each i is said to be in harmonic progression (H.P.) if the sequence 1/a1, 1/a2 , ... , 1/an , ... is in A.P. Note that an , the nth term of the H.P. , is given by an
1 1 1 1 where a and d . a (n 1)d a1 a2 a1
Note: (i) If a and b are two nonzero numbers, then the harmonic mean of a and b is number H such that the sequence a , H , b is an H.P. We have
ma r a h S . 82 K . 6 L 7 7 (ii) The n numbers H , H , ... , H are said to be harmonic a and 8be btheif 5 Er,.... ,8means 0are2between 1 a, H , H , .... , H , b are in H.P., that is, if 1/a, 1/H , 1/H 1/b in A.P. Let d 0 9 8015 common difference of this A.P. Then 39 8 1 1 (n 2 1)d 1 1 1 1 1 ... H n a1 a2 an 1
1
2
b
n
2
n
1
2
a
... Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
Page 64
d
1 1 1 ab n 1 b a (n 1)ab
Thus 1 1 ab , H1 a (n 1)ab 1 1 2(a b) 1 1 n(a b) ,..., H2 a (n 1)ab Hn a (n 1)ab
From here , we can get the values of H1 , H2 , ... , Hn.
(8)
Relation between means : If A, G, H are respectively A.M., G.M., H.M. between a and b both being unequal and positive then, G2 = AH A, G, H are in G.P.. Let a1, a2, a3,.......... an be n positive real numbers, then A.M. G.M. H.M. , where A.M.
H.M.
a1 a2 a3 ........ an , G.M. (a1.a2 .a3.a4.........an )1 / n and n n 1 1 1 . ..... a1 a2 an
Note: (i) The inequality of A.M. , G.M. and H.M. is only applicable for positive real numbers and the sign of equality holds for equal number.
A.M. G.M. H.M.
a1 a2 a3 ........ an.
(ii) If a, b , c R , then
(9)
ab c 3abc (abc)1 / 3 3 ab bc ac
ArithmeticoGeometric Series : A series each term of which is formed by multiplying the corresponding term of an A.P. and G.P. is called the ArithmeticoGeometric Series. e.g. 1 + 3x + 5x2 + 7x3 + ... Here 1, 3, 5, ........... are in A.P. and 1, x, x2, x3...... are in G.P... Sum of n terms of an ArithmeticoGeometric Series : Let Sn = a + (a + d)r + (a + 2d) r2 + ........ + [a + (n–1)d]rn–1 , then Sn
a dr(1 rn1 ) [a (n 1)d]rn , r 1. 1r 1r (1 r)2
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839
a dr Sum To Infinity : If r < 1 & n , then S 1 r (1 r)2
Mathematics Concept Note IITJEE/ISI/CMI
Page 65
(10) Important Results : n
(i)
n
(ar br )
r 1
n
ar
r 1
n
br
(ii)
r 1
n
kar k
r 1
a
r
r 1
n
(iii)
k k k k ...............n times nk ; where k is a constant r 1 n
r 1 2 3 ........ n
(iv)
r 1
n(n 1) 2
n
(v)
r
2
12 22 32 ......... n2
r 1
n(n 1)(2n 1) 6 2
n
n(n 1) r3 13 23 33 ......... n3 (vi) 2 r 1
(vii) 2.
ai.aj = (a + a + ......... + a )2 1 2 n
(a12 + a22 + ..... + an2)
1 i j n
n
(viii)
n
ar a1.a2.a3.........an
(ix)
r 1
r 1
n
(x)
n
kar kn.
r
a r 1
n
r n!
r 1
(xi)
n n (ar .br ) ar br r 1 r 1 r 1
(11) Method of Difference : If the differences of the successive terms of a series are in A.P. of G.P. , we can find nth term of the series by the following steps : Step 1: Denote the nth term by Tn and the sum of the series upto n terms by Sn. Step 2: Rewite the given series with each term shifted by one place to the right. Step 3: By subtracting the later series from the former, find Tn. Step 4: From Tn , Sn can be found by appropriate summation.
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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10. FUNCTIONS AND GRAPH (1) Definition : Function is a particular case of relation, from a non empty set A to a non empty set B, that associates each and every member of A to a unique member of B. Symbolically, we write f : A B. We read it as "f is a function from A to B".
Note: (i) Let f : A B, then the set A is known as the domain of f & the set B is known as codomain of f. (ii) If a member 'a' of A is associated to the member 'b' of B, then 'b' is called the fimage of 'a' and we write b = f(a). Further 'a' is called a preimage of 'b'. (iii) The set {f(a) : a A} is called the range of f and is denoted by f(A). Clearly f(A) B. (iv) Sometimes if only definition of f(x) is given (domain and s are not mentioned), then domain is set of those values of 'x' for which f(x) is defined, while codomain is considered to be ( , ). (v) A function whose domain and range both are sets of real numbers is called a real function. Conventionally the word "FUNCTION" is used only as the meaning of real function.
(2) Classification of Functions : On the basis of mapping functions can be classified as follows: (i) OneOne Function (Injective Mapping) A function f : A B is said to be a oneone function or injective mapping if different elements of A have different f images in B. Thus for x1, x2 A & f(x1), f(x2) B, f(x1) = f(x2) x1 = x2 or x1 x2 f(x1) f(x2). Diagrammatically an injective mapping can be shown as
A
B
A
B
OR (ii) Many  One function : A function f : A B is said to be a many one function if two or more elements of A have the same f image in B. Thus f : A B is many one iff there exists at least two elements x1, x2 A , such that f(x1) = f(x2) but x1 x2. Diagrammatically a many one mapping can be shown as
ma r a (i) If x , x A & f(x ), f(x ) B, f(x ) = f(x ) x = x or x x f(x.)S f(x ), then2 h K . 68 function is ONEONE otherwise MANYONE. L 7 7 . (ii) If there exists a straight line parallel to xaxis, which graph function 58 Ercuts the 02 of the 1 0 5 8 atleast at two points, then the function is MANYONE, otherwise ONEONE. 9 8 01 where 9 (iii) If either f'(x) 0, x complete domain or f'(x) 0 x complete domain, 83 equality can hold at discrete point(s) only, then function is ONEONE, otherwise Note:
1
2
1
2
1
2
1
2
1
2
1
2
MANYONE. Mathematics Concept Note IITJEE/ISI/CMI
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(iii)Onto function (Surjective mapping) If the function f : A B is such that each element in B (codomain) must have atleast one preimage in A, then we say that f is a function of A 'onto' B. Thus f : A B is surjective iff b B, there exists some a A such that f(a) = b. Diagrammatically surjective mapping can be shown as
A
B
A
B
OR
Note: If range codomain, then f(x) is onto, otherwise into (iv) Into function : If f : A B is such that there exists atleast one element in codomain which is not the image of any element in domain, then f(x) is into. Diagrammatically into function can be shown as
A
B
A
B
OR
Note: (i) function can be one of following four types :
(a) oneone onto (injective & surjective)
(b) oneone into (injective but not surjective)
(c) manyone onto (surjective but not injective)
(d) manyone into (neither surjective not injective) (ii) If f is both injective & surjective, then it is called a bijective mapping. The bijective functions are also named as invertible, nonsingular or biuniform functions. (iii) If a set A contains 'n' distinct elements then the number of different functions defined from A A is nn and out of which n! are one one.
(3) Various Types of Functions :
ma r a h S same image in B. Thus f : A B; f(x) = c, x A, c B is a constant function.82 . K L. 2776 8 . Note: r E 810 505 (i) Constant function is even and f(x) = 0 is the 9 801 only function which is both even and odd 39 8 (ii) Constant function is periodic with indeterminate
(i) Constant function : A function f : A B is said to be a constant function if every element of A has the
periodicity. Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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(ii) Identity function : The function f : A A defined by f(x) = x x A is called the identity function on A and is denoted by IA. It is easy to observe that identity function is a bijection.
(iii)Polynomial function : If a function f is defined by f(x) = a0xn + a1xn–1 + a2xn–2 + ... + an–1x + an where n is a non negative integer and a0, a1, a2, .........., an are real numbers and a0 0, then f is called a polynomial function of degree n. for example: linear functions, quadratic functions, cubic functions etc. (iv) Algebraic Function : y is an algebraic function of x, if it is a function that satisfies an algebraic equation of the form, P0(x) yn + P1 (x) yn–1 + ..... + Pn–1 (x)y + Pn(x) = 0 where n is a positive integer and P0(x), P1(x).....are polynomials in x. e.g. y = x is an algebraic functiontion, since it satisfies the equation y2 x2 = 0.
Note: (i) All polynomial functions are algebraic but not the converse. (ii) A function that is not algebraic is called Transcendental Function. (v) Fractional/Rational Function : g(x) A rational function is a function of the form y = f(x) = h(x) , where g(x) & h(x) are
polynomials and h(x) 0. (vi)Exponential Function : A function f(x)=ax=exlna (a 0, a 1, x R) is called an exponential function. Graph of exponential function can be as follows : For a > 1 For 0 < a < 1 f(x)
f(x) (0,1)
(0,1) 0
x
0
x
(vii) Logarithmic Function : f(x) = logax is called logarithmic function where a > 0 and a 1 and x > 0. Its graph can be as follows For a > 1 For 0 < a < 1
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839
Mathematics Concept Note IITJEE/ISI/CMI
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(viii) Absolute Value Function / Modulus Function: x
if
x 0
x if
x 0
The symbol of modulus function is f(x) = x and is defined as : y = x =
y y
=
–x
y
=
x
x
y = x
Note: (i)
x2  x  and  x  max{x, x}
(ii) Absolute value function is piecewise defined function and its domain and range are R and [0, ) respectively (iii) If 'a' and 'b' are nonnegative real numbers, then  x  a x [a, a];{i.e. a x a}  x  a x R /(a, a);{i.e. x a or x a} a  x  b x [b, a] [a, b] x   y   x y   x   y   x y   x y   x   y  xy 0.  x y   x   y  xy 0.
(iv) max{f(x), g(x)} (v) min{f(x), g(x)}
f(x) g(x) f(x) g(x) 2 2
f(x) g(x) f(x) g(x) 2 2
Y
(ix) Signum Function :
y = 1 if x > 0
A function f(x) = sgn (x) is defined as follows : 1 ; x0 f(x) = sgn (x) = 0 ; x 0 1; x 0
y = –1 if x > 0
y = sgn x
Note: (i)
x x 0 sgn x = x 0; x 0
(ii)
f(x) ; f(x) sgn f(x) = ; 0
f(x)0 f(x)0
(x) Greatest Integer Function or Step Up Function/Box function: The function y = f(x)=[x] is called the greatest integer function where [x] equals to the greatest integer less than or equal to x.
Note: (i) [x] x x I. (ii)
(iii) x 1 < [x] x
0; x I 1; x I
(iv) [x] + [ x] =
Mathematics for IITJEE Er. L.K.Sharma
y
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 –3 –2 –1
[x n] [x] n n I.
9810277682/8398015058
X
O
3 2 1
1 2 3 –1 –2 –3
(v) [x] + [y] [x + y] Page 70
x
.
y graph of y = {x}
(xi) Fractional Part Function :
1
y=1
It is defined as, y = {x} = x [x] and {x} [0,1)
x
Note:
–2
–1
0
1
2
3
2
f(x) x [x], x [x] or x [x] are periodic with period 1.
(4) Trigonometric Function : (i)
y sin x ; x R ; y [1, 1]
(ii) y cos x ; x R ; y [1, 1]
(iii) y tan x ; x R (2n 1) / 2, n I;y R
(iv) y cot x ; x R n, n I;y R
(v) y cosec x ;
(vi) y sec x ;
x R n, n I;y R /(1, 1)
x R (2n 1) / 2, n I;y R /(1, 1)
(5) Inverse Trigonometric Function : (i) y = sin–1 x, x 1, y , 2 2
(ii) y = cos–1 x, x 1, y [0, ]
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Mathematics Concept Note IITJEE/ISI/CMI
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(iii) y = tan–1 x, x R, y , 2 2
(iv)
(v) y = sec–1 x, x 1, y 0 , 2 2 ,
y = cot–1 x, x R, y (0, )
(vi) y = cosec–1 x, x 1,y 2 ,0 0 , 2
(6) Equal or Identical Function : Two functions f(x) and g(x) are said to be identical (or equal) iff : (i) The domain of f the domain of g. (ii) The range of f the range of g and (iii) f(x) = g(x), for every x belonging to their common domain, e.g. f(x) = g(x) =
x 2
x
are identical functions. But f(x) = x and g(x) =
x2 x
1 and x
are not identical functions.
(7) Bounded Function The function f(x) is said to be bonded above if there exists M such that y=f(x) M (i.e. not greater that M) for all x of the domain and M is called upper bound. Similarly f(x) is said to be bounded below if there exists in such that y=f(x) m(i.e. never less that m) for all x of the domain and m is called the lower bond. If however, there does not exist M and m as stated above, the function is said to be unbounded.
(8) Even and odd Functions :
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839
If f (–x) = f(x) for all x in the domain of 'f' then f is said to be an even function . If f(x) f(–x) = 0 f(x) is even.
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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If f(–x) = –f(x) for all x in the domain of 'f' then f is said to be an odd function. If f(x) + f(–x) = 0 f(x) is odd.
Note: (i) Even functions are symmetrical about the yaxis and odd functions are symmetrical about origin (ii) Any given function can be uniquely expressed as the sum of even function and odd function for example: f(x)
1 1 f(x) f(x) f(x) f(x) 2 2
(iii) For any given function f(x), g(x)=f(x)+f(–x) is even function and h(x)=f(x)–f(–x) is odd function. (iv) If an odd function is defined at x = 0, then f(0) = 0. (v) Let the definition of the function f(x) is given only for x 0. Even extension of this function implies to define the function for x < 0 assuming it to be even. In order to get even extension replace x by x in the given definition. Similarly, odd extension implies to define the function for x < 0 assuming it to be odd. In order to get odd extension, multiply the definition of even extension by 1.
(9) Periodic Function : A function f(x) is called periodic with a period T if there exists a positive real number T such that for each x in the domain of f the numbers x T and x + T are also in the domain of f and f(x) = f(x + T) for all x in the domain of 'f'. Domain of a periodic function is always unbounded. Graph of a periodic function with period T is repeated after every interval of 'T'. e.g. The function sin x & cos x both are periodic over 2 and tan x is periodic over . The least positive period is called the principal or fundamental period of f or simply the period of f.
Note: (i) sinn, cosnx, secnx, and cosecnx are periodic function with period when n is even and 2 when n is odd or fraction. e.g. period of sin2x is but period of sin3x, sin x is 2 . (ii) tannx and cotnx are periodic functions with period irrespective of 'n'. (iii) sin x, cos x, tan x, cot x, sec x, and cosec x are periodic functions with period . (iv) If f(x) is periodic with period T, then: (a) k. f(x) is periodic with period T. (b) f(x+b) is periodic with period T. (c) f(x)+c is periodic with period T. 1 (v) If f(x) has a period T, then f(x) and
f(x) also have a period T..
T (vi) If f(x) has a period T then f(ax + b) has a period a .
(vii)If f(x) has a period T1 and g(x) has a period T2 then period of f(x) g(x) or f(x). f(x) g(x) or g(x) is L.C.M of T1 & T2 provided their L.C.M. exists. However that L.C.M. (if
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839
exists) need not to be fundamental period. If L.C.M. does not exists f(x) g(x) or f(x).g(x) is not periodic. e.g. sin x has the period , cos x also has the period . sin x + cosx also has a period . But the fundamental period of sin x  + cos x is
Mathematics Concept Note IITJEE/ISI/CMI
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(10) Composite Function : Let f : X Y1 and g : Y2 Z be two functions and the set D = {x X : f(x) Y2}. If D then the function h defined on D by h(x) = g{f(x)} is called composite function of g and f and is denoted by gof. It is also called function of a function.
Note: (i) Domain of gof is D which is a subset of X (the domain of f). Range of gof is a subset of the range of g . If D = X, then f(x) Y2. (ii) In general gof fog (i.e. not commutative) (iii) The composite of functions are associative i.e. if three functions f, g, h are such that fo(goh) and (fog)oh are defined, then fo(goh)=(fog)oh. (iv) If f and g both are oneone, then gof and fog would also be oneone. (v) If f and g both are onto, then gof or fog may or may not be onto. (vi) The composite of two bijections is a bijection iff f & g are two bijections such that gof is defined, then gof is also a bijection only when codomain of f is equal to the domain of g. (vii) If g is a function such that gof is defined on the domain of f and f is periodic with T, then gof is also periodic with T as one of its periods.
(11) Inverse of a Function : Let f : A B be a function, then f is invertible iff there is a function g : B A such that gof(x) is an identity function on A and fog(x) is an identity function on B, g(x) is called inverse of f and is denoted by f–1. For a function to be invertible it must be bijective.
Note:
(0,1)
f( x)
=
a
,
x
a
>
1
(i) The graphs of f & g are the mirror images of each other in the line y = x. For example f(x) = ax and g(x) = loga x are inverse of each other, and their graphs are mirror images of each other on the line y = x as shown below.
a
g( x)
y
=
=
lo g
x
x
(1,0)
(ii) Normally points of intersection of f and f–1 lie on the straight line y = x. However it must be noted that f(x) and f–1 (x) may intersect otherwise also. (iii) In general fog(x) and gof(x) are not equal but if they are equal then in majority of cases f and g are inverse of each other or atleast one of f and g is an identity function.
ma r a h S 2 . (iv) If f & g are two bijections f : A B , g : B C then the inverse of gof exists 8and K . 6 L 7 (gof) = f o g . Er. 810271 5058 9 =8g'(x) 01 (v) If f(x) and g(x) are inverse function of each other then f'(g(x)) 9 83 –1
–1
–1
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(12) Functional relationships: If x, y are independent variables, then : (i) f(xy) = f(x) + f(y) f(x) = k ln x or f(x) = 0. (ii) f(xy) = f(x). f(y) f(x) = xn, n R (iii) f(x + y) = f(x). f(y) f(x) = akx. (iv) f(x + y) = f(x) + f(y) f(x) = kx, where k is a constant.
1 1 (v) f(x).f x =f(x)+f x f(x)=1 xn where n N. x y
(vi) f 2
f(x) f(y) f(x)=ax+b, a and b are constants. 2
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Mathematics Concept Note IITJEE/ISI/CMI
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11. LIMITS (1) Definition : Let f(x) be a function , if for every positive number there exists a positive number , such that 0 < x – a < and f(x) – L < , then f(x) tends to limit L as x tends to a and f(x) the limiting value of f(x) at location x = a is represented by xlim a
In above definition of limit if x tends to a from the values of x greater than a , then limiting value is termed as right hand limit (RHL.) and represented by lim f(x) or f(a+) x a
f(a ) lim f(a h) where h is infinitely small positive number.. h 0
In above definition of limit if x tends to a from the values of x less than a , then limiting f(x) or f(a–) value is termed as left hand limit (LHL.) and represented by xlim a
f(a ) lim f(a h) where h is infinitely small positive number.. h 0
Note: (i) Finite limit of a function f(x) is said to exist as x approaches a , if : lim f(x) lim f(x) some finite value.
x a
x a
(ii) If LHL and RHL both approaches to or , then limit of function is said to be infinite limit.
(2) Indeterminate Forms : Let f(x) be a function which is defined is same neighbourhood of location x = a , but functioning value f(a) is indeterminate because of the form of indeterminacy 0 , , , 0 , o , 0o and 1 . i.e. 0
For example : f(x)
x2 4 1 1 is at x 2 , f(x) 2 at x 0 , f(x) (1 cos x)x at x 0 , etc. x2 x sin2 x
To know the behavior of function at indeterminate locations , limiting values are calculated , f(x) , where x = a is the location of indeterminacy for function which is represented by xlim a f(x).
Note: f(x) f(a) , provided the limit If function f(x) is determinate at locations x = a , then xlim a exists (i.e. RHL = LHL). f(x) f(0) For example : If f(x) cos x , then xlim 0
a m r a x 4 h If f(x) , then lim f(x) f(2). S . 82 x 2 K . 6 L 7 Er. 81027 5058 Fundamental Theorems on Limits : 9 801 39 8 lim g(x) lim f(x) Let = L and = M. If L and M exists , then : 2
x 2
(3)
x a
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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(i) lim {f(x) g(x)} L M
{L M }
(ii) lim {f(x).g(x)} L.M
{L.M 0 }
x a
x a
(iii) lim
xa
f(x) L 0 L , provided m 0 , g(x) M M 0
(L)
(iv) lim (f(x))g(x) (L)M
M
xa
0o , o , 1
(v) lim k f(x) k lim f(x) ; where k is a constant. x a
x a
(vi) lim f(g(x)) f lim g(x) f(M) ; provided f is continuous at lim g(x) M. x a
x a
xa
(vii) lim log{f(x)} {lim f(x)} x a
(viii) lim
xa
x a
d d (f(x)) lim f(x) dx dx x a
1 0 x a f(x)
(ix) If lim f(x) or , then lim x a
(4) Standard Limits : sin x tan x 1 , lim 1 x 0 x0 x x
(a) If 'x' is measured in radians , then lim
xn an n(a)n 1 x a x a
cos x 0 x x
(c) lim
sin1 x 1 x 0 x
(e) lim
ex 1 1 x 0 x
(g) lim
loge (1 x) 1 x 0 x
(i) lim
(b) lim
tan1 x 1 x 0 x
(d) lim
ax 1 loge a x 0 x
(f) lim
loga(1 x) loga e x 0 x
(h) lim
(j) If m , n N and a0 , b0 0 , then a xn a xn 1 a xn 2 ....... an lim 0 m 1 m 1 2 m 2 x b x b1x b2 x .... bm 0
x (k) lim a x
1
; a1
0
;  a  1 ; a1
oscillates finitely
a0 ;mn b0 0
;nm
ma r a h S . 82 K . 6 L 7 (l) lim(1 Er. x)810e27 5058 9 801 839 1/x
; a 1
oscillates infinitely ; a 1
Mathematics Concept Note IITJEE/ISI/CMI
;nm
x 0
Page 77
x
1 (m) lim 1 e x x
lim g(x).loge (f(x))
g(x) ex a (n) lim(f(x)) x a
lim(f(x) 1)g(x)
g(x) 1 for x a , then lim(f(x))g(x) ex a (o) If (f(x)) xa
.
(5) Limits Using Expansion Let f(n) be a function which is differentiable for all orders throughout some interval containing location x = 0 as an interior point , then Taylor's series generated by f(x) at x = 0 ( i.e. Maclaurin's series ) is given by: f(x) f(0)
f '(0) f "(0) 2 f '''(0) 3 x x x ................ 1! 2! 3!
For example: If f(x) = ex , then ex = 1 +
x x2 x3 + + + ...... 1! 2! 3!
In solving limits sometimes standard series are more convenient than standard methods and hence following series should be remembered. (i) ex = 1 +
x x2 x3 + + + ...... 1! 2! 3!
(ii) ax = 1 +
x lna x2 ln2 a x3 ln3 a + + + ....... 1! 2! 3!
(iii) ln (1+x) = x–
(iv) sin x = x –
x2 x3 x4 + + ...... 2 3 4
x3 x5 x7 + + + ... 3! 5! 7!
(vi) tan x = x +
(–1 < x < 1)
(v) cos x = 1 –
x3 2x5 + + ..... 3 15
12 3 12.32 5 12.32.52 7 x + x + x + ...... 3! 5! 7!
(ix) sec–1 x = 1 +
x2 5x4 61x6 + + .......... 2! 4! 6! n(n1) 2!
x2 +
x2 x4 x6 + – + ... 2! 4! 6!
