FORMULAE OF MATHEMATICS
For IIT-JEE, ISI, AIEEE
& MCA ASPIRANT
Author
Prof. R. K. Malik
Director of Newton Classes
606, 6th Floor, Hariom Tower, Ranchi
Malik Publication
177, Rajdhani Enclave, Near Rani Bagh, Pitampura
Delhi - 110034
Ph : 011-27193845
, CONTENTS
Chapter Topic Page No.
1. Ratio and Proportion ………. 1
2. Set Theory ………. 2
3. Complex Numbers ………. 4
4. Quadratic Equations ………. 19
5. Determinants ………. 30
6. Matrices ………. 35
7. Sequence & Series ………. 45
8. Inequalities ………. 53
9. Permutation & Combination ………. 56
10. Mathematical Induction ………. 64
11. Binomial Theorem ………. 65
12. Trigonometric Ratios, Identities & Equations ………. 70
13. Inverse Trigonometric Function ………. 76
14. Properties & Solution of Triangle, Height & Distance ………. 84
15. Mensuration ………. 96
16. Function ………. 100
17. Limit ………. 112
18. Continuity ………. 116
19. Differentiation ………. 119
20. Application of Derivatives ………. 127
21. Indefinite Intergal ………. 134
22. Definite Intergal ………. 140
23. Differentail Equation ………. 149
24. Straight Line ………. 155
25. Pair of Straight Line ………. 165
26. Circle ………. 167
27. Conic Section ………. 179
28. Parabola ………. 181
29. Ellipse ………. 186
30. Hyperbola ………. 192
31. Vector ………. 201
32. 3-Dimensional Coordinate Geometry ………. 215
33. Parobability ………. 225
34. Measures of Centreal Tendency & Dispersion ………. 233
35. Correlation and Regression ………. 241
36. Statics ………. 245
37. Dynamics ………. 254
38. Methematical Logic ………. 266
39. Boolean Algebra ………. 271
40. Linear Programming ………. 277
41. Hyperbolic Function ………. 281
42. Numerical Methods ………. 287
43. Check your Intelligence ………. 292
44. Important Graphs ………. 295
, 1
Ratio and Proportion Chapter 1
ELEMENTARY LAWS AND RESULTS ON RATION AND PROPORTION
a a a+x
1. > 1 Implies > ( x > 0)
b b b+ x
a a a+x
2. <1 implies < ( x > 0)
b b b+ x
a1 a2 a
3. If , ,...., n are unequal fractions, of which the denominators are of the same sign, then the fraction
b1 b2 bn
a1 + a2 + ... + an
lies, in magnitude, between the greatest and least of them.
b1 + b2 + ... + bn
1
a1 a2 a α a n + α 2 a2n + ... + α k akn n
4. If = = ...= k , then each of these ratios is equal to 1 1n n
, where n is an
α1b1 + α 2b2 + ... + α k bk
n
b1 b2 bk
integer.
5. a, b, c, d are in proportion. Then
a c
(i) =
b d
a b
(ii) = (alternendo)
c d
a+b c+d
(iii) = (componendo)
b d
a −b c −d
(iv) = (dividendo)
b d
a+b c+d
(v) = (componendo and dividendo)
a −b c −d
b d
(vi) = (invertendo)
a c
, 2 R. K. Malik’s Formulae of Mathematics
Set Theory Chapter 2
SET
A well defined collection of distinct objects is called a set. When we say, ‘well defined’, we mean that
there must be given a rule or rules with the help of which we should readily be able to say that whether a
particular object is a member of the set or is not a member of the set. The sets are generally denoted by capital
letters A,B,C,….X,Y,Z.
The members of a set are called its elements. The elements of a set are denoted by small letters a, b,
c,..…,x, y, z.
If an element a belong to a set A than we write a ∈ A and if a does not belong to set A then we write a ∉ A.
LAW OF ALGEBRA OF SETS
1. Idempotent Laws : For any set A,
(i) A ∪ A = A (ii) A ∩ A =.A
2. Identity Laws : For any set A,
(i) A ∪ φ = A (ii) A ∩ U = A
i.e., φ and U are identity elements for union and intersection respectively.
