Category: Maths Notes PPT

The measure of an angle is identified by the amount of rotation from the original side to the end side. In radians, one complete counterclockwise rotation is 2π. Whereas, in degrees, one complete counterclockwise rotation is 360°. Thus, the degree measure and radian measure are linked by the equations

360°= 2π radians

and

180°= π radians

From the latter, we get the equation 1° = π180 radians.

This results in the rule to convert degree measure to radian measure. In order of conversion from degrees to radians, multiply the degrees by π180° radians.

 

How to Convert Degrees to Radians?

The value of 180° is equivalent to π radians. In order to convert any given angle from the measure of degrees to radians, the value needs to be multiplied by π/180. A standard formula for converting from degrees to radians is to simply multiply the number of degrees by π/180

Thus, Angle in Radian = Angle in degree × π / 180° 

Radian Measure Formula

In any circle of radius r, the ratio of the arc length ℓ to the circumference is equivalent to the ratio of the angle θ subtended by the arc at the midpoint and the angle in one rotation. Therefore, radian formula to measure the angles in radians, ℓ2πr=θ2π⟹ ℓ=rθ.

Degree Calculation Formula

The measure of the angle is considered in degrees (°). Radian is often considered while computing the angles of trigonometric functions or periodic functions. Radians are always denoted in terms of pi (π), where the value of pi is equivalent to 22/7 or 3.14.

A degree consists of sub-parts also, stated as minutes and seconds. This conversion is the crucial component of Trigonometry applications. Thus, the Degree formula is: – Radians × 180/π = Degrees

Solved Examples on How to Find Radian Measure From Degree

Example:

Convert 135° into radian measure

Solution:

Given = Angle of 135 degrees

135∘× π/180∘

=3π/4 radians

≈2.35 radians

Example:

Convert 210 degrees to radians.

Solution: Given = Angle of 210 degrees

Angle in radian = Angle in degree x (π/180)

= 210 x (π/180)

= 7π/6

≈3.67

Hence, 210 degrees is equal to 7π/6 (3.67) in radian.

If there is a bijection between two sets, they have the same number of elements (are equinumerous, or have the same cardinality). Arrangements: A mapping, also known as a function, is a rule that associates elements from one set with elements from another. This is how we write it:  f : X → Y , f is referred to as the function/mapping, the set X is referred to as the domain, and Y is referred to as the codomain. We specify the rule by writing f(x) =y or f : x 7→ y. e.g. X = {1, 2, 3}, Y = {2, 4, 6}, the map f(x) = 2x associates each element x ∈ X with the element in Y which means to double it.

In this article we are going to discuss cantor’s intersection theorem, state and prove cantor’s theorem, cantor’s theorem proof.

A bijection is a mapping that is injective as well as surjective.

• Injective (one-to-one): A function is injective if it takes each element of the domain and applies it to no more than one element of the codomain. It never maps more than one domain element to the same codomain element. Formally, if f is known to be a function between namely set X as well as set Y , then f is injective iff ∀a, b ∈ X, f(a) = f(b) → a = b 

• Surjective (onto): If a function maps something onto every element of the codomain, it is surjective. It can map multiple things to the same element in the codomain, but it must hit every element in the codomain.Formally, if f is known to be a function between set X and set Y , then f is surjective (if and only if)  iff ∀y ∈ Y, ∃x ∈ X, f(x) = y 

The Heine-Cantor theorem The cardinality of any set A is strictly less than the cardinality of A’s power set : |A| < |P(A)| 

Proof: To prove this, we will show (1) that |A| ≤ |P(A)| and then (2) that ¬(|A| = |P(A)|). This is equivalent to the strictly less than phrasing in the statement of the given theorem. (1) |A| ≤ |P(A)| : Now , to show this, we just need to produce a bijection between A as well as a subset of P(A). Then we’ll know A is the same size as that subset, which cannot be larger than P. (A).

Consider the set E = {{x} : x ∈ A}, the set of all single-element subsets of A. Clearly E ⊂ P(A), because it is made up of various subsets of A. Incidentally, it is a proper subset, since we know it doesn’t contain ∅. 

The map g : A → E defined by g(x) = {x} is one-to-one and onto. How do we know this? (This is laboured, but useful to be certain that you understand this!) 

