Question

Non perfect square number between square of 21 and 22 are:
A.42
B.44
C.441
D.404

Answer 1

Hint: In general, there are 2n non perfect square numbers between the square of the numbers n and (n+1). Non-perfect square numbers always end with 2, 3, 7 or 8. So, all other numbers ending with 0, 1, 4, 5, 6 or 9 are perfect square numbers.

Complete step-by-step answer:
Step1 we have to find the non-perfect squares between squares of 21 and 22.
Given number s are 21 and 22. Number 21 and 22 are consecutive integers. Because consecutive means following continuously unbroken. So means that consecutive integers follow a sequence where each number is one more than the previous number.
So, 22 is one more than the previous number 21. I.e. 21 + 1 =22.
Step 2 The formula is find consecutive integers n, n+1, n+2 ……………….
So, consider n =21, where n is denoted as natural number.
Therefore 21=21+1=n+1 where n =21
According to the question, we have to find the non-perfect square number between the square of 21 and 22. i.e. from step 2, we find the non perfect square number between $n^{2}and(n+1{)}^2$.
The square of n i.e. square of 21 is equal to 441. And the square of n+1 i.e. square of 22 is equal to 484.
Step 3 Consider ${21}^2$ and ${22}^2$
Let us assume any natural number n and (n+1) then :
$(n+1{)}^2-n^2=(n^2+2n+1)-n^2=2n+1$.
So, we find that between $n^{2}and(n+1{)}^2$ there are 2n numbers which is 1 less than the difference of two squares.
Step 4 here n =21
And n+1 = 22
Therefore, non perfect square numbers between 21 and 22= 2n= 2× 21 = 42.
So, the non-perfect square number between 21 and 22 is 42. Hence, option A is the correct answer.

Note: When we multiply an integer (a whole number, positive, negative or zero) times itself, the resulting product is called a square number , or a perfect square.
So, 0,1,4,9,16 and so on are all square numbers.
Squares of all integers are known as the perfect squares. All perfect squares end in 1,4,5, 6,9 or 0. Therefore, the number ends in 2, 3, 7, or 9 is not a perfect square.