3. Commutative Laws : For any two sets A and B,
(i) A∪ B = B ∪ A (ii) A ∩ B = B ∩ A
i.e. union and intersection are commutative.
4. Associative Laws : If A, B and C are any three sets, then
(i) ( A ∪ B ) ∪ C =A ∪ ( B ∪ C ) (ii) A ∩ ( B ∩ C ) = ( A ∩ B ) ∩ C
i.e. union and intersection are associative.
5. Distributive Laws : If A, B and C are any three sets, then
(i) A ∪ ( B ∩ C ) = ( A ∪ B) ∩ ( A ∪ C ) (ii) A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C )
i.e. union and intersection are distributive over intersection and union respectively.
6. De-Morgan’s Laws : If A and B are any two sets, then
(i) ( A ∪ B )′ =A′ ∩ B′ (ii) ( A ∩ B )′ =A′ ∪ B′
7. More results on operations on sets
If A and B are any two sets, then
(i) A − B = A ∩ B′ (ii) B − A = B ∩ A′
(iii) A − B = A ⇔ A ∩ B = φ (iv) ( A − B ) ∪ B = A ∪ B
(v) ( A − B) ∩ B = φ (vi) A ⊆ B ⇔ B′ ⊆ A′
(vii) ( A − B ) ∪ ( B − A) = ( A ∪ B ) − ( A ∩ B ) .
If A, B and C are any three sets, then
(i) A − ( B ∩ C ) = ( A − B) ∪ ( A − C ) (ii) A − ( B ∪ C ) = ( A − B) ∩ ( A − C )
(iii) A ∩ ( B − C ) = ( A ∩ B ) − ( A ∩ C ) (iv) A ∩ ( B ∆ C ) = ( A ∩ B ) ∆ ( A ∩ C )
8. Important results on number of elements in sets
If A, B and C are finite sets, and U be the finite universal set, then
(i) n ( A ∪ B )= n ( A ) + n ( B ) − n ( A ∩ B )
(ii) n ( A ∩ B )= n ( A ) + n ( B ) ⇔ A, B are disjoint non-void sets.
For IIT-JEE, ISI, AIEEE
& MCA ASPIRANT
Author
Prof. R. K. Malik
Director of Newton Classes
606, 6th Floor, Hariom Tower, Ranchi
Malik Publication
177, Rajdhani Enclave, Near Rani Bagh, Pitampura
Delhi - 110034
Ph : 011-27193845
, CONTENTS
Chapter Topic Page No.
1. Ratio and Proportion ………. 1
2. Set Theory ………. 2
3. Complex Numbers ………. 4
4. Quadratic Equations ………. 19
5. Determinants ………. 30
6. Matrices ………. 35
7. Sequence & Series ………. 45
8. Inequalities ………. 53
9. Permutation & Combination ………. 56
10. Mathematical Induction ………. 64
11. Binomial Theorem ………. 65
12. Trigonometric Ratios, Identities & Equations ………. 70
13. Inverse Trigonometric Function ………. 76
14. Properties & Solution of Triangle, Height & Distance ………. 84
15. Mensuration ………. 96
16. Function ………. 100
17. Limit ………. 112
18. Continuity ………. 116
19. Differentiation ………. 119
20. Application of Derivatives ………. 127
21. Indefinite Intergal ………. 134
22. Definite Intergal ………. 140
23. Differentail Equation ………. 149
24. Straight Line ………. 155
25. Pair of Straight Line ………. 165
26. Circle ………. 167
27. Conic Section ………. 179
28. Parabola ………. 181
29. Ellipse ………. 186
30. Hyperbola ………. 192
31. Vector ………. 201
32. 3-Dimensional Coordinate Geometry ………. 215
33. Parobability ………. 225
34. Measures of Centreal Tendency & Dispersion ………. 233
35. Correlation and Regression ………. 241
36. Statics ………. 245
37. Dynamics ………. 254
38. Methematical Logic ………. 266
39. Boolean Algebra ………. 271
40. Linear Programming ………. 277
41. Hyperbolic Function ………. 281
42. Numerical Methods ………. 287
43. Check your Intelligence ………. 292
44. Important Graphs ………. 295
, 1
Ratio and Proportion Chapter 1
ELEMENTARY LAWS AND RESULTS ON RATION AND PROPORTION
a a a+x
1. > 1 Implies > ( x > 0)
b b b+ x
a a a+x
2. <1 implies < ( x > 0)
b b b+ x
a1 a2 a
3. If , ,...., n are unequal fractions, of which the denominators are of the same sign, then the fraction
b1 b2 bn
a1 + a2 + ... + an
lies, in magnitude, between the greatest and least of them.