(vii) tan–1 x = x +
(viii) sin–1 x = x +
(x) (1 + x)n = 1 + nx +
(a > 0 , a 1)
x3 x5 x7 + – +...... 3 5 7
n(n1)(n2) 3 x + ................. ( x < 1 , n R ) 3!
ma r a h Let f(x) and g(x) are differentiable function of x at location x = a such that: S . 82 K . 6 L 7 lim f(x) lim g(x) 0 or lim f(x) lim g(x) Er. 81027 5058 Now , according to L' Hospital Rule 9 801 f(x) f '(x) 839 lim lim
(6) L' Hospital's Rule :
x a
x a
x a
g(x)
x a
x a
x a
g'(x)
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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Note: f '(x) 0 or and f '(x) , g'(x) satisfy the condiassumes the indeterminate form g'(x) 0 t ion s for L' Hospit al ru l e t h en appl i cat i on t he rul e can be repeated
If xlim a
f(x) f '(x) f "(x) lim lim . x a g(x) x a g'(x) x a g"(x) lim
(7) Sandwich Theorem (Squeeze Play Theorem): Let f(x) , g(x) and h(x) be functions of x such that g(x) f(x) h(x) x (a h , a h), g(x) lim f(x) lim h(x). where 'h' is infinitely small positive number , then xlim a x a x a g(x) lim h(x) L lim f(x) L . Now , if xlim a x a x a
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Mathematics Concept Note IITJEE/ISI/CMI
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12. CONTINUITY AND DIFFERENTIABILITY (1) Continuity of a function : A function f(x) is said to be continuous at x = c , if : lim f(x) lim f(x) f(c)
x c
x c
Geometrically is not if a function f(x) is continous at x = c the graph of f(x) at the corresponding point (c , f(c)) will not be broken , but if f(x) is discontinuous at x = c the graph have break at the corresponding point. A function f(x) can be discontinuous at x = c because of any of the following three reasons: (i)
lim f(x) does not exist ( i.e. lim f(x) lim f(x) ) x c x c
x c
(ii) f(x) is not defined at x = c f(x) f(c) (iii) xlim c Following graph illustrates the different cases of discontinuity of f(x) at x = c.
(2) Types of Discontinuity : (a) Discontinuity of First Kind: Let f(x) be a function for which left hand limit and right hand limit at location x = a are finitely existing. Now if f(x) is discontinous at x = a is termed as discontinuity of first kind which may be removable or nonremovable. Removable Discontinuity of 1st kind:
ma r a h S . 8a 2 K . 6 If case lim f(x) exists but is not equal to f(a) then the function is said to have L 7 27 0such 58 that Er. can81be0redefined removable discontinuity at x = a. In this case function 5 1 9 lim f(x) f(a) and hence function can become continous at x = a. 80 9 83 x a
x a
Removable type of discontinuity of 1st kind can be further classified as : Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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Missing Point Discontinuity : f(x) exists finitely but f(a) is not defined. Where xlim a
For example: f(x)
x2 4 has a missing point discontinuity at x = 2. x2
sin x has a missing point discontinuity at x = 0. x Isolated Point Discontinuity : f(x)
f(x) exists and f(a) also exists but lim f(x) f(a) . At locations x = a where xlim a x a x2 9 ; x3 For example: f(x) x 3 has isolated point discontinuity at x = 3 10 ; x3 0 ; xI f(x) [x] [x] is discontinous at integral x. 1 ; x I
NonRemovable discontinuity of Ist Kind: If LHL and RHL exists for function f(x) at x = a but LHL RHL , then it is not possible to make the function continous by redefining it and this discontinous nature is termed as nonremovable discontinuity of Ist kind. For example: (i) f(x) = sgn (x) at x = 0 (ii) f(x) = x – [x] at all integral x.
Note: In case of nonremovable discontinuity of the first kind the nonnegative difference between the value of the RHL at x = a and LHL at x = a is called the jump of discontinuity. Jump of discontinuity = RHL–LHL A function having a finite number of jumps in a given interval is called a Piece Wise Continuous or Sectionally Continuous function in this interval. For example: f(x) = {x} , f(x) = [x]. (b) Discontinuity of Second Kind: Let f(x) be a function for which atleast one of the LHL or RHL is nonexistent or infinite at location x = a , then nature of discontinuity at x = a is termed as discontinuity of second kind and this is nonremovable discontinuity. Infinite discontinuity: At location x = a , either LHL or RHL or both approaches to or – . For example: f(x)
1 1 or f(x) at x 3 . x3 (x 3)2
Oscillatory discontinuity: At location x = a , function oscillates rigorously and can not have a finite limiting
ma r a 1 h S For example: f(x) sin at x 0 . 82 K . 6 x L 7 Er. 81027 5058 Note: 1 kind. 9 8of0second Point functions which are defined at single point only are discontinous 9 83 For example: f(x) x 4 x 4 at x 4. Mathematics Concept Note IITJEE/ISI/CMI
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(3) Continuity in an Interval : (a) A function f(x) is said to be continuous in open interval (a,b) if f(x) is continuous at each and every interior point of (a,b). For example:
f(x) = tanx is continous in , 2 2
(b) A function f(x) is said to be continuous in a closed interval [a,b] if :
f(x) is continuous in the open interval (a,b) and f(x) is left continuous at x = a lim f(x) f(a) a finite quantity. y. x a
f(x) is right continuous at x = b lim f(x) f(b) a finite quantity. y. x b
(4) Continuity of Composition of Functions: (a) If f and g are two functions which are continuous at x = c then the functions defined by G(x) = f(x) g(x) ; H(x) = Kf(x) ; F(x) = f(x).g(x) are also continuous at x = c , where K is f(x) any real number. If g(c) is not zero , then Q(x) = g(x) is also continuous at x = c.
(b) If f(x) is continuous and g(x) is discontinuous at x = a then the product function (x) = f(x).g(x) may be continuous but sum or difference function (x) = f(x) g(x) will necessarily be discontinuous at x = a.
sin x 0 For example: f(x) = x and g(x) = x . 0 x 0 (c) If f(x) and g(x) both are discontinuous at x = a then the product function (x) = f(x).g(x) is not necessarily be discontinuous at x= a. For example:
1 x 0 . f(x) = g(x) = 1 x 0
(d) If f is continuous at x = c and g is continuous at x = f(c) then the composite g(f(x)) is continuous at x = c. For example:
(gof) (x) =
(5)
f(x) =
x sin x
x sinx x2 2
and g(x) = x are continuous at x = 0 , hence the composite
will also be continuous at x = 0.
ma r a h S Differentiability of a function at a point : . 82 K . 6 L 7 7 58 Er. 81+0h))2lies 0 Let y = f(x) be a function and points P (a , f(a)) and Q (a + h , f(a on the curve of 5 9 801 f(x). 839 f(a h) f(a) x2 2
Now , slope of secant passing through P and Q = Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
h
.
Page 82
If positive scalar h tends to zero , then point Q approaches to point P from right hand side and the secant becomes tangent to curve f(x). Slope of tan gent at P lim
h0
f(a h) f(a) f '(a ) h
Similar to the above discussion , if P(a , f(a)) and R(a – h , f(a–h)) are two points on the f(a h) f(a) . h Now , if positive scales h tends to zero , then point R approaches to point P from left hand side and the secant becomes tangent to curve f(x).
curve y = f(x) , then slope of secant through P and R =
Slope of tangent at P lim
h 0
f(a h) f(a) f '(a ) h
Conditions for differentiability of function at location x = a: (i) Necessary condition: function mast be continuous at x = a. lim f(x) lim f(x) f(a) x a
...(1)
x a
(ii) Sufficient condition: function must have finitely equal left hand and right hand derivative. (i.e. f ' (a–) = f ' (a+))
lim
h0
f(a h) f(a) f(a h) f(a) = some finite quantity.. lim h 0 a h
Differentiability of a function at x = a be confirmed if it satisfy both the conditions (i.e. necessary condition and sufficient condition).
ma r a h S . If function is differentiable at a location than it is always continuous at that location K . are7continous 682butbut L 7 the converse is not always true. Hence all differentiable .function all Er 8102 5058 continous functions are not differentiable. 9 at8x0= 10 For example: f(x) = e is continous at x = 0 but not differentiable If f(x) is differentiable at location x = c, then alternative formula 839for calculating differenNote:
–x
tiation is given by : f '(c) lim
x c
Mathematics Concept Note IITJEE/ISI/CMI
f(x) f(c) xc
Page 83
(6) Geometrical Interpretation of Differentiation: Geometrically a function is differentiable at x = a if there exists a unique tangent of finite slope at location x = a. Nondifferentiability of a function y = f(x) at location x = a can be visualized geometrically in the following cases: (i) Discontinuity in the graph of f(x) at location x = a.
(ii) a corner or sharp change in the curvature of graph at x = a , where the left hand and right derivatives differ.
(iii) A cusp , where the slope of tangent at x = a approaches from one side and – from the other side.
(iv) Location of vertical tangent at x = a , where the slope of tangent approaches from both side or – from both sides.
ma r a Note: h S 2 . 8corner K . 6 If the graph of a function is smooth curve or gradual curve without any sharp (or L 7 Er. 81027 5058 kink) then also it may be nondifferentiable at some location. 9 sharp8corner For example: f(x) = x represents a gradual curve without any 01 but it is non9 differentiable at x = 0. 83 If a function is nondifferentiable at some location then also it may have a unique 1/3
tangent. Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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For example: y = sgn (x) is discontinous at x = 0 and hence nondifferentiable but have a unique vertical tangent at x = 0. Geometrically tangent is defined as a line which joins two closed points on the curve.
(7) Differentiable Over an Interval : f(x) is said to be differentiable over an open interval if it is differentiable at each and every point of the interval and f(x) is said to be differentiable over a closed interval [a,b] if f(x) is continuous in [a , b] and : (i) for the boundary points f'(a+) and f'(b–) exist finitely , and (ii) for any point c such that a < c < b , f'(c+) and f '(c–) exist finitely and are equal.
(8) Differentiability of Composition of Functions (i) If f(x) and g(x) are differentiable at x = a then the functions f(x)g(x)) , f(x).g(x) will also be differentiable at x = a and if g(a) 0 then the function f(x)/g(x) will also be differentiable at x = a. (ii) If f(x) is not differentiable at x = a and g(x) is differentiable at x = a, then the product function f(x).g(x) may be differentiable at x = a For example: f(x) = x and g(x) = x2. (iii) If f(x) and g(x) both are not differentiable at x= a then the product function f(x).g(x) may be differentiable at x = a For example: f(x) = x and g(x) = x.
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839
(iv) If f(x) and g(x) both are nondifferentiable at x = a then the sum function f(x) g(x) may be a differentiable function. For example: f(x) = x & g(x) = x.
Mathematics Concept Note IITJEE/ISI/CMI
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13. DIFFERENTIAL COEFFICIENTS (1)
First Principle of Differentiation (abinitio Method): The derivative of a given function y = f(x) at a point x = a on its domain is represented by f '(a) or
dy f(a h) f(a) , provided the limit exists , alternatively the and f '(a) lim dx x a h0 h
derivative of y = f(x) at x = a is given by f '(a) lim f(x) f(a) , provided the limit exists. xa xa
If x and x + h belong to the domain of function y = f(x), then derivative of function is represented by f '(x) or
(2)
dy f(x h) f(x) and f '(x) lim , provided the limit exists. dx h0 h
Differentiation of Some elementary functions f(x)
f'(x)
1.
xn
nxn–1
2.
ax
ax
3.
nx
1 x
4.
logax
1 loga e x
5.
sin x
cos x
6.
cos x
sin x
7.
sec x
sec x tan x
8.
cosec x
cosec x cot x
9.
tan x
sec2x
10.
cot x
cosec2 x
11.
sin–1 x
12.
cos–1 x
13.
tan–1 x
14.
cot–1 x
15.
sec–1 x
16.
cosec–1 x
(x R, n R)
na
(a > 0 , a 1) (x 0)
1 1 x2
;
x < 1
;
x < 1
1 1 x2 1 1 x2 1 1 x2
; x R ; x R
1  x  x2 1
; x > 1
1
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
 x  x2 1
; x > 1
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Page 86
(3)
(4)
Basic Theorems of Differentiation: 1.
d (f(x) g(x))= f '(x) g '(x) dx
2.
d d (k(f(x)) = k f(x) dx dx
3.
d (f(x). g(x)) = f(x) g'(x) + g(x) f'(x) dx
4.
d dx
f(x) = g(x)
g(x)f '(x)f(x)g'(x) g2 (x)
Different methods of differentiation :
(4.1) Differentiation of function of a function: If f(x) and g(x) are differentiable functions , then fog(x) is also differentiable and d (f(g(x))) f '(g(x)).g'(x) is known as chain rule can be extended for two on move funcdx tions.
For example :
dy dy dz du . . . dx dz du dx
Note:
dy dy dx dy 1 1 . . dy dx dy dx dx dy
d2 y dx2
1 d2 x 2 dy
If y f(x) and x g(y) are inverse functions of each other , then g(f(x)) = x and
g'(f(x)).f '(x) 1 g'(f(x))
1 g'(x)
g'(y).f '(x) 1 . (4.2) Implicit differentiation : If f(x,y) = 0 is an implicit function , then
ma r a h S . 82 K . 6 L 7 r. 1027 058 E to x and y respectively. 98 8015 Note: 39 8 Partial differential coefficient of f(x , y) with respect to x means the ordinary differential
f x dy f f and , where are partial differential coefficients of f(x,y) with respect x x dx f y
coefficient of f(x , y) with respect to x keeping y constant. Mathematics Concept Note IITJEE/ISI/CMI
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(4.3) Logarithmic Differentiation : f1 (x).f2 (x)....... If function is of the form y = (f(x))g(x) or y g (x).g (x)...... , where gi (x) 0 and fi(x) , 1 2
gi(x) are differentiable functions i {1 , 2 , 3 ,.....,n}, then differentiations after taking log of LHS and RHS is convenient and this method of differentiation is termed as logarithmic differentiation. y (f(x))g(x) loge y g(x).loge f(x) y eg(x).loge f(x) f '(x) dy eg(x).loge f(x) .g(x) g'(x).loge f(x) dx f(x)
(4.4) Parametric Differentiation : dy dg If x = f( ) and y = g( ) , then dx df
dy dy g'() d . dx f '() dx d
Note: d2 y 2
dx
g "() . f "()
(4.5) Derivative of one function with respect to another : Let u = f(x) and v = g(x) be two functions of x , then derivative of f(x) w.r.t. g(x) is given by: du dx df(x) f '(x) dv dg(x) g'(x) dx
(4.6) Differentiation of infinite series: If y is given in the form of infinite series of x and we have to find out
dy then we remove dx
one or more terms , it does not affect the series (i) If y
2y
f(x) f(x) f(x) ...... , then y
dy dy dy f '(x) f '(x) , dx dx dx 2y 1 f(x)f(x)....
f(x) (ii) If y f(x)
then y f(x)y
1 dy y.f '(x) dy y2 f '(x) log f(x), y dx f(x) dx f(x)[1 y log f(x)]
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839
f(x) y y2 f(x) y
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1
(iii) If y f(x) f(x)
1 f(x) ....
then
dy yf '(x) dx 2y f(x)
(4.7) Derivative Of Inverse Trigonometric Functions : Inverse trigonometric functions can be differentiation directly by use of standard results and the chain rule , but is is always easier by use of the proper substitution however the conditions of substitution must be applied carefully. For example: (4.8) Differentiation using substitution : In case of differentiation of irrational function , method of substitution makes the calculations simpler and there the following substitution must be remember. Some suitable substitutions S. N. Functions Substitutions (i)
a2 x2
x 0 sin or a cos
(ii)
x2 a2
x a tan or a cot
(iii)
x2 a2
x a sec or a cosec
(iv)
a x a x
x a cos 2
a2 x2
(v)
a2 x2
x2 a2 cos 2
(vi)
ax x2
x a sin2
(vii)
x a x
x a tan2
(viii)
x ax
x a sin2
(ix)
(x a)(x b)
x a sec2 b tan2
(x)
(x a)(b x)
x a cos2 b sin2
(4.9) Differentiation of determinant : f(x) g(x) h(x) If F(x) = l(x) m(x) n(x) , where f, g, h, l, m, n, u, v, w are differentiable functions of u(x) v(x) w(x)
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 (4.10) Differentiation of definite integral : 839 at point x (a,b) If g (x) and g (x) both functions are defined on [a , b] and differentiable f'(x) g'(x) h'(x) f(x) g(x) h(x) f(x) g(x) h(x) x then F'(x) = l(x) m(x) n(x) + l'(x) m'(x) n'(x) + l(x) m(x) n(x) u'(x) v'(x) w'(x) u(x) v(x) w(x) u(x) v(x) w(x)
1
2
and f(t) is continuous for g1 (a) f(t) g2 (b) Mathematics Concept Note IITJEE/ISI/CMI
Page 89
Then
(5)
d dx
g2
g1
f(t) dt f[g2 (x)]g'2 (x) f[g1 (x)]g'1 (x) f[g2 (x)]
d d g2 (x) f[g1 (x)] g1 (x). dx dx
Derivatives of Higher Order: Let a function y = f(x) be defined on an open interval (a,b). It's derivative, if it exists on (a,b) is a certain function f'(x) [or (dy/dx) or y'] and is called the first derivative of y w.r.t. x. If it happens that the first derivative has a derivative on (a,b) then this derivative is
d2y called the second derivative of y w.r.t. x & is denoted by f''(x) or 2 or y''. dx Similarly, the 3rd order derivative of y w.r.t. x, if it exists, is defined by
d3y dx3
d = dx
d2y dx2 .
It is also denoted by f'''(x) or y'''.
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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14. TANGENT AND NORMAL (1)
Equation of Tangent and Normal : Let y = f(x) be the equation of a curve and poit P(x1 , y1) lies on the curve. For the curve y = f(x) , geometrically f '(x1) represents the slope to tongent to curve at location x = x1, hence the equations of tangent can be obtained by point slope form of line.
y y1 f '(x1 ) is the equation of tangent at point P(x , y ) to curve y = f(x). 1 1 x x1
1 Slope of noraml to cuver y = f(x) at point P(x1 , y1) is given by f '(x ) 1
y y1 1 is equation of normal at point P(x1 , y1) to curve y = f(x). x x1 f '(x1 )
Note: If slope of tangent is zero , then tangent is termed as 'horizontal tangent' and if slope of targent approaches infinite , then tangent is termed as 'vertical tangent'.
(2)
Length of Tangent and Normal : Let P(x1 , y1) be any point on curve y = f(x). If tangent drawn at point P meets xaxis at T and normal at point P meets xaxis at N , then the length PT is called the length of tangent and PN is called length of normal. Projection of segment PT on xaxis (i.e. QT) is called the subtangent and similarly projection of line segment PN on xaxis (i.e. QN) is called subnormal. (refer figure (1)).
dy Let m slope of tangent at P(x1 , y1) to curve y = f(x). dx P
dy m tan dx P
In PTQ , sin
y1 PT
PT  y1cosec  2
dx Length of tangent PT y1 1 dy P
Mathematics Concept Note IITJEE/ISI/CMI
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Page 91
In PQN , sin(90 )
y1 PN  y1sec  PN 2
dy Length of normal PN y1 1 dx P
y1 QT  y1 cot  QT y1 Length of subtangent dy dx p
In PTQ , tan
In PQN , tan(90 )
y1 QN  y1 tan  QN
dy Length of subnormal y1 dx P
(3)
Angle between the curves : Angle between two intersecting curves is defined as the angle between their tangents or the normals at the point of intersection of two curves.
m1 m2
tan = 1 m m , where is acute angle of intersection and ( ) is obtuse angle of 1 2 intersection.
Note: The curves must intersect for the angle between them to be defined. If the curves intersect at more than one point then angle between the curves is mentioned with references to the point of intersection. Two curves are said to be orthogonal if angle between them at each point of intersection is right angle (i.e. m1m2 = –1).
(4)
Shortest distance between two curves : If shortest distance is defined between two nonintersecting curves , then it lies along the common normal.
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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15. ROLLE'S THEOREM & MEAN VALUE THEOREM (1)
Rolle's Theorem: If a function f(x) is (i) continuous in a closed interval [a, b] (ii) derivable in the open interval (a, b) (iii) f(a) = f(b), then there exists at least one real number c is in (a, b) such that f '(c) = 0
(2)
Langrange's Mean Value Theorem: If a function f(x) is (i) continuous in the closed interval [a,b] and (ii) derivable in the open interval (a, b), then there exists at least one real number c in (a,b) such that f '(c)
f(b) f(a) . ba
Geometrical Illustration:
ma r a h S . 82 K . 6 L 7 f(b) f(a) The slope of chord AB is and that of the tangent point0 P2 is7 f'(c). These 58being Er.of 8 ba 1 0 5 9 the8tangent 01 at which is equal, by (i), it follows that there exists a point P on the curve, 9 parallel to the chord AB. 83
Let P be a point on the curve f(x) such that
Mathematics Concept Note IITJEE/ISI/CMI
f(b) f(a) f '(c) ba
...(i)
Page 93
There may exist more than one point between A and B, the tangents at which are parallel to the chord AB. Lagrange's mean value theorem ensures the existence of at least one real number c in (a,b) such that f '(c)
(3)
f(b) f(a) ba
Alternative Form of Langrange's Theorem: If a function f(x) is (i) continuous is a closed interval [a, a + h] (ii) differentiable in a open interval (a, a + h), then their exists at least one number lying between 0 and 1 such that f '(a h)
f(a h) f(a) h
i.e. f(a+h) = f(a) + hf' (a h)
(4)
(0 1)
Physical Illustration: [f(b) f(a)] is (b a) the average rate of change of the function over the interval [a,b]. Also f'(c) is the actual rate of change of the function for x = c. Thus, the theorem states that the average rate of change of a function over an interval is also the actual rate of change of the function at some point of the interval. In particular, for instance, the average velocity of a particle over an interval of time is equal to the velocity at some instant belonging to the interval. i.e. [f(b)–f(a)] is the change in the function f as x changes from a to b so that
their exists a tiem (t1 , t2 ) for which x '()
x(t2 ) x(t1 ) i.e. instantaneous velocity = t2 t1
average velocity.
(5)
ma r a h S . Intermediate Value Theorem (IVT) 82 K . 6 L 7 r. 1027 058 E If a function f is continuous in a closed interval [a,b] then 15at least one 98there8exists 0 f(a) f(b) 39 8 c (a,b) such that f(c) for all R . 1
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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16. MONOTONOCITY Basic Definitions (i) Strictly increasing function: A function f(x) is said to be strictly increasing in an interval 'D', if for any two locations x = x1 and x = x2, where x1 > x2 f(x1) > f(x2) x1 , x2 D . If function f(x) is differentiable in interval 'D', then for strictly increasing function f'(x) > 0. for example: f(x) = ex x R (ii) Strictly decreasing function: A function f(x) is said to be strictly decreasing in an interval 'D', if for any two locatious x = x1 and x = x2, where x1 > x2 f(x1) < f(x2) x1 , x2 D . If function f(x) is differentiable in interval 'D', then for strictly decreasing function f'(x) < 0. for example: f(x) = e–x x R
Note: For strictly increasing or strictly decreasing functions, f'(x) 0. For any point location or sub interval
A.
Monotonocity about a point
1.
A function f(x) is called an increasing function at point x = a. If in a sufficiently small neighbourhood around x = a. f(a h) < f(a) < f(a + h)
f(a+h) f(a)
f(a+h)
f(a)
f(a–h) f(a–h)
a
a–h
2.
a+h
a
A function f(x) is called a decreasing function at point x = a if in a sufficiently small neighbourhood around x = a. f(a h) > f(a) > f(a + h)
f(a–h) f(a)
f(a–h)
f(a+h) f(a)
f(a+h)
ma r a h S . 82 K . Note: 6 L 7 r.inequality 2to7test monotonocity If x = a is a boundary point then use the appropriate oneE sided 58 0 1 0 5 8 of f(x). 9 801 839 a–h
a
a+h
Mathematics Concept Note IITJEE/ISI/CMI
x=a
Page 95
f(a)
f(a+h)
f(a–h) f(a)
x=a f(a) > f(a–h) increasing at x = a 3.
x=a f(a+h) < f(a) decreasing at x = a
Test for increasing and decreasing functions at a point (i)
If f'(a) > 0 then f(x) is increasing at x = a.
(ii)
If f'(a) < 0 then f(x) is decreasing at x = a.
(iii)
If f'(a) = 0 then examine the sign of f'(a+) and f'(a ) (a)
If f'(a+) > 0 and f'(a ) > 0 then increasing
(b)
If f'(a+) < 0 and f'(a ) < 0 then decreasing
(c)
otherwise neither increasing nor decreasing.
B.
Monotonocity over an interval
1.
A function f(x) is said to be monotonically increasing for all such interval (a, b) where f'(x) 0 and equality may hold only for discreet values of x. i.e. f'(x) does not identically become zero for x (a,b) or any sub interval.
2.
f(x) is said to be monotonically decreasing for all such interval (a,b) where f'(x) 0 and equality may hold only for discrete values of x.{By discrete points, it mean that points where f'(x) = 0 don't form an interval}
Note:
C.
(i)
A function is said to be monotonic if it's either increasing or decreasing.
