1. Prove the following

For the real numbers \(m\) \(=\) 23 and \(n\) \(=\) 60, if \(m\) is increased by \(u\) \(=\) \(3\) and \(n\) is decreased by \(v\) \(=\) \(7\), then show that the  increment in product is always equal to \(un\) \(-\) \(mv\) \(-\) \(uv\).

 

 

Given, \(m\) \(=\) 23, \(n\) \(=\) 60, \(u\) \(=\) \(3\), and \(v\) \(=\) \(7\).

 

Their product \(mn\) \(=\) .

 

By the given condition, the new product is given by (m+v)(n-v)\((m-v)(n+v)\)\((m+v)(n-u)\).

 

The value of the new product \(=\)

 

The difference in products \(=\)

 

By the distributive property, the increment in product is given by un - mv + uvun - mv - uvun + mv - uv.

 

The increment in product using the obtained property \(=\) .

 

Difference in product is not equal tois equal to Increment in product

 

Hence, proved.