Problem from Geometry - Circles

In the figure given below, ABD is a right-angled triangle at B. Taking AB as diameter, a circle has been drawn intersecting AD at F. Prove that the tangent drawn at point F bisects BD.

Solution:

Trigonometry SN Dey Solved Problems - HS - Class XI

1. Trigonometric functions of Standard Angles [ 4 and 5 marks ]

2. Trigonometric functions of Associate Angles [ 4 and 5 marks ]

3. Trigonometric transformations of sums and products [ 4 and 5 marks]

4. Trigonometric functions of Compound Angles [ 4 and 5 marks]

5. Trigonometric functions of Multiple Angles [4 and 5 marks]

6. Trigonometric functions of Sub-multiple Angles [ 4 and 5 marks]

7. Trigonometric Equations  [ 4 and 5 marks]

Vectors SN Dey Solved Problems

 Introduction of Vectors Solved Problems SN Dey - Long Answer Type Questions 5 Marks

Product of Vectors Solved Problems SN Dey - Long Answer Type Questions 5 Marks

Set Theory Solved Problems : Class XI

The concept of set serves as a fundamental part of the present day mathematics. Today this concept is being used in almost every branch of mathematics. Sets are used to define the concepts of relations and functions. The study of geometry, sequences, probability, etc. requires the knowledge of sets. The theory of sets was developed by German mathematician Georg Cantor (1845-1918). He first encountered sets while working on “problems on trigonometric series”. In this Chapter, we discuss some basic definitions and operations involving sets.

Empty Sets

The set with no elements or null elements is called an empty set. This is also called a Null set or Void set. It is denoted by {}.

For example: Let, Set X = {x:x is the number of students studying in Class 6th and Class 7th}

Since we know a student cannot learn simultaneously on two classes, therefore set X is an empty set.

Singleton Set

The set which has only one element is called a singleton set.

For example, Set X = { 2 } is a singleton set.

Finite and Infinite Sets

Finite sets are the one which has a finite number of elements, and infinite sets are those whose number of elements cannot be estimated, but it has some figure or number, which is very large to express in a set.

For example, Set X = {1, 2, 3, 4, 5} is a finite set, as it has a finite number of elements in it.

Set Y = {Number of Animals in India} is an infinite set, as there is an approximate number of Animals in India, but the actual value cannot be expressed, as the numbers could be very large.

Equal Sets

Two sets X and Y are said to be equal if every element of set X is also the elements of set Y and if every element of set Y is also the elements of set X. It means set X and set Y have the same elements, and we can denote it as;

X = Y

For example, Let X = { 1, 2, 3, 4} and Y = {4, 3, 2, 1}, then X = Y

And if X = {set of even numbers} and Y = { set of natural numbers} the X ≠ Y, because natural numbers consist of all the positive integers starting from 1, 2, 3, 4, 5 to infinity, but even numbers starts with 2, 4, 6, 8, and so on.

Subsets

A set X is said to be a subset of set Y if the elements of set X belongs to set Y, or you can say each element of set X is present in set Y. It is denoted with the symbol as X ⊂ Y.

We can also write the subset notation as;

X ⊂ Y if a ∊ X

a ∊ Y

Thus, from the above equation, “X is a subset of Y if a is an element of X implies that a is also an element of Y”.

Each set is a subset of its own set, and a null set or empty set is a subset of all sets.

Power Sets

The power set is nothing but the set of all subsets. Let us explain how.

We know the empty set is a subset of all sets and every set is a subset of itself. Taking an example of set X = {2, 3}. From the above given statements we can write,

{} is a subset of {2, 3}

{2} is a subset of {2, 3}

{3} is a subset of {2, 3}

{2, 3} is also a subset of {2, 3}

Therefore, power set of X = {2, 3},

P(X) = {{},{2},{3},{2,3}}

Universal Sets

A universal set is a set which contains all the elements of other sets. Generally, it is represented as ‘U’.

For example; set X = {1, 2, 3}, set Y = {3, 4, 5, 6} and Z = {5, 6, 7, 8, 9}

Then, we can write universal set as, U = {1, 2, 3, 4, 5, 6, 7, 8, 9,}

Note: From the definition of the universal set, we can say, all the sets are subsets of the universal set. Therefore,

X ⊂ U

Y ⊂ U

And Z ⊂ U

Union of sets

A union of two sets has all their elements. It is denoted by ⋃.

For example, set X = {2, 3, 7} and set Y = { 4, 5, 8}

Then union of set X and set Y will be;

X ⋃ Y = {2, 3, 7, 4, 5, 8}

Properties of Union of Sets:

X ⋃ Y = Y ⋃ X ; Commutative law

(X ⋃ Y) ⋃ Z = X ⋃ (Y ⋃ Z)

X ⋃ {} = X ; {} is the identity of ⋃

X ⋃ X = X

U ⋃ X = U

Intersection of Sets

Set of all elements, which are common to all the given sets, gives intersection of sets. It is denoted by the symbol ⋂.

For example, set X = {2, 3, 7} and set Y = {2, 4, 9}

So, X ⋂ Y = {2}

Difference of Sets

The difference of set X and set Y is such that, it has only those elements which are in the set X and not in the set Y.

i.e. X – Y = {a: a ∊ X and a ∉ Y}

In the same manner, Y – X = {a: a ∊ Y and a ∉ X}

For example, if set X = {a, b, c, d} and Y = {b, c, e, f} then,

X – Y = {a, d} and Y – X = {e, f}

Disjoint Sets

If two sets X and Y have no common elements, and their intersection results in zero(0), then set X and Y are called disjoint sets.

