Statement 1 : If `cos^2\ pi/8` is a root of the equation `x^2+ax+b=0,` where `a,b in QQ,` then ordered pair `(a,b)` is `[-1, 1/8].` Statement 2: If `a + mb = 0` and m is irrational, then `a, b=0.`

Answer

`(-1.25)`

Explanation

`cos""(pi)/(4)=2cos^(2)""(pi)/(8)-1`
`impliescos^(2)""(pi)/(8)=((1)/(sqrt2)+1)(1)/(2)`
`impliescos^(4)""(pi)/(8)=(1)/(4)((1)/(2)+1+(2)/(sqrt2))=((3)/(2)+sqrt2)(1)/(4)`
`implies(1)/(4)((3)/(2)+sqrt2)+(a)/(2)((1)/(sqrt2)+1)+b=0`
`implies((3)/(8)+(a)/(2)+b)+sqrt2((1)/(4)+(a)/(4))=0`
Since a and b are rational,
`(1)/(4)+(a)/(4)=0,(3)/(8)+(a)/(2)+b=0`
`impliesa=-1,b =(1)/(8)` `:.x^(2)+8bx+a=x^(2)+x-1=(x+1//2)^(2)-5//4`
So, required least value is `-5//4`.