• One-to-one: Let’s suppose we have x, y ∈ A and g(x) = g(y). Then by the definition of injectiveness above, we want to be sure that this means x = y, if g is going to be one-to-one. g(x) = {x} and g(y) = {y}, so, g(x) = g(y) means {x} = {y}. These two one-element sets can only be equal if their members are equal, so x = y. Therefore g is one-to-one.

• Onto: Is it true that ∀y ∈ E, ∃x ∈ A, g(x) = y? Yes. We know that E = {{x} : x ∈ A} so ∀y ∈ E, ∃x ∈ A such that y = {x}. And that is because each element of E just is a set with an element from A as its sole member. And since g(x) = {x}, we have ∀y ∈ E, ∃x ∈ A, g(x) = y, so g is surjective.

Therefore |A| = |E| ≤ |P(A)| . 

Importance of Cantor’s Theorem

Cantor’s theorem had immediate and significant implications for mathematics philosophy. For example, taking the power set of an infinite set iteratively and applying Cantor’s theorem yields an infinite hierarchy of infinite cardinals, each strictly larger than the one before it.

Cantor was successful in demonstrating that the cardinality of the power set is strictly greater than that of the set for all sets, including infinite sets. (In fact, the cardinality of the Reals is the same as the cardinality of the Integers’ power set.) As a result, the power set of the Reals is larger than that of the reals.

If we talk about a company, gross profit is an income or we can say profit that a company makes after deducting all the costs associated with the production & selling of goods and services. On the other hand, for the individuals or households, it is an amount which is a summation of wages/salaries, profits or any interest payments, rents, and other kinds of earnings. It is an amount that we get before any payment of taxes & other deductions. It is exactly the opposite of net profit or net income where the amount we get is always after the payment of taxes & other deductions. In this article, we will cover all the formulas related to this concept such as gross margin formula or gross profit margin formula, gross profit ratio or gross margin percentage, etc.

Gross Profit Formula

The formula or gross profit equation through which we can calculate the gross profit or gross revenue is given below:

Gross Profit/Gross revenue formula = Revenue – Cost of Goods Sold

Here, revenue or profit is the amount that we get at the end after selling the products produced by the company at a specific time. The amount before any payment of taxes & other deductions is taken here. On the other hand, if we talk about COGS or Cost of Goods Sold, that includes all the direct costs which are associated with the production of the products & it does not include the costs of administration or marketing-related costs.  The costs it includes are depreciation, labour cost, factory overheads, cost of materials or storage.

Gross Margin Formula / Gross Profit Margin Formula

When we express gross profit in the form of a percentage, it is known as Gross Profit Margin or gross margin. The gross profit margin formula is given below:

Gross margin formula = (Revenue – Cost of Goods Sold)/Revenue x 100

Here, we get to know that gross profit can be used to find out other metrics such as gross profit margin. Sometimes the terms gross profit margin, as well as gross profit margin, are used interchangeably but they are not the same because the former is calculated as well as expressed in the form of currency whereas the latter is expressed in the form of a percentage. 

Gross Profit Ratio Formula

This can be calculated with the help of the following formula:

[text{Gross Profit Ratio = }frac{text{Gross Profit}}{text{Net Sales}}]

When we express the gross profit ratio in the form of percentage form, it is simply called gross profit margin or gross profit percentage. The gross profit percentage formula is mentioned below:

[text{Gross Profit Ratio = }frac{text{Gross Profit}}{text{Net Sales}}times100]

In both the above-mentioned formulas, the two required components are gross profit as well as net sales. Gross profit can be calculated by subtracting the cost of goods sold from revenue or net sales whereas net sales can be calculated by subtracting any returns inwards as well as discounts allowed from the gross sales. This information can be collected from the income statement of the company.

Examples

Let’s understand the GP formulas more clearly with the help of the following solved examples whose answers can be checked through the profit percentage calculator:

Problem 1. Calculate the Gross Profit using the gross profit rate formula, if the Cost of goods sold is Rs. 12,000 and revenue is Rs. 76000.

Solution: Using the formula:

Gross profit formula = Revenue – Cost of Goods Sold

= 76,000 – 12,000

= Rs. 64,000/-

Problem 2. Calculate the gross profit of the company if gross sales are Rs. 20,00,000, sales return is Rs. 2,50,000 and COGS is Rs. 1,50,000.

Solution: Gross profit can be calculated in the following way:

 

Problem 3. The following data relates to a small trading company. Compute the gross profit ratio and gross profit percentage of the company.

