Question

Consider the following two statements:
I. Any pair of consistent linear equations in two variables must have a unique solution.

II. There do $not$ exist two consecutive integers, the sum of whose squares is 365.

• A. both I and II are true
• B. both I and II are false
• C. I is true II is false
• D. I is false II is true

Answer 1

The correct option is B both I and II are false

Clearly I statement is false as they can have infinitely many solution also.

II. Solution:
Let the integers be $n,n+1$ then,
$n^2+(n+1{)}^2=365$
$⟹n^2+(n^2+1+2n)=365$
$⟹2n^2+1+2n=365$
$⟹2n^2+2n=364$
$⟹n^2+n-182=0$
$⟹(n+14)(n-13)=0⟹n=13or-14$

Hence there exist two integers $13,14$ sum of whose squares is $365$.
${13}^2+{14}^2=365$