Define inertia. Give its classification

The inherent property of a body to resist any change in its state of  rest or  the state of uniform motion, unless it is influenced upon by an external unbalanced force, is known  as ‘inertia’.

Types of Inertia

Inertia of rest: The resistance of a body to change its state of rest is called inertia of rest.

Example

When you vigorously shake the branches of a tree, some of the leaves and fruits are detached and they fall down, (Inertia of rest).

Inertia of motion: The resistance of a body to change its state of motion is called  inertia of motion.

Example

An athlete runs some distance before jumping. Because, this will help him jump longer and higher. (Inertia of motion)

Inertia of direction: The resistance of a body to change its direction of motion is called  inertia  of direction.

Example

When you make a sharp turn while driving a car, you tend to lean sideways, (Inertia of direction).

Video:https:https://youtu.be/yAZxgwkfvmw

What is function?

A relation f between two non-empty sets A and B is called a function from A to B if, for each a ∈ A there exists only one b ∈B such that (a,b) ∈ f .That is, f ={(a,b)| for all a ∈ B, b∈B }. A function is also called as a mapping or transformation

• A function is also called as a mapping or transformation
• The set A is called the domain of the function f and the set B is called its co-domain.
• If f (a) = b, then b is called ‘image’ of a under f and a is called a ‘pre-image’ of b.
• The set of all images of the elements of A under f is called the ‘range’ of f.
• Every function is a relation. Thus, functions are subsets of relationsand relations are subsets of cartesian product
• The range of a function is a subset of its co-domain.

Example 

 Let X = {1,2, 3, 4} and Y = {2, 4,6, 8,10} and R = {(1,2),(2,4),(3,6),(4,8)}.

Show that R is a function and find its domain, co-domain and range?

Solution: From the diagram, we see that for each x∈ X , there exists only one y ∈Y . 

Thus all elements in X have only one image in Y. Therefore R is a function.

Domain X = {1,2,3,4}; Co-domain Y = {2,3,6,8,10}; Range of f = {2,4,6,8}.

                                               

Prime Numbers

A natural number greater than 1, having only two factors namely 1 and the number itself, is called a prime number

The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, ... 

Smallest Prime Number

The smallest prime number   is 2. To be prime, a number must be divisible only by 1 and the number itself which is fulfilled by the number 2

Twin Primes

A pair of prime numbers whose difference is 2, is called twin primes.

For example, (5, 7) is a twin prime pair as is (17,19)

If three successive prime numbers differ by 2, then the prime numbers form a prime triplet. The only prime triplet is (3, 5, 7).

 Express 42 and 100 as the sum of two consecutive primes.

 42 = 19+23

100 = 47+53

Express 31 and 55 as the sum of any three odd primes. 

 31 = 5+7+19

55 = 3 + 23+29

Finding the Prime Numbers by Sieve of Eratosthenes Method

Sieve of Eratosthenes, is a simple method of elimination by which we can easily find the prime numbers upto a given number. This method given by a Greek Mathematician, Eratosthenes of Alexandria, follows some simple steps which are listed below, by which we can find the prime numbers.

 

Step 1: Write the numbers 1 to 100 in ten rows.

Step 2: Cross out 1 because 1 is not a prime.

Step 3: Circle 2 and cross out all multiples of 2. (2, 4, 6, 8, 10, ...)

Step 4: Circle 3 and cross out all multiples of 3. (3, 6, 9, 12, 15, ...)

Step 5: Circle 5 and cross out all multiples of 5. (5, 10, 15, 20, ...)

Step 6: Circle 7 and cross out all multiples of 7. (7, 14, 21, 28, ...)