Class X Session 2023-24 Question 1

  MATH SAMPLE QUESTION PAPER  

Class X Session 2023-24

   MATHEMATICS STANDARD (Code No.041)

Question 1

If two positive integers a and b are written as  \[ a = \ x^3y^2 \ and \ b =\ xy^3 \] , where x, y are prime

numbers, then the result obtained by dividing the product of the positive integers by the

LCM (a, b) is :

(a) xy (b) xy2 (c) x3y3 (d) x2y2

Explanation: 

First, understand the Prime number 

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself

Example of prime Numbers 2, 3 , 5,7,11  , Let's take the number 5 for example. It is only divisible by 1 and itself. Similarly, 11 is also only divisible by 1 and itself.   

Let's understand LCM (Least Common Multiple) of two numbers as an example

Let's find the LCM of 12 and 15. 

To find the LCM, we need to find the smallest multiple that both numbers share

how can you write the Prime factorization of 12: \[ 2\times 2\times 3\ ---------(A)  \]  

Similarly,   Prime factorization of 15:   \[ 3\times 5\ ---------(B)  \]  

Now, we take each prime factor to its highest power from (A)   \[  2^2 \]

we take each prime factor to its highest power from (B)  \[ 3^1 \ and \ 5^1 \] both have the same power.

Multiplying these together, we get: LCM(12, 15)

\[ \Rightarrow  3^1\times 5^1\times 2^2 \ = \ 60\] 

Hope above explanation has been cleared now come to the question :

Given  \[ a = \ x^3y^2 \ and \ b =\ xy^3 \]

\[ LCM(a, b) \Rightarrow x^3y^3 \]

Now, we need to find the result obtained by dividing the product of the positive integers by the LCM(a,b)

The product of the positive integers is  a and b \[ ab = \ x^3y^2\times xy^3 \ = \ x^4y^5\] 

So, the result obtained by dividing the product of the positive integers by the LCM(a,b)

\[\Rightarrow  \frac{ab}{LCM(a, b)}  \ = \frac{x^4y^5}{x^3y^3}\]  

OR

\[\Rightarrow  \frac{ab}{LCM(a, b)}  \ = \frac{x^4y^5}{x^3y^3} \ = x^(4-3)y^(5-3) \ = x^1y^2 \]

The correct option is .....?.... comment below