Gross sales: ＄1,00,000

Sales returns: ＄10,000

Cost of goods sold: ＄18,000

Solution:

Net sales = Gross sales – Sales returns

= ＄1,00,000 – ＄10,000

= ＄90,000 

Gross profit = Net sales – Cost of goods sold

= ＄90,000 – ＄18,000

= ＄72,000

Gross profit ratio / Gross margin ratio formula

 =[ frac{text{Gross Profit}}{text{Net Sales}}]

= [ frac{90,000}{72,000}]

= 0.8 

Gross Profit Percentage or Gross Margin Percentage Formula =  [frac{text{Gross Profit}}{text{Net Sales}}] * 100

= 0.8 * 100

= 8%

Conclusion

To conclude in the end, we can say that after reducing the costs related to the production of the goods and services from the total revenue that we generate after selling the goods and services, we can get gross profit but this is not the actual profit that we enjoy because taxes and other related deductions have not been deducted from this. Thus, when we deduct the taxes from this gross profit, then we get actual income that we can enjoy. Besides these, we also learned how to calculate other metrics such as gross margin ratio and gross revenue percentage, etc.

Types of Matrices

If you are searching for Matrix formulas for Class 12,  then it is very important that you should know the basic definitions also.

Different type of mattresses are there so some of them are:

1. Row Matrix – Row matrix is the matrix which can have any number of columns but it must have only one row.

2. Column Matrix – Column matrix is the matrix which can have any number of rows but it must have only one column.

3. Rectangular Matrix – A matrix in which the number of rows and number of columns is not equal is called a rectangular matrix.

4. Square Matrix – A matrix having the same number of rows and number of columns is called a square matrix.

5. Diagonal Matrix – A square matrix is called a diagonal Matrix if all the diagonal elements are non-zero, rest all are zero.

6. Scalar Matrix – A square Matrix in which all the elements except the diagonal elements are 0 and the diagonal elements are equal.

7. Unit Matrix – A square matrix is called a unit matrix if all the elements except the diagonal elements are 0 and the diagonal elements are one.

8. Equal Matrix – Two Matrices are said to be equal if the number of rows and columns in the first matrix is equal to the number of rows and columns of the second Matrix and the corresponding elements of both the matrices are equal.

9. Singular Matrix – A matrix is said to be a singular Matrix if the determinant of a is zero.

10. Non-Singular Matrix – A matrix is said to be a null singular Matrix and determinant of the matrix is not zero.

Algebra of Matrices

One of the most important topics in Matrices formula Class 12 is the algebra of matrices. Algebra of matrices includes the addition of matrices, subtraction of matrices,  multiplication of matrices, and multiplication of matrices by a scalar.

Two matrix A and  B can be added if and only if the order of matrix A is equal to the order of matrix  B.

[A = [a_{ij}]_{m times n]

[B = [b_{ij}]_{m times n]

[A + B = [a_{ij} + b_{ij}]_{m times n}]

Properties of Addition of Matrices

1. Commutative Law – Addition of mattresses follow commutative law.  

A + B = B+ A

2. Associative Law – Addition of matrices follow associative law.

A + (B + C) = (A + B) + C

3. Additive Inverse – If matrix A  is the matrix then Matrix (-A ) is the additive inverse for matrix A.

A + ( -A ) = 0

4. Identity of the Matrix – For a matrix A, 0 is the additive identity of the matrix when,

A + 0 = 0 + A

5. Cancellation Properties

For matrices A, B and C,

If A + B = A + C 

Then, B = C  (This is known as left cancellation)

Similarly,

For matrices A, B and C

If B + A = C + A

Then, B = C (This is known as Right Cancellation)

Two matrices A and B can be subtracted if and only if the order of matrix A is equal to the order of matrix B.

If A is a matrix of order M x N similarly be is a matrix of order M x N then,

[A – B = [a_{ij} – b_{ij}_{m times n}]

One of the most important concepts and formulas is the multiplication of matrices from the formulas of matrices Class 12.

For two matrices A and B,  multiplication of matrices can be done if the number of rows of the first matrix is equal to the number of columns of the second matrix.

