# How to be good at Mathematics?

Mathematics is one of the most important subjects we study. But, maximum students fear from it. I’ve experienced that students fear from it because they don’t know the best way to study it.

I’m just going to share some effective techniques to study math on the basis of my research and experience. I wish it will help you a lot whether you are a JEE or NEET aspirant or a student of any class.

**I. Understand the concepts**: The first step towards commanding mathematics is to study your text books line by line and to try to understand the theoretical part by the help of examples. To understand the concepts you may have to take help from your teacher.

**II. Applications of concepts : **The 2nd step is to grasp the applications of concepts. For doing so you must go through the example given before the exercises.

Maximum students skip the example section and attempt the exercise directly due to which they are not able to solve majority of questions. So, I always insist on solving example first. When you do examples you can observe the uses of different results and formulae.

**III. Attempt the exercise : **Once you go through all the examples, you should start to solve questions in the exercise. Try to solve questions on your own. It will help you to grow your thinking capacity.

If you are unable to solve any question, you can take help from examples or you teacher.

**IV. Set a daily target : **I always encourage students to follow the ‘**Target Based Study. **You can set a target of trying at least a definite number of questions per day. It generally helps not only in completing your syllabus in time but also it makes your study easier.

Suppose you have 5000 questions in your book and you have to finish it in a year i.e. 365 days. You just need to solve at least 15 days.

# Trigonometric Equations

**Trigonometric equation: **An equation having trigonometric functions is called a trigonometric equation.

Ex : sin θ = 1

sin θ + cos θ =1

tan θ = 1

**Trigonometric identity** : A trigonometric identity is also an equation which gets satisfied by any value of the unknown quantity.

Ex : sin^{2}θ + cos^{2}θ=1

**Key Point: **A trigonometric equation is satisfied by finite or infinite specific values of the unknown quantity not by any value of the unknown quantity.

**Solution of a trigonometric equation: **There are two types of solutions of a trigonometric equation:

**I. Principal solution: **A solution in which the values of the unknown quantity belong to the interval [0,2π] is called a principal solution.

**II. General solution: **A solution in which there are infinite values of the unknown quantity is called a general solution.

**Important results** :

I. sin θ = 0 ⇒ θ = nπ II. cos θ = 0 ⇒ θ = (2n+1)π/2 III. tan θ = 0 ⇒ θ = nπ

# Relations & Functions

**Cartesian Product—**

♦An ordered pair⟹(x,y)

♦An ordered triplet⟹(x,y,z)

♦Equality of ordered pairs ⟹ (x,y)=(α,β)⟺x=α,y=β

♦Equality of ordered triplets ⟹ (x,y,z)=(α,β,γ)⟺x=α, y=β, z=γ

Key Results: I.A (B C)=(A B)⋃(A C)

II.A (B⋂C)=(A B)⋂(A C)

III. A (B-C)=(A B)-(A C)

♦If A and B are two non-empty sets, then A B=B A⟺A=B

If A⊆B; then A A (A B)⋂(B A)

♦A⊆B⟹A C ⊆B C for any set C

♦A⊆B and C⊆D⟹A C ⊆ B D

♦(A B)⋃(C D)⊆(A C) (B⋃D)

♦(A B)⋂(C D)=(A⋂C) (B⋂D)

♦(A B)⋂(B A)=(A⋂B) (B⋂A)

♦Let A and B be two non-empty sets having n elements in common, then A B and B A have n^{2 }elements in common.

## Relations—

If A and B are any two non empty sets, then each subset of AxB is called a relation from A to B. It’s denoted by R:A⟶B

Key Point : I. If n(A)=p & n(B)=q, then the total number of relations from A to B =2^{pq }

♦ **Types of relations**:

**I.Empty Relation**: A relation having no element is called an empty relation.

Ex: R = {x : x+5=0 where x∊N}={}=ϕ

**II.Universal Relation**: If R:A⟶B is any relation such that R=AxB, then it’s called a universal relation.

Ex: A={1,2} & B={4,5}⟹AxB={(1,4),(1,5),(2,4),(2,5)}

R={(x,y):x+y<10 where x∊A, y∊B} = AxB

**III. Inverse Relation**: If R:A⟶B is any relation then its inverse relation R^{-1}:B⟶A is defined as follows: R^{-1} = {(y,x):(x,y)∊ R}

Ex: R={(1,2),(1,3),(2,2),(2,3)} ⟹ R^{-1}={(2,1),(3,1),(2,2),(3,2)}

**Functions— **

Name of the Functions |
Format |
Domain |
Range |

Constant Function | y=ƒ(x)=k, k∊R | R | {k} |

Identity Function⟹I | y=ƒ(x)=x | R | R |

Square Function | y=ƒ(x)=x^{2} |
R | [0,∞) |

Cube Function | y=ƒ(x)=x^{3} |
R | R |

Power Function | y=ƒ(x)=x^{n} |
R | R:when n is odd
[0,∞): when n is even |

Linear Function | y=ƒ(x)=ax+b, a≠0 | R | R |

Quadratic Function | y=ƒ(x)=ax^{2}+bx+c, a≠0 |
R | Calculate |

Polynomial Function | y=ƒ(x)=a_{0}x^{n}+a_{1}x^{n-1}+…a_{n},
where n,(n-1),…∊W & a |
R | Calculate |

Rational Function | y=ƒ(x)= ,h(x)≠0 | R-{x:h(x)=0} | Calculate |

Irrational Function | Having fractional powers of x | Calculate | Calculate |

Square Root Function | y=ƒ(x)= | [0,∞) | [0,∞) |

Cube Root Function | y=ƒ(x)= | R | R |

Signum Function | y=ƒ(x)= | R | {-1,0,1} |

Modulus Function | y=|x| | R | [0,∞) |

Exponential Function | y=a^{x },a>0,a≠1 |
R | (0,∞) |

Logarithmic Function | y=log_{a}x, a>0,a≠1,x>0 |
(0,∞) | R |

Greatest Integer Function | y=[x] | R | I |

Least Integer Function | y= | R | I |

Fractional Part Function | y={x} | R | [0,1) |

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