(ii)
The points for which f'(x) is equal to zero or doesn't exist are called critical points. Here it should also be noted that critical points are the interior points of an interval.
(iii)
The stationary points are the points where f'(x) = 0 in the domain.
Classification of functions Depending on the monotonic behaviour, functions can be classified into following cases. 1. 3.
Increasing functions Decreasing functions
2. 4.
Non decreasing functions Nonincreasing functions
This classification is not complete and there may be function which cannot be classified into any of the above cases for some interval (a, b).
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839
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12. MAXIMA & MINIMA (1)
Basic definitions: (i) Stationary points: For a given function f(x), stationary points are those locations of x (let it be 'x0') for which f'(x0) = 0, and functioning value f(x0) is termed as the stationary value of function. (ii) Critical points: For a given function f(x), critical points are those locations of x(let it be 'x') in the domain of f(x) for which f'(x0) = 0 or f'(x0) doesn't exist.
Note: At critical points for a function, local maxima, local minima
or point of inflection may
exist (iii) Points of extremum: critical points of a function at which either maxima or minima exists is termed as points of extremum. (iv) Point of inflection: critical locations of function at which neither maxima nor minima exists is termed as point of in flection, it is also known as point of no extremum. (v) Local maxima/regional maxima: A function f(x) is said to have a local maximum at critical point x= a if f(a) f(x) x (a h,a + h) , where h is a very small positive arbitrary number. If x = a is the point of local maxima, then f(a) lim f(x) and f(a) lim f(x) . xa
xa
following figure illustrates the point of local maxima in different situations:
(vi) Local minima/regional minima: A function f(x) is said to have local minimum at critical point x = a if f(a) f(x) x (a h, a + h), where h is a very small positive arbitrary number. If x = a is the point of local minima, if then f(a) lim f(x) and f(a) lim f(x) . xa
xa
following figure illustrates the point of local minima in different situations:
ma r a h S Note: . 82 K . 6 L 7 7 in the (i) local maxima of a function at x = a is the largest value function 8nearby 5 Er.of 8 02only 1 0 neighbourhood of critical point x = a. 9 8015 (ii) local minima of a function at x = a is the smallest value of function 39 only in the nearby 8 neighbourhood of critical point x = a. Mathematics Concept Note IITJEE/ISI/CMI
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(iii) If a function f(x) is defined on [a,b], then f(x) is having point of extremum at the boundary points. (iv)for function f(x), if local maxima on local minima doesn't exist at critical point x = a, then point of inflection exists. Following figure illustrates the point of inflection or point of no extremum
(2)
Test for Maxima/Minima 1. Ist derivative Test First derivative test is applicable only in those function for which the critical points are the points of continuity. (i) If f '(x) changes sign from negative to positive while passing through x = a from left to right then x = a is a point of local minima. (ii) If f '(x) changes sign from positive to negative while passing through x = a from left to right then x = a is a point of local maxima. (iii) If f '(x) does not changes its sign about x = a then x = a is neither a point of maxima nor minima. (i.e. x=a is point of inflection) following figure illustrates the Ist derivative test
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
Page 98
2. IInd derivative Test If f(x) is continuous function in the neighborhood of x = 0 such that f'(x) = 0 and f''(a) exists then we can predict maxima or minima at x = 0 by examining the sign of f''(a) (i) If f''(a) > 0 then x = a is a point of local minima. (ii) If f''(a) < 0 then x = a is a point of local maxima. (iii) If f''(a) = 0 then second derivative test does not give any use conclusive results. 3. nth derivative test Let f(x) be function such that f'(a) = f''(a) = f''(a) = ............. = f n1 (a) = 0 and n th f n (a) = 0, where f denotes the n derivative, then
(i) If n is even and f n (a) < 0, there is a local maximum at a, white if f n (a) > 0, there is a local minimum at a. (ii) If n is odd, there is no extremum at the point a.
(3)
Maxima and Minima of Parametrically defined Functions Let a function y = f(x) be defined parametrically as y = g(t) and x = h(t) where g and h both are twice differentiable functions for a certain interval of t and h'(t) 0. Now,
dy dy / dt g'(t) 0 g'(t) 0 dx dx / dt h'(t)
Let one of the root of g'(t) = 0 is say t0. d2 y d g'(t) d g'(t) 1 h'(t) g'(t)h "(t) . Now 2 3 dx g'(t) dt h'(t) dx / dt dx {h'(t)}
d2 y g "(t) because g'(t) = 0. Here {h'(t)}2 > 0 , 2 dx {h '(t)}2
Thus sign of
d2 y is same as that of the sign of g"(t) dx2
Now, if g "(t) t t0 0, f(x) has a maxm. at x = x0 = h(t0) if g "(t) t t0 0, f(x) has a minm. at x = x0 = h(t0) and so on.
(4)
Global Maxima/Minima (or Absolute Maxima/Minima) Global maximum or minimum value of f(x) x [a, b] refers to the greatest value and least value of f(x), mathematically (i) If f(c) f(x) for x [a, b] then f(c) is called global maximum or absolute maximum value of f(x).
ma r a h S . 82 K . 6 L 7 Note: Er. 81027 5058 For function f(s) defined on [a,b], is c , c , c , ...... c are 9 the locations 01of critical point, 8 9 then 83 Global maximum value = max{f(a), f(c ), f(c ), ...... f(c ), f(b)}
(ii) Similarly if f(d) f(x) x [a,b] then f(d) is called global minimum or absolute minimum value.
1
1
2
3
2
n
n
Global minimum value = min{f(a), f(c1), f(c2), ...... f(cn), f(b)} Mathematics Concept Note IITJEE/ISI/CMI
Page 99
(5)
(6)
Useful Formulae of Mensuration to Remember : 1.
Volume of a cuboid = bh.
2.
Surface area of cuboid = 2( b + bh + h ).
3.
Volume of cube = a3
4.
Surface area of cube = 6a2
5.
Volume of a cone =
6.
Curved surface area of cone = r ( = slant height)
7.
Curved surface of a cylinder = 2 rh.
8.
Total surface of a cylinder = 2 rh + 2 r2.
9.
Volume of a sphere =
10.
Surface area of a sphere = 4 r2.
11.
Area of a circular sector =
12.
Volume of a prism = (area of the base) (height).
13.
Lateral surface of a prism = (perimeter of the base) (height)
14.
Total surface of a prism = (lateral surface) + 2(area of the base)
15.
Volume of a pyramid =
16.
Curved surface of a pyramid =
1 r2h. 3
4 r3. 3
1 2 r , when is in radians. 2
1 (area of the base) (height). 3 1 (perimeter of the base) (slant height) 2
Curvature of function For continuous function f(x), If f''(x0) = 0 or doesnot exist at points where f'(x0) exists and if f''(x) changes sign when passing through x = x0 , then x0 is called a point of inflection . At the point of inflection the curve changes its concavity i.e. (i) If f''(x) < 0, x (a,b) then the curve y = f(x) is convex in (a,b)
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839
(ii) If f''(x) > 0, x (a,b) then the curve y = f(x) is concave in (a,b).
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
Page 100
18. INDEFINITE INTEGRAL If f and g are functions of x such that g'(x) = f(x), then
f(x)dx = g(x) + c
d dx {g(x) + c} = f(x), where c is called the constant of integration.
Note: (i) Integral of a function is also termed as primitive or antiderivative (ii) The indefinite integral of a function geometrically represents a family of curves having parallel tangents at their points of intersection with the line orthogonal to the axis of variable of integration.
(1)
Theorems on Integration (i)
c
(ii)
(f(x) g(x))dx
(iii)
d dx
(iv) (v)
(2)
f(x).d x = c f(x ).d x
f(x)dx g(x)dx
=
f(x)dx f(x)
d (f(x)).dx f(x) c . dx
f(x)dx g(x)+c
f(ax+b)dx
g(ax+b) +c a
Standard Formula :
2
xn1
(i)
x dx n 1 C , n 1
(ii)
x dx ln x + C
(iii)
e dx e
1
x
x
C
lna + C
x (iv) a dx ax/
( a x b ) n 1 + c, n a ( n 1)
(v) (ax+b)n dx = (vi)
dx 1 = a ax+b
(vii) eax+bdx =
ln (ax + b) + c 1 ax + b e +c a
1
(viii) apx+qdx = p (ix)
1
sin(ax+b)dx
apx q px+q a / na
lna + c; a > 0
=
1
a cos (ax + b) + c
Mathematics Concept Note IITJEE/ISI/CMI
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Page 101
(x)
1
cos(ax+b)dx = a
sin (ax + b) + c
(xi) tan(ax+b)dx =
1 a
ln sec (ax + b) + c
(xii) cot(ax+b)dx =
1 a
ln sin(ax + b) + c 1 tan(ax + b) + c a
(xiii) sec2(ax+b)dx =
(xiv) cosec2 (ax+b)dx =
1 cot(ax + b) + c a 1 sec (ax + b) + c a
(xv) sec(ax+b).tan(ax+b)dx =
(xvi) cosec(ax+b).cot(ax+b)dx = (xvii) sec x dx =
(xviii) cosec x dx =
dx
(xx) (xxi)
dx 2
=
2
x x a
(xxiii) (xxiv)
dx 2
ln tan 2x +c or ln(cosec x + cot x) +c
dx a x2 dx 2
2
x a
1 x sec–1 +c a a
x2 a2 + C
=
ln  x
x2 a2
dx
2
x +c a
ln  x
x a
x2 a2
ln (sec x – tan x) + c
=
2
ln tan 4 2x  + c
1 x tan–1 +c a a
=
a2 x2
(xxii)
(xxv)
= sin–1
a2 x2
or
ln (cosec x cot x) + c
or dx
1 cosec (ax + b) + c a
ln (sec x + tan x) + c or –
(xix)
=
1 2a
ln aaxx
+c
=
1 2a
ln xx aa
+c
x (xxvi) a2 x2 dx = 2 x (xxvii) x2 a2 dx = 2
x (xxviii) x2 a2 dx = 2
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
+C
ma r a h S . 82 K . x x a 6 L 7 ln a + E c r. 27 058 0 1 98 8015 839 x x a
x a2 –1 a2 x2 + 2 sin a + c
a2 x2 a2 + 2
2
2
2
x a
a2
2
ln
2
a
2
2
+ c
Page 102
(xxix) eax.sin bx dx =
eax 2
a b
(xxx) eax.cos bx dx =
(3)
2
b cos bx) + c or
eax
b sin bx tan1 C a a b 2
2
eax
eax 2
(a sin bx
2
a b
(a cos bx + b sin bx) + c or
b cos bx tan1 C a a b 2
2
Integration by Subsitutions If x = (t) is substituted in a integral, then
everywhere x is replaced in terms of t. dx gets converted in terms of dt.
(t) should be able to take all possible value that x can take.
Note: (i)
[f(x)n f'(x) dx =
(ii)
(iii)
f '(x) dx f(x) f '(x) f(x)
(f(x))n+1 +C n+1
lnf(x) + C
dx 2 f(x) C
(iv) f(x).f '(x)dx 2 f(x) f(x) C 3
(4)
Integration by Part : If f(x) and g(x) are the first and second function respectively then
f(x).g(x)dx f(x). g(x)dx f '(x). g(x)dx dx
The choice of f(x) and g(x) is decided by ILATE rule. The function which comes later is taken as second function (integral function). I Inverse function L Logarithmic function A Algebraic function T Trigonometric function E Exponential function
Note:
(5)
(i)
ex [f(x) f '(x)]dx ex.f(x) c
(ii)
[f(x) xf '(x)]dx xf(x) c
ma r a h S . 82 K . 6 L 7 r. 1027 058 E f(x) 15function of a 98 algebraic g(x) defines a rational 0 8 839
Integration of Rational Algebraic Functions by using Partial Fractions : PARTIAL FRACTIONS : If f(x) and g(x) are two polynomials, then rational function of x.
Mathematics Concept Note IITJEE/ISI/CMI
Page 103
f(x) If degree of f(x) < degree of g(x), then g(x) is called a proper rational function. f(x) If degree of f(x) degree of g(x then g(x) is called an improper rational function
f(x) If g(x) is an improper rational function, is divided f(x), by g(x) so that the rational function f(x) (x) g(x) is expressed in the form (x) + (g)x where (x) and (x) are polynomials such that f(x) the degree of (x) is less than that of g(x). Thus, g(x) is expressible as the sum of a
polynomial and a proper rational function. f(x) Any proper rational function g(x) can be expressed as the sum of rational functions, each
having a simple factor of g(x). Each such fraction is called a partial fraction and the process f(x) of obtaining them is called the resolution or decomposition of g(x) into partial fractions. f(x) The resolution of g(x) into partial fractions depends mainly upon the nature of the factors
of g(x) as discussed below. CASE I When denominator is expressible as the product of nonrepeating linear factors. Let g(x) = (x a1) (x a2)..........(x an). Then, we assume that A1 A2 A2 f(x) = + +..... + x a x a x an g(x) 1 2
where A1, A2, ...... An are constants and can be determined by equating the numerator on R.H.S. to the numerator on L.H.S. and then substituting x = a1, a2, ......., an. (Refer section 14,15,16,17 in A List of Evaluation Techniques )
Note: In order to determine the value of constants in the numerator of the partial fraction corresponding to the nonrepeated linear factor px + q in the denominator of a rational expression, we may proceed as follows : Replace x =
q
p (obtained by putting px + q = 0) everywhere in the given rational expres
sion except in the factor px + q itself. For example, in the above illustration the value of A is 3x 2 obtained by replacing x by 1 in all factors of (x 1)(x 2)(x 3) except (x 1) i..e 312 5 A = (12)(13) = 2 Similarly, we have
B=
32 1 = (12)(2 3)
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
8 and, C =
33 2 11 = (3 1)(3 2) 2
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Page 104
A List of Evaluation Techniques S.NO. Form of Integrate
1.
m
x cosn x dx where
sin
where m, n N
Value/Evaluation Techique
If m is odd put cos x = t If n is odd put sin x = t If both m and n are odd, put sin x = t if m n and cos x = t otherwise. If both m and n are even, use power reducting formulae sin2 x
1 (1 cos 2x) 2
1 (1 cos 2x) 2 If m + n is a negative even integer , put tan x = t cos2 x
2.
a sin x b cos x c
p sin x q cos x r dx aex be x
pe
x
dx
qex
a tan x b
p tan x q dx pe
2x
q
Change to
dx
or
(p , q 0, at least one of a , b 0) 4.
sin x and constant term (or ex, e–x) write the integral as d (DEN) dx dx dx dx DEN DEN
of a, b 0)
ae2x b
d (DEN) + dx
Find , and by comparing the eoefficients of cos x,
(p, q, 0 and at least one
3.
Write Num = (DEN)
dx
a sin x b cos x
a sin x b cos x
p sin x q cos x dx
aex be x
be
x
qex
dx
and use (8) above
Use a = r cos , b = r sin to put the interal in the form
1 r
(Both a, b 0)
5.
dx Q
or
dx
or
Q
or
L
Q dx
or
Q or
L
Q dx
L
Q
where L px q, p 0
dx
sin(x ) . Now, use formula for cosec x dx
c 2 b Write Q a x x a a
ma r a h S b . 82 K t x . and put 6 L 7 2a Er. 81027 5058 9 801 839
and Q = ax2 + bx + c, a 0 Mathematics Concept Note IITJEE/ISI/CMI
Page 105
S.NO. Form of Integrate 6.
7.
L1 dx L2
Value/Evaluation Techique Write as L1 dx L1L2
where L1 = ax+b, a 0
L2 = px + q , p 0
to reduce to form
Q dx L
Put L
L dx Q
1 to reduce to form t
dx Q
where L = px + q, p 0 and Q=ax2 + bx + c, a 0
8.
Q Rewrite the integrand as L Q
Q dx L
where L = px + q, p 0 Q = ax2 + bx + c, a 0
Divide Q by L to obtain Q = L1(L) + where L1 is a linear expression in x and is a constant. The integral reduces to
9.
Q1
dx
Q2
Write Q1 Q2
L1 Q
dx
dx
L
Q
.
d [Q2 ] dx
Find , , and write the integral as
where
Q1 = ax2+bx+c , a 0
d dx Q2 dx [Q2 ]dx y Q
dx Q2
Q2 = px2+qx+r , p 0 10.
L
dx Put L2 = t2
L2
1
where L1 = ax + b and L2 = px + q, with a , p 0, a p.
11.
Q
dx
where Q1=ax2+b, Q2=px2+q with a, p 0 , p 0 12.
1 / p, 1 / q,
R(x
ma r a h x=1/t and simplify then put S . 82 K . 6 p + qt = u L 7 Er. 81027 5058 Let l = lcm (p, q, r, ...) 9 01 8 9 83 and put x = t First put
Q2
1
x
x1 / t ...)dx
where R is a rational function. Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
2
2
l
Page 106
S.NO. Form of Integrate 13.
P(x)
Value/Evaluation Techique
Q(x) dx
If deg P(x) deg Q(x) first isolate the quotient A(x)
where P(x) and Q(x) are
when P(x) is divided by Q(x). Now,
polynomials in x
P(x) B(x) A(x) Q(x) Q(x) where deg B(x) < deg Q(x) Now use techniques 22 to 29
14.
P(x)
Q(x) dx
Write
where deg P(x)<deg Q(x)
A1 A2 An P(x) ... Q(x) x 1 x 2 x n
and Q(x) is a product of distinct linear factors, that is Q(x)=A(x 1 )(x– 2 )...
Evaluate Ai ' s and use
dx
x a log  x a 
(x– r ) where ai ' s are distinct and A 0 P(x)
15.
dx
(x a)
n
Put x–a = t,
where n 1 and P(x) is polynomial in x.
express P(x) in terms of ts and then integrate.
P(x)dx
16.
(x a)
m
(x b)n
where m , n 1, a b , and deg P(x) < m + n.
Write the integrand as A1 A2 Am B1 B2 Bn ... ... 2 m 2 x a (x a) x b (x b) (x a) (x b)n
Evaluate Ai's and Bj' and the integral.
P(x)dx
17.
(x a)(x
2
bx c)
where b2 – 4c < 0 and deg P(x) < 3 18.
19.
x x
x2 1 4
2
kx 1
x2 1 4
kx2 1
dx
dx
Write the integrand as
A Bx C x a x2 bx c
a k S 2 . 8 K Now, put x–1/x = t L. 2776 8 . r E 810 505 Divide the numerator by9 x to obtain 01 8 9 83 m hax r 1 / x
Divide the numerator by x2 to obtain
2
2
2
(1 1 / x2 )dx
x Mathematics Concept Note IITJEE/ISI/CMI
(1 1 / x2 )
2
1 / x2 k
Now , put x + 1/x = t Page 107
S.NO. Form of Integrate 20.
xP(x2 )
Q(x ) dx
Value/Evaluation Techique Put x2 = t
2
where P and Q are polynomials in x 21.
22.
23.
24.
P(x2 )
Q(x ) dx
Put x2 = t
where P and Q are polynomials in x
for partial fraction not for integration
2
1
(x
2
a2 )n
dx An
Begin with An–1 A n 1
R(sin x , cos x)dx
Universal substitution
where R is a rational function Special Cases (a) If R(–sinx , cosx) =–R(sinx , cosx) (b) If R(sinx , – cosx) =–R(sinx , cosx) (c) If R(–sinx, –cosx) =R(sinx , cosx)
tan (x/2) = t.
m
x
1. (x
1
where n > 1
2
a2 )n 1
dx
Put cos x = t Put sin x = t Put tan x = t
(a bxn )p dx
where m, n, p are rational numbers, (a) p is a positive integer Expand (a + bxn)n by using binomial theorem (b) p is a negative integer, Put x = tk m
(c)
a c ,n , b d
where k = L CM (b,d)
a, b, c, d I, b > 0, d > 0
where k = L CM (b,d)
m1 + p is an integer n
Put a + bxn = ts where p = r/s, r, s I, s > 0
(d)
m 1 p is an integer n
Put a + bxn = xnts where p = r/s, r , s I, s > 0
25.
Pn (x) 2
dx
ax bx c
where Pn(x) is a polynomial of degree n in x
ma r a h P (x) S 2dx . dx Q (x) ax K bx c k 8 ax bx c . 6 . L 0277 ax58bx c r E where Q (x) is polynomial of in x and k 1degree1(n–1) 50w.r.t. 98both is a constant. Differentiate the sides x and 0 8 9 3 multiplying by ax bx 8 c to get the identiy
Write
n
n 1
2
2
2
n–1
2
Pn(x) = Q'n–1 (x)(ax2 + bx+c) + 1/2 Qn–1 (x) (2ax+b) + k Compare coefficients to obtain Qn–1(x) and k. Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
Page 108
19. DEFINITE INTEGRAL (1)
NewtonLeibnitz formula Let f(x) be a continuous function defined on [a,b] and
f(x)dx = F(x) + c, then
b
f(x)dx = F(b) F(a) is called definite integral and this formula is known as Newton
a
Leibnitz formula.
Note: (i) NewtonLeibnitz formula is only applicable iff integrand f(x) is continous in [a,b]. b
(ii) Geometrically definite integral
f(x)dx represents the algebraic sum of the area bounded a
by integrand f(x) with the xaxis, lines x = a and x = b, bounded area above the xaxis is treated as positive and bounded area below the xaxis is treated as negative.
b
f(x)dx A
1
A 2.
a
(2)
Method of Substitution in a Definite Integral b
If evaluation of definite integral
f(x)dx ,
if variable x is substituted as (t), where
a
() a and () b, then b
f(x)dx
a
f((t)). '(t)dt.
Note: Substitution x = (t) must satisfy the following conditions.
ma r a h continuous derivative '(t); S . 82 K . 6 L 7 7 the5limits with t varying on [, ] the values of the function xr.(t) do not 8 of E 8102leave 0 [a, b]; 9 8015 () a and () b . 839
(t) is a continuous singlevalued function defined in [, ] and has in this interval a
Mathematics Concept Note IITJEE/ISI/CMI
Page 109
(3)
Properties of Definite Integral b
Property:1
b
f(x)dx =
f(t)dt
a
a
definite integral is independent of variable of integration. b
Property:2
a
f(x)dx =
f(x)dx
b
c
a
Property:3
b
b
f(x)dx = f(x)dx + a
a
f(x)dx, c
where c may lie inside or outside the interval [a,b]. b
Property:4
b
f ( x )d x =
f(a b x)dx
a
a a
a
f(x)dx =
0
f(a x)dx
0
Note: b
(i) If I
f(x)dx , than by property –4, a
b
I
b
f(a b x)dx 2I {f(x) f(a b x)}dx a
a
b
(ii) If I
f(x)
f(x) f(a b x) dx, then I a
a
Property:5
a
f(x)dx =
a
a
2a
Property:6
0
(f(x) f(x))dx
0
a
ba . 2
f(x)dx
a 2 f(x)dx ; if f(x) f(x) ( i.e. f(x) is even function) 0 ; if f(x) f(x) (i.e. f(x) is odd function) 0
a
f(x)dx = (f(x) f(2a x))dx 0
a m r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 Note: 9 is symmetrical about (i) If function f satisfy the condition f(2a–x) = f(x), then graph 8of3f(x) 2a
0
a 2 f(x)dx ; if f(2a x) f(x) f(x)dx 0 ; if f(2a x) f(x) 0
the line x = a.