If, $A = [a_{ij}]_{m times n$

[B = [b_{ij}]_{m times n]

[AB = c_{ij} = sum_{k=1}^{n} a_{ik}b_{kj}]

Other Related Links for Class 12 Maths Chapter 3

Easy Tips and Tricks for Memorising the Table of 4

• All the numbers in the multiplication table of 4 are even – they end with 0, 2, 4, 6, or 8.

• If the last 2 digits of a number is a multiple of 4, then the whole number is also a multiple of 4.

For example, look at the number 116. The last 2 digits, i.e 16 is a multiple of 4, so the number 116 is also a multiple of 4.

Look at the Below Method to Learn How to Remember Multiplication Table of 4

• Make a three-column table.

• Write Table of 2 in Column 1.

• Multiply the resultant product by two in order in Column 2 (Refer to table below).

• To get the 4 tables, write the resultant product in Column 3.

To do the above operation, look at the table below.

Solved Questions Based on Table of 4

The following are some questions based on the table of 4 that can come in the form of word problems, numericals, or multiple-choice questions.

Word Problems on Table of 4 with Practice Questions

1. A box contains four Apples. How many apples will be there in 3 boxes?

Ans: By using repeated addition, we will get 4 + 4 + 4 = 12

Then, 3 times 4, i.e., 3 × 4 = 12

Therefore, there are 12 Apples.

2. If there are 7 packets of biscuits having four biscuits in each of them, how many biscuits are there in total?

Ans: By using repeated addition, we get 4 + 4 + 4 + 4 + 4 + 4 + 4 = 28

Then, 7 times 4, i.e., 7 × 4 = 28.

Therefore, there are 28 biscuits.

Practice Questions

1. Using the table of 4, evaluate 4 times 16.

Answer: 64.

2. Find the 2 digit numbers, which are present in Table of 4 and they are less than 17.

Answer: 12 and 16.

MCQs Based on the Multiplication Table of 4 with Practice Questions

Question: 4 x __ = 24?

1. 3

2. 2

3. 6

4. 8

Ans: (c) , 4 x 6 = 24.

Practice Question

Question: What is 4 x 4?

1. 16

2. 20

3. 24

4. 36

Answer: (a)

Table Chart of 4 from 11 to 20

Multiplication table of 4 till 20 has been provided below. You can refer and download the image below to easily memorise the multiples of 4 from 11 to 20. 

For Parents/Teachers: How to Read the Table to Your Kid?

These simple methods for learning the 4 times tables allow kids to have fun while learning multiplication tables. A kid can easily remember his/her name due to repetition of the name multiple times throughout the day. Similarly, there is a way the table should be read to your kid. 

It can become a new rhyme/song that you repeat in the morning or play on your devices.

Read the table as mentioned below to your kids and ask them to repeat after you. 

Conclusion

Multiplication is a basic mathematical process that we use on a daily basis. Multiplication is an essential part of a child’s early education. Learning the multiplication tables is one of the simplest ways to learn how to multiply. Make use of the 4 times table chart above to help your child in memorising the multiplication table of 4.

Table Chart of 20 from 1 to 10

Tips and Tricks for 20 Times Table

Try the given tips and tricks to master the table of 20 quickly:

The Table of 20 is Just the 2 Times Table With a 0 at the End:

Here’s a quick technique to figure out the 20 times table. The 20 times table is simply the 2 times table with a zero at the end. So, if you’re comfortable with the two times tables, start there. Simply, add a zero to the end of each multiple of two to get the multiples of twenty. To help you understand, here are some instances.

Let’s look at Example 1 to see what 20 X 5 is.

To begin, write 2 X 5 on a piece of paper. Two times five equals ten.

To get a multiple of 20, add a zero to the end of the number 10. As a result, you receive a score of 100.

As a result, 20 X 5 Equals 100.

Let’s look at another scenario.

Let’s look at Example 2 to see what 20 X 8 is.

Find out what 2 X 8 is first. 2 x 8 equals 16.

Add a zero to the end of 16 to get the answer to 20 X 8. As a result, you receive 160.

As a result, 20 X 8 = 160.

Starting with 20 X 1, you can acquire all the multiples of 20 using this method.

Learn The Table of 20 By Doubling the 10 Times Table:

Another clever approach that can help you memorise the 20 times tables in no time is using the table of 10. And if you’re comfortable with the 10 times tables, you’ll have no trouble with this strategy.