It can be represented as; X ∩ Y = 0

HS Mathematics Question Paper Solved 2022 - CLASS XI

 HS Mathematics Question Paper Solved 2022 - CLASS XI

 Download the full question paper and solution of mathematics for the year 2022 under WBCHSE ( West Bengal Council of Higher Secondary Education ) HS.

HS (Class XI) 2022 Mathematics English Version

ICSE Class IX - Practice Set

Practice Set 32 Marks - Real Numbers and Compound Interest - Click Here

Practice Set 25 Marks -7 Questions- Real Numbers, Exponents, Compound Interest and Expansions - Set I

Practice Set 28 Marks-8 Questions- Real Numbers, Exponents, Compound Interest and Expansions - Set II

Practice Set 15 Questions - Real Numbers - Click Here

Harder Problems 22 Questions - Algebraic Identities - Click Here

Practice Set 39 Questions - Factorisations - Click Here

Practice Set 23 Questions - Expansion and Indices - Click Here

Practice Set 23 Questions - Real Numbers and Compound Interest - Click Here

HS MATHEMATICS PAPER English Version 2022 - XII-Question Paper and Solution - WBCHSE

 Download the full question paper and solution of mathematics for the year 2022 under WBCHSE ( West Bengal Council of Higher Secondary Education ) HS.

HS 2022 Mathematics English Version

WBJEE MATHEMATICS PAPER 2021 SOLVED

 The West Bengal Joint Entrance Examinations Board

The West Bengal Joint Entrance Examinations Board (WBJEEB) was established in 1962 by Government of West Bengal in exercise of the powers conferred under article 162 of the Constitution of India in pursuant to No. 828-Edn(T), dated 02.03.1962.

Subsequently in 2014, the Government of West Bengal enacted the West Bengal Act XIV of 2014 to form The West Bengal Joint Entrance Examinations Board and empowered it to conduct Common Entrance Examinations for selection of candidates for admission to undergraduate and postgraduate Professional, Vocational and General Degree Courses in the State of West Bengal and to conduct on-line counselling process or otherwise adopting a single-window approach.

WBJEEB has been instrumental in the admission process based on online application and allotment through e-Counselling since 2012. It advocates fairness and transparency, ensures no-error, and adopts state-of-the-art technology.

WBJEE 2022 Mathematics Syllabus

S.No.	Topics
1	Algebra Arithmetic Progression G.P., H.P Sets, Relations and Mappings Logarithms Complex Numbers Permutation and combination Polynomial equation Principle of mathematical induction Matrices Binomial theorem (positive integral index) Statistics and Probability
2	Trigonometry
3	Coordinate geometry of two dimensions
4	Coordinate geometry of three dimensions
5	Differential calculus Calculus Integral calculus Application of Calculus Differential Equations Vectors

WBJEE MATHEMATICS PAPER 2020 SOLVED

 The West Bengal Joint Entrance Examinations Board

The West Bengal Joint Entrance Examinations Board (WBJEEB) was established in 1962 by Government of West Bengal in exercise of the powers conferred under article 162 of the Constitution of India in pursuant to No. 828-Edn(T), dated 02.03.1962.

Subsequently in 2014, the Government of West Bengal enacted the West Bengal Act XIV of 2014 to form The West Bengal Joint Entrance Examinations Board and empowered it to conduct Common Entrance Examinations for selection of candidates for admission to undergraduate and postgraduate Professional, Vocational and General Degree Courses in the State of West Bengal and to conduct on-line counselling process or otherwise adopting a single-window approach.

WBJEEB has been instrumental in the admission process based on online application and allotment through e-Counselling since 2012. It advocates fairness and transparency, ensures no-error, and adopts state-of-the-art technology.

WBJEE 2022 Mathematics Syllabus

S.No.	Topics
1	Algebra Arithmetic Progression G.P., H.P Sets, Relations and Mappings Logarithms Complex Numbers Permutation and combination Polynomial equation Principle of mathematical induction Matrices Binomial theorem (positive integral index) Statistics and Probability
2	Trigonometry
3	Coordinate geometry of two dimensions
4	Coordinate geometry of three dimensions
5	Differential calculus Calculus Integral calculus Application of Calculus Differential Equations Vectors

Combinations - RD Sharma Solved Problems

 

What is a Combination?

A combination is a mathematical technique that determines the number of possible arrangements in a collection of items where the order of the selection does not matter. In combinations, you can select the items in any order.

Combinations can be confused with permutations. However, in permutations, the order of the selected items is essential. For example, the arrangements ab and ba are equal in combinations (considered as one arrangement), while in permutations, the arrangements are different.

Formula for Combination

Mathematically, the formula for determining the number of possible arrangements by selecting only a few objects from a set with no repetition is expressed in the following way:

 

 

Where:

• n – the total number of elements in a set
• k – the number of selected objects (the order of the objects is not important)
• ! – factorial

A few important results on combinations are as follows:

• The number of ways of selecting n objects out of n objects is:${}^n{C}_n=\frac{n!}{n!\left(n-n\right)!}=\frac{n!}{n!0!}=1$
• The number of ways of selecting 0 objects out of n objects is:${}^n{C}_0=\frac{n!}{0!\left(n-0\right)!}=\frac{n!}{0!n!}=1$
• The number of ways of selecting 1 object out of n objects is: ${}^n{C}_1=\frac{n!}{1!\left(n-1\right)!}=\frac{n\times \left(n-1\right)!}{\left(n-1\right)!}=n$
• ${}^n{C}_r{=}^n{C}_{n-r}$
• ${}^n{C}_r{+}^n{C}_{r-1}{=}^{n+1}{C}_r$