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
Page 110
(ii) If function f satisfy the condition f(2a–x) = –f(x), then graph of f(x) is symmetrical about point (a, 0) /2
(iii)
/2
loge (cos x)dx
0
0
Property:7 nT
(i)
If f(x) is a periodic function with period T and m, n I, then T
0
anT
T
f(x) dx = n f(x) dx, n z, a R
a
(iii)
0
nT
T
f(x) dx = (n m) f(x) dx ,m, n z
mT
0
a
anT
(iv)
f(x) dx =
z, a R
b
b nT
f(x) dx =
a nT
(4)
f(x) dx, n
0
nT
(v)
loge 1 / 2 2
f(x)dx = n f(x)dx, n z
0
(ii)
loge (sin x) dx
f(x)dx n z, a, b R a
Standard Results for Definite Integral b
(i) If f(x) 0 x [a,b] , then
f(x)dx 0 a
b
(ii) If f(x) 0 x [a,b] , then
f(x)dx 0 a
(iii) If (x) f(x) (x) for a x b, then
b
b
b
(x) dx
f(x) dx
a
a
(x) dx
a
b
(iv)If m f(x) M for a x b, then m(b a)
f(x) dx
M(b a)
a
Further if f(x) is monotonically decreasing in (a,b) then b
f(x) dx
ma r a b h S . 82 K f(a)(b a) < f(x) dx < f(b) (b a). . 6 L 7 a Er. 81027 5058 9 801 9 of equality holds if (v) If f(x) is continous in [a,b], then  f(x)  dx f(x)dx ; 8 the3sign f(b) (b a) <
< f(a)(b a) and if f(x) is monotonically increasing in (a,b) then
a
b
b
a
a
f(x) is nonnegative or nonpositive for all x [a,b] . Mathematics Concept Note IITJEE/ISI/CMI
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(vi)If f(x) is continous and increasing function in [a,b] and have concave graph, then b
(b a)f(a)
f(a) f(b) 2
f(x)dx (b a) a
(vii) If f(x) is continous and increasing function in [a,b] and have convex graph, then f(a) f(b) (b a) 2
b
f(x)dx (b a)f(b). a
(viii) If f(x) and g(x) are continous function for all x [a,b] and f2(x), g2(x) are integrable b
functions, then
b b f2 (x)dx g2 (x)dx a a
f(x). g(x) dx
a
b
(ix) If f(x) is continuous on the interval [a,b], then
f(x)dx f(c)(b a)
; where a < c < b.
a
(x) Leibnitz Rule: (x)
If F(x)
f(t)dt , then F'(x) = f(ψ(x)).ψ'(x)  f(φ(x)). φ'(x).
(x)
(5)
Definite Integral as a Limit of Sum. If f(x) is the continuous function in the interval [a,b] where a and b are finite and b > a and if the interval [a,b] be divided into n equal parts, each of width h so that n h = b – a , [h{f(a) f(a h) f(a 2h) ......... f(a (n 1)h}] is called the definite integral of Then hlim 0 f(x) between limits a and b, b
f(x)dx a
lt h[{f(a) f(a h) f(a 2h) ...... f(a (n 1)h)]
h0
n 1 lim h. f(a rh) h 0 r 0
If should be noted that as h 0 , 0 nh = b – a Putting a = 0 b = 1, so that h = 1/n. We get
1
0
f(x) dx lim
n
1 . n
n 1
r
f n
r 0
Working Rule
ma r a h S . 82 K . 6 L 7 . 27 058 1 r Er 0 lim . f 1 (ii) The limit when n is its sum n n Replace9r/n8by x, 1/n 1by5dx and 0 8 39 8 lim by the sign of integration . (i) Express the given series in the form of
1
r
n f n
n 1
n
r 0
n
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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(iii) The lower and upper limits of integration will be the value of r/n for the first and last term (or the limits of these values respectively).
Note: qn b
1 lim n n
(6)
r f n r pn a
q
f(x)dx.
p
Important Results: /2
If
In =
/2 n
sin xdx =
0
n
cos xdx , then
0
n1 n3 n5 ....... I0 or I1 n n2 n 4
In =
according as n is even or odd , I0 =
, I1 = 1 2
n5 1 n1 n3 ; n is even ... . n n2 n 4 2 2 Hence In = n1 n3 n5 2 n n2 n 4 ... 3 .1 ; n is odd /2
If Im,n =
n
n
sin x cos xdx m, n N then,
0
(n1)(n3)(m5)......(n1)(n3)(n5)..... (m1)(mn2)(mn 4)...... 2 Im,n = (m1)(m3)(m5)......(n1)(n3)(n5)..... (mn)(mn2)(mn 4)......
;
when both m,n are even
;
when both m,n are not even
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Mathematics Concept Note IITJEE/ISI/CMI
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20. AREA BOUNDED BY CURVES Following cases illustrates the method of computation of bounded area under different conditions:
b
(i)
A
f(x)dx. a
b
(ii)
A f(x)dx.
a
A A1 A2
(iii)
c
b
f(x)dx
A f(x)dx a
c
b
(iv)
A
f(x) g(x) dx. a
ma r a h S A f(x) g(x) dx. . 82 K . 6 L 7 Er. 81027 5058 9 801 839 b
(v)
a
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b
(vi)
A
f(x) g(x) dx. a
b
(vii)
A
f(x) g(x) dx. a
b
(viii)
A
f(y)dy a
b
(ix)
A f(y)dy
a
ma r a A A A.S h 82 K . 6 L 7 58 ErA. 8 f(y)dy 027 f(y)dy. 1 0 5 9 801 839 1
(x)
Mathematics Concept Note IITJEE/ISI/CMI
2
c
b
a
c
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b
(xi)
A
f(y) g(y) dy a
b
(xii)
A
f(y) g(y) dy a
b
(xiii)
A
f(y) g(y) dy a
b
(xiv)
A
f(y) g(y) dy a
ma r a h Curve Tracing : S . 82 K . 6 L 7 To find the approximate shape of a nonstandard curve, must 8 be 5 Er. the81following 027 steps 0 checked . 9 8015 (1)Symmetry about xaxis : 839 If all the powers of 'y' in the equation are even then the curve is symmetrical about the xaxis. Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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(2)Symmetry about yaxis : If all the powers of 'x' in the equation are even then the curve is symmetrical about the yaxis. (3) Symmetry about both axis : If all the powers of 'x' and 'y; in the equation are even, the curve is symmetrical about the axis of 'x' as well as 'y'. (4) Symmetry about the line y = x : If the equation of the curve remains unchanged on interchanging 'x' and 'y', then the curve is symmetrical about the line y = x.
a>0 E.g.: x2 +y3= 3axy. (5)Symmetry in opposite quadrants : If the equation of the curve remains unaltered when 'x' and 'y' are replaced by x and y respectively then there is symmetry in opposite quadrants. (6) Intersection points of the curve with the xaxis and the yaxis. (7) Locations of maxima and minima of the curve. (8) Increasing or decreasing behaviour of the curve. (9) Nature of the curve when x or x . (10) Asymptotes of the curve. Asymptoto(s) is (are) line(s) whose distance from the curve tends to zero as point on curve moves towards infinity along branch of curve. (i) If (ii) If
Lt f(x) =
x a
or xLt f(x) = a
then x= a is asymptote of y = f(x)
Lt f(x) = k or Lt f(x) = k, then y = k is asymptote of y = f(x) x
x
f(x) (iii) If xLt = m1 , xLt f(x) = m1(x) = c, then y = m1x + c1 is an asymptote. x (inclined to right)
(iv) If
Lt f(x) = m , Lt (f(x m x) = c , then y = m x + c is an asymptote 2 2 2 2 2 x x
x
(inclined to left).
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Mathematics Concept Note IITJEE/ISI/CMI
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21. DIFFERENTIAL EQUATION (1)
Introduction : An equation involving independent and dependent variables and the derivatives of the dependent variables is called a differential equation. There are two kinds of differential equation . Ordinary Differential Equation : If the dependent variables depend on one independent variable x, then the differential equation is said to be ordinary. For example:
d3y
dy + xy = sin x, dx
3
dx
+2
dy + y = ex, dx
3/2
2 2 dy dy dy k 2 = 1 + dx , y = x + k 1 + dx dx dx
d2y
Partial differential equation : If the dependent variables depend on two or more independent variables, then it is known as partial differential equation. For example 2z 2z z 2z +y = ax, + = xy.. y x y y2 x2 2
(2)
Linear and NonLinear Differential Equations: A differential equation is a linear differential equation if it is expressible in the form P0
dny n
dx
P1
dn 1y n 1
dx
P2
dn 2 y n2
dx
... Pn 1
dy Pn y Q dx
where P0, P1, P2, ... , Pn–1 , Pn and Q are either constants of function of independent variable x. Thus, if a differential equation when expressed in the form of a polynomial involves the derivatives and dependent variable in the first power and there are no product of these, and also the coefficient of the various terms are either constants or function of the independent variable, then it is said to be linear differential equation. Otherwise, it is a non linear differential equation. If follows from the above definition that a differential equation will be nonlinear differential equation if (i) its degree is more than one. (ii) any of the differential coefficient has exponent more than one (iii) exponent of the dependent variable is more than one. (iv)products containing dependent variable and its differential coefficients are present.
(3)
Order and Degree of a Differential Equation :
ma r a h S 2 . 8 K . 6 Degree: degree of differential equation is the highest exponent with which the highest L 7 27 0equation order derivative is raised in the differential equation, provided 58 is Er. 8the10differential 5 expressed in the polynomial form. 9 801 Note: 839
Order: Order of differential equation is the order of highest order derivative which is present in the equation.
If differential equation can't be expressed in the polynomial form, then its degree is not defined. Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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d2 y
2
d3 y x 3 y 0 is differential equation of 3 order and 2 degree. Example: dx2 dx
(4)
Geometrical Significance of Differential Equation: Geometrically, differential equation represents a family of curves which is having as many number of parameters as that of the order of differential equation. For example (i)
dy ex represents y ex c. dx
(ii) y
(iii)
(5)
dy x 0 represents x2 y2 c. dx
d2 y dx2
ex represents y ex c1x c2 .
Formation of Differential Equation: In order to formulate a differential equation for a given family of curves of n parameter (independent arbitrary constants), following steps is used. Step (1): differentiate the family of curves 'n' times to obtain n set of equations. Step (2): from the set of equations of step(1) and the equation of curve, all the arbitrary constants are eliminated to get the differential equation.
Note: Differential equation for a family of curves should not have any arbitrary constant and its order must be same as that of the number of parameters in family of curves.
(6)
Solution of a Differential Equation : Finding the dependent variable from the differential equation is called solving or integrating it. The solution or the integral of a differential equation is , therefore, a relation between dependent and independent variables (free from derivatives) such that it satisfies the given differential equation
Note: The solution of the differential equation is also called its primitive, because the differential equation can be regarded as a relation derived from it. There can be three types of solution of a differential equation : (i) General solution (or complete integral or complete primitive) :A relation in x and y satisfying the given differential equation and involving exactly same number of arbitrary constants as that of the order of differential equation. (ii) Particular solution : A solution obtained by assigning values to one or more than one arbitrary constant of general solution of a differential equation.
(7)
ma r a h S . 82 K . 6 L 7 Variables separable : If the differential equation can be r form,7f(x) dx = 8 (y) dy 2 05each E put. in8the1by0 we say that variables are separable and solution can be obtained integrating side 9 8015 separately. 39 8 f(x) dx (y) dy A general solution of this will be = + c,
Elementary Types of First Order and First Degree Differential Equations :
where c is an arbitrary constant. Mathematics Concept Note IITJEE/ISI/CMI
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Equations Reducible to the Variables Separable form : If a differential equation can be reduced into a variables separable form by a proper substitution, then it is said to be "Reducible to the variables separable type". Its general form is dy = f(ax + by + c) where a, b 0. dx
substitution, ax + by + c = t. reduces the general form in to variable separable form Homogeneous differential Equations : f(x,y) dy = g(x,y) where f and g are homogeneous dx function of x and y and of the same degree, is called homogeneous differential equation and can be solved by putting y = vx.
A differential equation of the from
Equations Reducible to the Homogeneous form Equations of the form
ax by c dy = Ax By C dx
.......... (1)
can be made homogeneous (in new variables X and Y) by substituting x = X + h and y = Y + k, where h and k are constants, we get
aX bY (ahbk c) dY = AX BY (AhBk C) ...........(2) dX
Now, h and k are chosen such that ah + bk + c = 0, and Ah + Bk + C = 0; the differential equation can now be solved by putting Y = vX.
Note: If the homogeneous equation is of the form : yf(xy) dx + xg(xy) dy = 0, the variables can be separated by the substitution xy = v. Exact Differential Equation : The differential equation M(x,y)dx + N(x,y)dy = 0 ..........(1) where M and N are functions of x and y is said to be exact if it can be derived by direct differentiation of an equation of the form f(x,y) = c
Note: M N (i) The necessary condition for exact differential equation is y = x
(ii) For finding the solution of exact differential equation, following exact differentials must be remembered: (a) xdy + ydx = d(xy)
(b)
xdy ydx y = d ln x xy
(e)
(d)
xdy ydx x2
xdy ydx 2
2
x y
y =d x
(c) 2(xdx + ydy) = d(x2 + y2)
1
= d tan
y x
(f)
xdy ydx = d(ln xy) xy
a m r a xy x y h S . 82 K . 6 L 7 r. 1027 058 E Linear differential equations of first order and first degree:8 9 8015 dy The differential equation + Py = Q, is linear in y.. 839 dx xdy ydx
(g)
2
2
1 = d xy
(h)
xdx ydy 2
2
1 d ln(x2 y2 ) 2
where P and Q are functions of x. Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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Integrating Factor (I.F.) : It is an expression which when multiplied to a differential equation converts it into an exact form. I.F for linear differential equation = e Pdx (constant of integration need not to be considered) after multiplying above equation by I. F it becomes ; dy . e Pdx + Py.. e Pdx = Q.. e Pdx dx d Pdx Pdx ) = Q.. e dx (y.. e
y.. e
Pdx
=
Q.e Pdx
+C
(General solution).
Bernoulli's equation : Equations of the form
dy + Py = Q.yn, n 0 and n 1 dx
where P and Q are functions of x, is called Bernoulli's equation 1
To convert Bernoulli's equation to linear form
n 1
y
is substituted as t.
dy dy f is termed as clairaut's dx dx equation. To get the solution of this form , the equation is differentiated as , explained in the next step:
Clairaut's Equation: Differential equation of the form y x
dy d2 y dy dy d2 y x f ' . 2 dx dx dx2 dx dx
d2 y dy x f ' 0. 2 dx dx d2 y 2
dx
0
or
dy Constant=c dx
general solution is given by y = cx + f(c).
dy x f ' 0. dx dy f ' x . gives dx
another solution, which is termed as singular solution
Differential equation reducible to the linear differential equation of first order and first degree. (i)
dx P(y)x Q(y). dy
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 39 8 dt P(x)t Q(x).
I.F. e P(y)dy and general solution is given by:
x(I.F.) (ii) f '(y)
Q(y)I.F.dy c
dy P(x).f(y) Q(x) dx
Put f(y) t
f '(y)
Mathematics Concept Note IITJEE/ISI/CMI
dy dt dx dx
dx
Page 121
(iii) f '(x)
dx P(y)f(x) Q(y). dy
Put f(x) t
(8)
f '(x)
dx dt dy dy
dt P(y)t Q(y). dy
Orthogonal Trajectory : An orthogonal trajectory of a given system of curves is defined to be a curve which cuts every member of the given family of curve at right angle.
Steps to find orthogonal trajectory : (i) Let f(x,y,c) = 0 be the equation of the given family of curves, where 'c' is an arbitrary constant. (ii) Differentate the given equation with respect to x and then eliminate c. (iii) Replace
dy by dx
dx
dy in the equation obtained in (ii).
(iv)Solving the differenetial equation obtained in step (iii) gives the required orthogonal trajectory.
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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22. BASICS OF 2 Dimensional GEOMETRY In 2dimensional coordinate system any point P is represented by (x,y), where x and y are the distances of 'P' from the yaxis and xaxis respectively
Note: (i) If both x and y I , point 'P' is termed as integral point. (ii) If both x and y Q , point 'P' is termed as rational point. (iii) If atleast one of x, y Q , point 'P' is termed as irrational point.
(1) Distance Formula : The distance between the points A(x1,y1) and B(x2,y2) is
(x1 x2 )2 (y1 y2 )2
(2) Section Formula : If P(x,y) divides the line joining A(x1,y1) and B(x2,y2) internally in the ratio m : n, then :
x
mx2 nx1 my 2 ny1 ;y and for external division P(x, y) is m n m n
x
mx2 nx1 my2 ny1 ;y m n m n
Note: x1 x2 y1 y2 , (i) The midpoint of AB is . 2 2
(ii) If P divides AB internally in the ratio m : n and Q divides AB externally in the ratio m : n then P and Q are said to be harmonic conjugates to each other with respect to A and B. Mathematically, 1 2 1 = + AQ AB AP
( AP, AB & AQ are in H.P.) .)
(3) Standard points of triangle. If A(x1,y1), B(x2,y2), C(x3,y3) are the vertices of triangle ABC, whose sides BC, CA, AB are of lengths a, b, c respectively, then the coordinates of the special points of triangle ABC are as follows : (i) Centroid G
x1 x2 x3 y1 y2 y3 , 3 3
(iii) Excentre I1
(ii)Incentre I
ax1 bx 2 cx 3 ay1 by 2 cy 3 , ab c ab c
ax1 bx2 cx3 ay1 by2 cy3 , ab c ab c
x (iv) Orthocentre H 1
x (v) Circumcentre C 1
Mathematics Concept Note IITJEE/ISI/CMI
ma r a hC 2 tan A x tanB x tan C y tan y tanB .yStan , K 8 . tan7C 76 tan A tanB tan C tan A.L tanB Er 8102 5058 sin2A x sin2B x sin2C y sin2A 9 y sin2B 01y sin2C , 8 9 sin 2A sin 2B sin C sin 2A 3 8 sin 2B sin2C 2
2
3
1
3
2
1
3
2
3
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Note: (i) Orthocentre, Centroid and Circumcentre are collinear and centroid divides the line joining orthocentre and circumcentre in the ratio 2 : 1. (ii) In an isosceles triangle G, H, I and C lie on the same line and in an equilateral triangle all these four points coincide with each other. (iii) Incentre and excentre are harmonic conjugate of each other with respect to the angle bisector on which they lie. (iv) In right angle triangle, orthocentre lies at the vertex of right angle and circumcentre lies at midpoint of hypotenuse.
(4) Area of a Triangle : If A(x1,y1), B(x2,y2), C(x3,y3) are the vertices of triangle ABC, then its area is equal to x1 1 x ABC = 2 2 x3
y1 1 y2 1 y3 1
Note: Area of nsided polygon formed by points (x1,y1); (x2,y2) ; ........ (xn,yn) is given by x1 x2 x2 x3 y y + .......... 1 2 y2 y3 (5) Change of Axes: 1 2
xn 1 xn yn 1 yn
xn x1 yn y2
(i) Rotation of Axes: If the axes are rotated through an angle in the anticlockwise direction keeping the origin fixed, then the coordinates (X,Y) of point P(x,y) with respect to the new system of coordinate are given by X = x cos +y sin and Y = y cos – x sin . (ii) Translation of Axes: The shifting of origin of axes without rotation of axes is called Translation of axes. If the origin (0, 0) is shifted to the point (h,k) without rotation of the axes then the coordinates (X,Y) of a point P(x,y) with respect to the new system of coordinates are given X = x – h, Y = y – k.
(6) Slope Formula : If is the angle at which a straight line is inclined to the positive direction of xaxis and 0 , / 2 , then the slope of the line denoted by m is defined by m = tan . If A(x1,y1) and B(x2,y2), x1 x2, are points on a straight line , then the slope m of the line is y y
given by x2 x 1 . 2 1
(7) Condition of collinearity of three points : Points A(x1,y1), B(x2,y2), C(x3,y3) are collinear if (i) mAB = mBC = mCA
(ii)
x1 x2 x3
y1 1 y2 1 = 0 y3 1
ma r a h Locus: S 2 . 8 K . 6 The locus of a moving point is the path traced out by that point under one or more given L 7 condition. Er. 81027 5058 1 points say Approach to find the locus of a point: Let (h,k) be the coordinates 9 of the 0moving 8 P. Now apply the geometrical condition on h, k. This gives a relation between h and k. Now 9 3 replace h by x and k by y in th eleminant and resulting equation8 would be the equation of the
(iii) AC = AB + BC or AB BC
(8)
locus. Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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23.STRAIGHT LINE (1) Equation of a straight Line in various forms : (i) PointSlope form : y y1 = m(x x1) is the equation of a straight line whose slope is m and which passes through the point (x1,y1). (ii) Slopeintercept form : y = mx + c is the equation of a straight line whose slope is m and which makes an intercept c on the yaxis. (iii) Two point form : y y1 =
y y1 2
x2 x1 (x x1) is the equation of a straight line which
passes through the points (x1,y1) and (x2,y2). (iv) Determinant form : Equation of line passing through (x1,y1) and (x2,y2) is x
y
x1 x2
y1 1 = 0 y2 1
1
(v) Intercept form :
x y + = 1 is the equation of a straight line which makes intercepts a b
a and b on Xaxis and Yaxis respectively. (vi) Perpendicular/Normal form : xcos +ysin = p (where p 0, 0 2 ) is the equation of the straight line where the length of the perpendicular from the origin on the line is p and this perpendicular makes an angle with positive x axis. x x
y y
1 1 (vii)Parametric form : cos = sin = r or P(r)=(x,y) (x1 + rcos , y1 + r sin ) is the equation of the line in parametric form , where 'r' is the parameter whose absolute value is the distance of any point (x, y) on the line from the fixed point (x1,y1) on the line. 'r' is take as positive for coordinate in increasing direction of 'y' and negative
for decreasing direction of 'y'. is angle of inclination of the line and [0, ) (viii) General Form : ax + by + c = 0 is the equation of a straight line in the general form a in this case, slope of line = b
(2) Angle between two straight lines: If m1 and m2 are the slopes of two intersecting straight lines (m1m2 1) and is the acute m m2
1 angle between them, then tan = 1+m . 1m2
Note: (i) Let m1, m2, m3 are the slopes of three lines L1 = 0; L2 = 0; L3 = 0 where m1 > m2 > m3 then the interior angles of the ABC formed by these lines are given by,,
ma r a h S 2 line . (ii) The equation of lines passing through point (x , y ) and making angle 8the with K . 6 L 7 y = mx + c are given by : r. 1027 058 E (y y ) = tan( )(x x ) and (y y ) = tan ( + ) (x x ), tan15= m. 98where 0 8 (iii) When two straight lines are parallel their slopes are equal. Thus any line parallel to 839 y = mx + c is of the type y = mx + k, where k is a parameter. m m2
m m3
1
m3 m1
2
tan A = 1m m ; tan B = 1m m & tan C = 1m m 3 1 1 2 2 3 1
1
Mathematics Concept Note IITJEE/ISI/CMI
1
1
1
1
Page 125
(iv) Two lines ax + by + c = 0 and a'x + b'y + c' = 0 are parallel if
a b c = . a' b' c'
Thus any line parallel to ax + by + c = 0 is of the type ax + by + k = 0, where k is a parameter. (v) The distance between two parallel lines with equations ax + by + c1 = 0 and ax + by + c2 = 0 is
c1 c2 a2 b2
.
(vi) The area of the parallelogram =
p1p2 , where p1 & p2 are sin
distances between two pairs of opposite sides & is the angle between any two adjacent sides. (vii)Area of the parallelogram bounded by the lines y = m1x + c1, y = m1x + c2 and y = m2x + d1, y = m2x + d2 is given by
(c1 c2 )(d1 d2 ) . m1 m2
(viii) When two lines of slopes m1 and m2 are at right angles, the product of their slopes is 1, i.e. m1 m2 = 1. Thus any line perpendicular to y = mx + c is of the form y=–
1 x + d where d is any parameter.. m
(ix) Two lines ax + by + c = 0 and a'x + b'y + c' = 0 are perpendicular if aa' + bb' = 0. Thus any line perpendicular to ax + by + c = 0 is of the from bx ay + k = 0, where k is any parameter.
(3) Position of the point (x1, y1) with respect to the line ax + by + c = 0 : If ax1 + by1 + c is of the same sign as c, then the point (x1,y1) lie on the origin side of ax + by + c = 0. But if the sign of ax1 + by1 + c is opposite to that of c, the point (x1, y1) will lie on the nonorigin side of ax + by + c = 0. In general two points (x1, y1) and (x2, y2) will lie on same side or opposite side of ax + by + c = 0 according as ax1 + by1 + c and ax2 + by2 + c are of same or opposite sign respectively.
Note:
If b is positive in line ax+by+c = 0, then point P(x1,y1) lies above the line if ax1 + by1 + c > 0 and point P lies below the line if ax1 + by1 + c < 0.