First, make a list of the 10 times table. Remember to multiply the numbers by 1 and add a zero to the end of the digit in the 10 times table. The number ten is half of the number twenty. Therefore, simply double the multiples of 10 to get the multiples of 20. To help you understand, here are some instances.

Let’s look at Example 1 to see what 20 X 2 is.

To begin, figure out what 10 X 2 is. 10 multiplied by two equals twenty.

Now multiply the answer by two. 20 plus 20 equals 40.

As a result, 20 X 2 = 40.

Let’s have a look at another example that uses a similar method.

Let’s look at Example 2 to see what 20 X 6 is.

To begin, figure out what 10 X 6 is. 60 is the result of multiplying 10 by 6.

Now multiply the answer by two. 60 plus 60 equals 120.

As a result, 20 X 6 = 120.

Adding 20 to the Previous Multiple To understand the 20 Multiplication Table

If you’re still having trouble multiplying by 20, try using the additional trick. Adding 20 to the previous answer is a quick approach to discovering the multiples of 20. To help you understand, here are some instances.

Let’s look at Example 1 to see what 20 X 2 is.

We know that 20 X 1 equals 20.

Add 20 to the answer for 20 X 1 to get 20 X 2.

As a result, 20 plus 20 equals 40.

As a result, 20 X 2 = 40.

Let’s look at another scenario.

Similarly, let’s see what 20 X 3 equals.

We already know that 20 + 2 = 40.

Add 20 to the answer for 20 X 2 to get 20 X 3.

As a result, 40 + 20 equals 60.

Therefore, 20 X 3 = 60.

Similarly, add 20 to each of the previous answers to get all the multiples of 10 till 20 X 10.

The Underlying Pattern in the 20 Times Table

The 20 times table has a very simple and cool pattern. Let us look at the table below.

Look at the one column’s digits; they’re all zeros. Look at the digits in the tens column; they’re all even and multiples of two. And the numbers are 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20 in order. You’ll be able to memorise the 20 times table quickly if you remember this pattern.

How to Solve Questions Based on Table 20?

In this section, we are going to solve a few questions that can appear in the form of word problems and multiple choice.

Word Problems on Table 20 with Practice Questions

Example 1: John is playing a video game in which he receives a credit of 20 points for finishing each level. Calculate the total points he earned after completing the 9th level using the table of 20?

Solution:

After each level, John will receive = 20 points in credit.

Credits earned at the 9th level = 20 x 9 = 180 points, according to the 20 times table.

As a result, when John completed the 9th level, he would have received 180 points.

Example 2: Find the value of 20 times 7 minus 5 using the table of 20.

Solution:

To begin, we’ll write 20 times 7 minus 5 mathematically.

We have: 20 times 7 minus 5 = 20 x 7 – 5 = 140 – 5 = 135 using the 20 times table.

As a result, 20 times 7 minus 5 equals 135.

 

Example 3: Mary would like to purchase 20 pencils. Estimate the cost of 20 pencils using a 20-times table if one pencil costs 5 cents.

Solution:

1 pencil costs 5 cents.

As a result of the 20 times table, the cost of 20 pencils is 20 x 5 = 100 cents = $1.

As a result, 20 pencils cost $1.

Practice Questions

1. Rama reads 3 pages a day, how many pages can he read in 20 days?

2. If 1 bag contains 3 apples, how many apples will be there in 20 bags?

MCQs on Table 20 with Practice Questions

Question: 

1. What is 4 times 20?
a) 60
b) 80
c) 90
d) 100

Solution: (B)

4 times 20 is 4 x 20 = 80

Practice Question

Question:

Rahul drives his bike 10 km per day to his office. What is the total distance he covers from his home to the office for the next 20 days?

a) 150 km
b) 200 km
c) 300 km

d) 410 km

Table Chart of 20 from 11 to 20

To keep the multiple of 20 on your tips, it is useful if you can remember the table from 11 to 20 as well. Refer to the image given below, the same is provided in the PDF, which can be easily downloaded and printed.

Reading and Understanding the Table 20

Learning the table of 20 by doubling the 10 times table, adding 20 to the preceding multiple to learn the 20 multiplication table, the underlying pattern in the 20 times table, and so on are some of the basic ways to learn the 20 times table.

We request parents/teachers to read this table with your kids. This will not only help them to learn the table quickly, but they will enjoy the learning activity. Learning the multiplication table of 20 is not a difficult job, and it will also help students to solve difficult problems quickly.