b1 + b2 + ... + bn
1
a1 a2 a α a n + α 2 a2n + ... + α k akn n
4. If = = ...= k , then each of these ratios is equal to 1 1n n
, where n is an
α1b1 + α 2b2 + ... + α k bk
n
b1 b2 bk
integer.
5. a, b, c, d are in proportion. Then
a c
(i) =
b d
a b
(ii) = (alternendo)
c d
a+b c+d
(iii) = (componendo)
b d
a −b c −d
(iv) = (dividendo)
b d
a+b c+d
(v) = (componendo and dividendo)
a −b c −d
b d
(vi) = (invertendo)
a c
, 2 R. K. Malik’s Formulae of Mathematics
Set Theory Chapter 2
SET
A well defined collection of distinct objects is called a set. When we say, ‘well defined’, we mean that
there must be given a rule or rules with the help of which we should readily be able to say that whether a
particular object is a member of the set or is not a member of the set. The sets are generally denoted by capital
letters A,B,C,….X,Y,Z.
The members of a set are called its elements. The elements of a set are denoted by small letters a, b,
c,..…,x, y, z.
If an element a belong to a set A than we write a ∈ A and if a does not belong to set A then we write a ∉ A.
LAW OF ALGEBRA OF SETS
1. Idempotent Laws : For any set A,
(i) A ∪ A = A (ii) A ∩ A =.A
2. Identity Laws : For any set A,
(i) A ∪ φ = A (ii) A ∩ U = A
i.e., φ and U are identity elements for union and intersection respectively.
3. Commutative Laws : For any two sets A and B,
(i) A∪ B = B ∪ A (ii) A ∩ B = B ∩ A
i.e. union and intersection are commutative.
4. Associative Laws : If A, B and C are any three sets, then
(i) ( A ∪ B ) ∪ C =A ∪ ( B ∪ C ) (ii) A ∩ ( B ∩ C ) = ( A ∩ B ) ∩ C
i.e. union and intersection are associative.
5. Distributive Laws : If A, B and C are any three sets, then
(i) A ∪ ( B ∩ C ) = ( A ∪ B) ∩ ( A ∪ C ) (ii) A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C )
i.e. union and intersection are distributive over intersection and union respectively.
6. De-Morgan’s Laws : If A and B are any two sets, then
(i) ( A ∪ B )′ =A′ ∩ B′ (ii) ( A ∩ B )′ =A′ ∪ B′
7. More results on operations on sets
If A and B are any two sets, then
(i) A − B = A ∩ B′ (ii) B − A = B ∩ A′
(iii) A − B = A ⇔ A ∩ B = φ (iv) ( A − B ) ∪ B = A ∪ B
(v) ( A − B) ∩ B = φ (vi) A ⊆ B ⇔ B′ ⊆ A′
(vii) ( A − B ) ∪ ( B − A) = ( A ∪ B ) − ( A ∩ B ) .
If A, B and C are any three sets, then
(i) A − ( B ∩ C ) = ( A − B) ∪ ( A − C ) (ii) A − ( B ∪ C ) = ( A − B) ∩ ( A − C )
(iii) A ∩ ( B − C ) = ( A ∩ B ) − ( A ∩ C ) (iv) A ∩ ( B ∆ C ) = ( A ∩ B ) ∆ ( A ∩ C )
8. Important results on number of elements in sets
If A, B and C are finite sets, and U be the finite universal set, then
(i) n ( A ∪ B )= n ( A ) + n ( B ) − n ( A ∩ B )
(ii) n ( A ∩ B )= n ( A ) + n ( B ) ⇔ A, B are disjoint non-void sets.
