Question

state whether following true/false. justify each of your answer..

1) √2/3 is a rational number.
2)there are infinitely many integers between two integers.
3) number of rational numbers between 15 and 18 is finite.
4) there are numbers which cannot be written in the form of p/q, q≠0, p,q both are integers.
5) the square of irrational number is always rational.
6) √12/√3 is not a rational number as √12 and √3 are not integers.
7) √15/√3 is written in the form p/q, q≠0 and so it is a rational number.



please help me..

dont forgot to write justification in each of the questions..

Answer 1

Heyà !!

==================================

1▶ √2/3 is a rational number.
[ ❌ False ]

REASON :- For a rational number p/q, q≠0 , p and q should be integers but in √2/3 , p(√2) is not an integer.

2▶ There are infinitely many integers between two integers.
[ ❌ False ]

REASON :- Taking an example of integers 1 and 2. There is no integer between 1 and 2.

3▶ Number of rational numbers between 15 and 18 is finite.
[ ❌ False ]

REASON :- Because between two rational numbers we can find infinity rational numbers.

4▶ There are numbers which cannot be written in the form of p/q, q≠0, p,q both are integers.
[ ✔ True ]

REASON :- Taking an example. √2 / √3 is in the form of p/q but here p and q are nit integers.

5▶ The square of irrational number is always rational.
[ ❌ False ]

REASON :- Taking an example. (³√3)² = (³√9) is an irrational number.

6▶ √12/√3 is not a rational number as √12 and √3 are not integers.
[ ❌ False ]

REASON :- Because √12 / √3 = √4 = 2 which is a rational number.

7▶ √15/√3 is written in the form p/q, q≠0 and so it is a rational number.
[ ❌ False ]

REASON :- Because √15 / √3 = √5 which is an irrational number.

==================================

Hope my ans.'s satisfactory.☺

Answer 2

1)  False because √2 is irrational and 3 is rational   and we know that on dividing irrational and rational is irrational 
2)  True because there are infinite numbers between any two any  numbers on the number line as we know that a line is made up of infinite dot .
3)  True because there are only two numbers between 15 and 18 and it is countable.
4)   it is True that we can write all the numbers in the form of p/q and where q≠0.
5)  False in some cases the square of √4 is rational number and on other hand the Square of√7 is irrational number.
6)  False on dividing √12/√3  we get√4  and the square of √4 is 2 which is a rational number.
7)   False on dividing √15/√3  we get √5 which is a irrational number