(4) Length of perpendicular from a point on a line :
The length of perpendicular from P(x1,y1) on ax + by + c = 0 is
ax1 by1 c a2 b2
(5) Reflection of a point about a line :
ma r a h S ax by c x x y y . 82 K = b = . 6 a L 7 a b 7 r. is given 58 02by: 1 0 The image of a point (x ,y ) about the line ax + by E + c = 08 5 9 801 ax by c x x y y 39 8 = = 2 a b a b
(i) Foot of the perpendicular from a point (x1 , y1 ) on the line ax+by+c = 0 is given by: 1
1
1
1
2
(ii)
1
1
1
1
9810277682/8398015058
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(6) Bisectors of the angles between two lines : Equations of the bisectors of angles between the lines ax + by + c = 0 and ax by c
a'x + b'y + c' = 0, (ab' a'b) are :
a2 b2
=
a'x b'y c' a' 2 b' 2
Note: (i) Equation of straight lines passing through P(x1, y1) and equally inclined with the lines a1x +b1y + c1 = 0 & a2x + b2y + c2 = 0 are those which are parallel to the bisectors between these two lines & passing through the point P. (ii) If be the angle between one of the lines and one of the bisectors and tan  < 1, then this bisector is the acute angle bisector, if tan  > 1, then the bisector is obtuse angle bisector. (iii) If aa' + bb' < 0, then the equation of the bisector of this acute angle is ax by c 2
=+
a'x b'y c'
(c and c´ constants are positive)
a' 2 b' 2
2
a b
If aa' + bb' > 0, the equation of the bisector of the obtuse angle is : ax by c
=+
a2 b2
a'x b'y c'
(c and c´ constants are positive)
a' 2 b' 2
(iv) To discriminate between the bisector of the angle containing the origin and that of the angle not containing the origin. the equations are written as ax + by + c = 0 and a'x + b'y + c' =0 such that the constant terms c, c' are positive. Then ax by c
=+
a2 b2
a'x b'y c'
ax by c
origin &
gives the equation of the bisector of the angle containing the
a' 2 b' 2
a2 b2
=
a'x b'y c' a' 2 b' 2
gives the equation of the bisector of the angle not
containing the origin. In general equation of the bisector which contains the point ( , ) is
ax by c a2 b2
=+
a'x b'y c'
or
a' 2 b' 2
ax by c
=
a2 b2
a'x b'y c' a' 2 b' 2
according as a +b +c
and a' +b' +c' having same sign or otherwise.
(7) Condition of Concurrency : Three non parallel lines a1x + b1y + c1 = 0, a2x + b2y + c2 = 0 & a3x + b3y + c3 = 0 are a1 b1 c1 a concurrent if 2 b2 c2 = 0 a3 b3 c3
a1 a a 2 3 b2 b3 b1
Alternatively : If three constants A, B and C (not all zero) can be found such that A(a1x + b1y + c1) + B(a2x + b2y + c2) + C(a3x + b3y + c3)=0, then the three straight lines are concurrent.
(8)
a m r a Family of Straight Lines : h S . 82 K . 6 L 7 7 The equation of a family of straight lines passing through of r.the point 2intersection E 0 1 0L5=8of0 the lines. L a x + b y + c = 0 and L a x + b y + c = 0 9 is 8 given by L 1+ i.e. 5 0 8 (a x + b y + c ) + (a x + b y + c ) = 0, where is an arbitrary real number. . 39 8 Note: 1
1
1
1
1
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1
2
2
2
2
2
2
1
2
2
(i) If u1 = ax + by + c, u2 = a'x + b'y + d, u3 = ax + by + c', u4 = a'x + b'y + d' Mathematics Concept Note IITJEE/ISI/CMI
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u 3= 0
C 0
0
D
A (ii)
u2 =
u4 =
then u1 = 0; u2 = 0; u3 = 0; u4 = 0 form a parallelogram. The diagonal BD can be given by u2u3 u1u4 = 0.
u1= 0
B
The diagonal AC is given by u1 + u4 = 0 and u2 + u3 = 0, if the two equations are identical for some real and .[For getting the values of & compare the coefficients of x, y & the constant terms].
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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24. PAIR OF STRAIGHT LINES (1) A Pair of Straight lines through origin : (i) A homogeneous equation of two degree ax2 + 2hxy + by2 = 0 always represents a pair of straight lines passing through the origin if : (a) h2 > ab lines are real and distinct. (b) h2 = ab lines are coincident. (c) h2 < ab lines are imaginary with real point of intersection i.e. (0,0)
(ii) If y = m1x and y = m2x be the two equations represented by ax2 + 2hxy + by2 = 0, then; m1 + m2 = 
2h a and m1 m2 = . b b
(iii) If is the acute angle between the pair of straight lines represented by 2 h2 ab ax + 2hxy + by = 0, then; tan = . ab 2
2
(iv) The condition that these lines are : (a) At right angles to each other is a + b = 0. i.e. coefficient of x2 + coefficient of y2 = 0. (b) Coincident is h2 = ab. (c) Equally inclined to the axis of x is h = 0. i.e. coeff. of xy = 0.
Note:
A homogeneous equation of degree n represents n straight lines passing through origin. (v) The equation to the pair of straight lines bisecting the angle between the straight lines, ax2 + 2hxy + by2 = 0 is
xy x2 y2 = . h ab
(2) General equation of second degree representing a pairs of straight lines:
ma r a h S a h g . 82 K . 6 L 7 h b f abc + 2fgh af bg ch = 0, i.e. if = 0. r. E 81027 5058 g f c 9 801 9 83equation The angle between the two lines representing by a general is the same
(i) ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represents a pair of straight lines if :
2
(ii)
2
2
as that between the two lines represented by its homogeneous part only.
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(3) Homogenization : The equation of a pair of straight lines joining origin to the points of intersection of the line L x + my + n = 0 and a second degree curve. S ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 2 x my x my x my + 2fy + c = 0. n n n
is ax2 + 2hxy + by2 + 2gx
The equation is obtained by homogenizing the equation of curve with the help of equation of line.
NOTE: Equation of any curve passing through the points of intersection of two curves C1 = 0 and C2 = 0 is given by C1 + C2 = 0 where & are parameters.
(4) Some Important Results (i) If ax2 + 2hxy + 2gx + 2fy + c = 0 represent a pair of parallel straight lines, then the distance between them is given by 2
g2 ac f 2 bc or 2 a(a b) b(a b)
(ii) Th e l i n e s joi n i n g t h e or igi n t o t h e po i n t s of i n t ersec t i on of t h e cu r ves ax 2 + 2hxy + by 2 + 2gx = 0 and a' x 2 + 2h' xy + b' y 2 + 2' gx = 0 will be mutually perpendicular, if g'(a + b). (iii) If the equation hxy + gx + fy + c = 0 represents a pair of straight lines, then fg = ch. (iv) The pair of lines (a2 – 3b2)x2 + 8abxy + (b2 – 3a2) y2 = 0 with the line ax+by+c = 0 form an equilateral triangle and its area
c2 3(a2 b2 )
.
(v) The area of a triangle formed by the lines ax2 + 2hxy + by2 = 0 and lx + my + n = 0 is given by
n2 h2 ab am2 2hlm bl2
(vi) The lines joining the origin to the points of intersection of line y = mx + c and the circle x2 + y2 = a2 will be mutually perpendicular, if a2(m2 + 1) = 2c2. (vii)If the distance of two lines passing through origin from the point (x1,y1) is d, then the equation of lines is (xy1 – yx1)2 = d2(x2 + y2) (viii) The lines represented by the equation ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 will be equidistant from the origin, if f4 – g4 = c(bf2 – ag2) (ix) The product of the perpendiculars drawn from (x1,y1) on the lines ax2 + 2hxy + by2 = 0 is given by
ax12 2hx1y1 by12 (a b)2 4h2
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839
(x) The product of the perpendiculars drawn from origin on the lines Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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c
ax2 + 2hxy + by2 + 2gx + 2fy = 0 is
(a b)2 4h2
(xi) If the lines represented by the general equation ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 are perpendicular, then the square of distance between the point of intersection and origin
f 2 g2 f 2 g2 or . h2 b2 h2 a2 (xii) The square of distance between the point of intersection of the lines represented by is
the equation ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 and origin is
c(a b) f 2 g2 . ab h2
(xiii) If one of the line given by the equation ax2 + 2hxy + by2 = 0 coincide with one of these given by a'x2 + 2h'xy + b'y2 = 0 and the other lines represented by them be perpendicular, then
ha'b ' h' ab 1 (aa'bb '). b ' a' b a 2
(xiv)The straight lines joining the origin to the points of intersection of the straight line kx + hy = 2hk with the curve (x – h)2 + (y – k)2 = c2 are at right angles, if h2 + k2 = c2.
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Mathematics Concept Note IITJEE/ISI/CMI
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25. CIRCLES (1) Equations of a Circle (i) An equation of a circle with centre (h, k) and radius r is (x–h)2 + (y–k)2 = r2 (ii) An equation of a circle with centre (0, 0) and radius r is x2+y2 = r2 (iii) An equation of the circle on the line segment joining (x1, x1) and (x2, y2) as diameter is (x – x1) (x – x2) + (y – y1) (y – y2) = 0 (iv) General equation of a circle is x2 + y2 + 2g x + 2f y + c = 0 where g, f and c are constants. Centre of this circle is (–g, –f)
Its radius is
g2 f 2 c, (g2 f 2 c).
(v) The intercepts made by the circle x2 + y2 + 2gx + 2fy + c = 0 on the xaxis and yaxis is given by 2 g2 c and 2 f 2 c respectively.. (vi) General equation of second degree ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 in x and y, represents a circle if and only if coefficient of x2 equals coefficient of y2, i.e., a = b 0. coefficient of x y is zero, i.e., h = 0.
g2 + f2 – ac 0
Note: (i) Equation of circle circumscribing a triangle whose sides are given by L1 = 0; L2 = 0 and L3 = 0 is given by ; L1L2 + L2L3 + L3L1 = 0, provided coefficient of xy = 0 and coefficient of x2 = coefficient of y2. (ii) Equation of circle circumscribing a quadrilateral whose side in order are represented by the lines L1 = 0, L2 = 0, L3 = 0 and L4 = 0 are u L1 L3 + L2L4 = 0 where values of u and can be found out by using condition that coefficient of x2 = coefficient of y2 and coefficient of xy = 0.
(2) Parametric Equations of a circle : Parametric equation of circle (x h)2 + (y k)2 = r2 is given by
x h y k r , where (h, k) cos sin
is the centre, r is the radius and is a parameter (0 2) Any point P() on the circle (x – h)2 + (y – k)2 = r2 can be assumed in parametric form as P (h r cos , k r sin )
Note: The equation of chord PQ to the circle x2 + y2 = a2 joining two points P( ) and Q( ) on it is given by x cos
(3)
+ y sin = a cos . 2 2 2
ma r a Position of a point with respect to a circle : h S . 82 K . 6 L 7 Let equation of the circle is given by S x + y + 2gx + c2 =7 0, then following 58 Er.+ 2fy 0 1 0 condition illustrates the relative position of point P. 5 8 9 801 (i) Point 'P' lies on the circle if: 839 2
2
x12 y12 2gx1 2fy1 c 0 . Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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(ii) Point 'P' lies inside the circle if:
x12 y12 2gx1 2fy1 c 0 (iii) Point 'P' lies outride the circle if:
x12 y12 2gx1 2fy1 c 0
Note: The greatest and the least distance of a point P from a circle with centre C and radius r is PC + r and PC r respectively..
(4) Line and circle If line y = mx + c intersects the circle x2 + y2 = a2 at point A and B, then the xcoordinate of the points of intersection is given by x2 + (mx + c)2 – a2 = 0. ........(1) (1 + m2) x2 + 2cm x + c2 – a2 = 0 In equation (1), if discriminant 'D' is positive, then two distinct real values of 'x' is obtained which confirms the intersection of line and circle at two distinct points (refer figure no. (1)).
D 0 c2 a2 (1 m2 ) p a.
If equation (1), if dircriminant 'D' is negative, then line is outside the circle (refer figure (2))
c2 a2 (1 m2 ) p a
If equation (1), if discriminant 'D' is zero, then line is tangential to the circle (refer figure(3)).
c2 a2 (1 m2 ) p a
(5) Tangent to a Circle: (a) Slope Form : Straight line y = mx + c is tangent to the circle x2 + y2 = a2 if c2 = a2(1 + m2). Hence, a2m a2
equation of tangent is y = mx a 1m2 and the point of contact is c , c .
Note:
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 point 8 = a at its (x0 ,y1 ) is given by 9 3 8
Equation of tangent of slope 'm' to the circle (x – h)2 + (y – k)2 = r2 is given by: (y – k) = m(x – h) r 1 m2 (b) Point form : (i) The equation of the tangent to the circle x2 + y2 x x1 + y y1 = a2.
2
1
1
(ii) The equation of the tangent to the circle x2 + y2 + 2gx + 2fy + c = 0 at its point Mathematics Concept Note IITJEE/ISI/CMI
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(x1, y1) is given by x x1 + y y1 + g(x + x1) + f(y + y1) + c = 0.
Note: In general the equation of tangent to any second degree curve at point (x1, y1) on it can be obtained by replacing x2 by xx1 , y2 by yy1 , x by
x x1 y y1 x y xy1 , y by , xy by 1 2 2 2
and c remains as c. (c) Parametric form : The equation of a tangent x cos + y sin = a.
to
circle x2 + y2 = a2 at
(a cos , a sin ) is
Note: a sin a cos 2 2 , The point of intersection of the tangents at the points P( ) & Q( ) is cos cos 2 2
.
(6) Normal to Circle If a line is normal/orthogonal to a circle then it must pass through the centre of the circle. Using this fact normal to the circle x2 + y2 + 2gx + 2fy + c = 0 at (x1 y1) is given by: y1 f y y1 = x g (x x1). 1
(7) Pair of Tangents from a Point : The equation of a pair of tangents drawn from the point P(x1, y1) to the circle x2 + y2 + 2gx + 2fy + c = 0 is given by SS1 = T2. Where S x2 + y2 + 2gx + 2fy + c ; S1 x12 + y12 + 2gx1 + 2fy1 + c T xx1 + yy1 + g(x + x1) + f(y + y1) + c.
Note: The locus of the point of intersection of two perpendicular tangents is called the director circle of the given circle. The director circle of a circle is the concentric circle having radius equal to
2 times the original circle.
(8) Length of a Tangent and Power of a Point : The length of a tangent from an external point (x1,y1) to the circle S x2 + y2 + 2gx + 2fy + c = 0 is given by L =
Note:
x 12 y 12 2g x 1 2 f1y c =
S1
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 39 8 are drawn from the point P(x ,y ) to the circle S x + y + 2gx
(i) Square of length of the tangent from the point P is called the power of point with respect to a circle. (ii) Power of a point with respect to a circle remains constant i.e. PA.PB = PT2.
(9) Chord of Contact :
If two tangents PT1 and PT2 2 1 1 + 2fy + c = 0, then the equation of the chord of contact T1 T2 is given by: T 0. Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0.
Note: (i) Chord of contact exists only if the point 'P' is outside the circle. 2LR
(ii) Length of chord of contact T1T2 =
R 2 L2
(iii) Area of the triangle formed by the pair of the tangents and its chord of contact is given by
RL3 R2 L2
.
(iv) If T1PT2 , then tan
2RL L R2 2
(v) Quadrilateral PT1OT2 is cyclic and the equation of the circle circumscribing the triangle PT1T2 is given by (x x1) (x + g) + (y y1) (y + f) = 0.
(10) Equation of the Chord with a given Middle Point : The equation of the chord of the circle S x2 + y2 + 2gx + 2fy + c = 0 in terms of its mid point M(x1,y1) is given by T = S1. xx1 + yy1 + g(x + x1) + f(y + y1) + c = x12 + y12 + 2gx1 + 2fy1 + c
(11) Common Tangents to two Circles : Direct (or external) common tangents meet at a point which divides the line joining centre of circles externally in the ratio of their radii. Transverse (or internal) common tangents meet at a point which divides the line joining centre of circles internally in the ratio of their radii. If (x – g1)2 + (y – f1)2 = r12 and (x – g2)2 + (y – f2)2 = r22 are two circles with centres C1(g1, f1) and C2(g2, f2) and radii r1 and r2 respectively, then 'P' and 'Q' are the respective points from which direct and Transverse common tangents can be drawn (as shown in figure).
ma r a h S . 82 K Note: . 6 L 7 Length of an external (or direct) common tangent & internal 58 Er.(or8transverse) 027 common 1 0 5 9 01 tangent to the circles are given by : L = d (r r ) & L = d (r8 r ) , 9 8r3, r are the radii of where d = distance between the centres of the two circles and ext
the two circles. Mathematics Concept Note IITJEE/ISI/CMI
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(12) Relative Position of two Circles Case
Number of Common Tangents
Condition
(i)
4 common tangents
r1 + r2 < c1 c2.
(ii)
3 common tangents.
r1 + r2 = c1 c2.
(iii)
2 common tangents.
r1 r2 < c1 c2 < r1 + r2
(iv)
1 common tangent
r1 r2 = c1 c2
(v)
No common tangent
c1 c2 < r1 r2
(13) Orthogonality Of Two Circles : Two circles S1 = 0 and S2 = 0 are said to be orthogonal or said to intersect orthogonally if the tangents at their point of intersection include a right angle. The condition for two circles to be orthogonal is : 2g1g2 + 2f1f2 = c1 + c2.
Note: (i) The centre of a variable circle orthogonal to two fixed circles lies on the radical axis of two circles. (ii) The centre of a circle which is orthogonal to three given circles is the radical centre provided the radical centre lies outside all the three circles.
(14) Radical Axis and Radical Centre : The radical axis of two circles is the locus of a point whose powers with respect to the two circles are equal. The equation of radical axis of the two circles S1 = 0 and S2 = 0 is given by S1–S2 = 0 i.e: 2(g1–g2)x + 2(f1–f2)y + (c1–c2) = 0.
ma r a h Note: S . 82 K . are equal. 6 (i) The length of tangents from radical centre to the three.circles L 7 r 27 058 E 0 1 (ii) If two circles intersect, then the radical axis is the common 15two circles. 98chord8of0the (iii) If two circles touch each other then the radical axis is the common 39 tangent of the 8 two circles at the common point of contact.
The common point of intersection of the radical axes of three circles taken two at a time is called the radical centre of three circles.
(iv) Radical axis is always perpendicular to the line joining the centres of the two circles. Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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(v) Radical axis will pass through the mid point of the line joining the centres of the two circles only if the two circles have equal radii. (vi) Radical axis bisects a common tangent between the two circles. (vii) A system of circles, every two of which have the same radical axis, is called a coaxial system. (viii) Pairs of circles which do not have radical axis are concentric circles.
(15) Family of Circles : (i) The equation of the family of circles passing through the points of intersection of two circles S1 = 0 and S2 = 0 is : S1 + S2 = 0 ; R {1} ( 1 if the coefficient of x2 and y2 in S1 and S2 are same)
Note:
If Coefficients of x2 and y2 are same in two intersecting circles S1 = 0 and S2 = 0, then the common chord of circles is given by: S1 – S2 = 0 (ii) The equation of the family of circles passing through the point of intersection of a circle S = 0 and a line L = 0 is given by S + L = 0 ; R (iii) The equation of a family of circles passing through two given points (x1,y1) and (x2,y2) can be written in the form : x x {(x x1) (x x2) + (y y1)(y y2)} + 1 x2
y 1 y1 1 = 0 where R . y2 1
(iv) The equation of a family of circles touching a fixed line ax+by+c = 0 at the fixed point (x1, y1) is {(x x1)2 + (y y1)2} + (ax+by+c) = 0 where is a parameter.. (v) The equation of family of circles touching a fixed circle S = 0 at fixed point (x1,y1) is given by: {(xx1 )2 +(yy1 )2 }+S=0 ; R .
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26. PARABOLA (1) Conic Sections : A conic section is the locus of a point which moves in a plane so that its distance from a fixed point (i.e. focus) is in a constant ratio (i.e. eccentricity) to its perpendicular distance from a fixed straight line (i.e. directrix). Let point P be (x,y) then equation of conic is given by:
PF e = constant ratio PM
(x )2 (y )2 e
px qy r p2 q2
ax2 2hxy by2 2gx 2fy c 0
Note: All conic sections can be derived from a right circular cone, as shown in the figure
(2) Recognisation of Conics: The equation of conics is represented by the general equation of second degree ax2 + 2hxy + by2 + 2gx + 2fy + c = 0..............(1) where = abc + 2fgh – af2 – bg2 – ch2 Different conics can be recognized by the conditions given in the tabular form. Case I: when 0 In this case equation (1) represents the Degenerate conic whose nature is given in the following table: Condition 0 and h2 –ab = 0 0 and h2 –ab 0 0 and h2 –ab > 0 0 and h2 –ab < 0
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
ma r a h S Nature of Conic . 82 K . 6 L 7 A pair of coincident r. lines 27 058 E 0 1 8 of straight Real or imaginary9 pair 15lines 0 8 9 line. A pair of intersecting3 straight 8 Point.
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Case II: When 0, In this case equation (1) represents the Nondegenerate conic whose nature is given in the following table: Condition
Nature of Conic
0, h = 0, a = b
a circle
0, h2 – ab = 0
a parabola
2
0, h – ab < 0
an Ellipse or empty set
0, h2 – ab > 0
a Hyperbola
0, h2 – ab > 0 and a + b = 0
a Rectangular hyperbola
PARABOLA (3) Basic Definitions A parabola is the locus of a point, whose distance from a fixed point (focus) is equal to perpendicular distance from a fixed straight line (directrix). Standard forms of the parabola are y2 = 4ax; y2 = 4ax; x2 = 4ay; x2 = 4ay
Focal Distance : The distance of a point on the parabola from the focus. Focal Chord : A chord of the parabola, which passes through the focus. y2 = 4ax. Double Ordinate : A chord of the parabola perpendicular to the axis of the symmetry Latus Rectum : A double ordinate passing through the focus or a focal chord perpendicular to the axis of parabola is called the latus Rectum (L.R.). For y2 = 4ax. Length of the latus rectum = 4a.
ends of the latus rectum are L(a, 2a) & L' (a,
2a).
Note: (i) Perpendicular distance from focus on directrix = half the latus rectum. (ii) Vertex is middle point of the focus & the point of intersection of directrix & axis. (iii) Two parabolas are said to be equal if they have the same latus rectum.
(4) Parametric Representation :
ma r a h S . Note: 82 K . 6 L 7 (i) The equation of a chord joining t and t is 2x (t + Etr).y +821at0t 2=70. 5058 (ii) If t and t are the ends of a focal chord of the parabola 9 y = 4ax then 01t t = 1. Hence 8 9 2a a 83 the coordinates at the extremities of a focal chord are (at ,2at) and , .
Parametric coordinates of a point on the parabola is (at2, 2at) i.e. the equations x = at2 and y = 2at together represents the parabola y2 = 4ax, t being the parameter and t R
1
2
1
2
1 2
2
1
2
1 2
2
t
2
t
2
(iii) Length of the focal chord making an angle with the xaxis is 4acosec . Mathematics Concept Note IITJEE/ISI/CMI
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(5) Position of a point Relative to a Parabola : The point (x1,y1) lies outside, on or inside the parabola y2 = 4ax according as the expression y12 4ax1 is positive, zero or negative respectively..
(6) Line and a Parabola : The line y = mx + c meets the parabola y2 = 4ax in two points real, coincident or imaginary according as c
a a a ,c or c respectively. Condition of tangency is c = a/m. m m m
(7) Tangents to the Parabola y2 = 4ax : (i) yy1 = 2a(x + x1) at the point (x1, y1); (ii) y = mx +
a 2a a (m 0) at 2 , m m m
(iii) yt = x + at2 at (at2, 2at).
Note:
Point of intersection of the tangents at the point t1 and t2 is (at1t2, a(t1 + t2))
(8) Normal to the parabola y2 = 4ax : y1 (i) y y1 = 2a (x x1) at (x1, y1) ; (ii) y = mx 2am am3 at (am2, 2am) (iii) y + tx = 2at2 + at3 at (at2 2at).
Note: (i) Point of intersection of normal at t1 and t2 are, (a (t12 + t22 + t1t2 + 2), at1t2 (t1 + t2)). (ii) If the normals to the parabola y2 = 4ax at the point t1 meets the parabola again at
2
1
the point t2 then t2= t1 t . (iii) If the normals to the parabola y2 = 4ax at the points t1 and t2 intersect again on the parabola at the point 't3' then t1t2 = 2; t3= (t1 + t2) and the line joining t1 and t2 passes through a fixed point ( 2a,0).
(9) Pair of Tangents : The equation to the pair of tangents which can be drawn from any point (x1,y1) to the parabola y2 = 4ax is given by : SS1 = T2 where : S y2 4ax ; S1 = y12 4ax1 ; T yy1 2a(x + x1).
