UPSC NDA Math Model Paper 3

Let α and β be roots of the equation x2−nx+m=0
Given, the roots of the equation x2−nx+m=0 differ by 1
⇒α−β=1
⇒β=α+1
Now,
The sum of the roots =α+β=−(−n)=n
⇒α+α+1=n
⇒2α+1=n
⇒α=(n−1)∕2
The product of the roots =αβ=m
⇒α(α+1)=m
⇒(

n−1

2

)(

n−1

2

+1)=m
⇒(

n−1

2

)(

n−1+2

2

)=m
⇒(n−1)(n+1)=4m
⇒n2−1=4m
⇒n2−4m−1=0
∴ Option 1 is correct.