Note: (i) From any point on the directrix perpendicular tangents can be drawn to the parabola and their chord of contact is the focal chord. (ii) If tangents are drawn at the extremity of a focal chord of parabola, then these tangents are perpendicular and meet on the line of directrix. (iii) Circle drawn with focal chord as the diameter always touches the directrix of parabola.
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 (10) Chord of Contact : 9 801 39 8 Equation to the chord of contact of tangents drawn from a point P(x ,y ) is 1
1
yy1 = 2a (x + x1). Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
Page 140
Note: The area of the triangle formed by the tangents from the point (x1,y1) and the chord of contact is (y12 4ax1)3/2/2a.
(11) Chord with a given middle point : Equation of the chord of the parabola y 2=4ax whose middle point is (x 1,y 1) is T = S1, where, S1 = y12 4ax1 ; T yy1 2a(x + x1).
(12) Important Results : (i) If the tangent and normal at any point 'P' of the parabola intersect the axis at T and G then ST=SG=SP where 'S' is the focus. In other words the tangent and the normal at a point P on the parabola are the bisectors of the angle between the focal radius SP and the perpendicular from P on the directrix. From this we conclude that all rays emanating from S will become parallel to the axis of the parabola after reflection. (ii) The portion of a tangent to a parabola cut off between the directrix and the curVe subtends a right angle at the focus. (iii) The tangents at the extremities of a focal chord intersect at right angles on the directrix, and hence a circle on any focal chord as diameter touches the directrix. Also a circle on any focal radii of a point P(at2,2at) as diameter touches the tangent at the vertex and intercepts a chord of length a 1 t2 on a normal at the point P.. (iv) Any tangent to a parabola & the perpendicular on it from the focus meet on the tangent at the vertex. (v) If the tangents at P and Q meet in T, then : TP and TQ subtend equal angles at the focus S. ST2= SP. SQ and The triangles SPT and STQ are similar. (vi) Semi latus rectum of the parabola y2 = 4ax, is the harmonic mean between segments of any focal chord of the parabola. (vii)The area of the triangle formed by three points on a parabola is twice the area of the triangle formed by the tangents at these points. (viii) If normal are drawn from a point P(h,k) to the parabola y2 = 4ax then k = mh 2am am3 i.e. am3 + m(2a h) + k = 0.
2ah k ; m1m2m3 = . a a where m1 m2 & m3 are the slopes of the three concurrent normals. m1 + m2 + m3 = 0; m1m2 + m2m3 + m3m1 =
Note:
algebraic sum of the slopes of the three concurrent normals is zero. algebraic sum of the ordinates of the three conormal points on the parabola is zero.
Centroid of the formed by three conormal points lies on the xaxis. Condition for three real and distinct normals to be drawn from a point P(h,k) is 4
h > 2a and k2 < 27a (h 2a)3
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839
(ix) Length of subtangent at any point P(x,y) on the parabola y2 = 4ax equals twice the abscissa of the point P and the subtangent is bisected at the vertex. (x) Length of subnormal is constant for all points on the parabola and is equal to the semi latus rectum.
Mathematics Concept Note IITJEE/ISI/CMI
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27. ELLIPSE Ellipse is the locus of a point which moves in such a way so that sum of its distances from two fixed points is always constant
Note: (i) PS + PS' = AA' (ii) Eccentricity (e) < 1
(1) Basic definitions: Standard equation of an ellipse referred to its principal axes along the coordinate axes is
x2 a
2
+
y2 b2
= 1, where a > b & b2 = a2 (1 e2)
Eccentricity : e = 1
b2 a2
, (0 < e < 1)
Foci : S (ae,0) and S' ( ae, 0). Equations of Directrix : x =
a a and x = . e e
Major Axis : The line segment AA' in which the foci S and S' lie is of length 2a and is called the major axis (a > b) of the ellipse. Point of intersection of major axis with directrix is called the foot of the directrix (Z and Z'). Minor Axis : The yaxis intersects the ellipse in the points B' (0, b) & B (0,b). The line segment BB' is of length 2b (b
x2 y2 1 are a+ex a2 b2
and a–ex Focal Chord : A chord which passes through a focus is called a focal chord. Double Ordinate : A chord perpendicular to the major axis is called a double ordinate. Latus Rectum : The focal chord perpendicular to the major axis is called the latus rectum. 2
Length of latus rectum (LL') =
(minor axis) 2b2 =2 = 2a(1 e2) major axis a
Centre : The point which bisects every chord of the conic drawn through it, is called the centre of the conic. C = (0,0) the origin is the centre of the ellipse
x2 a2
+
y2 b2
=1.
ma r a h If the equation of the ellipse is given as + =1 and nothing is.mentioned then S 2 the a b 8 K . 6 standard form (a rel="nofollow"> b) is assumed. r. L 10277 058 E If b > a is given, then the yaxis become major axis and xaxis 5 minor 1the 98 become 0 axis and all other points and lines change accordingly. (as shown 8 if figure) 39 8 x y
Note: (i) (ii)
2
vertical ellipse is given by
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
2
a
x2
y2
2
2
2
2
b
1 ; (a < b)
Page 142
Vertical Ellipse:
(2) Auxiliary Circle: A circle described on major axis of ellipse as diameter is called the auxiliary circle. Let Q be a point on the auxiliary circle x2 + y2 = a2 such that line through Q perpendicular to the xaxis on the way intersects the ellipse at P, then P & Q are called as the Corresponding Points on the ellipse and the auxiliary circle respectively. ' θ ' is called the Eccentric Angle of the point P on the ellipse (0 2) .
Note: (i)
Semi minor axis (PN) b = = Semi major axis (QN) a
(ii) If from each point of a circle perpendiculars are drawn upon a fixed diameter then the locus of the points dividing these perpendiculars in a given ratio is an ellipse of which the given circle is the auxiliary circle.
(3) Parametric Representation : The equations x = a cos θ & y = b sin θ together represent the ellipse
x2 a2
+
y2 b2
= 1.
Where θ is a parameter, if P( θ ) (acos θ , b sin θ ) is on the ellipse then, P( θ ) (acos θ , a sin θ ) is on the auxiliary circle. The equation to the chord of the ellipse joining two points with eccentric angles & is given by
(4)
x y cos 2 + sin 2 =cos 2 a b
a m r a Position of a Point with respect to an Ellipse : h S . 82 K . 6 L 7 7 58 Er. 8102 y50 x 9 as 8 01 1 is posiThe point P(x ,y ) lies outside, inside or on the ellipse according b a 9 3 8 tive, negative or zero respectively. 1
1
Mathematics Concept Note IITJEE/ISI/CMI
2 1
2 1
2
2
Page 143
(5) Line and an Ellipse : The line y = mx + c meets the ellipse
x2
a2 according as c2 is < = or > a2m2 + b2.
+
y2 b2
Hence y = mx + c is tangent to the ellipse
x2 a2
= 1 in two points real, coincident or imaginary
+
y2 b2
= 1 if c2 = a2m2 + b2.
(6) Tangents to Ellipse: x 2 y2 (a) Slope form : y = mx a2m2 b2 is tangent to the ellipse 2 + 2 = 1. a b
(b) Point form :
xx 1 2
a
+
yy 1 b
2
= 1 is tangent to the ellipse
x2 a
2
+
y2 b2
= 1 at (x1, y1).
x cos y sin x2 y2 + = 1 is tangent to the ellipse 2 + 2 = 1 at the a b a b point (a cos ,b sin ).
(c) Parametric form :
Note: (i) There are two tangents to the ellipse having the same slope m, i.e. there are two tangents parallel to any given direction. These tangents touches the ellipse at extremities of a diameter. (ii) Point of intersection of the tangents to the ellipse at the points a cos b sin 2 2 ' ' and ' ' is , cos cos 2 2
(iii) The eccentric angles of the points of contact of two parallel tangents differ by .
(7) Normals of Ellipse: (i) Equation of the normal at (x1, y1) to the ellipse
x2 a2
+
y2 b2
= 1 is
a2 x b2y = a2 b2. x1 y1
(ii) Equation of the normal at the point (acos ,bsin ) to the ellipse
x2 a2
+
y2 b2
= 1 is
axsec bycosec =(a2 b2).
Note: (i) Atmost four normals can be drawn to an ellipse from a point in its plane. (ii) In general, there are four points A, B, C and D on the ellipse the normals at which pass through a given point. These four points A, B, C, D are called the conormal points.
ma r a h S . 82 K . 6 L 7 The equation to the pair of tangents which can be drawn any 2 point y )8 to the 7 (x , 5 Er.from 0 ellipse is given by : SS = T where : 1 0 98 8015 xx yy x y x y 39 8 S + + + 1 ; S = 1 ; T 1. a b a b a b
(8) Pair of Tangents :
1
2
1
1
2
2
2
2
1
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
2 1
2 1
2
2
1
2
1
2
Page 144
Note: Locus of the point of intersection of the tangents which meet at right angles is called the Director Circle. The equation to this locus is x2 + y2 = a2 + b2 i.e. a circle whose centre is the centre of the ellipse & whose radius is the length of the line joining the ends of the major and minor axes.
(9) Chord of Contact : Equation to the chord of contact of tangents drawn from a point P(x1, y1) to the ellipse
x2 2
a
+
y2 2
b
= 1 is T = 0, where T =
xx1
+
2
a
yy1 b2
1
(10) Chord with a given middle point :
Equation of the chord of the ellipse
where S1 =
x 21 2
a
+
y 21 2
b
1 ; T
xx1 2
a
x2 a2
+
+
yy1 b2
y2 b2
= 1 whose middle point is (x1, y1) is T = S1,
1.
(11) Important results: Refering to the ellipse
x2 a2
+
y2 b2
=1
The tangent & normal at a point P on the ellipse bisect the external & internal angles between the focal distances of P. This refers to the reflection property of the ellipse which states that rays from one focus are reflected through other focus & viceversa. Hence we can deduce that the straight lines joining each focus to the foot of the perpendicular from the other focus upon the tangent at any point P meet on the normal PG and bisects it where G is the point where normal at P meets the major axis. The product of the lengths of the perpendicular segments from the foci on any tangent to the ellipse is b2 and the feet of these perpendiculars lie on auxiliary circle and the tangents at these feet to the auxiliary circle meet on the ordinate of P and that the locus of their point of intersection is a similar ellipse as that of the original one. The portion of the tangent to an ellipse between the point of contact and the directrix subtends a right angle at the corresponding focus. If the normal at any point P on the ellipse with centre C meet the major and minor axes in G and g respectively and if CF be perpendicular upon this normal then (i) PF. PG = b2 (ii) PF. Pg = a2 (iii) PG. Pg = SP. S'P (iv) CG. CT = CS2 (v) locus of the mid point of Gg is another ellipse having the same eccentricity as that of the original ellipse. [S and S' are the focii of the ellipse and T is the point where tangent at P meet the major axis] The circle on any focal distance as diameter touches the auxiliary circle. Perpendiculars from the centre upon all chords which join the ends of any perpendicular diameters of the ellipse are of constant length. If the tangent at the point P of a standard ellipse meets the axes in Q and R and CT is the perpendicular on it from the centre then, (i) QR. PT = a2 b2 (ii) least value of QR is a + b.
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839
Mathematics Concept Note IITJEE/ISI/CMI
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28. HYPERBOLA Hyperbola is the locus of a point which moves is such a ways so that difference of its distances from two fixed points is always constant.
Note: (i) PS–PS'=AA'; AA' is the length of transverse axis. (ii) eccentricity (e) > 1.
(1) Basic Definitions: Standard equation of the hyperbola is
x2 2
a
y2 b2
= 1,where b2 = a2(e2 1)
Eccentricity (e) :
e 1
b2 a2
Foci : S (ae, 0) & S' ( ae, 0). Equations Of Directrix : x=
a a and x = e e
Transverse Axis : The line segment AA' of length 2a in which the foci S and S' both lie is called the transverse axis of the hyperbola. Conjugate Axis : The line segment BB' between the two points B' (0, b) and B (0, b) is called as the conjugate axis of the hyperbola. Principal Axes : The transverse and conjugate axis together are called Principal Axes of the hyperbola. Vertices : A (a, 0) and A' (–a, 0) Focal distance: the focal distance of any point (x,y) on the hyperbola
x2 y2 1 are ex– a2 b2
a and ex+a. Focal Chord : A chord which passes through a focus is called a focal chord. Double Ordinate : A chord perpendicular to the transverse axis is called a double ordinate. Latus Rectum ( ) : The focal chord perpendicular to the transverse axis is called the latus rectum.
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 39 y 8 = 1.
2b2 = 2a (e2 1) = 2e (distance from focus to directrix) a Centre : The point which bisects every chord of the conic drawn through it is called the centre of the conic.
=
C (0,0) the origin is the centre of the hyperbola Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
x2 a2
2
b2
Page 146
Note: Since the fundamental equation to the hyperbola only differs from that to the ellipse in having b2 instead of b2 it is found that many propositions for the hyperbola are derived from those for the ellipse by simply changing the sign of b2.
(2) Conjugate Hyperbola : Two hyperbolas such that transverse and conjugate axes of one hyperbola are respectively the conjugate & the transverse axes of the other are called Conjugate Hyperbolas of each other. for example:
x2 a2
y2 b2
= 1 and
x2 a2
+
y2 b2
= 1 are conjugate hyperbolas of each other..
Note: (i) If e and e' are the eccentricities of the hyperbola & its conjugate then 1/e2 + 1/e'2 = 1. (ii) The foci of a hyperbola and its conjugate are concyclic and form the vertices of a square. (iii) Two hyperbolas are said to be similar if they have the same eccentricity. (iv) Two similar hyperbolas are said to be equal if they have same latus rectum. (v) If a hyperbola is equilateral then the conjugate hyperbola is also equilateral.
(3) Auxiliary Circle : A circle drawn with centre C and transverse axis as a diameter is called the Auxiliary Circle of the hyperbola. Equation of the auxiliary circle is x2 + y2 = a2. In the figure P and Q are called the "Corresponding Points" on the hyperbola and the auxiliary circle.
(4) Parametric Representation : y a m ara b h S 2 38 K. 7, 6 .[0, 2) 7 where is a parameter, which is termed as eccentric angle,. L Er 8102 2 520 58 9 point If P( ) (asec , btan ) is on the hyperbola then corresponding 01 8 9 Q( ) (acos , asin ) is on the auxiliary circle. 83 The equations x = a sec and y = b tan together represents the hyperbola
x2 2
2
2
=1
The equation to the chord of the hyperbola joining two points with eccentric angles & Mathematics Concept Note IITJEE/ISI/CMI
Page 147
x y cos 2 sin 2 = cos 2 . a b
is given by
(5) Position of Point 'P' with respect to Hyperbola : x12 y12 The quantity S1 2 2 1 is positive, zero or negative according as the point (x1,y1) b a lies inside, on or outside the curve respectively.
(6) Line And A Hyperbola :
The straight line y=mx+c is a secant, a tangent or passes outside the hyperbola
x2 2
a
y2 b2
=1
according as c2 > or = or < a2m2 b2 respectively..
(7) Tangent of Hyperbola (i) Slope Form : y = mx a2m2 b2
x2
hyperbola
a2
y2 b2
can
be
taken
xx1
yy1
a2
b2
the
x2 a2
y2 b2
to
the
= 1 at the point (x1, y1)
= 1.
(iii) Parametric Form : Equation of the tangent to the hyperbola point.
tangent
= 1 having slope 'm'.
(ii) Point Form : Equation of tangent to the hyperbola
is
as
(a sec , b tan ) is
x2 a2
y2 b2
= 1 at the
x sec y tan = 1. a b
Note: a cos 2 b sin 2 , (i) Point of intersection of the tangents at ' ' and ' ' is cos cos 2 2
(ii) If   = , then tangents at these points ' ' and ' ' are parallel.
(8) Normal of Hyperbola:
ma r a h a b S . 82 K . 6 L 7 Er. 81027 5058 9 801 point P(acos , btan ) on the hyperbola 839
(i) The equation of the normal to the hyperbola
is
a2 x b2y + = a2 + b2. x1 y1
(ii) The
equation
x2 2
a
y2 2
b
of the normal at the
= 1 is
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
x2 2
y2 2
= 1 at the point P(x1,y1) on it
ax by + = a2 + b2 or sec tan
Page 148
(9) Pair of Tangents : The equation to the pair of tangents which can be drawn from any point (x1,y1) to the
x2
hyperbola
S
x2 2
a
a2 y2
y2
= 1 is given by : SS1 = T2 where :
b2
1
2
b
;
S1 =
x 21 2
a
y 21
1
2
b
; T
xx1 a2
yy1 b2
1.
Note: The locus of the intersection point of tangents which are at right angles is known as the Director Circle of the hyperbola. The equation to the director circle is : x2 + y2 = a2 b2. If b2 < a2 this circle is real. If b2 = a2 (rectangular hyperbola) the radius of the circle is zero & it reduces to a point circle at the origin. In this case the centre is the only point from which the tangents at right angles can be drawn to the curve. If b2 > a2, the radius of the circle is imaginary, so that there is no such circle & so no pair of tangents at right angle can be drawn to the curve.
(10) Chord of Contact : Equation to the chord of contact of tangents drawn from a point P(x1,y1) to the hyperbola
x2 2
a
y2 2
b
= 1 is T = 0, where T
xx1
2
a
yy1 b2
1
(11) Chord with a given middle point : Equation of the chord of the hyperbola
where S1
x 21 2
a
y 21 2
b
1 ; T
xx1 2
a
yy1 b2
x2 a2
y2 b2
= 1 whose middle point is (x1, y1) is T = S1,
1.
(12) Diameter : The locus of the middle points of a system of parallel chords with slope 'm' of a hyperbola is called its diameter. It is a straight line passing through the centre of the hyperbola and has the equation y =
b2x a2m
, all diameters of the hyperbola passes through its centre.
(13) Asymptotes : Definition : If the length of the perpendicular let fall from a point on a hyperbola to a straight line tends to zero as the point on the hyperbola moves to infinity along the hyperbola, then the straight line is called the Asymptote of the hyperbola.
ma r a Equations of Asymptote : h S . 82 K . 6 L 7 x b x y x y r. 1027 058 + = 0 and =0 =0 E a b a b a a 98 8015 Note: 39 8 (i) A hyperbola and its conjugate have the same asymptote. 2
2
2
2
(ii) The equation of the pair of asymptotes differs from the equation of hyperbola Mathematics Concept Note IITJEE/ISI/CMI
Page 149
(or conjugate hyperbola) by the constant term only. (iii) The asymptotes pass through the centre of the hyperbola & are equally inclined to the transverse axis of the hyperbola. Hence the bisector of the angles between the asymptotes are the principle axes of the hyperbola. (iv) The asymptotes of a hyperbola are the diagonals of the rectangle formed by the lines drawn through the extremities of each axis parallel to the other axis.
(14) Rectangular Or Equilateral Hyperbola : The hyperbola in which the lengths of the transverse & conjugate axis are equal is called an Equilateral Hyperbola, the eccentricity of the rectangular hyperbola is Rectangular Hyperbola (xy = c2) :
2.
It is referred to its asymptotes as axis of coordinates. Vertices: (c, c) and ( c, c) ; Foci: ( 2 c,
2 c) and ( 2 c, 2 c)
Directrix: x + y =
2c
Latus Rectum: = 2 2 c Parametric equation x = ct, y = c/t , t R {0}
Note: (i) Equation of a chord joining the points P(t1) and Q(t2) is x + t1t2y = c(t1+t2). x
y
x
(ii) Equation of the tangent at P(x1,y1) is x + y = 2 and at P(t) is +ty = 2c. t 1 1 (iii) Equation of the normal at P(t) is xt3 yt = c(t4 1). (iv) Chord with a given middle point as (h,k) is kx + hy = 2hk.
(15) Important Results :
Locus of the feet of the perpendicular drawn from focus of the hyperbola
Light Ray
The ellipse
x2
+
y2
YTangent Q P
= 1 and the hyperbola
Xa S rm a x y h S = 1 (a > k > b > 0) are confocal . 82 K a k k b . 6 L 7 and therefore orthogonal. Er. 81027 5058 9tangent8meets The foci of the hyperbola and the points P and Q in which any 01 the tangents at the vertices are concyclic with PQ as diameter of the circle. 39 8 If from any point on the asymptote a straight line be drawn perpendicular to the transverse 2
2
y2
= 1 upon a2 b2 any tangent is its auxiliary circle i.e. x2+y2=a2 and the product of these perpendiculars is b2. The portion of the tangent between the point of contact and the directrix subtends a right angle at the corresponding focus. The tangent & normal at any point of a hyperbola bisect the angle between the focal radii. This spells the reflection property of the hyperbola as "An incoming light ray" aimed towards one focus is reflected from the outer surface of the hyperbola towards the other focus. It follows that if an ellipse and a hyperbola have the same foci, they cut at right angles at any of their common point.
Note:
x2
2
a
b2
S’
2
2
2
2
axis, the product of the segments of this line, intercepted between the point & the curve is Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
Page 150
always equal to the square of the semi conjugate axis. Perpendicular from the foci on either asymptote meet it in the same points as the corresponding directrix & the common points of intersection lie on the auxiliary circle.
The tangent at any point P on a hyperbola
x2 a2
y2 b2
= 1 with centre C, meets the asymp
totes in Q and R and cuts off a CQR of constant area equal to ab from the asymptotes & the portion of the tangent intercepted between the asymptote is bisected at the point of contact. This implies that locus of the centre of the circle circumscribing the CQR in case of a rectangular hyperbola is the hyperbola itself & for a standard hyperbola the locus would be the curve: 4(a2x2 + b2y2) = (a2 + b2)2.
If the angle between the asymptote of a hyperbola
x2 a2
y2 b2
= 1 is 2 then the eccentricity
of the hyperbola is sec .
A rectangular hyperbola circumscribing a triangle also passes through the orthocentre of
this triangle. If ct1 ,
c i = 1, 2, 3 be the angular points P, Q, R then orthocentre is t 1
c , ct1 t 2 t 3 . t1t 2 t3
If a circle and the rectangular hyperbola xy = c2 meet in the four points t1, t2, t3 & t4,then (i) t1 t2 t3 t4 = 1 (ii) the centre of the mean position of the four points bisects the distance between the centres of the two curves. (iii) the centre of the circle through the points t1, t2 & t3 is: 1 c c 2 t1 t2 t3 , t1t2 t3 2
1 1 1 t1 t 2 t3 t1 t2 t3
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29. VECTORS (1)
Basic Definitions : Vector is a physical quantity having magnitude and definite direction. A directed line seg ment AB represents a vector having initial point 'A' and the terminal point 'B'. Magnitude of vector AB is represented by  AB . In 3dimensional space if a point P is represented by (x1 , y1 , z1) , then position vector of 'P' If P(x , y , z ) and Q(x , y , z ) are two points in space , is given by OP x1i y1j z1k. 1 1 1 2 2 2 then : and OQ x i y j z k OP x1i y1j z1k 2 2 2 PQ = (position vector of Q)–(position vector of P) (z z )k PQ (x2 x1 )i (y2 y1 )j 2 1  PQ  (x2 x1 )2 (y2 y1 )2 (z2 z1 )2
(i) Zero Vector/Null Vector : A vector having zero magnitude and indeterminate direction is termed as zero vector. For example : AA 0 (i.e. vector having same initial and terminal point). (ii) Unit Vector :
A vector having unit magnitude is termed as unit vector. Unit vector in the direction of a is
represented by a. a a or a  a  a.  a (iii) Parallel Vector or Collinear Vectors : A set of vectors is termed as parallel (or collinear) vectors if they have common line of support or parallel line of support. Parallel vectors are termed as like vectors if they have same direction and further if like vectors have same magnitude , then vectors are termed as equal vectors. Parallel vectors are termed as unlike vectors if they have opposite sense of direction and negative of vector a is represented by – a
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 39 are all collinear (or parallel)8 vectors
Vectors a , b , c , d , e and f Vectors a , e , d and f are like vectors. Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
Page 152
Vectors a and b are unlike vectors , similarly b and d are unlike vectors. Let a and b are two nonzero collinear vectors , then a b , where R . I f a(a1 , a2 , a3 ) and b (b1 , b2 , b3 ) are col l i n ear or paral l el vect ors , t hen . (b i b j b k) (a1i a2j a3 k) 1 2 3
a1 a a 2 3. b1 b2 b3
Note:
Three points A(a) ,B(b) and c(c) are collinear , if there exists scalars x , y and z such that xa yb zc 0 , where x + y + z = 0 and x , y , z are not simultaneously zero.
(iv) Coplanar Vectors : Vectors which lie one the same plane of support or parallel plane of support are termed as coplanar vectors.
Note:
Two vectors are always coplanar and if three vectors a , b and c are coplanar then any y one of the vector can be expressed as a linear combination of other tow vectors.
If a , b and c are coplanar vectors , then (a b).c 0 If a(a1 , a2 , a3 ) , b (b1 , b2 , b3 ) and c (c1 , c2 , c3 ) are coplanar , then
a1 a2 a3 a b c b b b 0 1 2 3 c1 c2 c3
ma r a h 2 S . Three points in space are always coplanar and if A(a) , B(b) are four 8 , C(c) and6 D(d) K . L 7 7 . r 8 002and x5,0 coplanar points , then xa yb zc wd 0 , where x +E y + z + w1= y ,5 z , w are 8 9 801 scalars which are not simultaneously zero. 39 8 If A , B , C and D are four point lying on a plane , then AB AC AC 0 .
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(v) Coinitial Vectors : Vectors which are having same initial points are termed as coinitial vectors. For example : AB , AC , AD are coinitial vectors. (vi) Coterminous Vectors : Vectors which have same terminal point are termed as coterminous vectors. For example : PA , QA ,RA are coterminous vectors.
(2)
Linear Combination of Vectors : Linearly Independent Vectors : A set of vectors a1 , a2 , ..., an is said to be linearly independent iff x1 a1 x2 a2 ... xn an 0 x1 x2 ... xn 0 Linearly Dependent Vectors: A set of vectors a1 , a2 , ..., an is said to be linearly dependent iff there exists scalars x1 , x2 , ... , xn not all zero such that x1 a1 x2 a2 ... xn an 0
Note:
If two nonzero vectors a and b are linearly dependent , then a and b are collinear or parallel ( a b 0 ) If two nonzero vectors a and b are linearly independent , then a b 0 and xa yb 0 x y 0. 0 x y 0. For example : xi yj If three nonzero vectors a , b and c are linearly dependent , then any one of the
vector can be other two vectors which further implies that a , b and c are coplanar.. ( [a b c] 0) If three nonzero vectors a , b and c are linearly independent , then a b c c and
hence the vectors a , b and c are noncoplanar..
Any set of four or more nonorthogonal system of vectors is always linearly dependent
(3)
Addition of Vectors :
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 8 3c)9 (a b) c a (b
If the vectors a and b are represented by AB and AD , then a b is the vector addition which is represented by AC , where AC is the diagonal of parallelogram ABCD.
Note:
ab b a a o o a a
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Triangle inequality:  a b   a b 
 a b   a b 
Sign of equality holds if a and b are like vectors.  a b   a b  Angle between a and b is 90o. a b
(4)
 a 2  b 2 2  a   b  cos
Multiplication of a Vector by Scalar : Scalar multiplication of vector a and scalar quantity k , where K R , is represented by Ka . Ka is vector which is parallel to a and having magnitude K times the magnitude of vector a.
Note: If P , K R , then K(a) (a)K K(a) (K P)a Ka Pa
(5)
K(Pa) P(Ka) PK(a) K(a b) Ka Kb.
Basic Application of Vectors to Geometry: (i) If a and b are the position vectors of two points A and B , then position vector of points A and B , then position vector of point C (r) which divides AB in the ratio m : n is given by mb ba r mn mb ba r mn
(for internal division). (for external division).
(ii) If A(a) , B(b) and C(c) are the vertices of ABC , then :
ab c P.V. of Controid 3 b c  a  c a b  a b  c P.V. of incentre  a b  b c   c a (iii) If A(a) , B(b) , C(c) and D(d) are the vertices of a tetrahedron , then P.V. of centroid of ab c d tetrahedron is given by . r 4
(6)
ma r a h S . 82 K . 6 L 7 Scalar or Dot Product : Er. 81027 5058 9 801 9 Scalar product of two vector a and b is denoted by a.b and8 is3 defined as : a.b  a   b  cos , where is the angle (0 ) between vectors a and b .
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In OBL , OL  b  cos a.b a.b OL (Projection of b on a ) OL 2 a and LB b  a   a Similarly , In OAM , OM  a  cos a.b OM (Projection of a on b ) b a.b OM  b 2
a.b b and MA a  b 2
b.
Properties of dot Product:
i.i j.j k.k 1
i.j j.k k.i 0 and b b i b j b k , then a.b a b a b a b . If a a1i a2j a3 k 1 2 3 1 1 2 2 3 3 Angle between vectors a and b is given by :
a.b
cos  ab 
a1b1 a2b2 a3b3
a12 a22 a32
b
2 1
b22 b32
If a1b1 a2b2 a3b3 0, then is Acute angle. If a1b1 a2b2 a3b3 0, then is obtux angle.
If a1b1 a2b2 a3b3 0, then is 90o. If
a1 a a 2 3 , then 0 or . b1 b2 b3
a.a  b 2 a.b b.a (b.j)j (c.k)k a a.(b c) a.b a.c (a.i)i  a  b  a.b  a  b   a b 2  a 2  b 2 2 a.b  a b c 2  a 2  b 2  c 2 2(a.b b.c c.a)
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839
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(7)
Vector or Cross Product : Vector product of two vectors a and b is denot ed by a b an d is defin ed as: is the unit a b  a  b  sin n , where is the angle between vector a and b , and n vector perpendicular to both a and b . (direction of n is determined by right handed screw
rule)
Geometrically  a b  represents the area of parallelogram whose two adjacent sides are represented by a and b
Properties of Cross Product : 0 i i j j k.k a a 0
, j k i and k i j i j k
i and b b i b j b k , then a b a If a a1i a2j a3k 1 2 3 1
j
k
a2
a3
b1 b2 b3 a b b a a (kb) k(a b)
a (b c) (a b) a c a b 0 a and b are parallel
a b Unit vector perpendicular to both a and b is given by : n  a b  a b 2 (a.b)2  a 2  b 2 (Lagrange ' s Idantity) If A(a) , B(b) and C(c) are the vertices of ABC , then area of
1 a b c c a a b b c c a 0 A , B and C are collinear. 2 If A(a) , B(b) , C(c) and D(d) are the vertices of a quadrileteral , then area of Q uad ABC
(ABCD)
1 a b b c c d d a 2 1 AC BD 2
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8.
Scalar Triple Product : The scalar triple product of three vectors a , b and c is defined as : a b.c  a  b  c  sin θ cos where θ is the angle between a and b and is the angle between a b and c . It is also written as [a b c]
Scalar triple product geometrically represents the volume of the parallelopiped whose three coterminous edges are represented by a , b and c i.e. v [a b c]
Note: a1 a2 a3 (i) If a a1i a2 j a3k ;b b1i b2 j b3k and c c1i c2 j c3k then [a b c] b1 b2 b3 c1 c2 c3
(ii) In a scalar triple a .(b c) (a b).c or
product the position of dot and cross can be interchanged i.e. a b c b c a c b a (iii) a.(b c) a,(c b) i.e. [a b c] [a c b] (iv) If a , b , c are coplanar [a b c] 0.
(v) Scalar product of three vectors, two of which are equal or parallel is 0. (vi) If a , b , c are noncoplanar then [a b c] 0 for right handed system and [a b c] 0 for left handed system [K a b c] [K a b c] (vii) The volume of the tetrahedron OABC with O as origin and the pv's of A, B and C being 1 a , b and c respectively is given by V = 6 [a b c]
The position vector of the centroid of a tetrahedron if the pv's of its vertices are 1 a , b , c and d are given by [a b c d] 4
ma r a h vertices2 to the Note that this is also the point of concurrency of the lines joining the S . K . tetrahedron. 68In case the centroids of the opposite faces and is also called the centre ofL the 7 7 . r the four 8 tetrahedron is regular it is equidistant from the vertices faces of the5tetraheEand 02 1 0 5 8 dron. 9 801 839 Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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(9)
Vector Triple Product : Vector quantity a (b c) or (a b) c is termed as the vector triple product for any three vectors a , b and c . a (b c) (a.c)b (a.b)c (a b) c (a.c)b (b.c)a Let a (b c) R , then by definition of cross product , R is perpendicular to both a and b c . a.R 0 and R.(b c) 0 R is normal to a and R , b and c are coplanar. a (b c) represents a vector which is normal to a and lies in the plane containing b and c .
Note: a (b c) Unit vector normal to a and lying in plane of b and c  a (b c)  In general , a (b c) (a b) c.
(10) Scalar Product of four Vectors : If a , b , c , d are four vectors , the products (a b).(c d) is called scalar products of four vectors. a.b b .c i.e. , (a b).(c d) a.d b .d
(11) Vector product of four Vectors : If a , b , c , d are four vectors , the products (a b) (c d) is called scalar products of four vectors. i.e. , (a b) (c d) [ab d]c [ab c]d (a b) (c d) [a c d]b [a c c]a.
(12) Reciprocal System of Vector : If a a' , a.b '
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839
, b , c are three non coplanar vector such that [a b c] 0, then the system of vectors b ' , c whi ch sat isfy t h e con dit i on t h at a.a' = b.b' = c.c' = 1 an d a.c ' b.a' b.c ' c.a ' c.b ' 0.
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bc a' [ab c]
ca ab b ' and c ' [ab c] [ab c]
Properties of reciprocal system (a) [ab c] [a'b ' c] 1 b ' c ' c ' a' a' b ' b c (b) a [a'b ' c '] [a'b ' c '] [a' b ' c ']
is its own reciprocal (c) The system of unit vectors i , j , k ' k i.e. i' i j' j and k Not that any vector can be expressed in terms of a' , b ' and c ' as they also constitute a system of non coplanar vectors.
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30. THREE DIMENSIONAL GEOMETRY Coordinates of any point 'P' in the space is given by (x, y, z) and the point is located with reference to three mutually perpendiculer corordinates axes ox, oy and oz.
Position vector of point P is given by OP xi yj zk .
(1)
Distance formula : Distance between any two points A(x1,y1,z1) and B(x2,y2,z2) is given by:
(x1 x2 )2 (y1 y2 )2 (z1 z2 )2
Note:
If position vector of two points A and B are given as OA and OB , then
AB =  OB OA 
(2)
AB = (x2i + y2j + z2k) (x1i + y1j + z1k)
AB =
(x1 x2 )2 (y1 y2 )2 (z1 z2 )2
Section Formula : If point P divides the distance between the points A(x1, y1, z1) and B(x2, y2, z2) internally in the ratio of m : n , then coordinates of P are given by : mx2 nx1 my2 ny1 mz2 nz1 , , mn mn mn
(3)
Centroid of a Triangle ABC : x x2 x3 y1 y2 y3 z1 z2 z3 G 1 , , 3 3 3
(4)
Incentre of Triangle ABC :
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 39 cz 8
ax bx2 cx3 ay1 by2 cy3 az1 bz2 I 1 , , ab c ab c ab c
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(5)
Centroid of a tetrahedron : A (x1, y1, z1) , B(x2, y2, z2) , C (x3, y3, z3) and D(x4, y4, z4) are the vertices of a tetrahedron then coordinate of its centroid (G) is given by : x x2 x3 x 4 y1 y2 y3 y 4 z1 z2 z3 z 4 G 1 , , 4 4 4
(6)
Direction Cosines And Direction Ratios : Direction cosines: Let α , β , γ be the angles which a directed line makes with the positive directions of the axes of x, y and z respectively, then cos α , cos β , cos γ are called the direction cosines of the line. The direction cosines are usually denoted by ( , m, n).
cos , m cos , n cos .
Note: (i) If , m, n be the direction cosines of a line, then 2 + m2 + n2 = 1. (ii) If , m, n are direction cosines of line L then ˆi mˆj nkˆ is a unit vector parallel to the line L. Direction ratios : Let a, b, c be proportional to the direction cosines , m , n then a, b, c are called the direction ratios.
m m a b c
Note: (i) If , m, n be the direction cosines and a, b , c be the direction ratios of a vector,, then a b c ,m ,n a2 b2 c2 a2 b2 c2 a2 b2 c2
(ii) If the coordinates of P and Q are (x1, y1, z1) and (x2, y2, z2) respectively, then the direction ratios of line PQ can be assumed as: a = x2 x1, b = y2 y1 and c = z2 z1 , and the direction cosines of line PQ are
=
x 2 x1 PQ
, m=
y 2 y1
and n =
PQ
z2 z1 PQ
.
(iii) If a, b, c are the direction ratios of any line L then a ˆi bˆj ckˆ is a vector parallel to the line L. (iv) Direction cosines of axes: Direction cosines of xaxis are (1, 0,0) Direction cosines of yaxis are (0,1,0) Direction cosines of zaxis are (0,0,1)
(7)
ma r a h S . 82 K . 6 L 7 Angle Between Two Line Segments : Er. 81027 5058 9 8 If two lines have direction ratios a , b , c and a , b , c respectively then 01vectors parallel 9 to the lines are a i + b j + c k and a i + b j + c k and angle between them is given as : 83 1
1
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1
2
1
2
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2
2
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cos
a1a2 b1b2 c1c2 a12 b12 c12 a22 b22 c22
Note: (i) The line will be perpendicular if a1a2 + b1b2+ c1c2 = 0 a1 b1 c1 (ii) The lines will be parallel if a b c 2 2 2
(iii) Two parallel lines have same direction cosines i.e. 1 = 2 , m1 = m2 , n1 = n2.
A LINE (8)
Equation Of A Line (i) A straight line in space is characterised by the intersection of two planes which are not parallel and therefore, the equation of a straight line is a solution of the system constituted by the equations of the two planes, a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0. This form is also known as nonsymmetrical form. (ii) The equation of a line passing through the point (x1, y1, z1) and having direction ratios a, b, c is
x x1 y y1 z z1 . This form is called symmetric form. A a b c
general point on the line is given by (x1 a , y1 b , z1 c) , where is a scalar quantity (iii) Vector equation: Vector equation of a straight line passing through a fixed point with position vector a and parallel to a given vector b is r a b , where λ is a scalar quantity. (iv) The equation of the line passing through the points (x1, y1, z1) and (x2, y2, z2) is x x1 y y1 z z1 x2 x1 y2 y1 z2 z1
(v) Vector equation of a straight line passing through two points with position vectors a and b is r a (b a) . (vi) Reduction of cartesion form of equation of a line to vector form & vice versa x x1 y y1 z z1 a b c
(ai bj ck) r (x1i y1 j z1k)
Note: (a) If lines
x x1 y y1 z z1 a b c
and
x x2 y y2 z z2 intersect each others at a' b' c'
ma r a h a b c 0 unique point, then : S . 82 K . 6 a' b' c' L 7 r. 1027 058 E 98 b0 .1a5 a 0. (b) If lines r a b and r a b are intersecting, then : b 8 839 x2 x1 y2 y1 z2 z1
1
1
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(9)
Foot, Length Of perpendicular From A point To A Line : Let equation of the line be
x x1 y y1 z z1 a b c
.....(i)
and A(x' , y' , z') be the point. Any point on line (i) is P (a x1 , b y1 , c z1 )
....(ii)
If P is the foot of the perpendicular from A on the line, then AP is perpendicular to the line.
(a x1 x ')a ( b y1 y ')b ( c z1 z ')c 0
From above equation, value of ' ' is obtained and putting this value of in (ii), we get the foot of perpendicular from point A on the given line , since foot of perpendicular P is known, then the length of perpendicular is given by AP.
(10) To find image of a point w.r.t a line :
Image of point P(x' , y' , z') w.r.t. line
x x1 y y1 z z1 . a b c
Let image of P(x' , y' , z') be Q( , , ) , hence PQ is perpandiclor to the given line a( x ') b( y ') c(r z ') 0
......(i)
Now midpoint of PQ lies on the line x' y' r z' x1 y1 z1 2 2 2 a b c
......(ii)
from above equation, point Q( , , ) can be calculeted in terms of , which further satisfy the equation (i) and hence value of ' ' can be obtained After getting the value of ' ' point Q( , , ) cam ne calculeted from equation (ii).
(11) Skew Lines : The straight lines which are nonparallel and nonintersecting are termed as skew lines. If
x x1 y y1 z z1 a b c
and
x x2 y y2 z z2 a' b' c'
are
skewli n es
,
th en
x2 x1 y2 y1 z2 z1 a
b
c
a'
b'
c'
0
and the shortest distance (S.D.) between the skewlines is
x2 x1 a
y2 y1 z2 z1 b c
a m r a given by : . h S.D. S . 82 K . (bc ' b ' c) (ac ' a' c) (ab ' a'b) 6 L 7 Er. 81027 5058 Note: 9 801 (i) If r a b and r a b are the skewlines in vector 39form, then 8 a'
b'
2
1
b
1
1
c'
2
2
2
2
b2 . a2 a1 0 and the shortest distance between them is given by :
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b1 b2 a2 a1 S.D.  b1 b2 
(ii)
Shortest distance 'd' between the two parallel lines r a1 λb and
(a2 a1 ) b . r a2 μb is given by : d = b 
A PLANE If line joining any two points on a surface is perpendicular to some fixed straight line , then this surface is called a plane and the fixed line is called the normal to the plane.
(12) Equation Of A Plane (i) Normal form of the equation of a plane is x + my + nz = p, where, , m, n are the direction cosines of the normal to the plane and p is the distance of the plane from the origin. (ii) General form : ax + by + cz + d = 0 is the equation of a plane, where a, b , c are the direction ratios of the normal to the plane.
(iii) The equat ion of a plane passing t hrough t he point (x 1, y1, z1) is given by a(x x1) + b(y y1) + c(z z1) = 0 where a, b, c are the direction ratios of the normal to the plane. (iv) Plane through three points: The equation of the plane through three noncollinear points(x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) is given by : x x1
y y1
z z1
x2 x1 y2 y1 z2 z1 0 x3 x1 y3 y1 z3 z1
(iv) Intercept Form : The equation of a plane cutting intercept a, b, c on the axes is
x y z = 1. a b c (v) Vector form : The equation of a plane passing through a point having position vector a normal to vector n is (r a).n 0 or r .n a.n (vi) Vector equation of a plane normal to unit vector n and at a distance d from the origin is r .n d
Note: (a) Coordinate planes (i) Equation of yzplane is x = 0 (ii) Equation of xzplane is y = 0 (iii) Equation of xyplane is z = 0.
ma r a h S . 82 K (b) Planes parallel to the axes : . 6 L 7 r.of the10plane 27 parallel If a = 0, the plane is parallel to xaxis i.e. equation 58to the E 0 5 8 xaxis is by + cz + d = 0. 1 9 8l0 S i mi l arl y, equ at i on of pl anes parall el to yaxi s an d paral el t o zaxi s are 9 83 ax + cz + d = 0 and ax + by + d = 0 respectively. (c) Plane through origin : Equation of plane passing through origin is ax + by + cz = 0.
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(d) Any plane parallel to the given plane ax + by + cz + d = 0 is ax + by + cz + k = 0. Distance between two parallel planes ax + by + cz + d1 = 0 and ax + by + cz + d2=0  d1 d2 
is given as
a2 b2 c2
(e) Equation of a plane passing through a given point & parallel to the given vectors: The equation of a plane passing through a point having position vector a and parallel to b and c is r a b c (parametric form) where λ and μ are scalars. or r . (b c) a.(b c) (non parametric form) (f) A plane ax + by + cz +d = 0 divides the line segment joining (x1, y1, z1) and ax1 by1 cz1 d (x2, y2, z2) in the ratio ax by cz d 2 2 2
(g) The xyplane divides the line segment joining the points (x1, y1, z1) and (x2, y2, z2) in the ratio
x z1 y . Similarly yzplane in 1 and zxplane in 1 x2 z2 y2
(h) Coplanarity of four points The points A(x1 , y1 , z1), B(x2 , y2 , z2) C(x3 , y3 , z3) and D(x4 , y4 , z4) are coplanar , then x2 x1
y2 y1
z2 z1
x3 x1
y3 y1
z2 z1 0
x4 x1 y4 y1 z4 z1
(13) Volume Of A Tetrahedron : Volume of a tetrahedron with vertices A(x1, y1, z1), B(x2, y2, z2), C(x3, y3, z3) and D(x4, y4, z4) is given by V
x2 x1 y2 y1 z2 z1 1 x3 x1 y3 y1 z2 z1 6 x4 x1 y4 y1 z4 z1
(14) Sides of a plane: A plane divides the three dimensional space in two equal parts.Two points A (x1 , y1 , z1) and B(x2 , y2 , z2) are on the same side of the plane ax + by + cz + d = 0 if ax1 + by1 + cz1 + d and ax2 + by2 + cz2 + d are both positive or both negative and are opposite side of plane if both of these values are in opposite sign.
(15) A Plane and A Point :
ma r a h S . ax by cz d 82 K . 6 p L 7 . a b c Er. 81027 5058 9 80 1 (ii) The length of the perpendicular from a point having position3 9 a to plane 8 vector
(i) Distance of the point (x1 , y1 , z1) from the plane ax + by + cz + d = 0 is given by : 1
1
2
1
2
2
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r .n d is given by:
a.n d p n
(iii) The coordinates of the foot of perpendicular from the point (x1 , y1 , z1) to the plane ax + by + cz + d = 0 are given by : x x1 y y1 z z1 (ax1 by1 cz1 d) a b c a2 b2 c2
(iv) Reflection of a point w.r.t. a plane. The coordinate of the image of point (x1 , y1 , z1) w.r.t. the plane ax + by + cz + d = 0 are given by : x x1 y y1 z z1 (ax1 by1 cz1 d) 2 a b c a2 b2 c2
(16) Angle Between Two Planes : (i) If two planes are ax + by + cz + d = 0 and a'x + b'y + c'z + d' = 0, then angle between these planes is the angle between their normals. Since direction ratios of their normals are
a,b , c and a', b ', c ' respectively, hence , ' ' the angle betweeen cos
them, is given by
aa' bb' cc' a2 b2 c2 a'2 b'2 c'2
Planes are perpendicular if aa' + bb' + cc' = 0 and planes are parallel if
a b c = = a' b' c'
(ii) The angle θ between the planes r .n1 d1 and r . n2 d2 is given by, y, n1 .n2 cos  n . n  2 1
Planes are perpendicular if n1 .n2 0 and planes are parallel if n1 n2 .
(17) Angle Between A Plane And A Line : (i) If θ is the angle between line
x x1 y y1 z z1 and the plane m n
a bm cn ax + by + cz + d = 0 , then sin θ = 2 2 2 2 2 2 (a b c ) m n
.
(ii) Vector form : If θ is the angle between a line r a b and r .n d, then b.n sin  b  n 
ma r a h S . 82 K . 6 L 7 m n . 27 058 (iii) Condition for perpendicularity : = = or b E n r0 0 a 1 b c 8 015 9 (iv) Condition for parallel case : a + bm + cn = 0 or b.n 0 398 8
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(18) Condition For A Line To Lie in A Plane : x x1 y y1 z z1 will lie in plane ax + by + cz + d = 0, m n if ax 1 + by1 + cz1 + d = 0 and a + bm + cn = 0. holds simultanously (ii) Vector form :Line r a λb will lie in plane r .n d if b.n 0 and a.n d holds simultanously.
(i) Cartesian form: Line
(19) Angle Bisectors : (i) The equations of the planes bisecting the angle between two given planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 are : a1x b1y c1z d1
a12 b12 c12
a2x b2y c2z d2 a22 b22 c22
(ii) Equation of bisector of the angle containing origin: If both the constant terms d1 and d2 are positive ,then the positive sign in a1x b1y c1z d1 a12
b12
c12
a2x b2y c2z d2 gives the bisector of the angle which contains
a22 b22 c22
the origin. (iii) Bisector of acute/obtuse angle: First make both the constant terms positive , then a1a2 + b1b2 + c1c2 > 0 origin lies on obtuse angle a1a2 + b1b2 + c1c2 < 0 origin lies in acute angle
(20) Reduction Of NonSymmetrical Form To Symmetrical Form : Let equation of the line in nonsymmetrical form be a1x + b1y + c1z + d1 = 0, a2x + b2y + c2z + d2 = 0. To find the equation of the line in symmetrical form, we must know (i) its direction ratios (ii) coordinate of any point on it. Direction ratios : Let , m, n be the direction ratios of the line. since the line lies in both the planes, it must be perpendicular to normals of both planes. So a1 + b1m + c 1 n = 0 , a2 + b2m + c2n = 0. From these equations, proportional values of , m, n can be calculated by cross  multiplication as
m n b1c2 b2c1 c1a2 c2a1 a1b2 a2b1
Alternative method i j The vector a1 b1 a2 b2
k c1
= i(b1c2–b2c1) + j(c1a2–c2a1) + k(a1b2–a2b1) will be parallel to the
c2
line of intersection of the two given planes.
ma r a h S Point on the line : As , m, n cannot be zero simultaneously, so at least one coordinate 2 . 8 K 6 amust be nonzero. Let a b –a b 0, then the line cannot L be.parallel7 to7xy plane, so it intersect it. Let it intersect xyplane in (x , y , 0). Then 0 an d Era.x 8+1b0y2 + 5d 0=58 a x + b y + d = 0. solving these, we get a point on the line. equation becomes. 1 9Then its 80 9 3 xx yy z0 8 = = : m : n = (b1c2–b2c1) : (c1a2–c2a1) : (a1b2 –a2b1)
1
2
2
1
1
2
1
2
1
1
1
1
1
1
1
2
1
b1c2 b2 c1
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
1
c1a2 c2a1
a1b2 a2b1
Page 168
If 0, take a point on yzplane as (0, y1, z1) and if m 0, take a point on xzplane as (x1, 0, z1).
(21) Family of Planes : (i) Any plane passing through the line of intersection of nonparallel planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 is given by : (a1x + b1y + c1z + d1) + λ (a2x + b2y + c2z + d2) = 0 (ii ) The equ at i on of pl an e passin g t h rou gh t h e i nt ersect ion of t h e pl anes r .n1 d1 and r . n2 d2 is given by r .(n1 n2 ) d2 d1 where λ is arbitrary scalar quantity.
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Mathematics Concept Note IITJEE/ISI/CMI
Page 169
31. TRIGONOMETRIC RATIO AND IDENTITIES An angle is the amount of revolution which a line OP revolving about the point O has undergone in passing from its initial position OA into its final position OB.
If the rotation is in the clockwise sense, the angle measured is negative and it is positive if the rotation is in the anticlockwise sense. The two commonly used systems of measuring an angle are 1. Sexagesimal system is which 1 right angle = 90 degrees (90º) 1 degree = 60 minutes (60') 1 minute = 60 seconds (60") 2. Circular systems in which the unit of measurement is the angle subtended at the centre of a circle by an are whose length is equal to the radius and is called a radian. Relation between degree and radian radian = 180 degrees 1 radian =
180 degress
= 57º 17' 45" (approximately) The triagonometrical ratios of an angle are numberical quantities. Each one of them represent the ratio of the length of one side to another of a right angled triangle.
1.
3.
Basic Trigonometric Identities : (a)
sin2 + cos2 = 1
;
1 sin 1; 1 cos 1 R
(b)
sec2 tan2 = 1
;
sec  1 R 2 n 1 2 , n I
(c)
coses2 cot2 = 1 ;
cosec  1 R {n , n I}
Trigonometric Functions Of Allied Angles : If is any angle, then , 90 , 180 , 270 , 360 etc. are called ALLIED ANGLES. (a)
sin( ) = sin
;
cos( ) = cos
(b)
sin(90o ) = cos
;
cos(90o ) = sin
(c)
sin(90o + ) = cos
;
(d)
sin(180o ) = sin
;
(e)
sin(180o + ) = sin
;
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
ma r a h S 2 . K cos(90 + L )= sin 768 . Er. 81027 5058 cos(180 9 ) = cos0 1 8 9 3 cos(180 +8 ) = cos o
o
o
Page 170
(f)
sin(270o ) = cos
;
cos(270o ) = sin
(g)
sin(270o + ) = cos
;
cos(270o + ) = sin
(h)
tan(90o ) = cot
;
cot(90o  ) = tan
(4) Graphs of Trigonometric Function : (i)
(ii) y cos x ; x R ; y [1, 1]
y sin x ; x R ; y [1, 1]
(iii) y tan x ; x R (2n 1) / 2, n I;y R
(iv) y cot x ; x R n, n I;y R
(v) y cosec x ;
(vi) y sec x ;
x R n, n I;y R /(1, 1)
5.
x R (2n 1) / 2, n I;y R /(1, 1)
Trigonometric Functions of Sum or Difference of Two Angles : (a)
sin (A B) = sinA cosB cosA sinB
(b)
cos(A B) = cosA cosB sin A sin B
(c)
sin2A sin2B = cos2B cos2A = sin(A+B). sin(A B)
(d)
(e)
a m r a cos A sin B = cos B sin A = cos (A+B). cos(A B) h S . 82 K . 6 L 7 tan A tan B Er. 81027 5058 tan(A B) = 1 t an A tan B 9 801 839 2
2
Mathematics Concept Note IITJEE/ISI/CMI
2
2
Page 171
6.
7.
8.
(f)
cot A cot B 1 cot(A B) = cot B cot A
(g)
tan(A+B+C) =
tan A tanB tanC tan A tanB tanC 1 tan A tanB tanB tanC tanC tan A
Factorisation of the Sum of Difference of Two Sines or Cosines : C D C D cos 2 2
(a)
sinC + sinD = 2sin
(b)
sinC sinD = 2cos
(c)
cosC + cosD = 2cos
(d)
cosC cosD =
C D C D sin 2 2 C D C D cos 2 2 C D
C D
2sin 2 sin 2
Transformation of Products into Sum or Difference of Sines & Cosines: (a)
2 sinA cosB = sin(A+B) + sin(A B)
(b)
2 cosA sinB = sin(A+B) sin(A B)
(c)
2 cosA cosB = cos(A+B) + cos(A B)
(d)
2 sinA cosB = cos(A B) cos(A + B)
Multiple and Submultiple Angles : cos 2 2
(a)
sin 2A = 2sinA cosA ; sin = 2 sin
(b)
cos 2A = cos2A sin2A = 2cos2A 1 = 1 2 sin2A ; 2cos2
= 1 + cos 2 sin2 = 1 cos . 2 2
(c)
2 tan 2A = ; tan = 1 tan2 1 tan2 A 2
(d)
sin 2A =
(e)
sin 3A = 3 sin A 4 sin3A
(f)
cos 3A = 4 cos3A 3 cos A
(g)
tan 3A =
2 tan
2 tan A
2 tan A 1 tan2 A
3tan A tan3 A 13tan2 A
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
, cos 2A =
1tan2 A 1 tan2 A
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Page 172
9.
Important Trigonometric Ratios : (a)
sin n = 0 ; cos n = ( 1)n ; tan n = 0, where n I
(b)
sin 15o or sin
3 1 5 = = cos 75o or cos ; 12 12 2 2
cos 15o or cos
tan 15o =
(c)
10.
sin
3 1 5 = = sin 75o or sin 12 12 2 2
3 1 = 2 3 = cot 75o; tan 75o = 3 1
or sin 18o = 10
5 1 4
& cos 36o or cos
3 1 = = cot 15o 3 1 2 3 5 1
= 5
4
Conditional Identities : If A + B + C = then :
11.
(i)
sin2A + sin2B + sin2C = 4 sinA sinB sinC
(ii)
sin A + sin B + sinC = 4 cos
(iii)
cos 2A + cos 2B + cos 2C =
(iv)
cos A + cos B + cos C = 1 + 4 sin
(v)
tanA + tanB + tanC = tanA tanB tanC
(vi)
tan
A B B C C A tan + tan tan + tan tan =1 2 2 2 2 2 2
(vii)
cot
A B C A B C + cot + cot = cot . cot . cot 2 2 2 2 2 2
(viii)
cot A cot B + cot B cot C + cot C cot A = 1
(ix)
A+B+C=
2
A B C cos cos 2 2 2
1 4 cos A cos B cos C A B C sin sin 2 2 2
then tan A tanB + tanB tanC + tanC tanA = 1
Range of Trigonometric Expression: E = a sin + b cos E= =
b
a 2 b 2 sin ( + ), where tan = a a2 b 2
a cos ( ), where tan = b
Hence for any real value of ,
Mathematics Concept Note IITJEE/ISI/CMI
a2 b2 E
a2 b2
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Page 173
12.
Sine and Cosine Series : n 2 n1 sin + sin( + ) + sin ( + 2 ) + .......+ sin( + n 1 ) = sin 2 sin 2
sin
n 2 n1 cos + cos( + ) + cos( +2 ) +............+ cos( + r 1 ) = cos 2 s in 2 s in
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
Page 174
32. TRIGONOMETRIC EQUATIONS Trigonometric Equations Principal Solutions : The solutions of a trigonometric equation which lie in the interval [0,2 ) are called Principal solutions. General Solution : The expression involving an integer 'n' which gives all solutions of a trigonometric equation is called General solution. General solution of some standard trigonometric equations are given below : (i)
If sin = sin = n + ( 1)n where 2 ,2 , n I.
(ii)
If cos = cos = 2n where [0, ], n I.
(iii)
If tan = tan = n + where 2 ,2 , n I.
(iv)
If sin2 = sin2 a = n
(v)
(vi)
tan2 = tan2n = n
[Note : is called the principal angle]
cos2 = cos2 = n .
Types of Trigonometric Equations : v
(i)
Solutions of equations by factorising . Consider the equation : (2sinx cosx) (1 + cos x) = sin2 x.
(ii)
Solutions of equations reducible to quadratic equations. Consider the equation ; 3 cos2x 10 cos x + 3 = 0.
(iii)
Solving equations by introducing an Auxiliary argument. Consider the equation : sin x + cos x = 2
&
3 cos x + sin x = 2.
(iv)
Solving equations by Transforming a sum of Trigonometric functions into a product. consider the example; cos 3x + sin 2x sin 4x = 0.
(v)
Solving equations by transforming a product of trigonometric functions into a sum Consider the equation, sin5x. cos 3x = sin 6x. cos 2x.
(vi)
Solving equations by change of variable : (a)
Equations of the form P(sin x cos x , sin x, cos x) = 0, where P(y z) is a polynomial, can be solved by the change. cos x sin x = t 1 2 sin x . cos x = t2. Consider the equation; sinx + cos x = 1 + sin x . cos x.
(b)
Equations of the form of a . sinx + b . cos x + d = 0, where a , b & d are real number s and a, b 0 can be solved by changing sin x & cos x into their corresponding tangent of half the angle. consider the equation 3 cos x + 4 sin x = 5.
(c)
ma r a h S 2 . K .variable7e.g. 6the8equation, L 7 Many equations can be solved by introducing a.new r 02 5058 sin 2x + cos 2x = sin 2x . cos 2x changesE to 1 8 9 801 1 y 2 (y + 1) = 0 by substituting, sin 2x . cos 8392x = y.. 2 4
(vii)
4
Solving equations with the use of the Boundness of the functions sinx & cos x .
Mathematics Concept Note IITJEE/ISI/CMI
Page 175
Trigonometric Inequalitities : Solutions of elementary trigonometric inequalitities are obtained from graphs _____________________________________________________________ Inequality Set of solutions of Inequality (n z) _____________________________________________________________ sin x > a (a sin x < a (a cos x > a (a cos x < a (a tan x > a
< < < <
1) 1) 1) 1)
tan x < a
x (sin 1a + 2 n, sin 1a + 2 n) x ( sin 1a + 2 n, sin 1a + 2 n) x ( cos 1a + 2 n, cos 1 a + 2 n) x (cos 1 a + 2 n, 2 cos 1a + 2 n) x (tan 1a + n, /2 + n) 1 x n,tan an 2
___________________________________________________________ Inequalities of the form R(y) > 0, R(y)< 0, where R is a certain rational function and y is a trigonometric function (sine, cosine or tangent), are usually solved in two stages: first the rational inequality is solved for the unknown y and then follows the solution of an elementary trigonometric inequality.
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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33. PROPERTIES OF TRIANGLE The angles of a triangle are denoted by capital letters A, B, C and the sides opposite to them are denoted by small letters a, b, c In triangle A+B+C= also a + b > c and a – b < c b + c > a and b – c < a c + a > b and c – a < b
(1) Sine Rule : In any triangle ABC, the sines of the angles are proportional to the opposite sides i.e.
a b c = = = 2R ; 'R' is circumradius of ABC sin A sinB sinC
(2) Cosine Formula :
(i) cos A =
b2 c2 a2 or a2 = b2 + c2 2bc cosA = b2 + c2 + 2bc cos(B + C) 2bc
(ii) cos B =
c2 a2 b2 2ca
a2 b2 c2 2ab
(iii)
cos C =
(ii)
b = c cosA + a cos C
(ii)
tan
(3) Projection Formula : (i) a = b cosC + c cos B (iii) c = a cosB + b cosA
(4) Napier's Analogy  tangent rule: (i) tan
B C bc A = cot 2 b c 2
(iii) tan
A B ab C = cot 2 a b 2
C A ca B = cot 2 c a 2
(5) Trigonometric Functions of Half Angles :
(i) sin
A = 2
(ii)
cos
(iii) (iv)
B C (sb)(s c) (s c)(s a) ; sin = ; sin = 2 2 bc ca
A = 2
B s(s a) ; cos = 2 bc
C s(s b) ; cos = 2 ca
(s a)(s b) ab s(s c) ab
ma r a h (s b)(s c) A a b c S 2 . tan = is semi perimeter of8 triangle. = s(s a) where s = K . 6 2 s(s a) 2 L 7 r. 1027 058 E 2 2 98 8015 sin A = = bc s(sa)(sb)(s c) bc 839
Mathematics Concept Note IITJEE/ISI/CMI
Page 177
(6) Area of Triangle ( ) 1
1
1
= 2 ab sin C = 2 bc sin A = 2 ca sin B = s(s a)(s b)(s c)
(7) m  n Rule : (m + n) cot = m cot n cot = n cot B m cotC
A B
m
D
C
n
(8) Radius of Circumcircle : R=
a b c abc = = = 2 sin A 2 sinB 2 sinC 4
(9) Radius of The Incircle : (i) r =
s
(ii) r = (s a) tan
asin B sin C
(iii) r =
2
2
A
cos 2
A B C = (s b)tan = (s c)tan 2 2 2
(iv) r = 4R sin
A B C sin sin 2 2 2
(ii)
A B C ; r2 = s tan ; r3 = s tan 2 2 2
(10) Radius of The ExCircles : (i) r1 =
,r = ,r = s a 2 s b 3 s c
r1 = s tan
acos B cos C
(iii) r1
2
2
(iv) r1 = 4 R sin
A
cos 2
A B C . cos . cos 2 2 2
(11) Length of Angle Bisectors, Medians and Altitudes : A
ma
Aa a
B
D
E
F
C
(i) Length of an angle bisector from the angle A = a (ii) Length of median from the angle A = ma = Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
1 2
ma r a h S . 82 K . 6 L 7 Er2bc. cos81027 5058 9 = 01 b c 98 83 A 2
2b2 2c2 a2
Page 178
(iii) Length of altitude from the angle A = Aa = 2
Note:
2
2 a
3 2 (a + b2 + c2) 4
2
m a + mb + m c =
A and l1a = r1 2
(iii) Excentre (l1)
:
I1 A = r1 cosec
(iv) Orthocentre(H)
:
HA = 2R cos A and Ha = 2R cos B cos C
(v) Centroid (G)
:
GA =
1 3
2
2b2 2c2 a2 and Ga = 3a
(13) Orthocentre and Pedal Triangle : The triangle which is formed by joining the feet of the altitudes is called the Pedal Triangle. (i) Its angles are 2a, 2B and 2C. (ii) Its sides are a cosA = R sin 2A, b cosB = R sin 2B and c cosC = R sin 2C (iii) Circumradii of the triangles PBC, PCA, PAB and ABC are equal.
(14) Excentral Triangle : The triangle formed by joining the three excentres I1, I2 and I3 of ABC is called the excentral or excentric triangle.
I3
I2
A
B
C
B/2 B 2 2
90
o
I1
(i)
ABC is the pedal triangle of the I1 I2 I3 ,
(ii) Its angles are
2
A
B
C
2 , 2 2 and 2 2 .
(iii) Its sides are 4R cos 4R cos
A , 2
B C and 4R cos 2 2
(iv) I I1 = 4R sin
A B C ; I I2 = 4R sin ; I I3 = 4R sin . 2 2 2
Mathematics Concept Note IITJEE/ISI/CMI
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Page 179
34. INVERSE TRIGONOMETRIC FUNCTION (1) Domain and Range of Inverse Trigonometric/Circular Functions: No.
Domain
Range
1 x1
2 y 2
(ii) y = cos–1x
1 x1
0y
(iii) y = tan–1x
x R
(i)
Function y = sin–1x
2
(iv) y = cosec–1x
x
1 or x 1
2 y 2 , y 0
(v) y = sec–1x
x
1 or x 1
0y ; y 2
(vi) y = cot–1x
x R
0
Note: (a) 1st quadrant is common to the range of all the inverse functions. (b) 3rd quadrant is not used in inverse functions. (c) 4th quadrant is used in the clockwise direction i.e.
2 y 0.
(d) No inverse function is periodic.
(2) Properties of Inverse Trigonometric Functions : Property  (i) , –1 x 1 2
(i)
sin–1 x + cos–1 x =
(ii)
tan–1 x + cot–1 x =
(iii)
cosec–1 x + sec–1 x =
, x R 2 , x 1 2
Property  (ii) (i) sin(sin–1x)= x, 1 x 1 (iii) tan(tan–1x)= x, x R (v) sec(sec–1x) = x, x 1, x 1
(ii) (iv) (vi)
cos(cos–1x) = x, –1 x 1 cot(cot–1x) = x, x R cosec(cosec–1x) = x, x 1, x 1
(i) sin–1(sin x) = x, x 2 2
(ii)
cos–1(cos x) = x,
(iii) tan–1(tan x)= x, <x< 2 2
(iv)
Property  (iii)
(v) sec–1(sec x) = x, 0 x , x
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
2
(vi)
ma r a h S . 82 K . 6 L 7 cot (cot < 8 5 Er.x) =8x,10270 <5x 0 9 801 cosec (cosec x)9 = x; x 0, x 2 2 3 8 0x
–1
–1
Page 180
Property  (iv) (i) sin–1(–x) = –sin–1x, –1 x 1
tan–1(–x) = –tan–1 x, x R –1 –1 cot (–x) = –cot x, x R
(ii) (iv)
(iii) cos–1(–x) = –cos–1x, 1 x 1
Note: The functions sin–1x, tan–1x and cosec–1x are odd functions and rest are neither even nor odd Property(v) (i) cosec–1x = sin–1 (ii) sec–1 x = cos–1
1 ;x–1 , x 1 x
1 ; x –1, x 1 x
tan1 1 ; x 0 x (iii) cot–1 x = 1 1 tan x ; x 0 Property  (vi) (i) sin(cos–1 x) = cos (sin–1 x) = , (ii) tan(cot–1 x) = cot (tan–1x) =
1 x2 –1 x 1
1 , x R, x 0 x
x
(iii) cosec (sec–1 x) = sec (cosec–1 x) =
x2 1
, x > 1
(3) Identities of Addition and Substraction: Property(i) (i) sin–1x + sin–1y = sin–1 [x 1 y2 + y 1 x2 , x 0, y 0 and (x2 + y2) 1 = sin–1 [x 1 y2 + y 1 x2 ], x 0, y 0 and x2 + y2 >1
Note: x2 + y2 1 0 sin 1x + sin–1y 2 x2 + y2 > 1 < sin–1x + sin–1y < 2
(ii)
cos–1x + cos–1y = cos–1 [xy 1 x2
(iii)
x y tan–1x + tan–1y = tan–1 1 xy , x > 0, y > 0 & xy < 1
1 y2 ], x 0, y 0
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 39 8 < tan x + tan y <
x y = + tan–1 1 xy , x > 0, y > 0 & xy > 1
=
, x > 0, y > 0 & xy = 1 2
Note: xy < 1 0 < tan–1x + tan–1y < ; xy > 1 2
Mathematics Concept Note IITJEE/ISI/CMI
2
–1
–1
Page 181
Property(ii) (i) sin–1x sin–1y = sin–1 [x 1 y2 y 1 x2 ], x 0, y 0 (ii) cos–1x cos–1y = cos–1 [xy +
1 x2
1 y2 ], x 0, y 0, x y
x y (iii) tan–1x tan–1y = tan–1 1 xy , x 0, y 0
Note: For x < 0 and y < 0 these identities can be used with the help of properties 2(C) i.e. change x and y to x and y which are positive. Property(iii) 1 2 sin1 x if x 2 1 1 2 if x (i) sin–1 2x 1 x = 2 sin x 2 ( 2 sin1 x) if x 1 2
(ii) cos
–1
2 cos 1 x if (2x 1) = 1 2 2 cos x if 2–
0 x 1 1 x 0)
(iii) tan–1
2 tan1 x if x 1 2x 1 if x 1 = 2 tan x 1 x2 ( 2 tan1 x) if x 1
(iv) sin–1
2 tan1 x if x 1 2x 1 if x 1 = 2 tan x 1 x2 ( 2 tan1 x) if x 1
(v)
cos
–1
2 tan1 x if x 0 = 1 1 x2 2 tan x if x 0 1 x2
Property(iv)
x y z xyz If tan–1x + tan–1y + tan–1z = tan–1 1xy yz zx if, x > 0, y > 0, z > 0 & (xy+yz+zx)<1
ma r a h S . 82 K . (ii) If tan x + tan y + tan z = then xy + yz + zx = 1 6 L 7 2 Er. 81027 5058 (iii) tan 1 + tan 2 + tan 3 = 9 801 1 1 (iv) tan 1 + tan + tan = 839 2 3 2 Note:
(i) If tan–1 x + tan–1 y + tan–1 z = then x + y + z = xyz –1
–1
–1
–1
–1
–1
–1
–1
–1
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
Page 182
(4) Graphs of Inverse Trigonometric Function : (i) y = sin–1 x, x 1, y , 2 2
(ii) y = cos–1 x, x 1, y [0, ]
(iii) y = tan–1 x, x R, y , 2 2
(iv)
(v) y = sec–1 x, x 1, y 0 , 2 2 ,
y = cot–1 x, x R, y (0, )
(vi) y = cosec–1 x, x 1,y 2 ,0 0 , 2
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Mathematics Concept Note IITJEE/ISI/CMI
Page 183
(vii)y = sin (sin–1 x) = cos (cos–1 x) = x, x [ 1, 1], y [ 1, 1]; y is aperiodic
(viii)
y = tan (tan–1 x) = cot (cot–1 x) = x, x R, y R; y is aperiodic
(ix) y = cosec (cosec–1 x) = sec (sec–1 x) = x, x 1, y 1; y is aperiodic
(x) y = sin–1(sin x), x R, y , , is periodic with period 2 2 2
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
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(xi) y = cos–1 (cos x), x R, y [0, ], is periodic with period 2
(xii) y = tan–1 (tan x), x R (2n 1) 2 , n I , y 2 ,2 is periodic with period
(xiii)
y
sec –1
=
(sec
x),
y
is
periodi c
wi th
peri od
2;
x R (2n 1) 2 , n I ,y 0 , 2 2 ,
(xiv)
y = cosec–1(cosec x), y is periodic with period 2 π ; x R {n π , n I}, y 2 , 2
{0}
O
– 2
– 2
Mathematics Concept Note IITJEE/ISI/CMI
2
x
2 x
ma r a .Sh 2 82 K 2 . 6 L 7 Er. 81027 5058 9 801 839 y=
x y=
x+
y=
y=
x
– 2
(
– 2
y=
2
y 2
x
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(xv)
y = cot–1 (cot x), y is periodic with period ; x R {n , n I}, y 0, 2
, 2
Part  3(C)
y –1 2 (i) graph of y = sin
–1
2 x 1 x 2
–1 2
0
–1
(ii) graph of y = cos (2x
2
x
y
–1
1
2
1) –1
0
1 x
Note: In this graph it is advisable not to check its derivability just by the inspection of the graph because it is difficult to judge from the graph that at x = 0 there is a shapr corner or not.
y
(iii) graph of y = tan–1
2x
–1
0
1
x
1x 2
(iv)
graph of y = sin–1
2x
1 x 2
ma r a h y .S 82 K . 6 L 7 Er. 81027 5058 9 801 x –1 1 0 9 3 8
Mathematics for IITJEE Er. L.K.Sharma 9810277682/8398015058
Page 186
y
(v)
graph of y = cos–1
1 x
2
2
1 x 2
0
x
ma r a h S . 82 K . 6 L 7 Er. 81027 5058 9 801 839 Mathematics Concept Note IITJEE/ISI/CMI
Page 187